# Uncomputably Complex Renormalisation Group Flows

UUncomputably Complex RenormalisationGroup Flows

James D. Watson * , Emilio Onorati † , and Toby S. Cubitt ‡ Department of Computer Science, University College London, UK

Abstract

Renormalisation group (RG) methods provide one of the most im-portant techniques for analysing the physics of many-body systems,both analytically and numerically [Wil71; WK74]. By iterating an RGmap, which “course-grains” the description of a many-body system andgenerates a ﬂow in the parameter space, physical properties of interestcan often be extracted even for complex many-body models. RG ana-lysis also provides an explanation of physical phenomena such as uni-versality. Many systems exhibit simple RG ﬂows, but more complicated— even chaotic — behaviour is also known [MBK82; SKS82; DEE99;DT91; MN03]. Nonetheless, the general structure of such RG ﬂows canstill be analysed, elucidating the physics of the system, even if speciﬁctrajectories may be highly sensitive to the initial point. In contrast, re-cent work [CPGW15a; CPGW15b; BCW21] has shown that importantphysical properties of quantum many-body systems, such as its spectralgap and phase diagram, can be uncomputable, and thus impossible todetermine even in principle.In this work, we show that such undecidable systems exhibit anovel type of RG ﬂow, revealing a qualitatively different and moreextreme form of unpredictability than chaotic RG ﬂows. In contrast tochaotic RG ﬂows in which initially close points can diverge exponen-tially according to some Lyapunov exponent, trajectories under thesenovel uncomputable RG ﬂows can remain arbitrarily close together foran uncomputably large number of iterations, before abruptly diverging * † [email protected] ‡ [email protected] a r X i v : . [ qu a n t - ph ] F e b o different ﬁxed points that are in separate phases. The structure ofsuch uncomputable RG ﬂows — e.g. the basins of attraction of its ﬁxedpoints — is so complex that it cannot be computed or approximated,even in principle. To substantiate these claims, we give a mathemat-ically rigorous construction of the block-renormalisation-group (BRG)map for the original undecidable many-body system that appeared inthe literature [CPGW15a; CPGW15b]. We prove that each step of thisRG map is efﬁciently computable, and that it converges to the correctﬁxed points, yet the resulting RG ﬂow is uncomputable. Contents H T ⊗ ( H e ⊕ H q ) . . . . . . . . . . . . . . . . . 456.2 Renormalising H d . . . . . . . . . . . . . . . . . . . . . . . . . 556.3 Renormalising | (cid:105) . . . . . . . . . . . . . . . . . . . . . . . . . 556.4 The Overall Renormalised Hamiltonian . . . . . . . . . . . . . 566.5 Order Parameter Renormalisation . . . . . . . . . . . . . . . . 596.6 Uncomputability of RG ﬂows . . . . . . . . . . . . . . . . . . 59 Understanding collective properties and phases of many-body systems froman underlying model of the interactions between their constituent parts re-mains one of the major research areas in physics, from high-energy physicsto condensed matter. Many powerful techniques have been developed totackle this problem. One of the most far-reaching was the development byWilson [Wil71; WK74] of renormalisation group (RG) techniques, buildingon early work by others [BP53; GML54]. At a conceptual level, an RG ana-lysis involves constructing an RG map that takes as input a description ofthe many-body system (e.g. a Hamiltonian, or an action, or a partition func-tion, etc.), and outputs a description of a new many-body system (a new3amiltonian, or action, or partition function, etc.), that can be understoodas a “coarse-grained” version of the original system, in such a way that phys-ical properties of interest are preserved but irrelevant details are discarded.For example, the RG map may “integrate out” the microscopic detailsof the interactions between the constituent particles described by the fullHamiltonian of the system. This procedure produces a coarse-grained Hamilto-nian that still retains the same physics at larger length-scales [Kad66]. Byrepeatedly applying the RG map, the original Hamiltonian is transformedinto successively simpler Hamiltonians, where the physics may be far easierto extract. The RG map therefore describes a dynamical map on Hamiltoni-ans, and consecutive applications of this map generates a “ﬂow” in the spaceof Hamiltonians. Often, the form of the Hamiltonian is preserved, and theRG ﬂow can be characterised as a trajectory for its parameters.The development of RG methods not only allowed sophisticated theoret-ical and numerical analysis of a broad range of many-body systems. It alsoexplained phenomena such as universality , whereby many physical systems,apparently very different, exhibit the same macroscopic behaviour, even ata quantitative level. This is explained by the fact that these systems “ﬂow”to the same ﬁxed point under the RG dynamics.For many condensed matter systems – even complex strongly interactingones – the RG dynamics are relatively simple, exhibiting a ﬁnite number ofﬁxed points to which the RG ﬂow converges. Hamiltonians that convergeto the same ﬁxed point correspond to the same phase, so that the basinsof attraction of the ﬁxed points map out the phase diagram of the system.However, more complicated RG dynamics is also possible, including chaoticRG ﬂows with highly complex structure [MBK82; SKS82; DEE99; DT91;MN03]. Nonetheless, as with chaotic dynamics more generally, the struc-ture and attractors of such chaotic RG ﬂows can still be analysed, even ifspeciﬁc trajectories of the dynamics may be highly sensitive to the precisestarting point. This structure elucidates much of the physics of the sys-tem [GP83; ER85; SS14]. RG techniques have become one of the mostimportant technique in modern physics for understanding the properties ofcomplex many-body systems.On the other hand, recent work has shown that determining the macro-scopic properties of many-body systems, even given a complete underlyingmicroscopic description, can be even more intractable than previously an-4 a) Uncomputable RG ﬂow. (b)

Chaotic RG ﬂow.

Figure 1:

In both diagrams, k represents the number of RG iterations and η rep-resents some parameter characterising the Hamiltonian; the blue and red dots areﬁxed points corresponding to different phases. We see that in the chaotic case, theHamiltonians diverge exponentially in k , according to some Lyapunov exponent. Inthe undecidable case, the Hamiltonians remain arbitrarily close for some uncom-putably large number of iterations, whereupon they suddenly diverge to differentﬁxed points. ticipated. In fact, [CPGW15a; CPGW15b; BCW21] showed that this goalis unobtainable in general: they proved that the spectral gap of a quantummany-body system, as well as phase diagrams and any macroscopic propertycharacterising a phase, can be uncomputable.In this work, we show that the RG ﬂow of such undecidable systemsexhibits a novel type of behaviour, displaying a qualitatively new and moreextreme form of unpredictability than chaotic RG ﬂows. Speciﬁcally, tra-jectories under the RG ﬂow can remain arbitrarily close together for an un-computable number of iterations before abruptly diverging to different ﬁxedpoints that correspond to separate phases (see ﬁg. 1a). Thus, the structureof the RG ﬂow — e.g. the basins of attraction of the ﬁxed points — is socomplex that it cannot be computed or approximated, even in principle. Asimilar form of unpredictability has previously been seen in classical single-particle dynamics, in seminal work by Moore [Moo90; Moo91; Ben90]. Ourresults show for the ﬁrst time that this extreme form of unpredictability canoccur in RG ﬂows of many-body systems.The unpredictability of chaotic systems arises from the fact that even atiny difference in the initial system parameters — which in practice may notknown exactly — can eventually lead to exponentially diverging trajectories(see ﬁg. 1b). However, if the system parameters are perfectly known, it is in5rinciple possible to determine the long-time behaviour of the RG ﬂow. Andthe more precisely the initial parameters are known, the longer it is possibleto accurately predict it.The RG ﬂow behaviour exhibited in this work is more intractable still.Even if we know the exact initial values of all system parameters, its RGtrajectory and the ﬁxed point it ultimately ends up at is provably impossibleto predict. Moreover, no matter how close are two sets of initial paramet-ers, it is impossible to predict how long their trajectories will remain closetogether.To substantiate these claims, we give a fully rigorous mathematicallyproof and analysis of this qualitatively new RG behaviour, for the originalundecidable many-body model in the literature [CPGW15a; CPGW15b]. Wenote that our techniques can also be adapted to establish a rigorous proof ofchaotic RG dynamics (see discussion in Section 8). We give a rigorous con-struction of the block renormalisation group [Jul+78; JP79; PJP82; BS99](BRG) map for this model. We prove that the resulting RG ﬂow convergesto the correct ﬁxed points, and preserves the order parameters and phasesof the model. Moreover, each step of the RG ﬂow is computable (in fact,efﬁciently computable). Nonetheless, the RG ﬂow itself is uncomputable: Theorem 1 (Uncomputability of RG Flows – informal statement of Theor-ems 47 and 48) . We construct an RG map for the Hamiltonian of [CPGW15a]which has the following properties:1. The RG map is computable at each renormalisation step.2. The RG map preserves whether the Hamiltonian is gapped or gapless.3. The Hamiltonian is guaranteed to converge to one of two ﬁxed pointsunder the RG ﬂow: one gapped, with low energy properties similar tothose of an Ising model with ﬁeld; the other gapless, with low energyproperties similar to the critical XY-model.4. The behaviour of the Hamiltonian under the RG mapping, and whichﬁxed point it converges to, are uncomputable.

The paper is structured as follows: in Section 2 we introduce the ne-cessary notation and formalism, give a brief overview of real-space RG ﬂowprocedures, and review the undecidable model of [CPGW15b; CPGW15a].6n Section 3 we state our main results and give a high-level overview oftheir proofs. The full proof of the main results is given in sections 4, 5, and6. Sections 4 and 5 analyse the structure of real-space RG procedures ap-plied to the undecidable model of [CPGW15b; CPGW15a]; Section 6 provesthat this RG procedure exhibits the properties and behaviour claimed in themain results. In Section 7 we discuss the properties of the ﬁxed points ofthis resulting RG ﬂow, before Section 8 concludes.

Throughout we will denote the L × H square lattice by Λ( L × H ) . If L = H we will sometimes denote the lattice as Λ( L ) . For points i, j ∈ Λ( L × H ) ,we will sometimes use (cid:104) i, j (cid:105) to denote that they are nearest neighbours. Fora Hilbert space H , B ( H ) denotes the set of bounded linear operators on H . λ ( A ) will denote the minimum eigenvalue of an operator A ∈ B ( H ) , andmore generally λ k ( A ) will denote the ( k + 1) th smallest eigenvalue. Further-more, we denote the spectral gap of an operator A as ∆( A ) = λ ( A ) − λ ( A ) .Consider local interaction terms h row , h col ∈ B ( C d ⊗ C d ) and h (1) ∈ B ( C d ) which deﬁne a translationally invariant Hamiltonian on an L × L lattice, H Λ( L ) = (cid:80) Lj =1 (cid:80) L − i =1 h rowi,i +1 + (cid:80) Li =1 (cid:80) L − j =1 h colj,j +1 + (cid:80) Li,j =1 h (1) i,j , where thesums over i and j are over rows and columns respectively.We denote the renormalisation group map by R , and the k -fold iter-ation of this map by R ( k ) . We will denote renormalised quantities andoperators with R or R ( k ) preﬁx for the renormalised and k -times renorm-alised cases respectively. For example, denote the renormalised Hamilto-nians terms as R ( h row ) i,i +1 and R ( h col ) j,j +1 , and the local terms after k -fold iterations as R ( k ) ( h row ) i,i +1 and R ( k ) ( h col ) j,j +1 . We then denote theHamiltonian deﬁned over the lattice by the renormalised interactions as R ( H ) Λ( L ) , and for the k -times iteration as R ( k ) ( H ) Λ( L ) . We note that ingeneral R ( h rowi,i +1 ) (cid:54) = R ( h row ) i,i +1 , and similarly for the other terms.If the initial local Hilbert space is H , then the local Hilbert space after k iterations of the RG map is denoted R ( k ) ( H ) . Throughout, we will denote acanonical set of local basis states by B , and after the renormalisation map-ping has been applied k times it becomes B ( k ) , so that R ( k ) ( H ) = span {| x (cid:105) ∈ ( k ) } .It will occasionally be useful to distinguish h row acting on given row j .When this is important, we write h rowi,i +1 ( j ) to denote the interaction betweencolumns i and i + 1 in the j th row. Similarly h colj,j +1 ( i ) denotes the interactionbetween rows j and j + 1 in the i th column.Finally, following [CPGW15a], we adopt the following precise deﬁni-tions of gapped and gapless: Deﬁnition 2 (Gapped, from [CPGW15a]) . We say that H Λ( L ) of Hamiltoni-ans is gapped if there is a constant γ > and a system size L ∈ N such thatfor all L > L , λ ( H Λ( L ) ) is non-degenerate and ∆( H Λ( L ) ) ≥ γ . In this case,we say that the spectral gap is at least γ . Deﬁnition 3 (Gapless, from [CPGW15a]) . We say that H Λ( L ) is gapless ifthere is a constant c > such that for all (cid:15) > there is an L ∈ N so that forall L > L any point in [ λ ( H Λ( L ) ) , λ ( H Λ( L ) ) + c ] is within distance (cid:15) from spec H Λ( L ) . We note that these deﬁnitions of gapped and gapless do not character-ise all Hamiltonians; there are Hamiltonian which ﬁt into neither deﬁnition,such as systems with closing gap or degenerate ground states. However,[CPGW15a] showed that the particular Hamiltonians they construct alwaysfall into one of these clear-cut cases, allowing sharp spectral gap undecid-ability results to be proven.

The notion of what exactly constitutes a renormalisation group scheme issomewhat imprecise, and there is no universally agreed upon deﬁnition inthe literature. We therefore start from a minimal set of conditions that wewould like a mapping on Hamiltonians to satisfy, if it is to be considered areasonable RG map. The RG scheme we deﬁne for the Hamiltonian from[CPGW15a] will satisfy all these conditions as well as additional desirableproperties.

Deﬁnition 4 (Renormalisation Group (RG) Map) . Let { h i } i be an arbitraryset of r -local interactions h i ∈ B (( C d ) ⊗ r ) , for r = O (1) and d ∈ N . A renorm- lisation group (RG) map R ( { h i } ) = { h (cid:48) i } (2.1) is a mapping from one set of r -local interactions to a new set of r (cid:48) -local in-teractions h (cid:48) i ∈ B (( C d (cid:48) ) (cid:48)⊗ r (cid:48) ) , with r (cid:48) ≤ r and d (cid:48) ∈ Z , satisfying the followingproperties:1. R ( { h i } ) is a computable map.2. Let H and R ( k ) ( H ) be the Hamiltonian deﬁned by the original localterms and the k -times renormalised local terms respectively. If H is gap-less, then R ( k ) ( H ) is gapless, as per Deﬁnition 3. If H is gapped, then R ( k ) ( H ) is gapped, as per Deﬁnition 2.3. If the order parameter for the system has a non-analyticity between twophases of H , then there is a renormalised order parameter which alsohas a non-analyticity between the two phases for R ( k ) ( H ) .4. If the initial local Hamiltonian terms can decomposed into as h i = (cid:88) j α j O j , (2.2) for some operator { O j } j , then k -times renormalised local Hamiltonianterms are of the form R ( k ) ( h ) i = (cid:88) j α ( k ) j R ( k ) ( O ) j , (2.3) where α ( k ) i = f ( { α ( k − i } i ) for some function f . The motivation for points 2 and 3 of Deﬁnition 4 is that we want to preservethe quantum phase diagram of the system. Point 3 of Deﬁnition 4 requiresthat if we start in phase A, the system should remain in phase A under theRG ﬂow: a key property of any RG scheme. Furthermore, any indicators of aphase change still occur (e.g. non-analyticity of the order parameter). Point4 asks that the “form” of the Hamiltonian is preserved.Hamiltonians under RG ﬂows have “ﬁxed points” which occur where theHamiltonian is left invariant by the action of the RG procedure. If H ∗ is the9xed point a particular Hamiltonian is converging to under the RG ﬂow, and h ∗ is the corresponding local term, then the local terms away from the ﬁxedpoint can be rewritten in terms of their deviation from the ﬁxed point as: h = h ∗ + (cid:88) i β i O i (2.4)and after renormalisation R ( k ) ( h ) = h ∗ + (cid:88) i β ( k ) i O (cid:48) i , (2.5)where if β ( k ) i → as k → ∞ then O i is said to be an irrelevant operator ; if β ( k ) i → ∞ , then O i is a relevant operator ; and if β ( k ) i → c for a constant c ,then O i a marginal operator .We note that many well-known renormalisation group schemes ﬁt thecriteria given in Deﬁnition 4 when applied to the appropriate Hamiltoni-ans. In the following subsections, we review a number of these. However,in general, a given RG scheme may satisfy the conditions for the family ofHamiltonians it was designed for, but will not necessarily satisfy all the de-sired conditions when applied to an arbitrary Hamiltonian. We base our RG map on a blocking technique widely used in the literatureto study spin systems, often called the Block Spin Renormalisation Group(BRG) [Jul+78; JP79; PJP82; BS99]. Modiﬁcations and variations of thisRG scheme have also been extensively studied [MDS96; WKL02].The BRG is among the simplest RG schemes. The procedure works bygrouping nearby spins together in a block, and then determining the associ-ated energy levels and eigenstates of this block by diagonalisation. Havingdone this, high energy (or otherwise unwanted) states are removed result-ing in a new Hamiltonian.As an explicit example, suppose there exists a Hamiltonian on a 1D chain H = N − (cid:88) i =1 K (0) h (0) i,i +1 + C (0) N (cid:88) i =1 i . (2.6) This is also sometimes called the “quantum renormalisation group”. H = K (0) N − (cid:88) i odd h (0) i,i +1 + K (0) N − (cid:88) i even h (0) i,i +1 + C (0) N (cid:88) i =1 i . (2.7)We then diagonalise the operators for odd i . (In higher dimensional geomet-ries we group the terms into blocks of neighbouring qudits.) Having donethis, remove all “high energy states” within each block, either by introdu-cing an energy cut-off or just keeping a chosen subset of the lowest energystates. The produces a renormalised Hamiltonian R (1) ( H ) = K (1) N/ − (cid:88) i =1 h (1) i,i +1 + b (1) N/ (cid:88) i =1 h (1) i + C (1) N/ (cid:88) i =1 i . (2.8)For each further RG iteration the same process is repeated: the terms h i,i +1 for odd i are diagonalised and the high energy states are removed.After k iterations, the RG procedure returns a Hamiltonian of the sameform, but now with different coupling constants: R ( n ) ( H ) = K ( n ) N/ − (cid:88) i =1 h ( n ) i,i +1 + b ( n ) N/ (cid:88) i =1 h ( n ) i + C ( n ) N/ (cid:88) i =1 i . (2.9) Form of the RG Mapping

This BRG mapping can be reformulated in terms of a series of isometries (orunitaries and subspace restrictions). Given the local terms of some Hamilto-nian, h i,i +1 ∈ B ( C d ⊗ C d ) , we will consider renormalisation mappings of theform R : h i,i +1 → V † h i,i +1 V (2.10)where V : C d → C d (cid:48) is an isometry which will take a states in the initial setof basis states to a restricted new set of renormalised basis states.Equivalently we can formulate this in terms of a unitary U and a sub-space Γ , as: R : h i,i +1 → U † h i,i +1 U | Γ . (2.11)The unitary U maps the original basis states to the new set (called blocking ).11his is followed by a restriction to the subspace Γ which is the “low-energy”subspace: that is, all basis states which locally pick up too much energyare removed. This subspace restriction is called truncating . In our particu-lar variation of the BRG, the truncation step is not done entirely based onenergy truncation, but also on overlap with a particular state. Classical 1D Ising Model

A particularly famous RG scheme which satisﬁes Deﬁnition 4 is the decima-tion scheme for the classical 1D Ising model [Car96]. Here the ground statesare trivially either all σ i = 1 or − . Under the decimation RG procedure,half the spins are removed by “averaging out” the others. The order para-meter for the phase is the magnetisation: M = (cid:80) Ni =1 σ i and it can be seento undergo a non-analytic change between phases. This is true even afterrenormalisation, thus satisfying point 3 of Deﬁnition 4.The decimation mapping further gives a transformation of the form R : J (cid:88) σ i σ i +1 + h (cid:88) i σ i + CN → J (cid:48) (cid:88) σ i σ i +1 + h (cid:48) (cid:88) i σ i + C (cid:48) N, (2.12)thus satisfying condition 4. It can also be shown [Car96] that the RG pro-cedure preserves the phase of the Ising model, and hence satisﬁes condition3. MERA

A more recent and widely studied RG ﬂow scheme in the quantum in-formation literature is the multiscale entanglement renormalisation ansatz(MERA) developed in [Vid08]. This is implemented by iteratively apply-ing isometries to the local terms to produce new local Hamiltonian termsand density matrices. This (approximately) preserves expectation valuesand hence can often be made to satisfy 3. Whether conditions 2 and 4 aresatisﬁed is dependent on the Hamiltonian and isometries in question.12 .3 Properties of the Spectral Gap Undecidability Construction

Constructing a mathematically rigorous RG ﬂow for the undecidable Hamilto-nian exhibited in [CPGW15b; CPGW15a] presents particular challenges,since its properties are uncomputable. Nonetheless, we are able to by care-fully analysing the local structure and properties of this Hamiltonian, whichwe review here.We start by stating the main result in [CPGW15a], where the authorsconstruct a Hamiltonian depending on one external parameter, which isgapped iff a universal Turing Machine halts on an input related to the Hamilto-nian parameter. The spectral gap problem for this Hamiltonian is thereforeequivalent to the Halting Problem, hence undecidable.

