Single-Shot Decoding of Linear Rate LDPC Quantum Codes with High Performance
11 Single-Shot Decoding of Linear RateLDPC Quantum Codes with High Performance
Nikolas P. Breuckmann and Vivien Londe
Abstract —We construct and analyze a family of low-densityparity check (LDPC) quantum codes with a linear encoding rate,polynomial scaling distance and efficient decoding schemes. Thecode family is based on tessellations of closed, four-dimensional,hyperbolic manifolds, as first suggested by Guth and Lubotzky.The main contribution of this work is the construction ofsuitable manifolds via finite presentations of Coxeter groups,their linear representations over Galois fields and topologicalcoverings. We establish a lower bound on the encoding rate k/nof 13/72 = 0.180... and we show that the bound is tight for theexamples that we construct.Numerical simulations give evidence that parallelizable de-coding schemes of low computational complexity suffice toobtain high performance. These decoding schemes can deal withsyndrome noise, so that parity check measurements do not have tobe repeated to decode. Our data is consistent with a threshold ofaround 4% in the phenomenological noise model with syndromenoise in the single-shot regime.
Index Terms —Quantum codes, quantum error-correction,single-shot decoding, hyperbolic, quantum fault-tolerance, Cox-eter groups, cellular automata, belief-propagation.
I. I
NTRODUCTION Q UANTUM systems are susceptible to noise, which pro-vides a formidable challenge to designing functioningand scalable quantum computers. Noise prevents us frombuilding even more powerful computing devices known as ran-dom access machines. These are computers operating on ana-log signals and it can be shown that they can solve PSPACE-complete problems in polynomial time [1]. However, smallerrors can build up uncontrollably in any analog computer.This makes it impossible to scale these types of devices whennoise is present and control is imperfect. Shor showed thatquantum computers are fundamentally different from analogcomputers in this regard, by showing that quantum errorscan be dealt with by encoding the state of the quantumcomputer into a quantum code [2]. The accumulation of smallerrors is controlled by periodically performing measurementson the redundant degrees of freedom of the quantum code,thereby discretizing the error, and using the outcome of themeasurement to determine a recovery operation.A framework for the construction of quantum codes isprovided by algebraic topology: any manifold supporting atessellation can be turned into a quantum code via its homol-ogy. Well-knonw examples are the toric code, which is derivedfrom a square tessellation of a torus and the surface code,which corresponds to the square tessellation of a topologicaldisk [3], [4]. Properties of the code such as number of physicalqubits n , number of encoded qubits k and the code distance d N. P. Breuckmann, University College London, [email protected]. Londe, Team SECRET, INRIA, [email protected] are determined by the geometrical and topological propertiesof the tessellated manifold.In [5], [6] it was shown that the parameters of homologicalcodes derived from 2D manifolds (surfaces) necessarily obeythe bound kd ≤ const. × (log k ) n. (1)In [7] the author asked whether it is generally true thatparameters of homological codes will be constrained by thebound kd ∈ n o (1) . The work of Guth and Lubotzky [8]answered this question in the negative, by showing that codesderived from tessellations of four-dimensional hyperbolic man-ifolds have a linear encoding rate k ∼ n and polynomiallyscaling distance d ∈ Θ( n (cid:15) ) . Their work left open how toactually construct these codes.In this paper we discuss several approaches to this problemand explicitely construct closed, hyperbolic 4-manifolds fromwhich we derive quantum codes. We show that the codefamily has an asymptotic encoding rate k/n lower boundedby / . For the construction we consider regular tessel-lations of hyperbolic space. We will focus on a particulartessellation by a four-dimensional regular polytope called the . This polytope owes its name to the fact that itsthree-dimensional boundary consists of 120 dodecahedra. Theadvantage of considering regular tessellations is that they canbe described by their groups of symmetry, called Coxetergroups . The first construction is based on finite presentations,which has been previously used to construct 2D hyperboliccodes [9]. A disadvantage of this approach is that findingclosed manifolds is computationally expensive. This problemis overcome by considering faithful representations of theCoxeter groups as matrix groups with coefficients in thering Z [ φ ] , where φ is the golden ratio. We relate the process ofcompactifying the infinte hyperbolic space H to an algebraicprocedure in terms of the linear representation. It turns outthat under certain conditions the symmetry group of thecompactified space has a simple and well-known structure,allowing us to derive a formula for the size of the quantumcode. In order to obtain more examples of smaller size weuse finite coverings, allowing us to construct spaces withless symmetries compared to the group-based constructions.Finally, we perform Monte Carlo simulations to determine theperformance of these codes. We consider a decoder basedon cellular automata [10] as well as a decoder based ona message-passing algorithm, called belief-propagation. Bothdecoding procedures have the advantage that they can beimplemented using very simple classical control and are highlyparallelizable. The simulation results suggest that even whenmeasurements are subject to noise it is possible to decode a r X i v : . [ qu a n t - ph ] J a n without having to repeat the measurement (single-shot errorcorrection). Even more encouraging is that the performanceis higher than currently favoured quantum error correctingschemes. Our data is consistent with an asymptotic thresholdof p = 4% in the phenomenological X/Z -flip noise modelwith syndrome noise q = p . This performance includingmeasurement errors is higher than for a family of LDPC codeswith similar parameters called hypergraph product codes whenassuming perfect measurements [11]. A. Previous work
Quantum codes based on hyperbolic 4-manifolds were orig-inally proposed in [8] where it was shown that they possessa linear encoding rate and polynomially growing distance.In [12] a local decoding scheme was proposed and it wasshown that under this scheme logical errors are polynomiallysuppressed. Single examples of 4D hyperbolic codes were con-structed in [13] and [14]. Examples of hyperbolic 4-manifoldswith small volume were constructed in [15] and [16].
B. Summary
In Section II we review the homological construction ofquantum codes and the results obtained in [8]. In Section IIIwe introduce regular tessellations of four-dimensional, hyper-bolic space and their associated groups of symmetries and wederive the lower bound on the encoded rate for homologicalcodes derived from such tessellations. We then discuss theconstruction of closed, four-dimensional hyperbolic manifoldssupporting regular tessellations using finitely presented groupsand linear representations. The list of examples is extended byconsidering less symmetric manifolds which are obtained byfinite coverings. We conclude the section by discussing theconstructed examples in more detail. Finally, in Section IVwe introduce simple decoding schemes and perform numericalsimulations to determine the performance of the constructedcode family. II. D
EFINITION AND P ROPERTIES
A. Quantum Codes from Tessellated Manifolds
Throughout the paper we assume that the number of phys-ical qubits is n and that their states form a Hilbert space H = ( C ) ⊗ n . A quantum code C is a subspace of H ofdimension k that is interpreted as the Hilbert space of k logical qubits. Due to interactions with the environment erroroperators are applied randomly on the physical state. It isassumed that such error operators act locally, meaning thatthey only act non-trivially on a small number of physicalqubits.A convenient class of quantum codes are called stabilizercodes where the code space is the +1 -eigenspace of allelements of a subgroup S of the Pauli group P = (cid:104) X i , Y i , Z i | i ∈ { , . . . , n }(cid:105) . If the stabilizer group S can be generatedby operators which act as either purely X or Z then we callit a CSS stabilizer code . CSS codes are closely related tobinary linear codes from classical coding theory. Given twobinary linear codes of size n with parity check matrices H X and H Z we can define a CSS stabilizer code simply by takingeach row r of H X ( H Z ) and defining an operator which actsas X ( Z ) on qubit i if r i = 1 and as the identity I otherwise.Note that for the +1 -eigenspace of S to be non-trivial it isnecessary that all of its generators commute. This is achievedby demanding that H X · H TZ = 0 . (2)Random constructions, which are commonly used in theclassical setting, will generally not satisfy this constraint. Oneway to find suitable parity check matrices H X and H Z is byconsidering homology over F , the field with two elements:Given a closed manifold M of dimension D tessellated bypolytopes, let C be the F -vector space which is formallygenerated by all vertices of the tessellation. Similarly, wedefine C i as the vector space formally generated by all i -dimensional constituents of the tessellation (edges, faces, 3-cells,...). We can now define boundary operators ∂ i : C i → C i − . As each C i comes with a distinguished basis we willalways consider ∂ i as an F -matrix with entries ( ∂ i ) m,n equal to if and only if the i − -dimensional cell withlabel n is attached to the i -dimensional cell with label m .The elements of C i can be identified with subsets of i -cells.Applying ∂ i to such an element will map it onto a subsetof i − -cells. As contributions from neighboring i -cells willcancel modulo 2, we obtain that the result is the boundaryof the initial subset. An important observation is the factthat boundaries do not have boundaries themselves, which isequivalent to ∂ i ◦ ∂ i +1 = 0 for all i = 1 , . . . , D − .To define a CSS code we can simply define H X = ∂ i and H Z = ∂ Ti +1 . By doing so we have essentially identified i -cells with qubits, i − -cells with X -checks and i + 1 -cellswith Z -checks.An alternative view on this construction is given by con-sidering the tessellation as a partially-ordered set (poset). Theelements of the poset are all cells of the tessellation, wherecells x and y fulfill the relation x ≺ y if and only if x isa subcell of y of one dimension lower. The poset can bevisualized as a diagram, as illustrated in Figure 1, where cellsare nodes with two nodes x and y connected by an edge if andonly if x ≺ y . As only cells with dimension differing by 1are related the poset diagram forms a D + 1 -partite graph,where each partition is given by cells of a fixed dimension.Picking any three consecutive layers we obtain what is calledthe Tanner graph of a CSS code: the middle layer forming theset of qubits and the outer two layers forming X -checks and Z -checks, respectively. We note that the dual tessellation hasthe same poset diagram with the levels in reverse order. If i ischosen to be the middle dimension then X and Z are relatedby duality.The logical operators of a quantum code are characterizedas those operators which commute with all checks while notbeing generated by them. In particular, the logical Z -operatorscorrespond to closed i -dimensional submanifolds which arenot the boundary of an i +1 -dimensional volume as they corre-spond to elements in F n which are in the kernel of the bound-ary operator, but not in its image. They therefore correspond toelements of the homology groups H i = ker ∂ i / im ∂ i +1 , where (cid:13) (cid:13) (cid:13) (cid:13)(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)(cid:13) (cid:13) (cid:13) (cid:13)(cid:13) (cid:13) (cid:13) (cid:13)(cid:13) (cid:13) (cid:13) (cid:13) Fig. 1. Poset diagram of a tessellation. The elements are cells and two relatedelements are connected by an edge. By definition of the relation (see maintext) only cells with dimension differing by 1 are related. We can define aquantum code by picking three consecutive layers and define H X ( H Z ) asthe adjacency matrix between nodes in layers i and i − ( i + 1 ). The boxshows the case i = 2 . Note that any pair of an i − -cell and an i + 1 -cellhave an even number of i -cells that they are connected to in common, so thatEquation (2) is satisfied. The subgraph in the box is the Tanner graph of theCSS code. i -cells correspond to qubits. Assuming that i is the middledimension, the logical X -operators similarly correspond toclosed i -dimensional submanifolds which are not the boundaryof an i + 1 -dimensional volume in the dual tessellation.A familiar example of this construction is the toric codewhich is obtained by a torus with a square tessellation. Thequbits are identified with edges ( i = 1 ) so that faces give Z -checks and vertices give X -checks. The two non-contractibleloops of the primal (dual) tessellation are identified with thelogical Z ( X ) operators.It is common to be imprecise with the word code . It canrefer to a single instance, but also to a whole family of codes.For our purposes here, a code family will be obtained from asequence of manifolds with increasing volume which all comefrom the same tessellation, so that they all share the same localstructure. B. Single-Shot Decoding
Single-shot decoding was first discussed in [17] in thecontext of the 3D gauge color code, although the resultsimmediately apply to 4D homological codes as well. Themain idea is that the syndrome, which is extracted by themeasurement, contains redundancies. This makes it possibleto infer a recovery operation in the presence of syndromenoise, either by performing classical decoding on the syndromefirst and then feed the fixed syndrome into the quantum codedecoder. Alternatively, it is known that cellular automata arerobust against noise in the classical setting [18] and there isnumerical evidence that cellular automaton decoders appliedto higher-dimensional quantum codesClearly, the recovery operation will in general not correctback to a code state and leave a residual error. It is shownin [17] that there exists a threshold below which a recoveryis still possible by employing percolation type arguments tocontrol the spread of errors.