Deﬁnition 5 (From theorem 3 of [CPGW15a]) . For any given universal Tur-ing Machine (UTM), we can construct explicitly a dimension d , d × d matrices A, A (cid:48) , B, C, D, D (cid:48) , Π and a rational number β which can be as small as desired,with the following properties:1. A is diagonal with entries in Z .2. A (cid:48) is Hermitian with entries in Z + √ Z ,3. B, C have integer entries,4. D is diagonal with entries in Z ,5. D (cid:48) is Hermitian with entries in Z .6. Π is a diagonal projector.For each natural number n , deﬁne: h ( n ) = α ( n )Π ,h col ( n ) = D + βD (cid:48) , independent of nh row ( n ) = A + β (cid:16) A (cid:48) + e iπϕ B + e − iπϕ B † + e iπ −| ϕ | C + e − iπ −| ϕ | C † (cid:17) , where α ( n ) ≤ β is an algebraic number computable from n and | ϕ | denotesthe length of the binary representation of ϕ . Then:1. The local interaction strength is bounded by 1, i.e. max( (cid:107) h ( n ) (cid:107) , (cid:107) h row ( n ) (cid:107) , (cid:107) h col ( n ) (cid:107) ) ≤ . . If UTM halts on input n , then the associated family of Hamiltonians { H Λ( L ) ( n ) } is gapped with gap γ ≥ .3. If UTM does not halt on input n , then the associated family of Hamilto-nians { H Λ( L ) ( n ) } is gapless. We ﬁrst explain the overall form of the Hamiltonian and the Hilbert spacestructure, and later how the individual parts ﬁt together.

The Hamiltonian H u ( ϕ ) is constructed such that its ground state is com-posed of two components: a classical “tiling layer” and a highly entangled“quantum layer”. The local Hilbert space decomposes as: H u = H c ⊗ ( H q ⊕ | e (cid:105) ) , (2.13)where H c is the Hilbert space corresponding ot the classical tiling layer and H q ⊕ | e (cid:105) is the “quantum” layer. The local terms h u are constructed as h u = h ( i,i +1) T ⊗ ( i ) eq ⊗ ( i +1) eq + ( i ) c ⊗ ( i +1) c ⊗ h ( i,i +1) q + “coupling terms” . (2.14)Let h ( i,j ) u ∈ B ( C d ⊗ C d ) be the local terms of the Hamiltonian H u , h ( i,j ) d ∈B ( C ⊗ C ) be the local interactions of the 1D critical XY model, and let H d be the Hamiltonian composed of XY interactions along the rows of thelattice. This has a dense spectrum in the thermodynamic limit [LSM61]. h ( i,j ) u = h ( i,j ) u ( ϕ ) is designed so that H u ( ϕ ) = (cid:80) h u ( ϕ ) has a ground stateenergy which depends on whether a universal Turing Machine (UTM) haltswhen given on input ϕ supplied in binary. In particular, on a lattice of size L × L , the ground state energy is λ ( H Λ( L ) u ) = − Ω( L ) if UTM does not halt on input ϕ , +Ω( L ) if UTM does halt on input ϕ . (2.15)Since the halting problem is undecidable, determining which of the twoground state energies of H u ( ϕ ) occurs is undecidable.The local Hilbert space of the overall Hamiltonian can be decomposed14s: H = | (cid:105) ⊕ H u ⊗ H d . (2.16)Here | (cid:105) is a zero-energy ﬁller state, H d is the Hilbert space associated withthe d ense spectrum Hamiltonian h d , and H u is the Hilbert space associatedwith the Hamiltonian with u ndecidable ground state energy h u .The local interactions along the edges and on the sites of the lattice areact on this local Hilbert space as: h ( ϕ ) ( i,j ) = | (cid:105) (cid:104) | ( i ) ⊗ ( − | (cid:105) (cid:104) | ) ( j ) + h ( i,j ) u ( ϕ ) ⊗ ( i,j ) d + ( i,j ) u ⊗ h ( i,j ) d (2.17) h ( ϕ ) (1) = − (1 + α )Π ud , (2.18)where Π ud is a projector onto H u ⊗H d , and α = α ( | ϕ | ) is a constantdepending only on | ϕ | . Importantly, the spectrum of the overall latticeHamiltonian composed of these local interactions is spec H ( ϕ ) = { } ∪ { spec( H u ( ϕ )) + spec( H d ) } ∪ S, (2.19)for a set S with all elements > . This means that if λ ( H Λ( L ) u ) → −∞ thenthe overall Hamiltonian has a dense spectrum, while if λ ( H Λ( L ) u ) → + ∞ the overall Hamiltonian has a spectral gap > .In the λ ( H Λ( L ) u ( ϕ )) = +Ω( L ) case, the ground state of the entireHamiltonian is | (cid:105) Λ . In the λ ( H Λ( L ) u ( ϕ )) = − Ω( L ) case, the overall groundstate is | ψ u (cid:105) ⊗ | ψ d (cid:105) where | ψ u (cid:105) and | ψ d (cid:105) are the ground states of H u ( ϕ ) and H d = (cid:80) i ∈ Λ h i,i +1 d respectively.We now explain the terms h T and h q as well as the cumulative effects ofthe coupling terms. The Tiling Hamiltonian

Wang tiles are square tiles of unit length withmarkings on each side, together with rules stipulating that a pair of tiles canonly be placed next to each other if the markings on their adjacent sidesmatch. In [CPGW15a] the tile set is chosen to be a slightly modiﬁed versionthe Robinson tiles from [Rob71], shown in ﬁg. 2a. When placed on a 2Dgrid such that the tiling rules are satisﬁed, the markings on the tiles form an15 a) The modiﬁed Robinson tiles used in[CPGW15a]. (b)

Ground state | T (cid:105) c of the classicalHamiltonian. Figure 2 aperiodic tiling consisting of a series of nested squares of sizes n + 1 , for all n ∈ N , as shown in ﬁg. 2b.This set of tiles can then be mapped to a 2D, translationally invari-ant, nearest neighbour, classical Hamiltonian by simply mapping each tiletype to a state in the local Hilbert space and introducing local interac-tions that apply an energy penalty to neighbouring pairs which do not sat-isfy the tiling rules. That is, the local terms are deﬁned as ( h T ) i,i +1 := (cid:80) ( t α ,t β ) (cid:54)∈ A | t α , t β (cid:105)(cid:104) t α , t β | i,i +1 where A is the set of allowed neighbouring tiles.Then, the ground state of the entire 2D lattice, | T (cid:105) c , corresponds to theRobinson tiling pattern as shown in ﬁg. 2b. Any other conﬁguration mustviolate a tiling rule and thus receives an energy penalty. The Quantum Hamiltonian H u ( ϕ ) is constructed so that its ground stateenergy encodes the halting or non-halting of a computation. The funda-mental ingredient required is the “QTM-to-Hamiltonian” mapping [GI09;CPGW15a]. This takes a given quantum QTM and creates a correspondingHamiltonian which has a ground state which encodes its evolution. Thisquantum state is called a history state . Let | ψ t (cid:105) be the state describing theconﬁguration of the QTM after t steps. Then the history state takes thegeneral form | Ψ hist (cid:105) = 1 √ T T (cid:88) t =1 | ψ t (cid:105) | t (cid:105) , (2.20)16here | t (cid:105) is a state labelling which step of the computation | ψ (cid:105) correspondsto. It is then possible to add a local projector term to the Hamiltonian whichgives an additional energy penalty to certain outcomes of the computation.In particular, [CPGW15a] penalise the halting state, so that if the QTM haltsat some point, the Hamiltonian deﬁned by h q picks up an additional energycontribution. As a result, the energy of the ground state differs dependingon whether or not the QTM halts within time T .In particular, [CPGW15a] adapt the QTM-to-Hamiltonian mapping ori-ginally developed by Gottesman and Irani [GI09], which takes a QTM andmaps its evolution to 1D, translationally invariant, nearest neighbour, Hamilto-nian. By H q we denote this modiﬁed version of the Gottesman-Irani Hamilto-nian (cf. Subsection 2.4).The length of the computation encoded on a chain of length L is T ( L ) ∼ poly( L )2 L , and the associated ground state energy is λ ( H q ( L )) = if QTM is non-halting within time T ( L ) , θ (1 /T ) if QTM halts within time T ( L ) . (2.21)We give a more detailed analysis of the construction at the beginning ofSection 5. Combining h T , h q and the Coupling Terms The terms h u are designedso that all eigenstates of H Λ( L ) u are product states | T (cid:105) c ⊗ | ψ (cid:105) eq where | T (cid:105) ∈H ⊗ ( L × L ) c and | ψ (cid:105) ∈ H ⊗ ( L × L ) eq [CPGW15a, Lemma 51].Furthermore, the coupling terms are chosen such that the ground statehas the following properties:1. the classical part of the ground state | T (cid:105) c corresponds to a perfectRobinson tiling. The pattern created has a series of nested red Robin-son squares as per ﬁg. 2b.2. the quantum part of the ground state | ψ (cid:105) eq has the following structure:along the top of every red Robinson square there is a history state (asdeﬁned in eq. (2.20)); everywhere which is not along the top of asquare is in the zero energy ﬁller state | e (cid:105) e .The consequence of this is that ground states of H q ( (cid:96) ) of all lengths17ppear with a constant density across the lattice. If, for any length, the en-coded computation halts, then the ground state picks up a constant energydensity, so that the energy scales as Ω( L ) . However, if the encoded compu-tation never halts, then for all lengths the ground state of the Gottesman-Irani Hamiltonian has zero energy, and (due to boundary effects), the groundstate has energy − Ω( L ) [CPGW15a]. The particular circuit-to-Hamitonian mapping used in the previous sectionwill be important when it comes to renormalising the overall Hamiltonian.The overall structure used in [CPGW15a] is a modiﬁcation of the one usedin [GI09].We start by deﬁning one of the core concepts behind the construction anbehaviour of the Hamiltonian of [CPGW15a]: the Quantum Turing Machine(QTM).

Deﬁnition 6 (Quantum Turing Machine [BV97]) . A quantum Turing Ma-chine (QTM) is deﬁned by a triplet (Σ; Q ; δ ) where Σ is a ﬁnite alphabet withan identiﬁed blank symbol , Q is a ﬁnite set of states with an identiﬁed initialstate q and ﬁnal state q f (cid:54) = q , and δ is the quantum transition function δ : Q × Σ → C Σ ×Q× [ L,R ] (2.22) The QTM has a two-way inﬁnite tape of cells indexed by Z and a single read/writetape head that moves along the tape. A conﬁguration of the QTM is a com-plete description of the contents of the tape, the location of the tape head andthe state q ∈ Q of the ﬁnite control. At any time, only a ﬁnite number of thetape cells may contain non-blank symbols. The initial conﬁguration satisﬁesthe following conditions: the head is in cell , called the starting cell, and themachine is in state q .We say that an initial conﬁguration has input x ∈ (Σ \{ } ) ∗ if x is writtenon the tape in positions , , , . . . and all other tape cells are blank. The QTMhalts on input x if it eventually enters the ﬁnal state q f . The number of steps aQTM takes to halt on input x is its running time on input x .Let S be the inner-product space of ﬁnite complex linear combinations of con-ﬁgurations of the QTM M with the Euclidean norm. We call each element φ ∈ S a superposition of M . he QTM M deﬁnes a linear operator U M : S → S , called the time evolutionoperator of M , as follows: if M starts in conﬁguration c with current state p and scanned symbol σ , then after one step M will be in superposition of conﬁg-urations ψ = (cid:80) j α j c j , where each non-zero α j corresponds to the amplitude δ ( p ; σ ; τ ; q ; d ) of | τ (cid:105) | q (cid:105) | d (cid:105) in the transition δ ( p ; σ ) and c j is the new conﬁgur-ation obtained by writing τ , changing the internal state to q and moving thehead in the direction of d . Extending this map to the entire S through linearitygives the linear time evolution operator U M . Following [GI09], the QTM can be encoded into a 1D, translationally-invariant, nearest-neighbour Hamiltonian, which we refer to as a Gottesman-Irani Hamiltonian , denoted by H q ( L ) ∈ B (( C d ) ⊗ L ) . This is summarised bytheorem 32 of [CPGW15a]; we write out a slightly simpler version here asthe speciﬁc details are not important for our purposes. These constructionswill be needed in order to formulate Lemma 28 for the block-renormalisationof the quantum Hamiltonian. Theorem 7 (Informal Version of Theorem 32 of [CPGW15a]) . Let C d = C C ⊗ C Q be the local Hilbert space of a 1-dimensional chain of length L , with special marker states (cid:12)(cid:12) (cid:11) , (cid:12)(cid:12) (cid:11) . Denote the orthogonal complement of span( (cid:12)(cid:12) (cid:11) , (cid:12)(cid:12) (cid:11) ) in C d by C d − . Let d, Q and C all be ﬁxed.For any well-formed unidirectional Quantum Turing Machine M = (Σ , Q, δ ) and any constant K > , we can construct a two-body interaction h ∈ B ( C d ⊗ C d ) such that the 1-dimensional, translationally-invariant, nearest-neighbour Hamilto-nian H ( L ) = (cid:80) L − i =1 h ( i,i +1) ∈ B ( H ( L )) on the chain of length L has the fol-lowing properties:1. d depends only on the alphabet size and number of internal states of M .2. h ≥ , and the overall Hamiltonian H ( L ) is frustration-free for all L .3. Denote H ( L −

2) := ( C d − ) ⊗ L − and deﬁne S br = span( (cid:12)(cid:12) (cid:11) ) ⊗H ( L − ⊗ span( (cid:12)(cid:12) (cid:11) ) ⊂ H . Then the unique ground state of H ( L ) | S br is acomputational history state (cf. eq. (2.20) for a deﬁnition) encoding theevolution of M .Moreover, the action of M satisﬁes:1. The computational history state always encodes Ω(2 L ) time-steps. If M halts in fewer than the number of encoded time steps, exactly one | ψ t (cid:105) as support on a state |(cid:62)(cid:105) that encodes a halting state of the QTM. Theremaining time steps of the evolution encoded in the history state leave M ’s tape unaltered, and have zero overlap with |(cid:62)(cid:105) .2. If M runs out of tape within a time T less than the number of encodedtime steps, the computational history state only encodes the evolution of M up to time T . The remaining steps of the evolution encoded in thecomputational history state leave M ’s tape unaltered. We provide in the following a more detailed sketch of how the modiﬁedGottesman-Irani construction works, and refer the reader to [CPGW15a;GI09] for a detailed overview. We begin by considering the general setup.Our basis states for ( C d ) ⊗ L (i.e. the chain of length L ) will have the follow-ing structure: · · · Track 1: Clock oscillator · · ·· · ·

Track 2: Counter TM head and state · · ·· · ·

Track 3: Counter TM tape · · ·· · ·

Track 4: QTM head and state · · ·· · ·

Track 5: QTM tape · · ·· · ·

Track 6: Time-wasting tape · · ·

The local Hilbert space at each site is the tensor product of the local Hilbertspace of each of the six tracks H = (cid:78) i =1 H i .The outline of the construction is the following: tracks 1 encodes theevolution of an oscillator which goes back and forth along its track as perﬁg. 3 Tracks 2 and 4 contain the heads of a classical and quantum TM re- Figure 3:

Evolution of the Track 1 clock oscillator. C C ⊗ C Q . Back to claim 3 of Deﬁnition 4, we now discuss order parameters in moredetail. As noted in [BCW21], the two phases of the Hamiltonian (which welabel A and B for convenience) can be distinguished by an order parameter O A/B , deﬁned as O A/B = 1 | Λ | (cid:88) i ∈ Λ | (cid:105)(cid:104) | ( i ) . (2.23)In particular, upon moving from one phase to another, the expectation valueof the order parameter is expected to undergo a non-analytic change. Inthe case λ ( H Λ( L ) u ( ϕ )) = +Ω( L ) the ground state of the entire Hamiltonianis then | (cid:105) Λ and hence (cid:104) O A/B (cid:105) = 1 , and otherwise (cid:104) O A/B (cid:105) = 0 . This istrue even if we restrict O A/B to subsections of the lattice, hence O A/B is alocal order parameter (as opposed to the global order parameters required Phase in this context refers to the state of matter, not a quantum mechanical phase factor(of the form e iθ ).

21o distinguish topological phases). Thus O A/B undergoes a non-analyticchange between phases, which itself demonstrates a phase transition. Moregenerally for a ball B ( r ) of radius r , and for a state | ν (cid:105) ∈ H ⊗ Λ we can deﬁnea local observable O A/B ( r ) = 1 | B ( r ) | (cid:88) i ∈ B ( r ) | (cid:105)(cid:104) | ( i ) , (2.24)which acts as a local order parameter. Theorem 8 (Exact RG ﬂow for Undecidable Hamiltonian) . Let H ( ϕ ) be theHamiltonian deﬁned in [CPGW15a]. We construct a renormalisation groupprocedure for the Hamiltonian which has the following properties:1. R is computable.2. If H ( ϕ ) is gapless, then R ( k ) ( H ( ϕ )) is gapless, and if H ( ϕ ) is gapped,then R ( k ) ( H ( ϕ )) is gapped (where gapped and gapless are deﬁned inDeﬁnition 2 and Deﬁnition 3).3. For the order parameter O A/B ( r ) (as deﬁned in eq. (2.24) ) which dis-tinguishes the phases of H Λ( L ) and is non-analytic at phase transitions,there exists a renormalised observable R ( k ) ( O A/B ( r )) which distinguishesthe phases of R ( k ) ( H ) Λ( L ) and is non-analytic at phase transitions.4. Under an arbitrary number of iterations, the renormalised local interac-tions belong to a family F ( ϕ, τ , τ , { α i } i , { β i } i ) , and for any ﬁnite k allof the parameters are computable.5. If H ( ϕ ) initially has algebraically decaying correlations, then R ( k ) ( H ( ϕ )) also has algebraically decaying correlations. If H ( ϕ ) initially has zerocorrelations, then R ( k ) ( H ( ϕ )) also has zero correlations. Theorem 9 (Uncomputability of RG ﬂows) . Let h ( ϕ ) , ϕ ∈ Q , be the fulllocal interaction of the Hamiltonian from [CPGW15a]. H ( ϕ ) := (cid:80) h ( ϕ ) ( i,j ) is gapped if the UTM corresponding to h ( ϕ ) halts on input ϕ , and gapless ifthe UTM never halts, where gapped and gapless are deﬁned in Deﬁnition 2 and eﬁnition 3. Consider k iterations of the RG scheme (deﬁned later in Deﬁni-tion 42) acting on H ( ϕ ) , such that the renormalised local terms are given by R ( k ) ( h ( ϕ )) , which can be parameterised as part of the family F ( ϕ, τ , τ , { α i } i , { β i } i ) (as per Corollary 44). Then, if the UTM is non-halting on input ϕ , for all k >k ( ϕ ) , τ ( k ) = − k , for some computable k ( ϕ ) . If the UTM is halting on input ϕ , then there exists an uncomputable k h ( ϕ ) such that for k ( ϕ ) < k < k h ( ϕ ) , τ ( k ) = − k , and for all k > k h ( ϕ ) then τ ( k ) = − k + Ω(4 k − k h ( ϕ ) ) . A direct consequence of this is:

Corollary 10.

Determining which ﬁxed point the Hamiltonian ﬂows to underthis RG scheme is undecidable.

The overall RG scheme is explicitly given in Deﬁnition 42, and the family F ( ϕ, τ , τ , { β i } ) which the renormalised Hamiltonians belong to is given inCorollary 44. One of the consequences of Theorem 9 is that the Hamiltonianis guaranteed to ﬂow towards one of two ﬁxed points. However, determin-ing which ﬁxed point it ﬂows to for a given value of ϕ is undeciable.The undecidability of the ﬁxed point follows implicitly from undecidab-ility of the spectral gap [CPGW15a; CPGW15b], since the ﬁxed point de-pends on the gappedness of the unrenormalised Hamiltonian. However,Theorem 9 shows precisely how the trajectory of the Hamiltonian in para-meter space diverges in an uncomputable manner under RG ﬂow. The renormalisation group scheme we will employ will be a variant of theBRG described in Subsection 2.2.1, where we block × groups of spins toa single “super-spin” which preserves some of the properties of the originalset. Due to the complexity of the Hamiltonian in consideration, we willﬁrst renormalise the different parts h u , h d , | (cid:105) of the Hamiltonian separately,then combine these RG maps into the complete map. For a ﬁnite size lattice, h u has a ground state which is product between H C and H q ⊕ | e (cid:105) . This keyproperty allows us to essentially renormalise the tiling Hamiltonian and theGottesman-Irani Hamiltonian separately. Renormalising the Tiling Hamiltonian

Fig. 2b shows that the ground state of the tiling Hamiltonian corresponds23o a particular pattern; notably the Robinson tiling creates a self-similarpattern for across all sizes of squares, where smaller squares are nestedwithin larger ones. We design a blocking procedure which takes a set of × Robinson tiles, then maps them onto a single new tile which has the samemarkings and tiling rules as one in the original set of Robinson tiles. Doingthis we recover a set of tiles which recreate the Robinson tiling pattern, butnow with the smallest squares “integrated out”. Repeated iterations of thisprocess still preserve the Robinson tiling pattern. The details are give inSection 4.