C. 4D Hyperbolic Codes
What makes the homological construction of the previoussection appealing is that the properties of the code are deter- mined by the underlying tessellated manifold. In particular,the number of logical qubits k is determined by its topologyand the distance d is bounded by the minimum volume of anon-contractible submanifold.We will now review the results of [8] on the encoding rateand distance of quantum codes derived from families of 4-dimensional hyperbolic manifolds.
1) Encoding rate:
We will first discuss the number oflogical operators k . As mentioned in the introduction, hy-perbolic manifolds give rise to quantum codes which havea linear rate k ∼ n . The linear rate of hyperbolic codesfollows from the Chern–Gauß–Bonnet theorem, which relatesthe Euler characteristic χ ( M ) := D (cid:88) i =0 ( − i dim H i ( M ) (3)of a closed manifold M of even dimension D to the geometryof the manifold. The exact statement is that χ ( M ) = 1(2 π ) D (cid:90) M Pf (Ω) (4)where Pf (Ω) is the Pfaffian of the curvature form ofthe Levi-Civita connection. For a hyperbolic manifoldthe integral on the right-hand side is in fact equal to ( − D vol ( M ) / vol ( S D ) [19]. Note that we always assumethat M is connected, which implies that dim H = dim H D =1 . For D = 2 we can exactly solve for dim H : dim H = area ( M )2 π + 2 (5)By tessellating M with regular polygons we can define aquantum code with k = (cid:18) − r − s (cid:19) n + 2 (6)where r and s are the weights of the X -checks and Z -checks [9].For D = 4 and i = 2 we can not solve exactly for k =dim H , since we do not know the dimensions of the oddhomology groups. However, as they both have a negative signin the alternating sum we obtain the lower bound dim H ≥ vol ( M ) vol ( S ) − . (7)Since vol ( S ) = 8 π / this gives dim H ≥ . vol ( M ) − .This establishes that a quantum code defined on a tessellationwith uniform density of M will have linear rate k ∼ n . Thevalue of the encoding rate k/n will depend on the tessellation.In Section III-C we derive a lower bound for the encodingrate of a quantum code based on a particular tessellation of4D hyperbolic space. This lower bound turns out to be tightfor the examples we construct later (cf. Section III-G).
2) Distance:
For quantum codes derived from hyperbolicsurfaces ( D = 2 ) one can establish upper and lower bounds onthe distance which are logarithmic in the number of qubits n .For D = 4 a lower bound on the distance follows from aresult of systolic geometry by Anderson [20]. Let R be the greatest length such that any ball of radius R can be embeddedanywhere in M . This quantity is called the injectivity radiusof M . Anderson’s theorem states that any essential i -cycle γ of M has its volume lower-bounded by the volume of a ball ofradius R in i -dimensional hyperbolic space. It is shown in [8]that for a hyperbolic manifold M we have R ≥ c log vol ( M ) with a constant c > . Combining this with Anderson’s boundand the fact that the volume of a ball of radius r in H i growslike exp(( i − r ) we obtain thatvol ( γ ) ≥ vol ( B R ) = c (cid:48) exp (( i − R ) (8)where c (cid:48) > is a constant depending on i . Hence we obtainfor i = 2 that vol ( γ ) is lower bounded by c (cid:48) vol ( M ) c .III. C ONSTRUCTION AND E XAMPLES
The discussion in Section II leaves open the question ofhow to obtain concrete examples of tessellations of closedhyperbolic 4-manifolds. We will explain how we can describetessellations using
Coxeter groups , which are generated by re-flections along all hyperplanes of symmetry of the tessellation.We will review Coxeter groups in Section III-B. In particular,we will discuss how families of closed manifolds supportinga fixed tessellation are related to coverings of an infinitetessellation of H . We then give two separate constructionsto obtain concrete examples of tessellated, closed, hyperbolic4-manifolds as well as a method to obtain smaller, lesssymmetric manifolds from larger ones. A. Regular Tessellations A tessellation is a gapless covering of a manifold byregular polytopes such that each adjacent pair of polytopesoverlaps exactly on their facets. We can decompose the regularpolytopes into simplices by cutting them along their planesof symmetry. We say that a tessellation is regular if thesymmetry group of the tessellation operates transitively onthese simplices. This implies in particular that all polytopesare identical and that the same number of polytopes meet atevery vertex, edge, face, etc.Regular tessellations are classified by their Schl¨afli symbol { p, q, r, s, . . . } , which for a D -dimensional tessellation is asequence of D positive integers. It encodes the incidencenumbers of the cells: q is the number of faces incident toa vertex in a 3-cell, r is the number of 3-cells incident to anedge in a 4-cell and s is the number of 4-cells incident to aface and so on.Not every sequence of numbers corresponds to a validtessellation of space due to geometric constraints. For example,in 2D euclidean space the fundamental triangle of an { r, s } tessellation has internal angles π/ , π/r and π/s . Since all in-ternal angles have to add up to π the only valid tessellations arethe square tessellation { , } , the hexagonal tessellation { , } and the triangular tessellation { , } .The only possible regular tessellations of 4D hyperbolicspace H are:1) { , , , } tessellation by 120-cells, self-dual2) { , , , } tessellation by hypercubes3) { , , , } tessellation by 120-cells, dual to 2 (a) 120-Cell (b) Fundamental SimplexFig. 2. (a) A 4D regular polytope called the 120-cell projected into 3D.(b) A single cube of a cubic tessellation { , , } . The fundamental simplexis highlighted in red. It is bounded by the reflections a , a , a and a ,which are highlighted in blue, yellow, green and magenta, respectively. Eachcube contributes 48 fundamental simplices. { , , , } tessellation by 4-simplices5) { , , , } tessellation by 120-cells, dual to 4The 120-cell is a 4-dimensional regular polytope with Schl¨aflisymbol { , , } (see Figure 2a). It has 120 dodecahe-dra { , } at its boundary. Note that the dual tessellation hasits Schl¨afli symbol reversed. Compact 4-manifolds supportingthe { , , , } tessellation were constructed in [15] and [16].Quantum codes based on the { , , , } tessellation werediscussed in [13]. B. Coxeter groups
The group of symmetries of a regular tessellation is gen-erated by reflections along hyperplanes of symmetry. The hy-perplanes of symmetry subdivide the tessellation into identical D -dimensional simplices (see Figure 2b). The symmetry groupacts freely and transitively on the simplices, meaning that nosimplex is stabilized by the group action and every simplex canbe mapped onto any other. By fixing one arbitrary simplex andassigning it the identity element of the group, we have a one-to-one correspondence between the simplices and the groupelements.The Coxeter group is defined in terms of the generators andtheir relations. As each generator a i corresponds to a reflectionwe have a i = e , where e is the neutral element of the group.The relations between the generators are given by the Schl¨aflisymbol ( a i a j ) r i,j = e (9)where r i,j is the j th entry of the Schl¨afli symbol if j = i + 1 .Note that the reflection relation gives r i,i = 1 . All other pairsof generators (those with | i − j | > ) commute. Since they arereflections this can be expressed as ( a i a j ) = e , i.e. r i,j = 2 .In the following chapters we will describe how we canuse this description to obtain tessellations of compactificationsof H . Although the tools we present work for general tessel-lations, we will focus on the self-dual { , , , } tessellationto construct quantum codes. C. Encoding Rate
Before discussing the constructions in the next few sections,we derive a lower bound on the encoding rate k/n for codes derived from the { , , , } tessellations introduced inSection III-A.Instead of using the integral expression of the Chern-Gauß-Bonnet theorem of Equation (4) we will instead considerthe well-known combinatorial expression in terms of thenumber of cells in the tessellation. In order to obtain thisexpression, we note that the number of i -cells is the sameas the dimension of the vector space of i -chains C i . By therank-nullity theorem and the definition of the homology groups H i = ker ∂ i / im ∂ i +1 we have that dim C i = dim ker ∂ i + dim im ∂ i = dim H i + dim im ∂ i +1 + dim im ∂ i . (10)Putting this into the definition of the Euler charactistic (Equa-tion (3)) we obtain χ = D (cid:88) i =0 ( − i dim C i = D (cid:88) i =0 ( − i . (11)The number of cells can be expressed in terms of of the num-ber of fundamental simplicies. For the { , , , } tessellationthe number of fundamental simplices per vertex and 120-cellis both 14400, the number of fundamental simplicies per faceis 100 and the number of fundamental simplices per edgeand dodecahedron is both 240. Let S ( M ) be the total numberof fundamental simplices of the tessellated manifold M . Weobtain the following formula for the Euler characteristic: χ = 137200 S ( M ) (12)Together with Equation (3) we finally obtain the bound k ≥ n − (13)where the inequality is due to ignoring the negative contri-butions of dim H ( M ) and dim H ( M ) . The constant termcomes from dim H ( M ) = dim H ( M ) = 1 .We note that Equation (7) and Equation (13) are consistentwith one another, as the volume of a 4D hyperbolic man-ifold M is related its Euler characteristic via the equationvol ( M ) = 4 π χ ( M ) / (see [21]). D. Construction based on FP-groups
We can use the identification between the fundamentalsimplices and the group elements to obtain tessellations ofclosed manifolds. The idea is to consider finite quotients ofthe infinite group, which leave the local structure of the groupinvariant. Geometrically, the procedure essentially consists offinding translations and identifying points which differ bythese translations. For example, on the 2D euclidean plane wecan take an arbitrary translation and by identifying all pointsdiffering by this translation we obtain a cylinder of infinitelength. Taking a second translation, which is not co-linear withthe first one, we obtain a torus.This process is less straightforward in curved spaces wheretranslations generally do not commute. In [9] this has beendone for 2D hyperbolic surfaces by enumerating normalsubgroups and their quotients up to a certain size. The Todd-Coxeter algorithm can be used to enumerate normal subgroups. Some faster adaptions using the KnuthBendix completionalgorithm are also known (see Chapter 5.6 in [22]).By Equation (9), the group of the infinite { , , , } tessel-lation is (cid:104) a, b, c, d, e | a , ( ab ) , ( ac ) , ( ad ) , ( ae ) , b , ( bc ) , ( bd ) , ( be ) , c , ( cd ) , ( ce ) , d , ( de ) , e (cid:105) . (14)For readability we have written the generators as a, . . . , e instead of a i for i = 0 , . . . , . Trying to find normal subgroupsof this group by exhaustive search yielded only two examplesin a reasonable amount of time. One example has 14,400fundamental simplices and the other 72,000. In the { , , , } tessellation there are 100 simplices per face, so that we obtainquantum codes with 144 and 720 physicsl qubits, respectively(see Table I and discussion in Section III-G).We found larger examples by considering the followingrandomized procedure: we can take a random word in thegenerators of a specified length w . We then obtain a normalsubgroup by taking its normal closure N of this group elementand check if the resulting group is finite. One additionallyneeds to check that N operates fixed-point free which is thecase if w does not correspond to a reflection or a rotation [23].This procedure gave two more examples with 18,432 and19,584 physical qubits. E. Construction Based on Matrix Representations
The second construction is based on matrix representationsof the symmetry groups. The main idea is to obtain a faithfulmatrix representation of the infinite tessellation. Let us assumethat we are able to find a representation with a distinguishedbasis such that all of the generators and their inverses aremapped onto matrices which have integer entries. Clearly, inthis case all group elements are represented by integer matri-ces. To obtain a finite group we could naively try to reduce theentries of all matrices modulo some positive integer p . Thiswould ensure that we are left with a finite set of matrices.There are some obvious problems with this approach: It isgenerally not possible to have purely integer entries. We willaddress these issues in what follows.
1) Hyperboloid Model:
The matrix representation is ob-tained by the hyperboloid model of hyperbolic space: In D +1 -dimensional Minkowski space R ,D we can identify the D -dimensional hyperbolic plane with the set H D = (cid:40) x ∈ R ,D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ◦ x = − x + D (cid:88) i =1 x i = − , x > (cid:41) (15)by defining the distance between any two points x, y ∈ H D asdist ( x, y ) = cosh − ( − x ◦ y ) where ◦ denotes the Lorentzianinner product. The group of invertible ( D + 1) × ( D + 1) -matrices which leave the Lorentzian inner product invariant iscalled O (1 , D, R ) . The isometry group of H D , i.e. the groupof transformations which preserve the distance function distis isomorphic to the subgroup O + (1 , D, R ) of index 2 whichsends the upper sheet to itself and the lower sheet to itself(cf. Figure 3). It is also known as the “orthochronous Lorentzgroup” in D + 1 -dimensions. x x x Fig. 3. The hyperboloid model of hyperbolic space. The equation x ◦ x = − defines a hyperboloid consisting of two disconnected sheets. We identify theupper sheet ( x > ) with the hyperbolic plane.