Renormalising the Gottesman-Irani Hamiltonian

The Gottesman-Irani Hamiltonian h q is a 1D Hamiltonian which serves as aQTM-to-Hamiltonian map. As noted in section Subsection 2.3, in the groundstate of (cid:80) h u , ground states of Gottesman-Irani Hamiltonians appear alongthe top edge of the Robinson tiles. We aim to design an RG scheme suchthat the energy of the Gottesman-Irani ground state attached to a squareremains the same even when the square size is halved. To do this, we mappairs of spins to a new “combined spin” which now has local Hilbert spacedimension d if the original dimension is d . As with the BRG, we considerthe new 1-local terms and diagonalise them. Since we know the form ofthe ground state explicitly, it is possible to identify states which pick up toomuch energy to have overlap with the ground state. We can truncate thelocal Hilbert space by removing these states and hence reduce the dimen-sion of the combined spin to something < d (but still > d ). This blockingprocedure will preserve whether the Hamiltonian has a zero energy groundstate or a ground state with energy > .In mathematical terms, the procedure is implemented by a series of iso-metries which are used to map the original states to the new blocked states,and then subspace restrictions which remove the high energy states. This issummarised in Lemma 28. We refer the reader to Section 5 for full details. Renormalising h u Since h u = h ( i,i +1) T ⊗ ( i ) eq ⊗ ( i +1) eq + ( i ) c ⊗ ( i +1) c ⊗ h ( i,i +1) q + coupling terms, to renormalise it, we do the following:• Choose a × block of spins.• Renormalise the classical tiling part of the Hamiltonian as above.24 To renormalise the quantum part of the Hamiltonian, break the × block into two × blocks. Renormalise these two sections as theabove renormalisation for the Gottesman-Irani Hamiltonian. The × block is now a × block.• Trace out part of the Hilbert space such the × block is now a singlesite in the renormalised Hilbert space such that we are left with 1-local and 2-local projector terms which introduce an energy shift. Thisenergy shift exactly compensates for any energy lost in the integratingout operation.The above can be shown to preserve the ground state energy in the desiredway. See Deﬁnition 36 in Subsection 6.1 for the complete description. Renormalising the Entire Hamiltonian

We renormalise h d and | (cid:105)(cid:104) | in atrivial way such that their properties are preserved. Thus the overall renor-malisation scheme acts on h u as above, and essentially leaves h d and | (cid:105)(cid:104) | unchanged.Since h u , h d and | (cid:105)(cid:104) | have their respective ground state energies pre-served (approximately), whether the ground state is | (cid:105) Λ or the more com-plex ground state of the tiling+quantum Hamiltonian, is preserved. Im-portantly it can be shown the spectral gap of both cases is preserved. TheRG process can then be iterated arbitrarily many times: we show the rel-evant properties are preserved throughout. Determining the properties ofthe ground state and spectral gap are undecidable for the unrenormalisedHamiltonian, and since these properties are preserved by the RG mapping,it is also undecidable for the renormalised Hamiltonians.The renormalisation of the entire Hamiltonian is given in detail in Sec-tion 6. In the following we will construct an RG map under which the two graphsrepresenting respectively the adjacency relations (roughly speaking, the rulestelling us what tiles can stay above / below / left / right of a given tile) forthe Robinson tiles and for a speciﬁc subset of × supertiles are isomorphic.25his implies that the pattern produced by the tiling of the 2D plane usingRobinson tiles is scale-invariant. This property is crucial in order to ensurethat the density of the Gottesman-Irani ground states (corresponding to thetop edges of the squares appearing in the pattern) which encode the QTMis preserved under the renormalisation procedure.More formally, we have that Theorem 11. (Adjacency Rules Isomorphism) Let T be the set of Robinsontiles and A be the corresponding adjacency rules. Let T be the set of × supertiles, obtained from all combinations allowed by A of four Robinson tilesplaced in a × square, and A be the adjacency rules of T , derived fromthe principle that two supertiles can be placed next to each other only if theRobinson tiles on the edges that are put adjacent respect A . Then there existsa subset T (cid:48) ⊂ T , | T (cid:48) | = | T | = 56 , with tiling rules A (cid:48) = A | T (cid:48) , and abijection T (cid:48) → T under which A and A (cid:48) are equivalent. From this result it follows that (cf. Appendix A)

Corollary 12. (Scale Invariance of the Robinson Tiling) Under the bijection inTheorem 11, the Robinson tiling pattern is preserved under the × → × renormalisation of the grid. We can then translate this scale invariance into a statement about theproperties of the Hamiltonian which describes the Robinson tiling, i.e.,

Theorem 13 ((Informal) Robinson Tiling Hamiltonian Renormalisation) . Let h T ∈ C T ⊗ C T be the local interactions which describe the Robinson tilingHamiltonian. Then there exists a renormalisation group mapping R T satisfy-ing R T ( h T ) = h (cid:48) T , where h (cid:48) T ∈ C T ⊗ C T , such that R T ( h T ) preserves both theground state energy and the tiling pattern. Before the explicit construction of the re-scaling transformation, we shallrecall the Robinson tiles and their adjacency rules.

Two tiles can be placed adjacent to each other only if their arrows are com-patible. That is, the head(s) of the arrow(s) in one tile and the tail(s) of the26 cross tiles × ×× × ×× × ×× × ×× × ×× × ×× × ×× × ×× × ×× × ×× × ×× × ××

Figure 4 arrow(s) in the other tile must match exactly on the edges put into contact.We refer to [Rob71] for a complete description.Recall that there are 28 different arrow markings in the Robinson tilesset, which we list in ﬁg. 4. Following Robinson, these arrow markings areaugmented with 4 parity tiles in a way that gives rise to 56 total differenttiles. More precisely, we consider the coloured tiles given in ﬁg. 5, whichfollowing Robinson we call the parity tiles , satisfying the tiling rules statingthat only borders with the same colour can be placed next to each other.Each parity tile can be thought of as being attached to a Robinson tile inanother layer. Thus, tiles are only allowed to be placed next to each other ifboth their Robinson markings and parity markings match along the edge incontact.We will use the following terminology from [Rob71]: cross tiles matched will be matched with horizontal tileswill be matched with cross tiles will be matched with any tilewill be matched with vertical tiles

Figure 5 a) The four parity structures of a × cell (b) pattern of the parity structure of theplane Figure 6 with the red/blue parity tile will be called “parity crosses”; horizontal tilescoupled with green/blue tiles will be denoted as “parity horizontal” andanalogously vertical tiles linked to red/yellow parity will be called “parityvertical”. Conversely, any tile associated to the green/yellow tile will becalled “free”, so we will have “free crosses / horizontal / vertical” tiles.When building adjacency rules, both arrow and parity rules must be obeyed.Parity tiles will force the following structure. Considering the plane asa grid of cells where the tiles are to be placed, then parity cross tiles willappear in alternating rows and alternating columns. The same applies forparity horizontal and parity vertical tiles. Thus, if we consider a grid of × blocks over the plane, each × supertile will have the same inner paritystructure. Depending on where we place the grid, we will obtain one of theconﬁgurations illustrated in ﬁg. 6a, repeated over the whole plane. In this section we will provide the proof of Theorem 11. When changing thegrid size, we go from × Robinson tiles to × supertiles. As we notedabove, depending on the positioning of the grid, we will obtain one of theinner parity structures given in ﬁg. 6a . From this point we will consider theﬁrst supertile on the top-left in ﬁg. 6a, that is, the one with the parity crosson the bottom-left. The parity structure of the plane will then look as shown28n ﬁg. 6b.With this parity structure in mind, we generate all × supertiles permit-ted by the arrows rules. There are a total of 68 such supertiles, that we willcall allowed supertiles . Our aim is to identify a bijection between a subsetof these 56 supertiles and the Robinson tiles that leads to Theorem 11. Inother words, we consider the adjacency relations of the × tiles: they willgenerate a directed graph . We want to prove that this graph is isomorphic tothe one describing the relations of the original Robinson tiles.Interestingly, from the approach that aims to replicate the Robinson pat-tern with supertiles described in Appendix A, we observe that we can for-mulate the projection from × to × tiles by looking at the two tiles thatoccupy the bottom-left and the top-right position of the supertile. Indeed,once the tiles on the bottom-left and top right position are placed, there isonly one possible choice for the two remaining tiles, which must also obeythe inner parity structure of the supertile (see ﬁg. 6a). This fact leads to thefollowing deﬁnition of the renormalisation map. Deﬁnition 14 (Renormalisation Map) . Given an allowed × supertile, weconsider its top-right tile with free parity, that we denote by T , and the paritycross on the bottom-left position, that we call C . The associated Robinsontile under the renormalisation map has the same marking as T and paritycharacterised by C according to the correspondence given in ﬁg. 7. Figure 7

We have veriﬁed in a

Mathematica notebook that, under the map inDeﬁnition 14, the adjacency relations of the Robinson tiles and the ones of The notebook is included in the supplementary material of the arXiv submission

29 subset of 56 allowed supertiles are equivalent, which proves Theorem 11.Refer to Appendix B for more details.Under this projection the supertiles that do not appear in the tiling of theplane, illustrated in Subsection 4.3, are not mapped to any Robinson tile.The reason for this is that there does not exist a Robinson tile with matchingof arrows and parity: the supertiles of type 1 in ﬁg. 8 would be mapped tovertical arms with horizontal parity, and conversely type 2 supertiles wouldcorrespond to horizontal arms with vertical parity.

Consider the subset of supertiles allowed by the adjacency rules which havea × parity cross in the bottom-left. There are 68 such tiles, however,there are only 56 Robinson tiles. The result of this is that 12 tiles cannotbe mapped under the renormalisation procedure. These have two distinctstructures, as shown in ﬁg. 8, where we have used the abbreviated notationused in [Rob71], indicating only the direction of the arms. Figure 8

For each of these two structures, we have 6 possible combinations. Thosesupertiles cannot appear in any tiling of the plane. Consider the structure1. By imposing the parity rules for supertiles that we described previously, asupertile of type 1 there must have one of the parity cross supertiles aboveit, which has the (abbreviated) form shown in ﬁg. 9. Clearly, no supertilewith the structure 1 can be placed below a parity cross supertile because ofthe arrow rules. Analogously, by parity rules, on the right of a supertile withstructure 2 must lie a parity cross supertile. Again, it is clear that this is notallowed by the arrow rules. 30 igure 9

The previous analysis was done by placing a × grid over the Robinsonpattern on the plane with the parity cross lying on the bottom-left of each × cell. Naturally, there are 4 possible ways that we could place our × grid. The same investigation has been performed for all other threecases when shifting the × grid one cell right, upwards, and diagonally,so that the parity cross will occupy the bottom-right, top-left and top-rightposition of the supertiles, respectively. The analysis for these other settingsis completely equivalent to the case we have discussed. For any of the fourpositioning of the parity cross in the × grid, there exist 68 allowed su-pertiles, and a subset of 56 will tile the plane and build adjacency relationsisomorphic to the Robinson tiles. Remark 15.

The four possible placements of the × grid give rise to fourcompletely disjoint sets of 68 supertiles each. This is a direct consequence of thedifferent inner parity structure of the supertiles illustrated in ﬁg. 6a. Let the set of possible renormalised supertiles, but restricted to thosewhich actually appear, be denoted: T ⊕ T ⊕ T ⊕ T . (4.1)which each T i corresponding to a different supertile parity structure. Thenthe set of supertiles that occurs for our choice of basis, T (cid:48) , is equal to one ofthese four disjoint sets. Which set occurs depends on where the × grid isplaced. 31 .5 Renormalising the Classical Hamiltonian We are now in a position to show that there is an RG transformation onthe tiling Hamiltonian which preserves the ground state. In terms of theHamiltonian, the RG scheme takes the form of restricting to sets of allowed × blocks, and then applying an isometry mapping these × blocks tonew supertiles belonging to the set T (cid:48) . The Initial Tiling Hamiltonian

Let h rowT = (cid:80) ( t i ,t j ) (cid:54)∈ A H | t i t j (cid:105) (cid:104) t i t j | and h rowT = (cid:80) ( t i ,t j ) (cid:54)∈ A V | t i t j (cid:105) (cid:104) t i t j | be thelocal interaction terms of the tiling Hamiltonian, where A H and A V are, re-spectively, the horizontal and vertical adjacency rules for tiles in T . Thenthe ground state of H = (cid:80) i ∈ Λ( L ) h rowT,i,i +1 + (cid:80) j ∈ Λ( L ) h colT,j,j +1 has a zero en-ergy ground state which is given by the tiling of the plane according to theRobinson pattern.We now consider the RG scheme for the Hamiltonian: Deﬁnition 16 (Tiling Renormalisation Isometry) . Let T (cid:48) be one of the disjointsubsets of × Robinson tiles which appear in the previously described renor-malisation scheme. Let V ( i,i +1) , ( j,j +1) : T ⊗ × → T (cid:48) be the isometry mappingthese × blocks to allowed supertiles which appear in the Robinson pattern, V ( i,i +1) , ( j,j +1) = (cid:88) | T α (cid:105)∈ T (cid:48) | T α (cid:105) I,J (cid:104) t a | i,j (cid:104) t b | i +1 ,j (cid:104) t k | i,j +1 (cid:104) t l | i +1 ,j +1 , (4.2) where | t m (cid:105) ∈ T , with the set of T (cid:48) tiles that maps to the Robinson tiles asdescribed in Deﬁnition 14. Deﬁnition 17 (Tiling Hamiltonian Renormalisation) . Let h colT , h rowT ∈ B ( C T ⊗ C T ) be the local interactions describing the tiling Hamiltonian. Let h rowi,i +1 ( j ) de-note the row interaction between sites ( i, j ) , ( i + 1 , j ) and similarly let h colj,j +1 ( i ) be the interaction between ( i, j ) , ( i, j + 1) . Let the × supertiles be assignedat ( i, j ) , ( i + 1 , j ) , ( i, j + 1) , ( i + 1 , j + 1) and sites consistent with it. Then therenormalised Hamiltonian has local terms R ( h colT ) , R ( h colT ) ∈ B ( C T ⊗ C T ) . ( h colT ) (cid:100) j/ − (cid:101) , (cid:100) j/ − (cid:101) +1 = V ( i,i +1) , ( j +2 ,j +3) V ( i,i +1) , ( j,j +1) (cid:16) h colT,j +1 ,j +2 ( i ) + h colT,j +1 ,j +2 ( i + 1) (cid:17) (cid:12)(cid:12) T (cid:48) (4.3) × V † ( i,i +1) , ( j,j +1) V † ( i,i +1) , ( j +2 ,j +3) (4.4) R ( h rowT ) (cid:100) i/ − (cid:101) , (cid:100) i/ − (cid:101) +1 = V ( i +2 ,i +3) , ( j,j +1) V ( i,i +1) , ( j,j +1) (cid:0) h rowT,i +1 ,i +2 ( j ) + h rowT,i +1 ,i +2 ( j + 1) (cid:1) (cid:12)(cid:12) T (cid:48) (4.5) × V † ( i,i +1) , ( j,j +1) V † ( i +2 ,i +3) , ( j,j +1) (4.6) In the above we have used the standard abbreviation that each local term isimplicitly tensored with the appropriate identity terms, e.g. h colT,j +1 ,j +2 ( i ) isactually i,j ⊗ i +1 ,j ⊗ h colT,j +1 ,j +2 ( i ) ⊗ i,j +3 ⊗ i +1 ,j +3 . Note that this renormalisation map is computable, as each V simplydescribes the mapping of tiles in the initial set to those in the new set inthe way illustrated previously.We now prove that the local Hamiltonian terms are mapped back ontothemselves when this RG transformation is applied. Lemma 18.

The matrix form of the initial and renormalised Hamiltonian arethe same, i.e., R ( h rowT ) i,i +1 = h rowT,i,i +1 and R ( h colT ) j,j +1 = h colT,j,j +1 . (4.7) Proof.

We consider two neighbouring × blocks ( i, j ) , ( i +1 , j ) , ( i, j +1) , ( i +1 , j + 1) and ( i + 2 , j ) , ( i + 3 , j ) , ( i + 2 , j + 1) , ( i + 3 , j + 1) , and determinehow the row and column interactions transform under this renormalisationprocess. We can then write h rowi,i +1 ( j ) = (cid:88) ( t k ,t l ∈ H ) | t k (cid:105) i,j | t l (cid:105) i +1 ,j (cid:104) t k | i,j (cid:104) t l | i +1 ,j (4.8)and, with | T α (cid:105) ∈ T (cid:48) , then V ( i,i +1) , ( j,j +1) = (cid:88) | T α (cid:105)∈ T (cid:48) | T α (cid:105) (cid:104) t a | i,j (cid:104) t b | i +1 ,j (cid:104) t k | i,j +1 (cid:104) t l | i +1 ,j +1 , (4.9)where | t a (cid:105) i,j | t b (cid:105) i +1 ,j | t k (cid:105) i,j +1 | t l (cid:105) i +1 ,j +1 is an allowed × supertile, and wesum over all such allowed × blocks.33ow consider the two blocks: there are 6 relevant row interactions: h rowi,i +1 ( j ) + h rowi,i +1 ( j + 1) (4.10) + h rowi +1 ,i +2 ( j ) + h rowi +1 ,i +2 ( j + 1) (4.11) + h rowi +2 ,i +3 ( j ) + h rowi +2 ,i +3 ( j + 1) . (4.12)Restrict to the set of appearing × supertiles, T (cid:48) , which are centred on the × blocks ( i, j ) , ( i +1 , j ) , ( i, j +1) , ( i +1 , j +1) and ( i +2 , j ) , ( i +3 , j ) , ( i +2 , j +1) , ( i +3 , j +1) . In this case we see that by enforcing only allowed supertiles,then ( h rowi,i +1 ( j )+ h rowi,i +1 ( j +1)) | T (cid:48) = 0 and ( h rowi +2 ,i +3 ( j )+ h rowi +2 ,i +3 ( j +1)) | T (cid:48) = 0 .Finally we need to consider the terms V ( i +2 ,i +3) , ( j,j +1) V ( i,i +1) , ( j,j +1) ( h rowi +1 ,i +2 ( j ) + h rowi +1 ,i +2 ( j + 1)) | T (cid:48) (4.13) × V † ( i,i +1) , ( j,j +1) V † ( i +2 ,i +3) , ( j,j +1) . (4.14)The application of the isometries maps the tiles to supertiles. Hence we canwrite R ( h row ) i/ ,i/ ( j ) = V ( i,i +1) , ( j,j +1) V ( i +2 ,i +3) , ( j,j +1) (cid:0) h rowi +1 ,i +2 ( j ) + h rowi +1 ,i +2 ( j + 1) (cid:1) | T (cid:48) × (4.15) × V † ( i,i +1) , ( j,j +1) V † ( i +2 ,i +3) , ( j,j +1) . (4.16)Note that R ( h row ) i,i +1 ( j ) acts on T (cid:48) and we see that there is an energyassigned to a particular term in R ( h row ) i,i +1 ( j ) iff there is a correspond-ing term in h rowi,i +1 . Furthermore R ( h row ) i,i +1 ( j ) is the same for all j , hence R ( h row ) i,i +1 = h rowi,i +1 . Corollary 19.

The Hamiltonian with local terms R ( k ) ( h rowT ) , R ( k ) ( h colT ) ∈B ( C T ⊗ C T ) , has the same ground state energy and excited state energies asthe unrenormalised Hamiltonian, for any k ≥ . In this section we will deal with the renormalisation of the quantum Hamilto-nian. For this, we will need a number of deﬁnitions from [CPGW15a].34 eﬁnition 20 (Standard Basis States) . Let the single site Hilbert space be H = ⊗ i H i and ﬁx some orthonormal basis for the single site Hilbert space.Label the set of single site basis states for site i as B ( i ) q . Then a standard basisstate for H ⊗ L are product states over the single site basis. Deﬁnition 21 (Penalty Terms and Transition Rules) . The two-local quantumHamiltonian will contain two types of terms: penalty terms and transition rule terms. Penalty terms have the form | ab (cid:105)(cid:104) ab | where | a (cid:105) and | b (cid:105) are standard basisstates. This adds a positive energy contribution to any conﬁguration containingthe state | ab (cid:105) , which we call an illegal pair . Transition rule terms take the form ( | ab (cid:105) − | cd (cid:105) )( (cid:104) ab | − (cid:104) cd | ) with | ab (cid:105) (cid:54) = | cd (cid:105) , where | ab (cid:105) and | cd (cid:105) act on the samepair of adjacent sites. Deﬁnition 22 (Legal and Illegal States) . We call a standard basis state legal if it does not contain any illegal pairs, and illegal otherwise

We then deﬁne a standard form Hamiltonian on the joint system H C ⊗ H Q := ( C C ⊗ C Q ) ⊗ L = ( C C ) ⊗ L ⊗ ( C Q ) ⊗ L . (5.1) Deﬁnition 23 (Standard-Form Hamiltonian [CPGW15a; Wat19]) . We saythat a Hamiltonian H = H trans + H pen + H in + H out acting on H C ⊗ H Q isof standard form if it takes the form H trans,pen,in,out = L − (cid:88) i =1 h ( i,i +1) trans,pen,in,out (5.2) where the local interactions h trans,pen,in,out satisfy the following conditions:1. h trans ∈ B (cid:0) ( C C ⊗ C Q ) ⊗ (cid:1) is a sum of transition rule terms, where allthe transition rules act diagonally on C C ⊗ C C in the following sense.Given standard basis states a, b, c, d ∈ C C , exactly one of the followingholds: • there is no transition from ab to cd at all; or • a, b, c, d ∈ C C and there exists a unitary U abcd acting on C Q ⊗ C Q together with an orthonormal basis { (cid:12)(cid:12) ψ iabcd (cid:11) } i for C Q ⊗ C Q , bothdepending only on a, b, c, d , such that the transition rules from ab to cd appearing in h trans are exactly | ab (cid:105) (cid:12)(cid:12) ψ iabcd (cid:11) → | cd (cid:105) U abcd (cid:12)(cid:12) ψ iabcd (cid:11) or all i . There is then a corresponding term in the Hamiltonian ofthe form ( | cd (cid:105) ⊗ U abcd − | ab (cid:105) )( (cid:104) cd | ⊗ U † abcd − (cid:104) ab | ) .2. h pen ∈ B (cid:0) ( C C ⊗ C Q ) ⊗ (cid:1) is a sum of penalty terms which act non-trivially only on ( C C ) ⊗ and are diagonal in the standard basis, suchthat h pen = (cid:80) ab illegal | ab (cid:105) (cid:104) ab | C ⊗ Q , where | ab (cid:105) are members of a dis-allowed/illegal subspace.3. h in = (cid:80) ab | ab (cid:105) (cid:104) ab | C ⊗ Π ab , where | ab (cid:105) (cid:104) ab | C ∈ ( C C ) ⊗ is a projectoronto ( C C ) ⊗ basis states, and Π ( in ) ab ∈ ( C Q ) ⊗ are orthogonal projectorsonto ( C Q ) ⊗ basis states.4. h out = | xy (cid:105) (cid:104) xy | C ⊗ Π xy , where | xy (cid:105) (cid:104) xy | C ∈ ( C C ) ⊗ is a projector onto ( C C ) ⊗ basis states, and Π ( in ) xy ∈ ( C Q ) ⊗ are orthogonal projectors onto ( C Q ) ⊗ basis states. Importantly the Gottesman-Irani Hamiltonian we will be consideringwill be of standard form.The 1D Gottesman-Irani Hamiltonian H q ( L ) ∈ B ( C d ) ⊗ L is a standard-form Hamiltonian according to the above deﬁnition, and is given by H q = H trans + H in + H pen + H halt , (5.3)where H trans contains transition rule terms, H pen is a set of penalty termswhich penalise states that should not appear in correct history states, H in penalises states which are incorrectly initialised, and H halt penalises stateswhich encode a halting computation. Moreover, it has a six-fold tensorproduct form H q = (cid:79) j =1 ( H q ) j . (5.4)where each ( H q ) j is identiﬁed with a different track.Lemma 43 of [CPGW15a] identiﬁes three subspaces of states, which areclosed under the action of H q .1. Illegal Subspace , S : All | x (cid:105) ∈ S ⊂ B ⊗ L are in the support of H pen and hence (cid:104) x | H | x (cid:105) ≥ . By [CPGW15a] Lemma 43, the minimum36igenvalue of these subspaces is λ ( H | S ) ≥ . (5.5)2. Evolve-to-Illegal Subspace , S : All standard basis states | x (cid:105) ∈ S ⊂ B ⊗ L will evolve either forwards or backwards in time to an illegalstate in O ( L ) steps under the transition rules. As per lemma 5.8 of[Wat19], the minimum eigenvalue of these subspaces is λ ( H | S ) = Ω( L − ) . (5.6)3. Legal Subspace , S : all standard basis states in S are legal and donot evolve to illegal states. By [CPGW15a] lemma 43, they have zerosupport on H pen or H in .In our renormalisation procedure we seek to preserve only the low en-ergy subspace, hence at any point where we can locally identify states asbeing in subspace S or S , we will remove them from the state space in therenormalisation step.However, we note that in the general case we cannot locally identify allsuch states in S . That is, determining the whether a state evolves to anillegal under the action of the transitions may be impossible if we only lookat what the state looks like on a O (1) -subset of the sites. From [CPGW15a] we know that there are two cases we need to consider:the QTM encoded in H q ( L ) halts or does not halt. Lemma 24.

Let a given UTM be encoded in the Gottesman-Irani Hamiltonian H q ( L ) . Then H q ( L ) has a ground state energy that is either if the UTM doesnot halt within time T ( L ) or − cos (cid:0) π T (cid:1) if the UTM does halt within T ( L ) . T ( L ) is a ﬁxed, predetermined function. In the non-halting case, the groundstate is | Ψ hist ( L ) (cid:105) = 1 √ T T ( L ) (cid:88) t =1 | t (cid:105) | ψ t (cid:105) , (5.7)37 nd in the halting case it is | Ψ halt ( L ) (cid:105) = T ( L ) (cid:88) t =1 (cid:18) (2 t + 1) πt T (cid:19) sin (cid:16) π T (cid:17) | t (cid:105) | ψ t (cid:105) , (5.8) where | t (cid:105) is the state of the clock register and | ψ t (cid:105) = (cid:81) tj =1 U j | ψ (cid:105) and | ψ (cid:105) isthe initial state of the computational register and the { U t } represent the actionof the QTM at time step t .Proof. Combine the standard form property of H q from [CPGW15a] withLemma 5.10 of [Wat19]. In this section we will construct a renormalisation scheme for the Gottesman-Irani Hamiltonian. For a given spin at site i , we write each possible conven-tional basis state (i.e. basis state before the RG procedure has started) as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) aα (cid:43) ( i ) ∈ C C ⊗ C Q , where the top cell indicates the classical tracks of theconstruction encoded in [CPGW15a], while the bottom cell indicates thequantum tracks (see Subsection 2.4).We then deﬁne a pair of operations: the blocking operation B q and thetruncation operation T q . Given a line of qudits B q will essentially combinetwo lattice sites into a single site with a larger local Hilbert space dimension,while T q will remove any of the new single site states which can be locallydetected to have non-zero overlap with the ground state. Thus T q reducesthe local Hilbert space dimension.We note that we do not truncate all high energy states since in the haltingcase this would remove the ground state of the Gottesman-Irani Hamilto-nian. Instead, we removed states based on a combination of high energyand a priori knowledge of the ground state. Blocking B q The blocking part of the renormalisation procedure is deﬁned as follows.

Deﬁnition 25 (Gottesman-Irani Blocking, B q ) . Let | ψ (cid:105) ∈ H ( i ) q ⊗H ( i +1) q , i ∈ N .The blocking operation, B q : H ( i ) q × H ( i +1) q → H (cid:48) ( i/ q , is given by the action of he unitary U i,i +1 : H ( i ) q × H ( i +1) q → R ( H q ) (cid:48) as B ( i,i +1) q : | ψ (cid:105) (cid:55)→ U i,i +1 | ψ (cid:105) (5.9) where U i,i +1 = (cid:88) | x (cid:105) , | y (cid:105)∈ B | xy (cid:105) i/ (cid:104) x | i (cid:104) y | i +1 . (5.10) We extend this to | χ (cid:105) ∈ H ⊗ Lq as B q : | χ (cid:105) (cid:55)→ U | χ (cid:105) , (5.11) where U = (cid:78) i ≤ L/ i ∈ N U i,i +1 . This can be expressed more intuitively in terms of basis states B ( i,i +1) q : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) aα (cid:43) ( i ) ⊗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bβ (cid:43) ( i +1) −→ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a bαβ (cid:43) ( i/ . (5.12)Note that B q is just a relabelling of the space, so the local Hilbert spacedimension is now C d and part of the tensor product structure is lost. Wedenote by H (cid:48) q this new local Hilbert space spanned by the basis B (cid:48) (1) . Truncation T q The truncation part of the RG map truncates the local Hilbert space to dis-card those states which locally have support on the penalty terms.

Deﬁnition 26 (Gottesman-Irani Truncation Mapping, T q ) . Let B (1) be theset of basis states deﬁned by B q such states with a preimage | a (cid:105) | b (cid:105) , such that | a (cid:105) , | b (cid:105) ∈ B cannot be locally identiﬁed as being in subspace S or S . That is (cid:104) a | (cid:104) b | h i,i +1 pen | a (cid:105) | b (cid:105) = (cid:104) a | (cid:104) b | h i,i +1 in | a (cid:105) | b (cid:105) = 0 , (5.13) (cid:104) a | (cid:104) b | h ( i,i +1) trans h ( i,i +1) pen h ( i,i +1) trans | a (cid:105) | b (cid:105) = 0 . (5.14) The truncation mapping is then T ( i,i +1) q : R ( H q ) (cid:48) → R ( H q ) for R ( H q ) =span { B (1) q } ⊂ R ( H ) (cid:48) q . Then the full restriction is T q : H (cid:48)⊗ L/ q → R ( H q ) ⊗ L/ .

39e now combine the unitary and subspace restriction to give an isometrywhich implements T q ◦ B q . Lemma 27 (Renormalisation Unitary Structure) . Let the renormalisation iso-metry V GIi,i +1 be the unitary map follow by subspace restriction previously de-scribed. Deﬁne V GI : H ⊗ Lq → R ( H q ) ⊗ L/ to implement the mapping T q ◦ B q on a state in H ⊗ Lq , as T q ◦ B q : | χ (cid:105) (cid:55)→ U | χ (cid:105) | R ( H q ) ⊗ L/ =: V GI | χ (cid:105) . (5.15) where U is deﬁned in Deﬁnition 25 and R ( H q ) is deﬁned in Deﬁnition 26.Then V GI can be deﬁned as and decomposed as V GI := i ≤(cid:98) L/ (cid:99) (cid:79) i ∈ N V GIi,i +1 = i ≤(cid:98) L/ (cid:99) (cid:79) i ∈ N (cid:79) j =1 V GI ( j ) i,i +1 , (5.16) with V GIi,i +1 : H ⊗ q → R ( H q ) (5.17) and where each part of the decomposition acts on one of the six different tracks, V GI ( j ) i,i +1 : H ⊗ q,j → R ( H q ) j . (5.18) Proof.

The decomposition V GI = (cid:78) i ≤(cid:98) L/ (cid:99) i ∈ N V GIi,i +1 is evident from the blockprocedure. The decomposition V GIi,i +1 = (cid:78) j =1 V GI ( j ) i,i +1 arises from the factthat the procedure keeps each basis state as a product across the differenttracks and hence the different H q,j .We now need to deﬁne how the Hamiltonian acts with respect to the RGprocedure. We want to break down the Hamiltonian into different subspacesand renormalise them separately while preserving the ground state (in boththe halting and non-halting cases) and its energy. Lemma 28 (Renormalised Gottesman-Irani Hamiltonian) . Let h q be the localterms of a nearest neighbour, translationally invariant Hamiltonian H q ( L ) = L (cid:88) i =1 h ( i,i +1) q = H trans + H pen + H in + H out , (5.19) such that H ( L ) is standard form. Let V : C d ⊗ C d → C f ( d ) , be the isometry rom Lemma 27. Then the renormalised Hamiltonian, deﬁned as R ( H q ( L )) = V GI H q ( L ) V GI † = L/ (cid:88) i =1 V GI h ( i,i +1) q V GI † = R ( H q )( L ) , (5.20) is a translationally invariant, nearest-neighbour Hamiltonian with local in-teractions R ( h q ) ( i/ ,i/ = V GI ( h ( i − ,i ) q + h ( i +1 ,i +2) q ) V GI † and R ( h q ) i/ = V GI h ( i,i +1) q V GI † . Furthermore, R ( H q )( L ) has the following properties:1. R ( H q )( L ) is a standard form Hamiltonian.2. R ( H trans ) encodes a transition V GI ( | ab (cid:105) | ψ abcd (cid:105) ) → V GI ( | cd (cid:105) U abcd | ψ abcd (cid:105) ) iff H trans encodes the transition | ab (cid:105) | ψ abcd (cid:105) → | cd (cid:105) U abcd | ψ abcd (cid:105) .3. R ( H pen ) , R ( H in ) , R ( H out ) have support on a renormalised basis state V GI ( | ab (cid:105) | ψ (cid:105) ) iff H pen , H in , H out respectively have non-zero support on | ab (cid:105) | ψ (cid:105) .4. λ ( H q ( L )) = λ ( R ( H q )( L/ (the ground state energy is preserved).5. R ( H q ) maintains the six-fold tensor product structure of the originalHamiltonian H q in eq. (5.4) , that is, R ( H q ) = (cid:78) j =1 R ( H q ) j .Proof. First note that for all i ∈ N , V GIi,i +1 h ( i,i +1) V GI † i,i +1 ∈ B ( C f ( d ) ) is now a -local term in the new renormalised Hamiltonian. However, V GIi +2 ,i +3 V GIi,i +1 h ( i +1 ,i +2) V GI † i,i +1 V GI † i +2 ,i +3 ∈B ( C f ( d ) ⊗ C f ( d ) ) Claims 1 and 2

From the linearity of V GI , we see that R ( H q ( L )) = R ( H trans )+ R ( H pen )+ R ( H in ) + R ( H out ) . It is trivial to see that R ( H trans ) = V GI H trans V GI † = (cid:80) ab → cd ( V GI | cd (cid:105) ⊗ U abcd − V GI | ab (cid:105) )( (cid:104) cd | ⊗ U † abcd V GI † − (cid:104) ab | V GI † ) , andhence encodes transitions between the renormalised states. This also shows R ( H trans ) satisﬁes Claim 2. Due to the decompositional properties of V GI ,as shown in Lemma 27, we preserve that H trans acts diagonally on the statesin C C . Likewise, it preserves the form of H pen , H in , H out as projectors ontoa subset of states. 41 laim 3: Consider the penalty terms: given a renormalised state V GI | ψ (cid:105) ,it is clear that ( (cid:104) ψ | V GI † ) V GI H pen V GI † ( V GI | ψ (cid:105) ) = (cid:104) ψ | H pen | ψ (cid:105) = 1 , hence V GI | ψ (cid:105) is penalised by the renormalised Hamiltonian iff | ψ (cid:105) is pen-alised by the unrenormalised Hamiltonian. The same applied to H in and H out . Claim 4:

First note that any state | Ψ { a t }(cid:105) = τ (cid:88) t =1 a t ( | t (cid:105) | ψ t (cid:105) ) . (5.21)which encodes a valid evolution is in the kernel of H in , H pen , and is con-tained in subspace S . Thus, V GI | Ψ { a t }(cid:105) ∈ R ( H ) ⊗ L/ , and after the RGprocedure T q ◦ B q the corresponding renormalised state is (cid:12)(cid:12) Ψ (cid:48) { a t } (cid:11) = τ (cid:88) t =1 a t V GI ( | t (cid:105) | ψ t (cid:105) ) . (5.22)To see the energy of such states is preserved note (cid:10) Ψ (cid:48) { a t } (cid:12)(cid:12) V GI H q ( L ) V GI † (cid:12)(cid:12) Ψ (cid:48) { a t } (cid:11) = (cid:104) Ψ { a t }| H q ( L ) | Ψ { a t }(cid:105) . (5.23)From Lemma 24 the ground states are of the form | Ψ { a t }(cid:105) . We know thatthe state V GI | Ψ { a t }(cid:105) has the same energy. Since the minimum eigenvalueis given by λ ( H q ( L )) = min x ∈H ⊗ Lq (cid:104) x | H q ( L ) | x (cid:105)(cid:104) x | x (cid:105) (5.24) = min x ∈H ⊗ Lq (cid:104) x | U U † H q ( L ) U † U | x (cid:105)(cid:104) x | U † U | x (cid:105) (5.25) ≤ min x ∈H ⊗ Lq V GI | x (cid:105)(cid:54) =0 (cid:104) x | V GI V GI † H q ( L ) V GI † V GI | x (cid:105)(cid:104) x | V GI † V GI | x (cid:105) (5.26) = λ ( R ( H q )( L/ , (5.27)where going from eq. (5.25) to eq. (5.26) we have used the fact that we have42estricted the subspace to remove the states that are integrated out by V GI .Since λ ( R ( H q )( L/ λ ( H q ( L/ , then we can conﬁrm V GI | ψ halt (cid:105) and V GI | ψ hist (cid:105) are the appropriate ground states after the renormalisation pro-cedure. Claim 5:

The preservation of the structure in eq. (5.4) follows directlyfrom the tensor product form of the isometry given in eq. (5.16) appliedaccording to the renormalisation method described by eq. (5.20).

Consecutive steps of the RG procedure can be derived straightforwardly. TheHilbert space obtained after k -th RG steps of can be constructed by induction ( T q ◦ B q ) ◦ ( k ) = T q ◦ B q ◦ ( T q ◦ B q ) ◦ ( k − (5.28)We ﬁrst combine two basis elements in the space B ( k − into a new state,i.e., (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a α a α · · ·· · · a ( k − α ( k − (cid:43) ⊗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b β b β · · ·· · · b ( k − β ( k − (cid:43) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a α a α · · ·· · · a ( k − α ( k − b β b β · · ·· · · b ( k − β ( k − (cid:43) We then truncate the basis set according to the criteria described in the pre-vious section. This will generate the set of renormalised local basis states B ( k ) . The local Hilbert space after k RG iterations is denoted by R ( k ) ( H ) .We note that this can still be decomposed it as R ( k ) ( H ) = (cid:78) i =1 R ( k ) ( H ) i corresponding to the 6 tracks of the original construction.We can thus concatenate multiple renormalisations of the Gottesman-Irani Hamiltonian in one isometry, V GI ( k ) : R ( k − ( H q ) ⊗ L → R ( k ) ( H q ) ⊗ L ,given by V GI [ k ] = Π kj =1 V GIL/ j (5.29)where V GIL/ j is the isometry outlined in Lemma 27, but now acting on theappropriate local Hilbert space, and the subscript L/ j indicates that theoperator is acting on a 1D chain of L/ j sites. We note the use of squarebrackets [] is to distinguish the isometry from V GI ( j ) which will denote the43sometry V GI acting on the j th row of a × lattice.Accordingly, the renormalised Hamiltonian is then R ( k ) ( H q ( L )) = V GI [ k ] H q ( L ) V GI † [ k ] . (5.30)It follows immediately from Lemma 28 is that this RG mapping takesstandard form Hamiltonians to standard form Hamiltonians while preservingthe energy of the ground state. Thus: Corollary 29.