2) Representaion of the infinite tessellation group:
We canconstruct the representation using the Gram-matrix g of thenormal vectors of the hyperplanes of reflection. Let e , . . . , e denote the standard basis vectors. The inner product betweenthem is defined by g ( e i , e j ) = − (cid:18) πr i,j (cid:19) (16)where r i,j are the exponents in the relations which definethe Coxeter group (see Section III-B). Since we considerregular tessellations g will be tridiagonal with entries α i = − π/r i,i +1 ) on the first diagonals. g = α α α α α
00 0 α α α (17)If the Schl¨afli-symbol belongs to a hyperbolic tessellationthen g has signature ( − , + , + , + , +) , i.e. g is equivalent, upto a change of basis, to the Lorentzian inner product ◦ .The matrix representation ρ : G → O + (1 , D, R ) can nowbe defined by their action on the basis vectors. For eachgenerator a i of the Coxeter group its representation ρ ( a i ) isdefined by its action on the standard basis: ρ ( a i ) · e j = e j − g i,j g i,i e i = e j − g i,j e i . (18)Let us verify that this is indeed a representation by explicitlychecking the group relations. To not clutter our notation wewill write r i := ρ ( a i ) (not to be confused with the matrix r from Section III-B, which defines the relations). Let us firstcheck that we are indeed mapping onto reflections. r i ( e j ) = r i · ( e j − g i,j e i )= e j − g i,j e i − g i,j ( e i − g i,i e i )= e j − g i,j e i + 2 g i,j e i = e j (19) Hence, r i is indeed a reflection.We are left to show that the r i satisfy the “off-diagonalrelations” of Equation (9). Let | i − j | > and define v ⊥ g as the space of all vectors orthogonal to v with respect to g .Since dim e ⊥ g i = dim e ⊥ g j = D and e ⊥ g i (cid:54) = e ⊥ g j we have dim (cid:16) e ⊥ g i ∩ e ⊥ g j (cid:17) = D − . (20)Since e i and e j do not belong to e ⊥ g i ∩ e ⊥ g j we can complete abasis of e ⊥ g i ∩ e ⊥ g j with e i and e j to form a basis B of R D +1 .Let us express r i and r j in B : r i, B = I D − − g i,j r j, B = I D − g i,j − (21)Their product is r i, B r j, B = I D − g i,j ) − − g i,j g i,j − . (22)We can focus on the bottom-right two by two submatrix A .Its determinant is det( A ) = 1 and its trace istr ( A ) = ( g i,j ) −
2= 4 cos ( π/r i,j ) −
2= 2 cos(2 π/r i,j ) . (23)Therefore A has two distinct eigenvalues λ + = exp( i π/r i,j ) and λ − = exp( − i π/r i,j ) and hence satisfies A r i,j = I . Wehave thus shown that ( r i, B r j, B ) r i,j = I n .We refer to Theorem 3A10 in [24] for a proof that therepresentation ρ is faithful, i.e. the reflections r i do not satisfyother relations than the ones satisfied by the generators a i ofthe Coxeter group, so that im ( ρ ) (cid:39) G . Note that im ( ρ ) isisomorphic a subgroup of O (1 , D, R ) as g is equivalent to theLorentzian inner product ◦ .
3) Matrix entries:
Our initial goal, which we stated at thebeginning of this section, was to obtain a matrix representationthat allows us to take all elements modulo a large number.Here we will see that this is generally not possible and wewill show how to amend the idea to make it work.What are the entries of the elements of im ( ρ ) ? – Let us con-sider the self-dual tessellation wit Schl¨afli symbol { , , , } .From Equation (18) it is clear that all matrices have entriesthat are integer polynomials of the entries of g . The entries onthe first diagonals of g (see Equation (17)) are α = α = − (cid:16) π (cid:17) = 1 + √
52 =: φ (24)and α = α = − (cid:16) π (cid:17) = − . (25)The matrix entries α and α take integer values. However, α and α are equal to the golden ratio φ , which is not aninteger. We can account for this by simply extending the ring of integers by φ and obtain Z [ φ ] . Note that φ still fulfillsthe relation φ − φ − . The associated polynomial h = x − x − is called the minimal polynomial of φ . Allgenerators are self-inverse and hence all matrices in im ( ρ ) have entries in Z [ φ ] .To be able to use a computer algebra system we constructthe ring Z [ φ ] from polynomials. This can be achieved byconsidering all polynomials up to arbitrary multiples of h . Theset of all multiples of h are called the ideal generated by h and denoted (cid:104) h (cid:105) = { p · h | p ∈ Z [ x ] } . (26)Note that (cid:104) h (cid:105) is by definition closed under linear combinations.The quotient ring Z [ x ] / (cid:104) h (cid:105) contains elements of the form p + (cid:104) h (cid:105) with p ∈ Z [ x ] . In particular, if p is a multiple of h wehave p + (cid:104) h (cid:105) = 0 + (cid:104) h (cid:105) . This means that x fulfills the samerelations as φ in Z [ x ] and hence we have Z [ φ ] (cid:39) Z [ x ] / (cid:104) h (cid:105) . (27)In the remainder of the paper we will abuse notation anddirectly identify Z [ φ ] with Z [ x ] / (cid:104) h (cid:105) .
4) Quotient:
In the previous paragraphs we have obtaineda faithful matrix representation ρ of the symmetry group ofthe infinite { , , , } tessellation of H . We have seen thateach element of im ( ρ ) has coefficients in Z [ φ ] .Our strategy to obtain symmetry groups of closed hyper-bolic four-manifolds is to factor out suitable ideals of Z [ φ ] toeffectively obtain representations of G over F q . We need toshow that factoring out ideals of the matrix entries does pre-serve the local structure, which means that the result should bethe symmetry group of a closed, tessellated manifold that looksidentical to the infinite tessellation in a large neighborhood.Our goal is to obtain a family of quantum codes withgrowing distance. To show that the distance increases itsuffices to show that the tessellation on the closed manifoldis indistinguishable from the infinite one in a large neigh-borhood, as no logical operator can have support inside thisneighborhood.Let l be a positive integer. We call a representation l -locally faithful if no non-identity element g ∈ G \ { e } , whichcan be written as a sequence of at most l generators of G ,is mapped to the identity matrix. The following theorem isadpated from [8] and [25]. Theorem 1.