Multiple iterations of the RG map applied to H q ( L ) preserve theproperties (1-5) in Lemma 28. In this section we combine the renormalisation group schemes for the sep-arate parts of the Hamiltonian. First recall Lemma 51 of [CPGW15a] whichcharacterises the ground state of the Hamiltonian deﬁned by the local terms h u : Lemma 30 (Tiling + quantum layers, Lemma 51 of [CPGW15a]) . Let h row c , h col c ∈B ( C C ⊗ C C ) be the local interactions of a 2D tiling Hamiltonian H c , with twodistinguished states (tiles) | L (cid:105) , | R (cid:105) ∈ C C . Let h q ∈ B ( C Q ⊗ C Q ) be the localinteraction of a Gottesman-Irani Hamiltonian H q ( r ) , as in Section 5. Thenthere is a Hamiltonian on a 2D square lattice with nearest-neighbour interac-tions h row u , h col u ∈ B ( C C + Q +1 ⊗ C C + Q +1 ) with the following properties: For anyregion of the lattice, the restriction of the Hamiltonian to that region has aneigenbasis of the form | T (cid:105) c ⊗ | ψ (cid:105) q , where | T (cid:105) c is a product state representing aclassical conﬁguration of tiles. Furthermore, for any given | T (cid:105) c , the lowest en-ergy choice for | ψ (cid:105) q consists of ground states of H q ( r ) on segments between sitesin which | T (cid:105) q contains an | L (cid:105) and an | R (cid:105) , a 0-energy eigenstate on segmentsbetween an | L (cid:105) or | R (cid:105) and the boundary of the region, and | e (cid:105) ’s everywhereelse. The | L (cid:105) and | R (cid:105) tiles are identiﬁed in [CPGW15a] with the right-downand left-down red cross in the Robinson tiles respectively (see Section 4).The ground state can then be shown to be the ground state of the Robin-son tiling Hamiltonian plus a “quantum layer” in which the Gottesman-Irani44round states appear only over the tops of the Robinson squares. Every-where else in the quantum layer is a ﬁller state | e (cid:105) .A key point is that the eigenstates are all product states across H c and H eq . We wish for the RG mapping to preserve this property. This restrictsthe type of renormalisation isometries we use, as detailed in the followinglemma. Lemma 31 (Separable Eigenstates) . Let H Λ(2 L ) u denote the Hamiltonian inLemma 30. Then for an isometry Z = Z c ⊗ Z eq where Z c : H ⊗ × c → R ( H c ) and Z eq : H ⊗ × eq → R ( H eq ) , the operator ZH Λ(2 L ) u Z † also has eigenstates ofthe form | T (cid:48) (cid:105) c ⊗ | ψ (cid:105) eq for | T (cid:48) (cid:105) c ∈ R ( H c ) ⊗ Λ( L ) and | ψ (cid:105) eq ∈ R ( H eq ) ⊗ Λ( L ) .Proof. As per Lemma 30, the eigenstates of H Λ(2 L ) u decompose as productstates | T c (cid:105) ⊗ | ψ i (cid:105) eq , hence we can write H Λ(2 L ) u = (cid:88) i λ i | T i (cid:105)(cid:104) T i | ⊗ | ψ i (cid:105)(cid:104) ψ i | eq . (6.1)Applying the renormalisation isometry Z gives ZH Λ(2 L ) u Z † = (cid:88) i λ i Z C | T i (cid:105)(cid:104) T i | c Z † C ⊗ Z eq | ψ i (cid:105)(cid:104) ψ i | eq Z † eq (6.2) =: (cid:88) i λ i (cid:12)(cid:12) T (cid:48) i (cid:11)(cid:10) T (cid:48) i (cid:12)(cid:12) c (cid:48) ⊗ (cid:12)(cid:12) ψ (cid:48) i (cid:11)(cid:10) ψ (cid:48) i (cid:12)(cid:12) eq (cid:48) . (6.3)Thus the product structure across the two subspaces is preserved.In Section 4 we showed that the ground state of the renormalised tilingHamiltonian preserves the tiling pattern of the unrenormalised Hamilto-nian. Here we show that renormalising the full Hamiltonian preserves thisRobinson tiling plus Gottesman-Irani ground state structure.We start by considering how to renormalise the Gottesman-Irani Hamilto-nian in the presence of ﬁller states on a 2D lattice (as opposed to the 1Dchain considered previously). After this we show the ground state energyHamiltonian is preserved under the RG map. H T ⊗ ( H e ⊕ H q ) From Lemma 31, we know the eigenstates of the Hamiltonian deﬁned by h u are product states across the classical-quantum Hilbert space partition and45his structure is preserved under a tensor product of isometries on the twosubspace separately. Thus we can consider the basis states of H T and H eq separately and then later show this preserves the desired properties. Blocking Operation B u We know that V C from Lemma 18 will renormal-ise the classical state space by mapping sets of × tiles to new tiles whichrecreate the tiling pattern at all but the lowest level. We use this isometryunchanged, acting on the classical part of the Hilbert space.Consider the quantum Hilbert space H eq . First note that the Gottesman-Irani Hamiltonian to be renormalised is a standard form Hamiltonian, andso can be renormalised as per Subsection 5.1. However, the blocking pro-cedure from Subsection 5.1 is not sufﬁcient for our purposes as it (a) takesa set of × lattice sites to a single lattice site and so is not appropriate fora 2D lattice, and (b) does not include the ﬁller state | e (cid:105) e . To remedy this weneed an isometry which acts as: V eq ( i,i +1)( j,j +1) : H ( i,j ) eq ⊗H ( i +1 ,j ) eq ⊗H ( i,j +1) eq ⊗H ( i +1 ,j +1) eq → ( H (cid:48) eq ⊗H (cid:48) eq ) ( i/ ,j/ . (6.4)We will ﬁnd it useful to deﬁne the following notation: Deﬁnition 32 ( k -times Blocked Basis States) . Let | x (cid:105) , | x (cid:105) , . . . , | x k (cid:105) ∈ B ∪ | e (cid:105) e , then we denote the corresponding renormalised basis state after k applications of the RG mapping as | x x . . . x k (cid:105) . Now deﬁne V q ( i,i +1) ( j ) as follows, where V GIi,i +1 is the isometry used inLemma 28: V q ( i,i +1) ( j ) = V GIi,i +1 + | ee (cid:105) i/ ,j/ (cid:104) e | i,j (cid:104) e | i +1 ,j (6.5) + | xe (cid:105) i/ ,j/ (cid:104) x | i,j (cid:104) e | i +1 ,j + | ex (cid:105) i/ ,j/ (cid:104) e | i,j (cid:104) x | i +1 ,j . (6.6)This deﬁnes a new set of quantum basis states which now reﬂect the fact | e (cid:105) e is part of the Hilbert space. Denote this C (1) := B (1) ∪ | ee (cid:105) (cid:91) x ∈ B | ex (cid:105) (cid:91) x ∈ B | xe (cid:105) . (6.7)46hese isometries essentially apply the same mapping as V GI , but now ac-count for the additional | e (cid:105) e state we have present. However, V q only maps × spins to a single spin. We need an operator which maps a × spinto a single spin. Deﬁne W : H (cid:48) ( i/ ,j ) eq ⊗H (cid:48) ( i/ ,j +1) eq → ( H (cid:48) eq ⊗H (cid:48) eq ) ( i/ ,j/ , assimply W ( i,i +1)( j,j +1) = (cid:88) ( | x (cid:105) q ⊗ | y (cid:105) q ) i/ ,j/ (cid:104) x | i/ ,j ⊗ (cid:104) y | i/ ,j +1 . (6.8)This unitary acts to map the × set of sites to a single lattice site in therenormalised lattice.The isometry: V eq ( i,i +1)( j,j +1) := W ( i,i +1)( j,j +1) (cid:16) V q ( i,i +1) ( j ) ⊗ V q ( i,i +1) ( j + 1) (cid:17) , (6.9)then maps × spins to a single spin.The overall blocking map B u is then given by: Deﬁnition 33 (Blocking Isometry, V b , B u ) . Let V C and V eq be the isometriesfrom Deﬁnition 17 and eq. (6.9) respectively. Then the blocking isometry for H u is given by V b ( i,i +1)( j,j +1) = V C ( i,i +1)( j,j +1) ⊗ V eq ( i,i +1)( j,j +1) . (6.10)We now need to consider the full renormalisation process: the isometrydeﬁned above will map a certain subset of states to states on the renorm-alised lattice. However, some parts of the Hilbert space will be “integratedout”. For convenience we will sometimes use indices I, J to indicate rowand column indices on the new lattice after the RG transformation.Let h ( i,i +1) q ( j ) , h ( i,i +1) q ( j + 1) be the local terms of the quantum Hamilto-nian before renormalisation, then we see that V eq ( i,i +1)( j,j +1) (cid:16) h ( i,i +1) q ( j + 1) + h ( i,i +1) q ( j ) (cid:17) V eq † ( i,i +1)( j,j +1) = h (1) (cid:48) ( I,J ) q ⊗ q + q ⊗ h (1) (cid:48) ( I,J ) q (6.11)47nd V eq ( i +2 ,i +3)( j,j +1) V eq ( i,i +1)( j,j +1) (cid:16) h ( i +1 ,i +2) q ( j ) + h ( i +1 ,i +2) q ( j ) (cid:17) V eq † ( i +2 ,i +3)( j,j +1) × V eq † ( i,i +1)( j,j +1) = h (cid:48) ( I,I +1) q ⊗ ( I,J ) q ⊗ ( I +1 ,J ) q + ( I,J ) q ⊗ ( I +1 ,J ) q ⊗ h (cid:48) ( I,I +1) q . (6.12) Truncation Operation T u The operator W has essentially merged twosites into a single site. We now wish to integrate out one of these sitesand restrict to the set of “allowed states” in the other. We will implementthis using the -local projector Π gs ( k ) Deﬁnition 34 (Truncation Operation T u ) . Let | ψ (cid:105) ∈ H c ⊗H eq , then T u : | ψ (cid:105) (cid:55)→ ( c ⊗ q ⊗ Π gs ( k )) | ψ (cid:105) , (6.13) where Π gs ( k ) = (cid:12)(cid:12)(cid:12) e × k (cid:69)(cid:68) e × k (cid:12)(cid:12)(cid:12) k even (cid:12)(cid:12)(cid:12) ψ hist (4 n + 1) e × k − n − (cid:69)(cid:68) ψ hist (4 n + 1) e × k − n − (cid:12)(cid:12)(cid:12) if k odd, k − < n + 1 < k ,and non-halting (cid:12)(cid:12)(cid:12) ψ halt (4 n + 1) e × k − n − (cid:69)(cid:68) ψ halt (4 n + 1) e × k − n − (cid:12)(cid:12)(cid:12) if k odd, k − < n + 1 < k ,and halting , (6.14) and where | ψ hist ( L ) (cid:105) and | ψ halt ( L ) (cid:105) are deﬁned in Lemma 24. This extends tostates | χ (cid:105) ∈ ( H c ⊗H eq ) ⊗ Λ( L ) , as T u : | χ (cid:105) (cid:55)→ (cid:79) ( I,J ) ∈ Λ( L ) ( ( I,J ) c ⊗ ( I,J ) q ⊗ Π ( I,J ) gs ( k )) | χ (cid:105) . (6.15) Deﬁnition 35 (Renormalisation Isometry, V u ) . Let V b ( i,i +1)( j,j +1) and Π gs beas deﬁned in Deﬁnition 33 and eq. (6.14) respectively. We deﬁne the isometryimplementing the entire renormalisation scheme as V u ( i,i +1)( j,j +1) := ( c ⊗ Π gs ) V b ( i,i +1)( j,j +1) . (6.16)48o see why this is appropriate note that the Hamiltonian after the applic-ation of the blocking isometries has two sets of local terms: a 1-local termand a 2-local term (see Deﬁnition 36 and the discussion following). Firstconsider the 1-local term h (1) (cid:48) ( I,J ) q ⊗ q + q ⊗ h (1) (cid:48) ( I,J ) q and examine how ittransforms under T u and Π gs . The idea is that Π gs will “integrate out” the q subspace by removing all states which are not the ground state whilemaintaining the energy contribution from this subspace. If the site is largeenough to contain a full history state of length n + 1 , for some n ∈ N , thenwe keep only that state and the relevant renormalised | e (cid:105) states. Otherwisewe keep only the renormalised | e (cid:105) states. Hence Π ( I,J ) gs ( k )( h (1) (cid:48) ( I,J ) q ⊗ ( I,J ) q + ( I,J ) q ⊗ h (1) (cid:48) ( I,J ) q )Π ( I,J ) gs ( k ) (6.17) = h (1) (cid:48) ( I,J ) q ⊗ Π ( I,J ) gs ( k ) + tr (cid:16) Π ( I,J ) gs ( k ) h (cid:48) ( I,J ) q (cid:17) ( I,J ) q ⊗ Π ( I,J ) gs ( k ) . (6.18)Since Π gs is a projector onto a 1-dimensional subspace, we will often omitit when writing the Hamiltonian. Thus obtain the term h (1) (cid:48) ( I,J ) q + Tr (cid:16) Π gs ( k ) h (cid:48) ( I,J ) q (cid:17) q . (6.19)Now examine how the 2-local terms transform: Π gs ( k ) ( I,J ) ⊗ Π gs ( k ) ( I +1 ,J ) (cid:0) h (cid:48) ( I,I +1) q ⊗ ( I,J ) q ⊗ ( I +1 ,J ) q (6.20) + ( I,J ) q ⊗ ( I +1 ,J ) q ⊗ h (cid:48) ( I,I +1) q (cid:1) Π gs ( k ) ( I,J ) ⊗ Π gs ( k ) ( I +1 ,J ) (6.21) = h (cid:48) ( I,I +1) q ⊗ Π gs ( k ) ( I ) ⊗ Π gs ( k ) ( I +1) (6.22) + tr (cid:16) h (cid:48) ( I,I +1) q Π gs ( k ) ( I ) ⊗ Π gs ( k ) ( I +1) (cid:17) ( I,J ) q ⊗ Π gs ( k ) ( I ) ⊗ Π gs ( k ) ( I +1) . (6.23) Importantly tr (cid:16) h (cid:48) ( I,I +1) q Π gs ( k ) ( I ) ⊗ Π gs ( k ) ( I +1) (cid:17) only picks up a non-zerocontribution from the terms proportional to ( I ) ⊗ ( I +1) (we also note thatthis latter term is zero for interactions going along columns). Again the sub-space spanned by Π gs ( k ) ( I ) ⊗ Π gs ( k ) ( I +1) is a 1-dimensional subspace andhence we will often omit writing it explicitly. Thus the 2-local terms effect-ively become h (cid:48) ( I,I +1) q + tr (cid:16) h (cid:48) ( I,I +1) q Π gs ( k ) ( I ) ⊗ Π gs ( k ) ( I +1) (cid:17) ( I,J ) q ⊗ ( I +1 ,J ) q . Multiple Iterations

The above is the RG transformation for a single itera-tion; in the following we construct the further iterations of the RG mapping49nalogously to the above.First deﬁne the set of local basis states in the quantum part of the Hilbertspace, C (cid:48) ( k ) := B ( k ) (cid:91) j =0 K (cid:91) | x i (cid:105)∈ B ∪| e (cid:105) | x ...x k (cid:105) , (6.24)such that | x . . . x k − (cid:105) ∈ C ( k − . From this we can deﬁne H (cid:48) ( k ) eq = span (cid:8) | x (cid:105) (cid:12)(cid:12) | x (cid:105) ∈ C (cid:48) ( k ) (cid:9) .Then B u : ( R ( k − H eq ) ⊗ × → H (cid:48) ( k ) eq . Finally we truncate the basis stateswhich are either bracketed or can immediately be identiﬁed as being illegalor evolving to an illegal state using T u . This leaves us with the basis C ( k ) asthe set of basis states and the renormalised local quantum Hilbert space as R ( k ) ( H eq ) = span {| x (cid:105) | | x (cid:105) ∈ C ( k ) } .The T u ◦ B u operation can be implemented analogously to the previouslydescribed transformation: we apply V b — now deﬁned on R ( k − ( H u ) —across the lattice which blocks and truncates part of the Hilbert space. Wethen apply Π gs ( k ) , as deﬁned in eq. (6.14), to project out the local groundstate (which may pick up energy).We formalise the overall RG mapping in the following deﬁnition: Deﬁnition 36 ( h u Renormalisation Mapping) . Let h col ( i,i +1) u , h row ( j,j +1) u ∈B ( C d ⊗ C d ) and V u ( i,i +1)( j,j +1) be as in Deﬁnition 35. Then the renormalisedlocal terms are given by R : h row ( i +1 ,i +2) u ( j ) + h row ( i +1 ,i +2) u ( j + 1) → V u ( i +2 ,i +3)( j,j +1) V u ( i,i +1)( j,j +1) × (6.25) (cid:16) h row ( i +1 ,i +2) u ( j ) + h row ( i +1 ,i +2) u ( j + 1) (cid:17) V u † ( i,i +1)( j,j +1) V u † ( i +2 ,i +3)( j,j +1) (6.26) =: R ( h rowu ) ( i,i +1) (6.27) R : h col ( j +1 ,j +2) u ( i ) + h col ( j +1 ,j +2) u ( i + 1) → V u ( i +2 ,i +3)( j,j +1) V u ( i,i +1)( j,j +1) × (6.28) (cid:18) h col ( j +1 ,j +2) u ( i ) + h col ( j +1 ,j +2) u ( i + 1) (cid:19) V u † ( i,i +1)( j,j +1) V u † ( i +2 ,i +3)( j,j +1) , (6.29) =: R ( h colu ) ( i,i +1) (6.30)50 : h row ( i,i +1) u ( j ) + h row ( i +1 ,i +2) u ( j + 1) + (cid:88) k =0 , (cid:96) =1 , (cid:16) h (1)( i + k,j + (cid:96) ) u (cid:17) → (6.31) V u ( i,i +1)( j,j +1) (cid:18) h row ( i,i +1) u ( j ) + h row ( i +1 ,i +2) u ( j + 1) + (cid:88) k =0 , (cid:96) =1 , (cid:16) h (1)( i + k,j + (cid:96) ) u (cid:17) (cid:19) V u † ( i,i +1)( j,j +1) (6.32) =: R ( h (1) u ) ( i ) . (6.33) R ( k ) ( h rowu ) , R ( k ) ( h colu ) ( i,i +1) , R ( k ) ( h (1) u ) ( i ) are deﬁned in the same way but withthe appropriate isometries for the k th iteration of the RG mapping. Remark 37. R ( k ) ( h (1) u ) ( i ) and R ( k ) ( h rowu ) ( i,i +1) have local projector terms ofthe form (cid:80) km =1 m κ ( m ) ( i ) and (cid:80) km =1 m γ ( m ) ( i ) ⊗ ( i +1) , where γ ( k ) and κ ( k ) are given by κ ( k ) := Tr (cid:16) Π gs ( k ) h (cid:48) ( I,J ) q (cid:17) (6.34) γ ( k ) := tr (cid:16) h (cid:48) ( I,I +1) q Π gs ( k ) ( I ) ⊗ Π gs ( k ) ( I +1) (cid:17) . (6.35)We now examine the properties of the full Hamiltonian under this map-ping, and show that its ground state energy and other properties are pre-served. Lemma 38 ( H u Renormalisation) . Let H u ( L ) = (cid:80) h row ( j,j +1) u + (cid:80) h col ( i,i +1) u , here h col j,j +1 = h col c ⊗ ( j ) eq ⊗ ( j +1) eq (6.36a) h row i,i +1 = h row c ⊗ ( i ) eq ⊗ ( i +1) eq (6.36b) + ( i ) c ⊗ ( i +1) c ⊗ h q (6.36c) + | L (cid:105)(cid:104) L | ( i ) c ⊗ ( eq − (cid:12)(cid:12) (cid:11)(cid:10) (cid:12)(cid:12) ) ( i ) ⊗ ( i +1) ceq (6.36d) + ( c − | L (cid:105)(cid:104) L | c ) ( i ) ⊗ (cid:12)(cid:12) (cid:11)(cid:10) (cid:12)(cid:12) ( i ) ⊗ ( i +1) ceq (6.36e) + ( i ) ceq ⊗ | R (cid:105)(cid:104) R | ( i +1) c ⊗ ( eq − (cid:12)(cid:12) (cid:11)(cid:10) (cid:12)(cid:12) ) ( i +1) (6.36f) + ( i ) ceq ⊗ ( c − | R (cid:105)(cid:104) R | ) ( i +1) c ⊗ (cid:12)(cid:12) (cid:11)(cid:10) (cid:12)(cid:12) ( i +1) (6.36g) + ( i ) c ⊗ | (cid:105)(cid:104) | ( i ) e ⊗ | R (cid:105)(cid:104) R | ( i +1) c ⊗ ( i +1) eq (6.36h) + | L (cid:105)(cid:104) L | ( i ) c ⊗ ( i ) eq ⊗ ( i +1) c ⊗ | (cid:105)(cid:104) | ( i +1) e (6.36i) + ( i ) c ⊗ | (cid:105)(cid:104) | ( i ) e ⊗ ( c − | L (cid:105)(cid:104) L | ) ( i +1) c ⊗ ( eq − | (cid:105)(cid:104) | ) ( i +1) e (6.36j) + ( c − | R (cid:105)(cid:104) R | ) ( i ) c ⊗ ( eq − | (cid:105)(cid:104) | ) ( i ) e ⊗ ( i +1) c ⊗ | (cid:105)(cid:104) | ( i +1) e , (6.36k) + ( i ) ceq ⊗ ( i +1) ceq (6.36l) h (1) i = − (1 + α ( ϕ )) ( i ) ceq , (6.36m) where α ( ϕ ) := (cid:88) n +7 > | ϕ | − n − λ ( H q (4 n )) , (6.37) as deﬁned in Proposition 53 of [CPGW15a]. Then the k times renormalisedHamiltonian R ( k ) ( H u ) Λ( L × H ) has the following properties:1. For any ﬁnite region of the lattice, the restriction of the Hamiltonian tothat region has an eigenbasis of the form | T (cid:105) c ⊗ | ψ i (cid:105) where | T (cid:105) c is aclassical tiling state (cf. Lemma 51 of [CPGW15a]).2. Furthermore, for any given | T (cid:105) c , the lowest energy choice for | ψ (cid:105) q con-sists of ground states of R ( k ) ( H q )( r ) on segments between sites in which | T (cid:105) c contains an (cid:12)(cid:12) R ( k ) ( L ) (cid:11) and an (cid:12)(cid:12) R ( k ) ( R ) (cid:11) , a 0-energy eigenstate onsegments between an (cid:12)(cid:12) R ( k ) ( L ) (cid:11) or (cid:12)(cid:12) R ( k ) ( R ) (cid:11) and the boundary of theregion, and | e (cid:105) ’s everywhere else. Any eigenstate which is not an eigen-state of R ( k ) ( H q )( r ) on segments between sites in which | T (cid:105) c containsan (cid:12)(cid:12) R ( k ) ( L ) (cid:11) and an (cid:12)(cid:12) R ( k ) ( R ) (cid:11) has an energy > (cf. Lemma 51 of[CPGW15a]). . The ground state energy is contained in the interval (cid:20) ( g ( k ) − k α ( ϕ )) LH − − k H + (cid:98) log ( L/ (cid:99) (cid:88) n =1 (cid:18) (cid:22) H n +1( k mod (cid:23) (6.38) × (cid:18)(cid:22) L n +1 − ( k mod (cid:23) − (cid:19) (cid:19) λ ( R ( k ) ( H q )(4 n −(cid:98) ( k mod / (cid:99) )) , (6.39) ( g ( k ) − k α ( ϕ )) LH − − k H + (cid:98) log ( L/ (cid:99) (cid:88) n =1 (cid:18) (cid:18)(cid:22) H n +1 − ( k mod (cid:23) + 1 (cid:19) (6.40) × (cid:22) L n +1 − ( k mod (cid:23) (cid:19) λ ( R ( k ) ( H q )(4 n −(cid:98) ( k mod / (cid:99) )) (cid:21) (6.41) where g ( k ) = 4 k (cid:88) n +1 < k − n − λ ( H q (4 n )) , (6.42) (cf. Lemma 52 of [CPGW15a]).Proof. We prove this in Appendix C.

Lemma 39.

Let S br ( k ) be the subspace spanned by states for which the left-most site is of the form (cid:12)(cid:12)(cid:12) e × p { x } × k − p − (cid:69) for a ﬁxed integer ≤ p ≤ k andthe right-most site is of the form (cid:12)(cid:12)(cid:12) { y } × k − q − e × q (cid:69) for ﬁxed integer ≤ q ≤ k . Then λ ( R ( k ) ( H q )( L ) | S br ( k ) ) = min k − L +1 ≤ x ≤ k L λ ( H q ( x )) (6.43) Proof. R ( k ) ( h q ) is block-diagonal with respect to the subspaces of R ( k ) ( H eq ) ⊗ spanned by products of (cid:12)(cid:12)(cid:12) e × p { x } × k − p − (cid:69) and (cid:12)(cid:12)(cid:12) { y } × k − q − e × q (cid:69) for ﬁxed p, q , together with the orthogonal complementthereof, while acting as identity on R ( k ) ( H c ) ⊗ .Thus the ground state energy is equal to min k − L +1 ≤ x ≤ k L λ ( H q ( x )) . Corollary 40. If lim L →∞ λ ( H Λ( L ) u ) = + ∞ , then lim L →∞ λ ( R ( k ) ( H u ) Λ( L ) ) = + ∞ for all k ≥ k ( | ϕ | ) , and k ( | ϕ | ) is the smallest integer such that k > | ϕ | +7 . If lim L →∞ λ ( H Λ( L ) u ) = −∞ , then lim L →∞ λ ( R ( k ) ( H u ) Λ( L ) ) = −∞ for all k ≥ k ( ϕ ) . roof. Consider applying the RG mapping k > k ( ϕ ) times, then we see that g ( k ) = 4 k (cid:88) n +1 < k − n − λ ( H q (4 n )) (6.44) = 4 k (cid:88) n +1 < k − n − λ ( H q (4 n )) + 4 k (cid:88) k < n +1 < k − n − λ ( H q (4 n )) (6.45) = 4 k α ( ϕ ) + 4 k (cid:88) k < n +1 < k − n − λ ( H q (4 n )) . (6.46)From Lemma 38, the interval the ground state energy is contained in is (cid:20) LH (cid:88) k < n +1 < k − n − k λ ( H q (4 n )) − − k H + (cid:98) log ( L/ (cid:99) (cid:88) n =1 (cid:18)(cid:22) H n +1 − ( k mod (cid:23) (cid:18)(cid:22) L n +1 − ( k mod (cid:23) − (cid:19)(cid:19) λ ( R ( k ) ( H q )(4 n −(cid:98) ( k mod / (cid:99) )) ,LH (cid:88) k < n +1 < k − n − k λ ( H q (4 n )) − − k H + (cid:98) log ( L/ (cid:99) (cid:88) n =1 (cid:18)(cid:18)(cid:22) H n +1 − ( k mod (cid:23) + 1 (cid:19) (cid:22) L n +1 − ( k mod (cid:23)(cid:19) λ ( R ( k ) ( H q )(4 n −(cid:98) ( k mod / (cid:99) )) (cid:21) . (6.47)From Lemma 39, if λ ( H q (4 n +1)) = 0 for all n , then λ ( R ( k ) ( H q )(4 n +1)) =0 for all n . In this case the ground state energy becomes λ ( R ( k ) ( H ) Λ( L ) ) = − − k L L →∞ −−−−→ −∞ .We see that if for any n , λ ( H q (4 n + 1)) > , then λ ( R ( k ) ( H q )(4 n +1)) > ∀ n ≥ n (cid:48) ( n (cid:48) not necessarily equal to n ). Deﬁne g ( k ) = η ( k ) +4 k α ( ϕ ) then η ( k ) ≥ , and we see that the lower bound of the ground stateis L η ( k ) − − k L + (cid:98) log ( L/ (cid:99) (cid:88) n =1 (cid:18)(cid:22) L n +1 − ( k mod (cid:23) (cid:18)(cid:22) L n +1 − ( k mod (cid:23) − (cid:19)(cid:19) × λ ( R ( k ) ( H q )(4 n −(cid:98) ( k mod / (cid:99) )) L →∞ −−−−→ + ∞ . (6.48)For k ≤ | ϕ | + 7 the above relationship is not necessarily preserved. To54ee why, note that for lengths (cid:96) ≤ | ϕ | + 7 the Gottesman-Irani Hamilonianwill not encode the correct computation and hence will pick up some energy.Since λ ( R ( k ) ( H q )( L ) | S br ) = min k − L +1 ≤ x ≤ k L λ ( H q ( x )) rather than λ ( R ( k ) ( H q )( L ) | S br ) = λ ( H q ( x )) , the energies in the summation term andthe α term will not exactly cancel out until we reach higher order steps ofthe RG ﬂow. This is only rectiﬁed once we reach k > | ϕ | + 7 as the energyintegrated out by the projector Π gs , as given in Deﬁnition 34, is exactly λ ( H q ( x )) , not λ ( R ( k ) ( H q )( L ) | S br ) . H d The only part of the Hamiltonian acting on H d is H d ; there is no couplingto other parts of the Hilbert space and so we can renormalise this part in-dependently. For concreteness, following [CPGW15a], we will let H d bethe critical XY-model with local terms X i ⊗ X i +1 + Y i ⊗ Y i +1 + Z i ⊗ ( i +1) + ( i ) ⊗ Z i +1 , which can be written as: h row ( i,i +1) d = X i ⊗ X i +1 + Y i ⊗ Y i +1 , (6.49) h col ( i,i +1) d = 0 , (6.50) h (1)( i ) d = 2 Z i . (6.51)However any Hamiltonian with a dense spectrum in the thermodynamiclimit could be substituted. Since the critical XY model is critical, it forms aﬁxed point in any reasonable RG scheme. Thus we expect any reasonableRG procedure to map the model to itself. Deﬁnition 41 (Renormalisation Unitary for h d , V d ) . We deﬁne the isometryimplementing the renormalisation operation as V d ( i,i +1) , ( j,j +1) (cid:16) h row ( i +1 ,i +2) d ( j ) + h row ( i +1 ,i +2) d ( j + 1) (cid:17) V d † ( i,i +1) , ( j,j +1) = h row ( i/ ,i/ d ( j/ , (6.52) V d ( i,i +1) , ( j,j +1) (cid:16) h row ( i,i +1) d ( j ) + h row ( i,i +1) d ( j + 1) (cid:17) V d † ( i,i +1) , ( j,j +1) = 2 Z i/ . (6.53) | (cid:105) If we wish to preserve the form of the possible ground states depending, itis straightforward to see that this can be done if the states | (cid:105) simply get55apped to themselves | (cid:105) ⊗ (2 × → | (cid:105) under the RG operation. This can beimplemented using the isometry V i,i +1) , ( j,j +1) := | (cid:105) ( i/ ,j/ (cid:104) | ( i,j ) (cid:104) | ( i +1 ,j ) (cid:104) | ( i,j +1) (cid:104) | ( i +1 ,j +1) . (6.54) Accounting for the renormalisation of all the different parts of the Hamilto-nian, we can now deﬁne renormalisation group mapping for the entireHamiltonian. Recall that the original local terms are h ( ϕ ) ( i,j ) = | (cid:105) (cid:104) | ( i ) ⊗ ( − | (cid:105) (cid:104) | ) ( j ) + ( − | (cid:105) (cid:104) | ) ( i ) ⊗ | (cid:105) (cid:104) | ( j ) (6.55) + h ( i,j ) u ( ϕ ) ⊗ ( i,j ) d + ( i,j ) u ⊗ h ( i,j ) d (6.56) h ( ϕ ) (1) = − (1 + α ( ϕ ))Π ud , (6.57)where α ( ϕ ) is deﬁned in Lemma 38. Deﬁnition 42 (Full Renormalisation Group Mapping) . Let V u , V , V d bethe isometries deﬁned in Deﬁnition 35, eq. (6.54) , and eq. (6.52) respectively.Deﬁne V r ( i,i +1) , ( j,j +1) := V i,i +1) , ( j,j +1) ⊕ (cid:16) V u ( i,i +1) , ( j,j +1) ⊗ V d ( i,i +1) , ( j,j +1) (cid:17) . (6.58) Then the overall RG mapping of local Hamiltonian terms is given by R : h ( ϕ ) ( i,i +1) (cid:55)→ V r † ( i,i +1) , ( j,j +1) h ( ϕ ) ( i,i +1) V r ( i,i +1) , ( j,j +1) (6.59) R : h ( ϕ ) ( i +1 ,i +2) (cid:55)→ V r † ( i +2 ,i +3) , ( j,j +1) h ( ϕ ) ( i +1 ,i +2) V r ( i,i +1) , ( j,j +1) V r ( i +2 ,i +3) , ( j,j +1) (6.60) Lemma 43.

Applying the RG mapping from Deﬁnition 42 to the terms ineq. (6.55) we see that the renormalised 1- and 2-local terms become R ( k ) ( h ( ϕ )) ( i,j ) =2 k ( | (cid:105)(cid:104) | ( i ) ⊗ Π ( j ) ud + Π ( i ) ud ⊗ | (cid:105)(cid:104) | ( j ) ) (6.61) + R ( k ) ( h u ( ϕ )) ( i,j ) ⊗ ( i,j ) d + ( i,j ) u ⊗ h ( i,j ) d (6.62) R ( k ) ( h ( ϕ )) (1) =( g ( k ) − k α ( ϕ ) − k )Π ( i ) ud + R ( k ) ( h (1) u ) ( i ) (6.63) where g ( k ) is deﬁned in Lemma 38. All the terms are computable. roof. Note that the RG isometry acts block-diagonally with respect to thesubspaces spanned by | (cid:105) ⊗ (2 × and those spanned by states in ( R ( k ) ( H u ) ⊗H d ) ⊗ (2 × . Furthermore, any state are not in one of the two subspaces isprojected out. The h u ( ϕ ) , h d and 1-local terms transform as they would inthe absence of the | (cid:105) state, thus giving the terms seen above. The explicitcoefﬁcients are calculated in Lemma 53 in the appendix. The term g ( k ) iscomputable for any k by calculating the λ ( H q )(4 n + 1) for all n ≤ k + 1 .Since this is a ﬁnite dimensional matrix for any ﬁnite n , this is a computablequantity.The form of the overall renormalisation isometry means the | (cid:105)(cid:104) | ( i ) ⊗ Π ( j ) ud term must be preserved in form, however, we note that because all statesof × blocks in different subspaces in the previous RG step must be in | (cid:105) ⊗ (2 × or ( R ( k ) ( H u ) ⊗ R ( k ) ( H ) d ) ⊗ (2 × , then two neighbouring blocks mustpick up an energy penalty of × of the previous local terms. Corollary 44.

The local terms of the initial Hamiltonian h ( ϕ ) and all furtherrenormalised local terms belong to a family of Hamiltonians F ( ϕ, τ , τ , { α i } i , { β i } i ) , which all take the form R ( k ) ( h ( ϕ )) ( i,j ) = τ ( | (cid:105)(cid:104) | ( i ) ⊗ Π ( j ) ud + Π ( i ) ud ⊗ | (cid:105)(cid:104) | ( j ) ) (6.64) + R ( k ) ( h u ( ϕ, { β t } t )) ( i,j ) ⊗ ( i,j ) d + ( i,j ) u ⊗ R ( k ) ( h d ) ( i,j ) (6.65) R ( k ) ( h ( ϕ )) (1) = τ Π ud + R ( k ) ( h u ( ϕ, { α t } t )) (1) , (6.66) where the sets { α t } t , { β i } characterises the parameters of the renormalisedGottesman-Irani Hamiltonian. Furthermore, for any k ∈ N , the coefﬁcients τ ( k ) , τ ( k ) , { α t ( k ) } t and { β t ( k ) } t are computable.Proof. Follows immediately from Lemma 43.

Lemma 45.

Let R ( k ) ( h ( ϕ )) ( i,j ) , R ( k ) ( h ( ϕ )) (1) be the local terms deﬁned by theRG mapping in Deﬁnition 42 for any k > k ( | ϕ | ) . The Hamiltonian R ( k ) ( H ) deﬁned by these terms then has the following properties:1. If the unrenormalised Hamiltonian H ( ϕ ) has a zero energy ground statewith a spectral gap of 1/2, then R ( k ) ( H ) also has a zero energy groundstate with zero correlations functions, and has a spectral gap of ≥ k . . If the unrenormalised Hamiltonian H ( ϕ ) has a ground state energy −∞ with a dense spectrum above this, then R ( k ) ( H ) also a ground state en-ergy of −∞ with a dense spectrum, and has algebraically decaying cor-relation functions.Proof. First examine the spectrum of the renormalised Hamiltonian fromLemma 43: for convenience let R ( k ) ( h ) ( i,j ) := 2 k ( | (cid:105)(cid:104) | ( i ) ⊗ Π ( j ) ud + | (cid:105)(cid:104) | ( i ) ⊗ Π ( j ) ud ) . (6.67)Further let R ( k ) ( H Λ( L )0 ) := (cid:88) (cid:104) i,j (cid:105) R ( k ) ( h ) ( i,j ) , (6.68) R ( k ) ( ˜ H u ) Λ( L ) := (cid:88) (cid:104) i,j (cid:105) ( i,j ) d ⊗ R ( k ) ( h u ) ( i,j ) (6.69) R ( k ) ( ˜ H d ) Λ( L ) := (cid:88) (cid:104) i,j (cid:105) ( i,j ) u ⊗ R ( k ) ( h d ) ( i,j ) (6.70)We note R ( k ) ( H ) Λ , R ( k ) ( ˜ H d ) Λ , R ( k ) ( ˜ H u ) Λ all commute. Further note that spec R ( k ) ( H ) Λ ⊂ k Z ≥ . (6.71)If λ ( H ( ϕ )) = 0 , then it implies λ ( H u ( ϕ )) → +Ω( L ) (see Subsec-tion 2.3). By Corollary 40, this implies λ ( R ( k ) ( H u ( ϕ ))) → +Ω( L ) too.Hence the ground state is the zero-energy | (cid:105) Λ( L ) state. Since spec R ( k ) ( H ) Λ ⊂ k Z ≥ , then the ﬁrst excited state (provided L is sufﬁciently larger) has en-ergy at least k . Finally, the state | (cid:105) Λ( L ) has zero correlations.If λ ( H ( ϕ )) = − Ω( L ) , then λ ( H u ( ϕ )) → − Ω( L ) (see Subsection 2.3).By Corollary 40, this implies λ ( R ( k ) ( H )) → − Ω( L ) . Since spec( R ( k ) ( H )) ⊂ k Z ≥ , then the ground state is the ground state of R ( k ) ( ˜ H d ) Λ( L ) + R ( k ) ( ˜ H u ) Λ( L ) .Since spec( R ( k ) ( ˜ H d ) Λ( L ) ) becomes dense in the thermodynamic limit, we seethat the Hamiltonian has a dense spectrum in the thermodynamic limit. Let | ψ (cid:105) u and | φ (cid:105) d be the ground states of R ( k ) ( H u ) Λ( L ) and R ( k ) ( H d ) Λ( L ) re-spectively, then the ground state of R ( k ) ( ˜ H d ) Λ( L ) + R ( k ) ( ˜ H u ) Λ( L ) is | ψ (cid:105) u | φ (cid:105) d Since R ( k ) ( H d ) Λ( L ) is just the critical XY-model and its ground state has al-gebraically decaying correlations [LSM61], hence the overall ground statehas algebraically decaying correlations.58 .5 Order Parameter Renormalisation In Subsection 2.5 we saw that the observable O A/B ( r ) functioned as an or-der parameter which distinguished the two phases. Deﬁning V r := V i,i +1) , ( j,j +1) ⊕ (cid:16) V u ( i,i +1) , ( j,j +1) ⊗ V d ( i,i +1) , ( j,j +1) (cid:17) , and V r [ k ] as the corresponding isometry forthe k th step of the RG process, then deﬁne R ( k ) ( O A/B )( r ) := V r [ k ] O A/B (2 k r ) V r † [ k ] . (6.72)The following lemma then holds: Lemma 46.

Let | ψ gs (cid:105) be the ground state of H u . The expectation value of theorder parameter satisﬁes: (cid:104) ψ gs | R ( k ) ( O A/B )( r ) | ψ gs (cid:105) = if λ ( R ( k ) ( H )) = 00 if λ ( R ( k ) ( H )) = Ω( L ) . (6.73) Proof. If λ ( R ( k ) ( H )) → − Ω( L ) , then the ground state is that of H (Λ( L )) u ,and hence the state | (cid:105) does not appear anywhere in the ground state.If λ ( R ( k ) ( H )) = 0 , the ground state is | (cid:105) Λ( L ) . Since, under V r [ k ] , | (cid:105) ⊗ k × k (cid:55)→| (cid:105) , the lemma follows.Thus the renormalised order parameter still acts as an order parameterfor the renormalised Hamiltonian. In particular, it still undergoes a non-analytic change when moving between phases. We ﬁnally have all the ingredients for the proof of our two main results.

Theorem 47 (Exact RG ﬂow for undecidable Hamiltonian) . Let H be theHamiltonian deﬁned in [CPGW15a]. The renormalisation group procedure,deﬁned in Deﬁnition 42, has the following properties:1. R ( h ) is computable.2. If H ( ϕ ) is gapless, then R ( k ) ( H ( ϕ )) is gapless, and if H ( ϕ ) is gapped,then R ( k ) ( H ( ϕ )) is gapped. . For the order parameter of the form O A/B ( r ) which distinguished thephases of H Λ( L ) , there exists a renormalised observable R ( k ) ( O A/B )( r ) which distinguishes the phases of R ( k ) ( H ) Λ( L ) and is non-analytic atphase transitions.4. For k iterations, the renormalised local interactions of R ( k ) ( H ) are com-putable and belong to the family F ( ϕ, τ , τ , { β i } ) , as deﬁned in Corol-lary 44.5. If H ( ϕ ) initially has algebraically decaying correlations, then R ( k ) ( H ( ϕ )) also has algebraically decaying correlations. If H ( ϕ ) initially has zerocorrelations, then R ( k ) ( H ( ϕ )) also has zero correlations.Proof. Claim 1 follows from Deﬁnition 42, where the renormalisation iso-metries and subspace restrictions are explicitly written down and are mani-festly computable, and hence for any k the coefﬁcients in Lemma 43 arecomputable. Claim 2 follows from Lemma 45: we see that, for all k > k the spectrum below energy k − is either dense with a ground state with en-ergy at −∞ , or is empty except for a single zero energy state, correspondingto the gapped and gapless cases of H ( ϕ ) . Claim 3 follows from Lemma 46.Claim 4 follows from Corollary 44. Claim 5 follows from the properties ofthe ground states in the cases λ ( H Λ( L ) u ) → ±∞ and by Lemma 45. Theorem 48 (Uncomputability of RG ﬂow) . Let h ( ϕ ) , ϕ ∈ Q , be the full localinteraction of the Hamiltonian from [CPGW15a]. Consider k iterations of theRG map from Deﬁnition 42 acting on H ( ϕ ) , such that the renormalised localterms are given by R ( k ) ( h ( ϕ )) , which can be parameterised as per Corollary 44.If the UTM is non-halting on input ϕ , then for all k > k ( ϕ ) we have that τ ( k ) = − k , for some computable k ( ϕ ) . If the UTM halts on input ϕ , thenthere exists an uncomputable k h ( ϕ ) such that for k ( ϕ ) < k < k h ( ϕ ) we have τ ( k ) = − k , and for all k > k h ( ϕ ) then τ ( k ) = − k + Ω(4 k − k h ( ϕ ) ) .Proof. Consider the expression for τ from Lemma 43: τ ( k ) = 4 k (cid:88) n +1 < k − n − λ ( H q (4 n )) + 4 k α ( ϕ ) − k (6.74) τ ( k ) = 4 k (cid:88) n +1 < k − n − λ ( H q (4 n )) + 4 k α ( ϕ ) − k . (6.75)60 igure 10: A schematic picture of the ﬂow of Hamiltonians in parameter space. σ ( k ) is deﬁned in Orange represents some value of ϕ = ϕ for which the QTM doesnot halt on input, while purple represents ϕ = ϕ + (cid:15) for any algebraic number (cid:15) for which the QTM halts. For small k , the orange and purple lines coincide. Thenat a particular value of k , σ ( k ) becomes non-zero and then increases exponentially. From the deﬁnition of α ( ϕ ) , we see that there is a k ( ϕ ) such that g ( k ( ϕ )) = α ( ϕ ) , and hence we get τ ( k ) = − k + 4 k (cid:88) k ϕ ) < n +1 < k − n − λ ( H q (4 n )) . (6.76)If the encoded QTM never halts, then by Lemma 24 λ ( H q (4 n )) = 0 for all n such that n + 1 > k ( ϕ ) . If the encoded UTM halts then by Lemma 24 thereexists an n such that λ ( H q (4 n )) > for all n > n . Then k h ( ϕ ) is deﬁnedas the minimum k such that n + 1 < k h ( ϕ ) . Thus determining k h ( ϕ ) isat least as hard as computing the halting time and thus is an uncomputablenumber. Theorem 47 shows that our RG scheme satisﬁes the expected properties. Wenow qualitatively examine the Hamiltonian for large values of k . Here we show that for gapped instances the Hamiltonian becomes “Ising-like”, for appropriately small energy scales. From Corollary 44 the renorm-61lised Hamiltonian is R ( k ) ( h row ( ϕ )) ( i,j ) =2 k ( | (cid:105)(cid:104) | ( i ) ⊗ Π ( j ) ud + | (cid:105)(cid:104) | ( i ) ⊗ Π ( j ) ud ) (7.1) + R ( k ) ( h rowu ( ϕ ) (cid:48) ) ( i,j ) ⊗ ( i,j ) d + ( i,j ) u ⊗ R ( k ) ( h d ) ( i,j ) (7.2) +2 k Π ( i ) ud ⊗ Π ( j ) ud (7.3) R ( k ) ( h col ( ϕ )) ( i,j ) =2 k ( | (cid:105)(cid:104) | ( i ) ⊗ Π ( j ) ud + | (cid:105)(cid:104) | ( i ) ⊗ Π ( j ) ud ) (7.4) + R ( k ) ( h colu ( ϕ ) (cid:48) ) ( i,j ) ⊗ ( i,j ) d (7.5) R ( k ) ( h ( ϕ )) (1) =( g ( k ) − k α ( ϕ ) − k )Π ud + R ( k ) ( h (1) u ( ϕ )) , (7.6)where here we have explicitly separated out Π ( i ) ud ⊗ Π ( j ) ud from the term R ( k ) ( h rowu ( ϕ )) ( i,j ) = R ( k ) ( h rowu ( ϕ ) (cid:48) ) ( i,j ) + Π ( i ) ud ⊗ Π ( j ) ud .Deﬁne the Ising-like Hamiltonian with local terms: h (cid:48) rowIsing ( k ) ( i,j ) := 2 k (cid:16) | (cid:105)(cid:104) | ( i ) ⊗ Π ( j ) ud + Π ( j ) ud ⊗ | (cid:105)(cid:104) | ( i ) + Π ( i ) ud ⊗ Π ( j ) ud (cid:17) (7.7) h (cid:48) colIsing ( k ) ( i,j ) := 2 k (cid:16) | (cid:105)(cid:104) | ( i ) ⊗ Π ( j ) ud + Π ( j ) ud ⊗ | (cid:105)(cid:104) | ( i ) (cid:17) (7.8) h (cid:48) Ising ( k ) (1) := B ( k )Π ud . (7.9)This is reminiscent of the Ising interaction with both an ferromagnetic | (cid:105)(cid:104) | ( i ) ⊗ | (cid:105)(cid:104) | ( j ) + | (cid:105)(cid:104) | ( i ) | (cid:105)(cid:104) | ( j ) along the rows and columns and an anti-ferromagnetic | (cid:105)(cid:104) | ( i ) ⊗ | (cid:105)(cid:104) | ( j ) term along just the rows, with local ﬁeld B ( k ) = ( g ( k ) − k α ( ϕ ) − k ) | (cid:105)(cid:104) | , but with the orthogonal projector Π ud playing the role of the projector onto the | (cid:105)(cid:104) | state. However, note that Π ud projects onto a larger dimensional subspace than | (cid:105)(cid:104) | , so e.g. the partitionfunction of this Ising-like Hamiltonian is not identical to that of an Isingmodel.We now show the following: Proposition 49.