For any positive integer l there exists an l -locallyfaithful representation of G .Proof. Let π I : Z [ ξ ] → Z [ ξ ] /I be the quotient map for anideal I ⊂ Z [ ξ ] . Here we will only consider maximal ideals I so that Z [ ξ ] /I is in fact a field F q of characteristic p . We notethat I is of the form (cid:104) p (cid:105) or (cid:104) p, g ( ξ ) (cid:105) , where g is an irreduciblefactor of the minimal polynomial of ξ in F p (see Theorem 2in Appendix A).Let ρ be the representation of the infinite tessellation groupdefined by Equation (18). We can extend π I to act on thecoefficients of matrices over Z [ ξ ] . Since G is generated byreflections it is easy to see that the matrices in im ( ρ ) havedeterminant ± . Since / ∈ I it follows that π I ( im ( ρ )) only contains invertible matrices and hence we have that thefunction π I ◦ ρ : G → GL ( D + 1 , F q ) is well-defined.We will now show that for a suitable choice of the ideal I the representation π I ◦ ρ is l -locally faithful. Let g ∈ G \{ e } bea Coxeter group element which can be written as the productof u ≤ l generators, i.e. g = a i · · · a i u . We need to showthat π I ◦ ρ ( g ) is not the identity matrix. Clearly ρ ( g ) is notthe identity matrix as ρ is faithful. Furthermore, by choosingthe prime p in the ideal I to be suitably large the image of ρ ( g ) under π I is not the identity matrix.
5) Group structure:
It turns out that the group obtained bythe procedure outlined above can have a particularly simplestructure. Assume that I = (cid:104) p (cid:105) and that π I ( g ) is non-singular.As the elements of O (1 , , Z [ φ ]) preserve g we have that theirimages preserve π I ( g ) . This means that im ( π I ) is a subgroupof GO ( q ) , the orthogonal group over F q . By diagonalizing g it is easy to see that π I is in fact surjective, so that we have ρ ◦ π I (cid:39) GO ( q ) .The structure of GO ( q ) for odd q is well-known [26]: itdecomposes into three simple groups asGO ( q ) (cid:39) Ω ( q ) (cid:111) ( Z × Z ) . (28)The group Ω ( q ) is also known as the Chevalley group B ( q ) in the literature. Remark 1.
Note that this decomposition is similar to thefamiliar one of the Lorentz group in D = 3 into four connectedcomponentsO (1 , , R ) (cid:39) SO + (1 , , R ) (cid:111) ( Z × Z ) (29) where SO + (1 , , R ) is the proper, orthochronous Lorentz groupand Z × Z is generated by a time-like reflection (timereversal) and a space-like reflection.6) Size of the quantum code: The number of physicalqubits n is given by the number of faces in the lattice.We can count the number of faces (and cells of any otherdimension) by counting the number of fundamental simplicesand divide by the number of simplices per cell. The numberof fundamental simplices is the same as the order of thesymmetry group of the lattice, which for odd q is given bythe polynomial | Ω ( q ) (cid:111) Z | = q − q − q + q [26].For the { , , , } tessellation there are 100 fundamentalsimplices per face and thus the formula for the size of aquantum code based on this construction is: n ( q ) = q − q − q + q (30)Note that we had to assume that q is odd. We will later discussexamples with q even for which Equation (30) fails.The golden ration φ has minimal polynomial x − x − .For p such that x − x − is irreducible in F p we obtain Z [ φ ] / (cid:104) p (cid:105) (cid:39) F p and thus n ∈ O ( p ) in agreement with [8]. The orthogonal groups over finite fields in odd dimensions are all isomor-phic [26]. R S (a) Infinite covering of circle S (blue) by real line R (green). (b) 4-fold covering of a × -torus (blue) by a × -torus(green).Fig. 4. (a) The circle S is covered by the real line. The covering can beconstructed by identifying S with the set of complex numbers with unit 2-norm and defining p = exp : R → S ⊂ C , t (cid:55)→ e it . This is an exampleof an infinite covering, as the pre-image of any point has infinite cardinality.The deck transformation group is the abelian group Z . (b) The small torus iscovered by the larger torus. The larger torus has 4 times the area of the smallerone. The covering is constructed by taking translations in the larger torusmodulo the corresponding length in the smaller torus, in this case modulo 20in the x -direction and modulo 10 in the y -direction. The preimage of anypoint contains four elements on which the deck transformation group Z × Z operates. F. Coverings
In order to obtain more examples from the ones generated inprevious sections we will now introduce topological coverings.They will allow us to construct less symmetric examples asthe group-based constructions.
1) Definition:
In addition to the previous two methodsfor obtaining finite manifolds we employ a third method toconstruct small examples. This method is based on coverings :If X and C are topological spaces we say that C is coveringspace if there exists a continuous surjective map p : C → X such that for any point x ∈ X we have that there exists anopen neighborhood U such that the pre-image p − ( U ) is adisjoint union of open sets in C each homeomorphic to U .If the number of these copies is fixed it is called the degree of the covering. We will call a covering of degree n an n -fold covering . A famous example from physics is the 2-foldcover of SO (3) by SU (2) . In Figure 4 we show two furtherexamples of coverings. The first (Figure 4a) is an infinitecover of the circle by the real line, depicted by putting thereal line in a spiral over the circle so that p can be thoughtof as a projection along the vertical axis. The covering anbe realized by identifying S with the unit circle in C andconsider p = exp : R → S ⊂ C , t (cid:55)→ e it . The secondexample (Figure 4b) is a 4-fold cover of a torus by anothertorus. The covering is realized by taking translations in x - and y -direction modulo 20 and 10, respectively.Coverings can be equipped with a group structure: Homeo-morphisms operating on the covering space φ : C → C suchthat p ◦ φ = p are called deck transformations . They form agroup under composition called the deck transformation group .There is a natural group operation of the deck transformationgroup of a covering p on the pre-image of a given point x ∈ X ,as it permutes the elements of p − ( x ) .What are the deck transformation groups in the two exam-ples of Figure 4? – For the real line and the circle we canperform shifts by multiples of π , i.e. t (cid:55)→ t + 2 πx , leavingthe image of t under exp invariant. The deck transformationgroup is hence isomorphic to the infinite abelian group Z . For the covering of the torus it is clear that we can performtranslations by 20 in the x -direction and translations by 10 inthe y -direction leaving the modulus invariant. Hence, the decktransformation group is isomorphic to Z × Z .