Let E be a ﬁxed energy cut-off and H (cid:48) Ising ( k ) = (cid:80) (cid:104) i,j (cid:105) h (cid:48) Ising ( k ) ( i,j ) .Then (cid:13)(cid:13)(cid:13) R ( k ) ( H ( ϕ )) | ≤ E − H (cid:48) Ising ( k ) | ≤ E (cid:13)(cid:13)(cid:13) op ≤ (cid:18) E k (cid:19) . (7.10) Proof.

Consider the local interaction term h = | (cid:105)(cid:104) | ⊗ Π ud + Π ud ⊗ | (cid:105)(cid:104) | .This commutes with all other terms in both the R ( k ) ( H ( ϕ )) Hamiltonianand the Ising-like Hamiltonian, and hence the eigenstates of both of the62verall Hamiltonians are also eigenstates of | (cid:105)(cid:104) | ⊗ Π ud + Π ud ⊗ | (cid:105)(cid:104) | . As aresult, for each eigenstate, a given site p ∈ Λ either has support only on | (cid:105) p or only on R ( k ) ( H ud ) . Therefore, an eigenstate deﬁnes regions (domains) ofthe lattice where all points in the domain are in H ud .For a given eigenstate | ψ (cid:105) , let D := (cid:110) i ∈ Z | tr (cid:16) | (cid:105) (cid:104) | ( i ) | ψ (cid:105) (cid:104) ψ | (cid:17) = 0 (cid:111) denote the region of the lattice where the state is supported on R ( k ) ( H ud ) ,and ∂D be the set of sites on the boundary of D . Then we see that the termsin eq. (7.2) act non-trivially only within D , and that the boundaries of D receive an energy penalty of k | ∂D | from terms in eq. (7.1) and eq. (7.4).Note that (cid:13)(cid:13) R ( k ) ( h d ) ( i,j ) (cid:13)(cid:13) op , (cid:13)(cid:13) R ( k ) ( h u ( ϕ ) (cid:48) ) ( i,j ) (cid:13)(cid:13) op , (cid:13)(cid:13)(cid:13) R ( k ) ( h (1) u ( ϕ )) (cid:13)(cid:13)(cid:13) op ≤ .For (cid:13)(cid:13) R ( k ) ( h d ) ( i,j ) (cid:13)(cid:13) op this is straightforward to see. For (cid:13)(cid:13) R ( k ) ( h u ( ϕ ) (cid:48) ) ( i,j ) (cid:13)(cid:13) op ,any states which pick up non-zero energy, other than those which receivea penalty due to halting, are removed from the local Hilbert space (as perSection 5).Let m ∈ N be a cut-off such that | ∂D | ≤ m , hence | D | ≤ m / . Sincefor each boundary term we get an energy penalty of at least k from h , wecan relate m to the energy cut-off E to m as E := 2 k m . If we consider theHamiltonians restricted to a subspace with energy ≤ E := 2 k m , then (cid:13)(cid:13)(cid:13) R ( k ) ( H ( ϕ )) | ≤ E − H (cid:48) Ising ( k ) | ≤ E (cid:13)(cid:13)(cid:13) op (7.11) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) (cid:104) i,j (cid:105) (cid:16) R ( k ) ( h u ( ϕ ) (cid:48) ) ( i,j ) ⊗ ( i,j ) d + ( i,j ) u ⊗ R ( k ) ( h d ) ( i,j ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) ≤ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) op (7.12) ≤ m (cid:18)(cid:13)(cid:13)(cid:13) R ( k ) ( h u ( ϕ ) (cid:48) ) ( i,j ) (cid:13)(cid:13)(cid:13) op + (cid:13)(cid:13)(cid:13) R ( k ) ( h d ) ( i,j ) (cid:13)(cid:13)(cid:13) op + (cid:13)(cid:13)(cid:13) R ( k ) ( h (1) u ( ϕ )) (cid:13)(cid:13)(cid:13) op (cid:19) (7.13) ≤ m (7.14) < (cid:18) E k (cid:19) . (7.15)Going from eq. (7.12) to eq. (7.13) we have used the fact that the terms inthe sum are only non-zero within domains, and | D | ≤ m / . Going fromeq. (7.13) to eq. (7.15) we have used the bound on the individual norms ofthe local terms. 63hus, for appropriately small energies, we expect only small deviationsfrom the "Ising-like" Hamiltonian. And these deviations vanish as the RGprocess is iterated. In particular, the spectrum will look like ﬁg. 11. Figure 11:

The energy level diagram of R ( k ) ( H ) . The blue levels represent excita-tions of the k ( | (cid:105)(cid:104) | ( i ) ⊗ Π ( j ) ud + Π ( i ) ud ⊗ | (cid:105)(cid:104) | ( j ) ) term, while the red area representsthe excited states of R ( k ) ( h u ( ϕ ) (cid:48) ) ( i,j ) , R ( k ) ( h d ) ( i,j ) , and R ( k ) ( h (1) u ( ϕ )) . The size ofthe red region increases as the domains get larger, and hence there are more highenergy states. The ground state has no associated red region due to the presence ofthe spectral gap. The blue lines have an energy spacing of integer multiples of k . For a ϕ for which H ( ϕ ) is gapless, R ( k ) ( H ( ϕ )) is also gapless and we seethat the ground state is that of R ( k ) ( H u ( ϕ )) . If we restrict to a low energysubspace, one can see that excited states are either the excited states of theGottesman-Irani Hamiltonians or the excited states of the critical XY-model.Indeed, let E ( k ) be the subspace of states with energy less than k , then forsufﬁciently large k we see that R ( k ) ( H ) Λ | E ( k ) = R ( k ) ( H u ( ϕ )) Λ | E ( k ) ⊗ Λ d + Λ ⊗ R ( k ) ( H d ) Λ | E ( k ) . (7.16)64ince R ( k ) ( H d ) Λ | E ( k ) has the same spectrum as H d , the spectrum of R ( k ) ( H ) Λ | E ( k ) is also dense in the thermodynamic limit. Furthermore, R ( k ) ( H ) Λ | E ( k ) hasalgebraically decaying correlations since R ( k ) ( H d ) Λ | E ( k ) also has algebraic-ally decaying correlations [LSM61]. We have seen under the renormalisation group procedure constructed here,the Hamiltonian ﬂows towards either an Ising-like Hamiltonian or an XY-likeHamiltonian. Which case occurs depends on the parameter τ in eq. (6.66).Let k be the number of iterations of the RG procedure, then from The-orem 48 we see that there are two cases: τ = − k always, or τ = − k initially, and once a sufﬁciently large value of k is reached it begins to di-verge as τ > − k + Ω(4 k ) . Determining which case occurs is undecidable.Moreover, the value of k at which we go from the ﬁrst case to the second isuncomputable. Thus, determining the trajectory of the system for an arbit-rary value of ϕ is uncomputable. Even if ϕ were known exactly, we see thatthe Hamiltonian’s path in parameter space would be unpredictable.Contrast this with chaotic behaviour: for chaotic systems, a tiny differ-ence in the initial system parameters can lead to large diverges in traject-ories later. Here the difﬁculty in predicting behaviour arises as it is usuallydifﬁcult to determine the initial system parameters exactly. However, if thesystem parameters are known exactly, it should theoretically be possible toascertain the long-time system. RG ﬂows which undergo chaotic behaviourhave been demonstrated before [MBK82; SKS82; DEE99; DT91; MN03].The behaviour of the RG trajectory shown here is stronger than this inthat even if the initial parameters characterising the microscopic interactionsare known exactly , determining which ﬁxed point the system may ﬂow to isnot possible to determine. We compare this to a similar uncomputabilityresult in [Moo90] which showed that computing the trajectory of a particlein a potential is uncomputable.The Hamiltonian discussed in this work is highly artiﬁcial and the RGscheme reﬂects this. Indeed, this Hamiltonian has an enormous local Hil-bert space dimension and its matrix elements are functions of both ϕ andthe binary length of ϕ , | ϕ | . Both of these factors are unlikely to be present innaturally occurring Hamiltonians. Thus an obvious route for further work is65o consider RG schemes for more natural Hamiltonians which display unde-cidable behaviour.Furthermore, although the RG scheme is essentially a simple BRG scheme,the details of its construction and analysis rely on knowledge of the struc-ture of the ground states. Due to the behaviour of this undecidable model,any BRG scheme will have to exhibit similar behaviour to the one we haveanalysed rigorously here. But it would be nice to ﬁnd a simpler RG schemefor this Hamiltonian (or other Hamiltonians with undecidable properties)which is able to truncate the local Hilbert space to a greater degree, withoutusing explicit a priori knowledge of the ground state, for which it is stillpossible to prove this rigorously.The Hamiltonian and RG scheme constructed here could also be used toprove rigorous results for chaotic (but still computable) RG ﬂows. Indeed, ifwe modify the Hamiltonian H ( ϕ ) so that instead of running a universal Tur-ing Machine on input ϕ , it carries out a computation of a (classical) chaoticprocess (e.g. repeated application of the logistical map), then two inputswhich are initially very close may diverge to completely different outputsafter some time. By penalising this output qubit appropriately, the Hamilto-nian will still ﬂow to either the gapped or gapless ﬁxed point depending onthe outcome of the chaotic process under our RG map, but the RG ﬂow willexhibit chaotic rather than uncomputable dynamics.Given the RG scheme here, it is also relevant to ask is whether we canapply a similar scheme to the Hamiltonians designed in [Bau+20; BCW21].Although we do not prove it here, we expect to be able to apply the modiﬁedBRG developed in this work to these Hamiltonians in an analgous way. Theonly additional consideration is the so-called "Marker Hamiltonian" com-ponent of both of these constructions which would need additional care ina rigorous proof. Since the Marker Hamiltonian has a similar ground statestructure to the circuit-to-Hamiltonian mapping — consisting of superposi-tions of a particle propagating along a line — we expect a similar RG processto sufﬁce. As a result, we do not expect an fundamentally different beha-viour in the RG ﬂow from the Hamiltonian analysed here.66 cknowledgements E.O. and T.S.C. are supported by the Royal Society. J.D.W. is supported bythe EPSRC Centre for Doctoral Training in Delivering Quantum Technologies(grant EP/L015242/1). This work has been supported in part by the EPSRCProsperity Partnership in Quantum Software for Simulation and Modelling(grant EP/S005021/1), and by the UK Hub in Quantum Computing andSimulation, part of the UK National Quantum Technologies Programme withfunding from UKRI EPSRC (grant EP/T001062/1).

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Phys. Rev. B ε expansion”. In: Physics Reports ppendix

A Recontructing Robinson pattern of 2D plane

A ﬁrst interesting fact is that × supertiles having a parity cross on thebottom left pointing up-right must have the following structure. A parityleft tile on top-left corner, a parity down tile in the bottom-right corner, andconsequently there must be a free cross on the top-right of the supertile.The orientation of the free cross will uniquely determine the type of left tileand down tile in the same supertile. Thus, there are only 4 supertiles with aparity cross pointing up-right: ×× ×× ×× ×× ×× ×× ×× ×× We make a ﬁrst educated guess: each of these four supertiles corres-ponds to the × parity cross having the same orientation as the free crosscontained in the top-right of the supertile.If we want to tile the plane according to the Robinson pattern, thesesupertiles must then appear in alternate positions in alternate rows. To thisaim, we will assign parity rules to the supertiles according to the orientationof the × parity cross on the bottom-left of each supertile (please note:this is a parity associated to the supertile as a whole and it is different to itsinner parity structure). That is,Thus, × supertiles with a up-right × parity cross in the bottom71eft must be interleaved in the vertical direction with a supertile with a × parity cross pointing bottom-right and in the horizontal direction with asupertile having a × up-left parity cross. Finally, supertiles with a down-left cross will alternate on the diagonal with the supertiles having a up-rightcross.Using these parity rules and the usual arrow heads/tails constraints,we shall obtain the adjacency relations for the supertiles which have to beobeyed. We make a point here: the only constraints that we will use in thetiling of the 2D plane are the ones set by these adjacency rules.We shall now reconstruct the basic 3-square in Robinson’s argument,this time using × supertiles. At the corners of these 3-squares there mustbe the four supertiles that we have identiﬁed as parity crosses. By strictlyfollowing the adjacency rules, we will end up with exactly four possible3-squares, that we will relate to the 3-squares with × tiles. No otherconﬁguration of a 3-square is allowed! ×× ×× ×××× ×× ××××× ××× ××××××× ××××× ×××××××× ×× ×× ×××× ×× ×× ×××× ×× ××××× ××× ××××××× ××××× ×××××××× ×× ×× ×××× ×× ×× ×××× ×× ××××× ××× ××××××× ××××× ×××××××× ×× ×× ×××× ×× ×× ×××× ×× ××××× ××× ××××××× ××××× ×××××××× ×× ×× ×××× We remark that the position of the × parity crosses at the corner ofthe 3-square are ﬁxed, and that the central × supertile of these 3-squares73 corresponding to one of the the × free crosses – uniquely determinesthe remaining ones.At this time we have recognised the ﬁrst 20 tiles. Among these are theparity cross and free cross tiles. The remaining ones will be determinedby those supertiles placed between the 3-squares, again in analogy to theRobinson pattern.We consider the 3-squares in the illustration above, labelled from 1 to4. When we pick the ﬁrst 3-square, we note that only the second 3-squarecan be placed at its right, and they must be interleaved with a string madeof three supertiles put in vertical order: only three conﬁgurations for thesestrings are allowed. Below we illustrate their arrow markings as well astheir renormalisation onto Robinson tiles. Note that the central Robinsontile in each renormalised string has free parity and is hence different fromthe parity vertical tile having the same arrow markings appearing in the 3-squares. Note also that the tile at the top of each string is the identical tothe one at its bottom (both in markings and parity). ××××× ××××××××××× ××××× ××××××××××× ××××× ××××××××××× ××××× ××××××××××× ××××× ××××××××××× ××××× ××××××××××× Vertical Robinson tiles with free parity and a single horizontal line cor-respond to supertiles between 3-squares whose free crosses do not face eachothers. More precisely, the mapping is given by75 ×× ×× ×× ×× ×× × × ×× ×× ×× ×× ×× ×

We have at this point a correspondence between 38 tiles and 38 super-tiles; in addition to parity crosses and free crosses, now all vertical armsassociated with both parities have been identiﬁed. The remaining 18 tilesare horizontal arms. To ﬁnd them, we proceed in analogous way.Below the ﬁrst 3-square we can place only the third 3-square, interleavedwith one of the following strings of three supertiles put in horizontal order.Again, we note that the left and right supertiles of each string coincide andare thus mapped to the same Robinson tile and that the central tile has freeparity. 76 ×× × ×× ×× ××× × × ×××××× × ×× ×× ××× × × ×××××× × ×× ×× ××× × × ×××

We have other three strings of supertiles that are allowed to stay betweenthe second three square placed above the fourth 3-squares. ××× × ×× ×× ××× × × ×××××× × ×× ×× ××× × × ×××××× × ×× ×× ××× × × ×××

It remains to associate the last 6 free horizontal Robinson tiles to thesupertiles that are still unmatched. These must be placed between 3-squareswhose free crosses do not face each other.77 ×× ×× ×× ×× ×× × × ×× ×× ×× ×× ×× ×

All 56 Robinson tiles have ﬁnally been identiﬁed with a subset of theallowed × supertiles. With this renormalisation, one can verify thatit is possible to reproduce the Robinson pattern of the plane with × supertiles. More importantly, one can ascertain that the adjacency rules forthe × supertiles, under this projection, correspond exactly to the rulesfor the Robinson tiles (cf. Appendix B). Stated in another way, the directedgraphs representing respectively the adjacency rules of the 56 supertiles andthe rules for the Robinson tiles are isomorphic. Thus, we have achieved acomplete renormalisation under which Theorem 11 and Corollary 12 hold. B Mathematica notebook

Available in the arXiv submission folder is a

Mathematica notebook for theexplicit construction of the renormalisation map in Deﬁnition 14, in the casewhen the parity cross occupies the bottom-left position of the grid.We begin inserting by hand the adjacency rules for the Robinson tiles,where the tiles are numbered according to the order given in ﬁg. 4, fromleft to right, top to bottom. For each tile we list what are the ones thatcan stay above (variable adjup1x1 in the notebook) or on its right (variable78 djright1x1 ), respectively. Using these rules, we then construct all allowedsupertiles with a parity cross in the bottom-left position; the total numberof those new elements is 68. In the notebook, each supertile is representedby a × matrix whose entries are numbers from 1 to 56 correspondingto the Robinson tiles which are composing it. We then construct adjacencyrules for these supertiles by obeying arrow markings and parity constraintson the shared edge.The renormalisation map is represented by the variable labelrenormaliz-ation : the number at the position j corresponds to the Robinson tile associ-ated to the supertile j . The supertiles that are not appearing in the Robinsonpattern discussed in Subsection 4.3 are mapped to numbers from 57 to 68;these are not connected to any Robinson tile, and subsequently removedfrom the bijection. Finally, we re-write the adjacency rules for supertilesunder this bijection and conﬁrm that the graph is isomorphic to the one ofthe Robinson tiles. C Proof of Lemma 38

For convenience we state Lemma 51 of [CPGW15a].