2) Finite coverings of hyperbolic 4-manifolds:
We con-struct further instances of hyperbolic 4-manifolds by enumer-ating all conjugacy classes of subgroups of the symmetrygroup of a given tessellated finite hyperbolic 4-manifold. Notall subgroups preserve the local structure of the tessellation,as they may contain elements which have fixed points, suchas reflections or rotations. However, since we consider tessel-lations we have the more stringent restriction that the decktransformation group should respect the local structure ofthe tessellation. This restriction can be formulated in group-theoretic language and we call it the non-local subgroupcondition which we derive in Appendix B.The deck transformation groups of the 4D hyperbolic man-ifolds constructed by coverings can be found in Table I under“structure”.
G. Examples of { , , , } -Codes Using the constructions of Sections III-D to III-F we havefound examples which are small enough to perform MonteCarlo simulations. Here we will discuss the properties ofthese examples in more detail. A summary can be found inTable I, where the properties of the tessellated manifolds andthe associated quantum codes are listed. The column labeled“structure” contains either the structure description of theassociated symmetry group or, if the example was constructedfrom a finite covering, the number of the covering manifoldand the deck transformation group.
1) Based on FP-groups:
The construction based on finitelypresented groups of Section III-D gave us three examples: thesmallest with 144 physical qubits, one with 18,432 physicalqubits and one with 19,584 physical qubits. All are obtainedby factoring out a single translation. They are (in the sameorder as above): • ababacbdedcbabacedcbaedced • bedcbabedcbabedcbabedcbabedcbabedcba • baedcbedcbabacbdcedcbabcedcbabacbded For readability we have written the generators as a, . . . , e instead of a i for i = 0 , . . . , (cf. Section III-D).The first example is known as the Davis manifold and wasfirst described in [27]. It can be constructed from a single120-cell (cf. Figure 2a) by taking opposing dodecahedra atthe boundary and identifying them. Note that doing so wedo not obtain a proper { , , , } tessellation, as for examplethe number of 3 cells incident to the (unique) 4-cell is 60instead of 120. However, the incidence numbers involving 2-cells is the same as for a proper { , , , } tessellation, sothat stabilizer weights and qubit degrees are unaffected. Byconstruction, the number of 3-cells in the Davis manifold is 60. The same procedure in 3D corresponds to identifying opposing faces of adodecahedron. Note that as opposed to 4D, in 3D these faces do not allign anddifferent rotations to make the faces match give rise to topologically different3-manifolds: the Poincar´e homology sphere, the Seifert-Weber space and the3D real projective space [28].
TABLE IE
XAMPLES OF { , , , } - CODES . n k ideal structure Euler characteristic χ ( SL (5) (cid:111) A ) (cid:111) Z
262 720 184 – 125-fold covered by 10, Z × Z × Z Z Z × Z Z (cid:104) (cid:105) Ω (4) Z Z × (cid:111) [( A (cid:111) A ) (cid:111) Z ] Ω (4) (cid:111) Z (cid:104)√ (cid:105) (cid:104)(cid:16) Z × (cid:111) SL (5) (cid:17) (cid:111) A (cid:105) (cid:111) Z (cid:104) (cid:105) Ω (9) (cid:111) Z (cid:104) (cid:105) Ω (11) (cid:111) Z (cid:104) (cid:105) Ω (19) (cid:111) Z The number of faces, edges and vertices is 144, 60 and 1and thus its Euler characteristic (cf. Section II-C) is χ = 26 .We note that the Davis manifold gives rise to a trivial errordetection code of encoding rate / and distance 2.The two larger examples are not proper { , , , } tessel-lations either as their 4-cells contain only 60 3-cells as well.They have, as far as we are aware, not appeared in previousliterature.
2) Based on Linear Representations:
The constructionbased on matrix representations (see Section III-E) yieldedseveral more examples. Let us first consider the simplestexample in which we reduce the matrices modulo 2, i.e. wefactor out the ideal (cid:104) (cid:105) . Since x − x − is irreducible in F we obtain a matrix group with coefficients in F . This givesrise to a quantum code with 9,792 physical qubits and 2,200logical qubits. We note that the underlying group is isomorphicto Ω (4) . We can change the set of generators from reflectionsto rotations by taking products a i a j as a new set of generators.Factoring out (cid:104) (cid:105) from the group generated by rotations givesthe group Ω (4) (cid:111) Z which defines a quantum code with19,584 physical qubits and 4,324 logical qubits. This is thesame group that we found previously using finitely presentedgroups.Next, we will consider an example where the minimal poly-nomial does become reducible: Consider the ideal generatedby √ φ − ∈ Z [ φ ] . The quotient Z [ φ ] / (cid:104) φ − (cid:105) turnsout to be isomorphic to F . The resulting quantum code has90,000 physical qubits and 18,024 logical qubits.The next largest examples are the ideals generated by 3, 11and 19 (see Table I). However, these were too large to deter-mine the number of encoded qubits. We note that the encodingrate is close to the upper bound given in Equation (13).
3) Based on Finite Coverings:
Further small examples canbe obtained by the covering procedure (see Section III-F). Twoof the coverings we found appeared in previous literature: a5-fold covering using the n = 90 , manifold (number 10 inTable I) had been found in [29]. It was also observed in [29] that the n = 90 , manifold is a -covering space of theDavis manifold. Further examples are enumerated in Table Iwhere the covering and the deck transformation group arespecified.