Lemma 50 (Tiling + quantum layers, Lemma 51 of [CPGW15a]) . Let h row c , h col c ∈B ( C C ⊗ C C ) be the local interactions of a 2D tiling Hamiltonian H c , with twodistinguished states (tiles) | L (cid:105) , | R (cid:105) ∈ C C . Let h q ∈ B ( C Q ⊗ C Q ) be the localinteraction of a Gottesman-Irani Hamiltonian H q ( r ) , as in Section 5. Thenthere is a Hamiltonian on a 2D square lattice with nearest-neighbour interac-tions h row , h col ∈ B ( C C + Q +1 ⊗ C C + Q +1 ) with the following properties: For anyregion of the lattice, the restriction of the Hamiltonian to that region has aneigenbasis of the form | T (cid:105) c ⊗ | ψ (cid:105) q , where | T (cid:105) c is a product state representing aclassical conﬁguration of tiles. Furthermore, for any given | T (cid:105) c , the lowest en-ergy choice for | ψ (cid:105) q consists of ground states of H q ( r ) on segments between sitesin which | T (cid:105) q contains an | L (cid:105) and an | R (cid:105) , a 0-energy eigenstate on segmentsbetween an | L (cid:105) or | R (cid:105) and the boundary of the region, and | e (cid:105) ’s everywhereelse. For the rest of this section we denote (cid:12)(cid:12) R ( k ) ( L ) (cid:11) and (cid:12)(cid:12) R ( k ) ( R ) (cid:11) to be thestates in the set of k -time renormalised Robinson tiles with a down-left anddown-right red cross marking on them, respectively. For simplicity we break79own Lemma 38 into two separate parts: the ﬁrst deals with the ﬁrst twoclaims and the last deals with the third claim. Lemma 51 (Restatement of Claim 1 and 2 in Lemma 38) . Let H Λ( L ) u = (cid:80) h row ( j,j +1) u + (cid:80) h col ( i,i +1) u , where h col j,j +1 = h col c ⊗ ( j ) eq ⊗ ( j +1) eq (C.1a) h row i,i +1 = h row c ⊗ ( i ) eq ⊗ ( i +1) eq (C.1b) + ( i ) c ⊗ ( i +1) c ⊗ h q (C.1c) + | L (cid:105)(cid:104) L | ( i ) c ⊗ ( eq − (cid:12)(cid:12) (cid:11)(cid:10) (cid:12)(cid:12) ) ( i ) ⊗ ( i +1) ceq (C.1d) + ( c − | L (cid:105)(cid:104) L | c ) ( i ) ⊗ (cid:12)(cid:12) (cid:11)(cid:10) (cid:12)(cid:12) ( i ) ⊗ ( i +1) ceq (C.1e) + ( i ) ceq ⊗ | R (cid:105)(cid:104) R | ( i +1) c ⊗ ( eq − (cid:12)(cid:12) (cid:11)(cid:10) (cid:12)(cid:12) ) ( i +1) (C.1f) + ( i ) ceq ⊗ ( c − | R (cid:105)(cid:104) R | ) ( i +1) c ⊗ (cid:12)(cid:12) (cid:11)(cid:10) (cid:12)(cid:12) ( i +1) (C.1g) + ( i ) c ⊗ | e (cid:105)(cid:104) e | ( i ) e ⊗ | R (cid:105)(cid:104) R | ( i +1) c ⊗ ( i +1) eq (C.1h) + | L (cid:105)(cid:104) L | ( i ) c ⊗ ( i ) eq ⊗ ( i +1) c ⊗ | e (cid:105)(cid:104) e | ( i +1) e (C.1i) + ( i ) c ⊗ | e (cid:105)(cid:104) e | ( i ) e ⊗ ( c − | L (cid:105)(cid:104) L | ) ( i +1) c ⊗ ( eq − | e (cid:105)(cid:104) e | ) ( i +1) e (C.1j) + ( c − | R (cid:105)(cid:104) R | ) ( i ) c ⊗ ( eq − | e (cid:105)(cid:104) e | ) ( i ) e ⊗ ( i +1) c ⊗ | e (cid:105)(cid:104) e | ( i +1) e (C.1k) + ( i ) ceq ⊗ ( i +1) ceq (C.1l) h (1) i = − (1 + α ( ϕ )) ( i ) ceq , (C.1m) for a constant α ( ϕ ) .Then the k times renormalised Hamiltonian under the RG mapping ofDeﬁnition 35, R ( k ) ( H u ) Λ( L × H ) , has the following properties:1. For any ﬁnite region of the lattice, the restriction of the Hamiltonianto that region has an eigenbasis of the form | T (cid:105) c ⊗ | ψ i (cid:105) where | T (cid:105) c ∈ R ( k ) ( H c ) Λ( L × H ) is a classical tiling state, | ψ i (cid:105) ∈ R ( k ) ( H eq ) Λ( L × H ) .2. Furthermore, for any given | T (cid:105) c , the lowest energy choice for | ψ (cid:105) q con-sists of ground states of R ( k ) ( H q )( r ) on segments between sites in which | T (cid:105) c contains an (cid:12)(cid:12) R ( k ) ( L ) (cid:11) and an (cid:12)(cid:12) R ( k ) ( R ) (cid:11) , a 0-energy eigenstate onsegments between an (cid:12)(cid:12) R ( k ) ( L ) (cid:11) or (cid:12)(cid:12) R ( k ) ( R ) (cid:11) and the boundary of theregion, and (cid:12)(cid:12)(cid:12) e × k (cid:69) ’s everywhere else. Any eigenstate which is not an ei-genstate of R ( k ) ( H q )( r ) on segments between sites in which | T (cid:105) c containsan (cid:12)(cid:12) R ( k ) ( L ) (cid:11) and an (cid:12)(cid:12) R ( k ) ( R ) (cid:11) has an energy > . roof. Claim 1

The fact the eigenstates of the unrenormalised Hamiltonian are a productstate across H c and H eq , | T c (cid:105) | ψ (cid:105) eq (cid:48) is from Lemma 50 (Lemma 51 of [CPGW15a]).The structure of the eigenstates of the renormalised Hamiltonian is then pre-served as per Lemma 31. Claim 2

Start by considering what each of the local terms looks like after applyingthe renormalisation isometries. We treat each term in the above lemma insuccession. Start with local interactions encoding the classical tiling, termsC.1a and C.1b. The isometry decomposes as V u ( i,i +1)( j,j +1) = (1 ⊗ Π gs ) V c ( i,i +1)( j,j +1) ⊗ V eq ( i,i +1)( j,j +1) , hence the classical Hamiltonian termstransform as per Lemma 18.We next consider the renormalisation of the Gottesman-Irani Hamilto-nian h q . All of these states are mapped by V eq ( i,i +1)( j,j +1) to a × chain,which V q ( i,i +1) acts on as per Lemma 28. Thus h q transforms as per Lemma 28. Coupling Terms

We ﬁrst note that given a × block, we wil get two sets of coupling terms:one between c and eq and another set between c and eq . Thus the termswill have the structure h i,i +1 a ⊗ h i,i +1 eq ⊗ h i,i +1 eq , where h i,i +1 eq and h i,i +1 eq areidentical except they act on different parts of the local Hilbert space.We will then “integrate out” eq in the next stage of the renormalisationprocedure leaving us with only a single set. Thus for the purposes of the RGprocedure, we need only consider how the coupling terms transform for aparticular ( i, i + 1; j ) set (as we will integrate out the other set anyways).We now consider the terms coupling the classical and quantum parts ofthe Hilbert space. Consider term C.1d. In any × block in the restrictedsubspace, at most one free | L (cid:105) or | R (cid:105) may appear (i.e. not parity cross),and under the classical renormalisation mapping, we see that a × blockwith a free cross is mapped to a cross supertile of the same colour and withrelevant orientation. Any parity cross is removed in the renormalisationstep, as per Subsection 4.2. Then we realise that the × block only receives81he penalty iff | L (cid:105) is not combined with (cid:12)(cid:12) (cid:11) . Since under the RG operations | L (cid:105) → | R ( L ) (cid:105) , and (cid:12)(cid:12) (cid:11) | x (cid:105) → (cid:12)(cid:12) R ( , x ) (cid:11) we see that the new term mustpenalise states which do not satisfy these states being paired. The parity | L (cid:105) tiles will be integrated out, however, these are associated with history statesthat will be integrated out in the same step, and hence can be ignored. Thusterm C.1d becomes | R ( L ) (cid:105)(cid:104) R ( L ) | ( i ) ⊗ (cid:0) eq (cid:48) − (cid:12)(cid:12) x (cid:11)(cid:10) x (cid:12)(cid:12) − (cid:12)(cid:12) e (cid:11)(cid:10) e (cid:12)(cid:12)(cid:1) ⊗ ( i +1) ceq (cid:48) , (C.2)where | x (cid:105) ∈ B are single site states of the original Hamiltonian. By similarreasoning, after k applications of the RG mapping, we get (cid:12)(cid:12)(cid:12) R ( k ) ( L ) (cid:69)(cid:68) R ( k ) ( L ) (cid:12)(cid:12)(cid:12) ( i ) ⊗ (cid:32) eq (cid:48) − (cid:88) m (cid:88) x t ∈ B (cid:12)(cid:12)(cid:12) e × m { x t } × k − m − (cid:69)(cid:68) e × m { x t } × k − m − (cid:12)(cid:12)(cid:12)(cid:33) ⊗ ( i +1) ceq (cid:48) . (C.3)The term C.1f transforms analogously.Now consider term C.1e. Again, × blocks in the restricted subspacewith the free tile being | L (cid:105) get renormalised to | R ( L ) (cid:105) . We see that this termpenalises anything but (cid:12)(cid:12) (cid:11) being combined with it, and hence we see it ismapped to ( c − | R ( L ) (cid:105)(cid:104) R ( L ) | c ) ( i ) ⊗ (cid:16)(cid:12)(cid:12) x (cid:11)(cid:10) x (cid:12)(cid:12) ( i ) + (cid:12)(cid:12) e (cid:11)(cid:10) e (cid:12)(cid:12) ( i ) (cid:17) ⊗ ( i +1) ceq . (C.4)By similar reasoning, after k iterations we get ( c − (cid:12)(cid:12)(cid:12) R ( k ) ( L ) (cid:69)(cid:68) R ( k ) ( L ) (cid:12)(cid:12)(cid:12) c ) ( i ) ⊗ (cid:32)(cid:88) m (cid:88) x t ∈ B (cid:12)(cid:12)(cid:12) e × m { x t } × k − m (cid:69)(cid:68) e × m { x t } × k − m (cid:12)(cid:12)(cid:12)(cid:33) ( i ) ⊗ ( i +1) ceq . (C.5)The C.1g transforms analogously.We now consider term C.1i. If we consider the term acting between × blocks, then this is only violated if there is a | L (cid:105) c at site ( i, j ) and atthe neighbouring site ( i + 1 , j ) is in state | e (cid:105) e . The renormalised basis states82hich get penalised by this are then: | R ( L ) (cid:105)(cid:104) R ( L ) | ( i ) c ⊗ ( i ) eq ⊗ ( i +1) c ⊗ | ee (cid:105)(cid:104) ee | e + (cid:88) | x (cid:105)∈ B | ex (cid:105)(cid:104) ex | q (cid:48) ( i +1) . (C.6)After k iterations this becomes (cid:12)(cid:12)(cid:12) R ( k ) ( L ) (cid:69)(cid:68) R ( k ) ( L ) (cid:12)(cid:12)(cid:12) ( i ) c ⊗ ( i ) eq ⊗ ( i +1) c ⊗ (cid:32)(cid:88) m (cid:88) x t ∈ B (cid:12)(cid:12)(cid:12) e × m { x } × k − m (cid:69)(cid:68) e × m { x } × k − m (cid:12)(cid:12)(cid:12)(cid:33) ( i +1) . (C.7)Term C.1h transforms analogously.We now consider term C.1j. This term forces a non- | e (cid:105) e to the left of anyother non-blank in the q -layer, except when a non-blank coincides with an | L (cid:105) in the c-layer. Again, we see that this penalty term is zero within any × blocks in the restricted subspace κ i,j , so we need only consider theinteractions between such states. If there is a | e (cid:105) e state next to a | x (cid:105) state inthe blocks, then we see that the quantum part of this tile must get mappedto | e (cid:105) e or | R ( x ) (cid:105) . The new term in the Hamiltonian becomes ( i ) c ⊗ | ee (cid:105)(cid:104) ee | e + (cid:88) | y (cid:105)∈ B | ye (cid:105)(cid:104) ye | q ( i ) ⊗ ( c − | R ( L ) (cid:105)(cid:104) R ( L ) | ) ( i +1) c ⊗ ( eq − | ee (cid:105)(cid:104) ee | ) ( i +1) e . (C.8)After k iterations of the RG map the term becomes ( i ) c ⊗ (cid:32) (cid:88) m =1 (cid:88) x t ∈ B (cid:12)(cid:12)(cid:12) { x t } × k − m , e × m (cid:69)(cid:68) { x t } × k − m , e × m (cid:12)(cid:12)(cid:12)(cid:33) ( i ) ⊗ ( c − | R ( L ) (cid:105)(cid:104) R ( L ) | ) ( i +1) c (C.9) ⊗ ( eq − (cid:88) m =1 (cid:88) x t ∈ B (cid:12)(cid:12)(cid:12) e × m , , { x t } × k − m − (cid:69)(cid:68) e × m , , { x t } × k − m − (cid:12)(cid:12)(cid:12) ) ( i +1) e . (C.10)Term C.1k transforms analogously. Identity Terms

Finally we need to consider how terms of the form ( i ) ceq and ( i ) ceq ⊗ ( i +1) ceq ( i,j ) ⊗ ( i +1 ,j ) : (cid:16) ( i,j ) ⊗ ( i +1 ,j ) + ( i,j +1) ⊗ ( i +1 ,j +1) (cid:17) → ( i/ ,j/ . (C.11)Similarly, consider (cid:16) ( i +1 ,j ) ⊗ ( i +2 ,j ) + ( i +1 ,j +1) ⊗ ( i +2 ,j +1) (cid:17) → ( i/ ,j/ ⊗ ( i/ ,j/ . (C.12)Consider the ( i,j ) terms, then ( i,j ) + ( i +1 ,j ) + ( i,j +1) + ( i +1 ,j +1) → ( i/ ,j/ . (C.13)Combining these terms, we see that these create new 1-local terms which,after k iterations have coefﬁcients: ( − k + k (cid:88) m =0 (4 m × m − k )) ( i/ ,j/ = − − k ( i/ ,j/ , (C.14)and 2-local terms of the form: k ( i/ ,j/ ⊗ ( i/ ,j/ . (C.15)Note that these 2-local terms only occur in the row interactions, and remainzero for the column interactions.So far we have shown that all terms in the Hamiltonian transform to ananalogous term to one in the original Hamiltonian. Now note the fact theHamiltonian can be block-decomposed into subspaces with respect to statescontaining and , and into a classical and quantum part. Then real-ise that the local quantum Hilbert space can be decomposed as R ( k ) ( H e ) ⊕ R ( k ) ( H q ) . These properties allow the proof from Lemma 51 of [CPGW15a]to be applied (we refer the reader to this proof for brevity) which alsoshows that states which are not R ( k ) ( H q ) eigenstates between (cid:12)(cid:12) R ( k ) ( L ) (cid:11) and (cid:12)(cid:12) R ( k ) ( R ) (cid:11) markers have energy at least 1.With this, we now wish to prove claim 3 of Lemma 38 and hence need84o ﬁnd the ground state energy for the renormalised Hamiltonian. To do sowe need the concept of tiling defects : Deﬁnition 52 (Tiling Defect) . A pair | t a (cid:105) i,j , | t b (cid:105) i +1 ,j ∈ H c form a tiling de-fect if they violate the local term between them: (cid:104) t a | (cid:104) t b | h i,i +1 c | t a (cid:105) | t b (cid:105) = 1 .Similarly, | t a (cid:105) i,j , | t b (cid:105) i +1 ,j ∈ R ( k ) ( H c ) form a tiling defect if they violate therenormalised local term between them: (cid:104) t a | (cid:104) t b | R ( k ) ( h c ) ( i,i +1) | t a (cid:105) | t b (cid:105) = 1 . In the following lemma we show the ground state is a state with no tilingdefects, and as a result the only energy contribution comes from groundstates of the Gottesman-Irani Hamiltonians.

Lemma 53 (Restatement of Claim 3 in Lemma 38) . Let h row c , h col c ∈ B ( C C ⊗ C C ) be the local interactions of the tiling Hamiltonianassociated with the modiﬁed Robinson tiles, let R ( k ) ( h rowc ) i,i +1 , R ( k ) ( h colc ) j,j +1 be the local interactions after k RG iterations, and let h row , h col ∈ B ( C C + Q +1 ⊗ C C + Q +1 ) be the local interactions deﬁned in Lemma 51. For a given ground state con-ﬁguration (tiling) of R ( k ) ( H c ) , let L denote the set of all horizontal line seg-ments of the lattice that lie between down/right-facing and down/left-facingred crosses (inclusive) in the Robinson tiling after k RG mappings.Then the renormalised Hamiltonian on a 2D square lattice of width L and height H with nearest-neighbour interactions R ( k ) ( h row ) , R ( k ) ( h col ) hasa ground state energy λ ( R ( k ) ( H ) Λ( L × H ) ) contained in the interval (cid:20) ( g ( k ) − k α ( ϕ )) LH − − k H + (cid:98) log ( L/ (cid:99) (cid:88) n =1 (cid:18) (cid:22) H n +1( k mod (cid:23) (C.16) × (cid:18)(cid:22) L n +1 − ( k mod (cid:23) − (cid:19) (cid:19) λ ( R ( k ) ( H q )(4 n −(cid:98) ( k mod / (cid:99) )) , (C.17) ( g ( k ) − k α ( ϕ )) LH − − k H + (cid:98) log ( L/ (cid:99) (cid:88) n =1 (cid:18) (cid:18)(cid:22) H n +1 − ( k mod (cid:23) + 1 (cid:19) (C.18) × (cid:22) L n +1 − ( k mod (cid:23) (cid:19) λ ( R ( k ) ( H q )(4 n −(cid:98) ( k mod / (cid:99) )) (cid:21) (C.19) where g ( k ) = 4 k (cid:88) n +1 < k − n − λ ( H q (4 n )) . (C.20)85 roof. We identify the red down-left and down-right cross tiles from the k -times renormalised tile set with the (cid:12)(cid:12) R ( k ) ( L ) (cid:11) and (cid:12)(cid:12) R ( k ) ( R ) (cid:11) state respect-ively. For convenience, assume k ∈ N (we will deal with the other caseseparately k ∈ N + 1 ). From Lemma 51 the ground state of the Hamilto-nian is a product state | T (cid:105) c ⊗ | ψ (cid:105) eq has a (cid:12)(cid:12)(cid:12) e × k (cid:69) state combined with everytile except those between (cid:12)(cid:12) R ( k ) ( L ) (cid:11) and (cid:12)(cid:12) R ( k ) ( R ) (cid:11) , where instead there isa ground state of a R ( k ) ( H q ) Hamiltonian between the two markers. Forsuch states, the terms C.1d-C.1k give zero energy contribution and we needonly consider the terms C.1a, C.1b, and C.1c. The terms C.1l and C.1m areconstant offsets, and so we will ignore them initially and consider them atthe end.We now consider the energy of the tiling + quantum; from lemma 48 of[CPGW15a] the number of segments is lower bounded by ≥ (cid:98) H − n − (cid:99) ( (cid:98) L − n − − (cid:99) ) and upper bounded by ≤ (cid:98) H − n − + 1 (cid:99) ( (cid:98) L − n − (cid:99) ) .In the case we have d defects in the tiling, the energy is at least E ( d defects ) = d + LH ( g ( k ) − k α ( ϕ )) + (cid:88) (cid:96) ∈L λ ( R ( k ) ( H q )( | (cid:96) | )) (C.21) ≥ d + LH ( g ( k ) − k α ( ϕ )) (C.22) + (cid:98) log ( L/ (cid:99) (cid:88) n =1 (cid:32)(cid:22) H n +1 (cid:23) (cid:18)(cid:22) L n +1 (cid:23) − (cid:19) − d (cid:33) λ ( R ( k ) ( H q )(4 n )) , (C.23)where in the second line we have used the result from lemma 49 of [CPGW15a]to bound the number of segments of size n is at least (cid:4) H n +1 (cid:5) (cid:0)(cid:4) L n +1 (cid:5) − (cid:1) − d . Note, that lemma 49 of [CPGW15a] still applies to the renormalisedHamiltonian terms as the tiling rules for the renormalised tile set are identicalto the original tile set, as per Lemma 18.It can be shown from deﬁnition 50 of [CPGW15a] that (cid:80) ∞ n =1 λ ( H q (4 n +1)) < / , and since each defect carries an energy penalty of at least we seethe ground state is always achieved in the case where there are no defectsand hence the Robinson tiling is correct. Thus we see that the ground stateis given by E = LH ( g ( k ) − k α ( ϕ )) + (cid:88) (cid:96) ∈L λ ( R ( k ) ( H q )( | (cid:96) | )) . (C.24)86gain we use the bound on the number of segments allowed from lemma48 of [CPGW15a] to show that the ground state energy lies in the bounds (cid:88) (cid:96) ∈L λ ( R ( k ) ( H q )( | (cid:96) | )) ∈ (cid:20) (cid:98) log ( L/ (cid:99) (cid:88) n =1 (cid:18)(cid:22) H n +1 (cid:23) (cid:18)(cid:22) L n +1 (cid:23) − (cid:19)(cid:19) λ ( R ( k ) ( H q )(4 n )) , (C.25) (cid:98) log ( L/ (cid:99) (cid:88) n =1 (cid:18)(cid:18)(cid:22) H n +1 (cid:23) + 1 (cid:19) (cid:22) L n +1 (cid:23)(cid:19) λ ( R ( k ) ( H q )(4 n )) (cid:21) (C.26)Finally consider the constant energy offset from the terms C.1l and C.1m.After k iterations of the RG mapping, from the deﬁnition of g ( k ) in eq. (C.20),the coefﬁcient of the ( i ) term is b := 4 k (cid:88) n +1 < k − n − λ ( H q (4 n )) + 4 k (1 − α ( ϕ )) − k k (cid:88) m =1 − m (C.27) = 4 k (cid:88) n +1 < k − n − λ ( H q (4 n )) + 4 k (1 − α ( ϕ )) − k (1 − − k ) , (C.28)where the − k (cid:80) km =1 − m term arises due to part of the 2-local terms beingintegrated into the 1-local terms. The coefﬁcient in front of the 2-local term ( i ) ⊗ ( i +1) is then b := − k . The energy contribution from these term is b LH + b ( L − H = ( b + b ) LH − b H (C.29) = k (cid:88) n +1 < k − n − λ ( H q (4 n )) + 4 k (1 − α ( ϕ )) − k (1 − − k ) − − k LH + b H (C.30) = k (cid:88) n +1 < k − n − λ ( H q (4 n )) − k α ( ϕ ) LH − k H (C.31) = ( g ( k ) − k α ( ϕ )) LH − k H, (C.32)where g ( k ) is deﬁned in the lemma statement. Adding this to the en-ergy contribution from the renormalised Gottesman-Irani segments givesthe value in the lemma statement. 87or k ∈ N + 1 all of the above goes through with L/ n +1 → L/ n +1 − ( k mod , (C.33) H/ n +1 → H/ n +1 − ( k mod , (C.34) λ ( H q (4 n )) → λ ( H q (4 n −(cid:98) ( k mod / (cid:99) )) ..