4) Further remarks:
Since we do not have an expression forthe number of encoded qubits k we constructed the boundaryoperators and obtained the dimension of the second homologygroup (see Table I). We observe that for the examples weconstructed the encoding rate k/n is close to the lower boundof Equation (13). Unlike for 2D homological codes for whichone can efficiently determine the distance [30] we are notaware of any efficient procedure to obtain the distance ofhigher-dimensional homological codes. A randomized search-ing procedure yielded logical operators of weight 2 for the n = 144 code, a logical operator of weight 12 for the n = 720 -code and a logical operator of weight 6 for the n = 3 , code. Note that these are upper bounds as logical operators ofsmaller weight may still exist.IV. D ECODING AND P ERFORMANCE
The stabilizer checks of a { , , , } code correspond to do-decahedra in the primal tessellation ( Z -checks) and in the dualtessellation ( X -checks). Each check acts on all of its adjacentqubits which correspond to the pentagonal faces. The stabilizerchecks of the code to fullfill non-trivial linear dependencies:The boundary of a 120-cell contains dodecahedra and as theboundary itself is boundaryless it follows that the product ofall checks belonging to a 120-cell has to vanish. This canalso be understood when we interpret the poset diagram inFigure 1 as a Tanner graph. The three levels in the middle formthe quantum code, while levels 0 and 4 determine the lineardependencies of the checks. Due to the linear dependenciesthe syndrome in a 4D code consists of closed loops.Assuming that errors occur independently and homoge-niously a good decoding strategy is minimum weight decod-ing. Unfortunately, there is no known efficient algorithm which given a collection of loops in 4D lattice returns a minimum-weight surface which has these loops as its boundary. Regard-less, we can settle for a less optimal, but efficient solution.It was observed in [10] that a 4D code can be decoded by“shrinking” the syndrome loops. In [12] it was shown that in4D hyperbolic space a linear time decoding procedure exists.In this paper we will consider two decoding stategies: thefirst decoder is based on a cellular automaton and the secondon belief-propagation. A. Cellular Automaton Decoder1) Background:
Cellular automata can be used to imple-ment a primitive decoding algorithm. This type of decoder hasseveral desirable features. It can be implemented using verysimple classical control, as it essentially performs a majorityvote on a small number of input signals and sends a signalto perform a bit- or phase-flip based on the outcome. Thisis extremely fast and dissipates little heat when comparedto other decoding schemes, such as minimum-weight perfectmatching. This is important as it makes it possible to imple-ment the classical control close to the qubits which, dependingon the specific hardware implementation, have to be kept attemperatures of a few Kelvin down to hundreds of milli-Kelvin.Using cellular automata to decode quantum codes was firstsuggested in [10]. Two different update rules have been usedin the literature. The first is a majority-vote rule which simplyperforms a bit-/phase-flip if more than half of the Z-/X-checksincident to the qubit are violated. The second rule is called
Toom’s rule . It only performs bit-/phase-flips if parity checksin a specified direction are violated. Toom’s rule was firstintroduced in the classical setting as non-equilibrium dynamicsfor the 2D Ising model, exhibiting the unusual property of astable memory phase at non-zero temperature in the presenceof a magnetic field [18]. Toom’s rule has been shown toperform better than the majority-vote rule [31] when appliedto the 4D toric code with a hypercubic tessellation. It hasbeen generalized to other euclidean tessellations in [32]. It is,however, not clear how to apply Toom’s rule in hyperbolicspace. The reason for this is that to be well-defined it needsa distinguished direction and hence a notion of parallel lineswhich is consistent throughout the system. In hyperbolic spacea single line does not uniquely define a parallel line throughany other point, as Euclid’s fifth postulate famously does nothold in hyperbolic space. We will therefore only consider theisotropic majority-vote rule.
2) Monte Carlo:
We consider the independent bit-/phase-flip model, where each qubit is acted upon by Pauli- X andPauli- Z each with probability p . We then run the CA decoderuntil the weight of the syndrome stops decreasing. If thesyndrome weight is non-zero we declare the trial a failure.If the syndrome weight is zero we are back in a code state.In this case we check whether the error together with therecovery given by the CA decoder contains a non-trivial logicaloperator.The results of the simulation can be found in Figure 5.In [31] the same decoder under the same error model was .
00 0 .
01 0 . . p P n (a) linear − − − − − − − p P n (b) log-logFig. 5. (a) Performance of CA decoder (b) Same data plotted with log-logaxes. The dashed lines indicate the error probability if we were to take k unencoded qubits − (1 − p ) k . applied to the 4D toric code, which is defined on the eu-clidean { , , , } hypercubic tessellation. We notice that theperformance of the hyperbolic codes is better: for the 4D toriccode the threshold error rate is below . while for thehyperbolic codes here errors are suppressed for physical errorrates below . This is despite the fact that the hyperboliccode has higher stabilizer weight (12 instead of 6).A quantity of interest is the pseudothreshold , which isthe physical error probability below which the logical errorprobability is below the physical error probability, i.e. the errorprobability below which encoding is benificial over havingbare qubits. The error probability of k unencoded qubits is − (1 − p ) k and marked in dashed lines in Figure 5b. We seethat the n = 144 code does not have a pseudothreshold whichis expected, as it has distance 2 (cf. Section III-G). All othercodes have a pseudothreshold of around 1%. B. Belief-Propagation Decoder
The cellular automaton decoder of the previous sectionmakes decisions based on a very limited amount of informa-tion: it can only see parity-check violations in its immediatevicinity. This limitation is overcome by considering belief-propagation which is a commonly used decoding algorithmfor classical LDPC codes [33]. Belief-rpropagation has beenpreviously applied to quantum codes in [34] and [35].
1) Background:
For Tanner graphs which are trees theBelief Propagation (BP) decoder corresponds to maximumlikelihood decoding. As the Tanner graph of a { , , , } quantum code is not a tree, BP gives a heuristic decodingalgorithm in this setting.In order to define BP, let X ( j ) be random variables cor-responding to qubits. With the assumed noise model, theyare independently and identically distributed like Bernoullivariables with parameter p ∈ [0 , . Let Y ( k ) be randomvariables corresponding to check nodes defined as Y ( k ) = (cid:77) j neighbour of k X ( j ) . (31)The values of Y ( k ) are what we observe when extracting thesyndrome information and we denote them by y ( k ) obs .We want to compute marginals of the random variables X ( j ) conditioned on the observations y ( k ) obs . If the Tanner graph were a tree, we could set one of the qubits to be the root of thistree. We will use the notation k > j to denote that k is adescendant of j . For each qubit j , we define the followingfunction whose domain is { , } : p ( j ) ( x ) = Pr (cid:16) X ( j ) = x | { Y ( k ) = y ( k ) obs } k>j (cid:17) (32)For each check node k , denote by j its parent qubit, wedefine the following function whose domain is { , } : q ( k ) ( x ) = Pr (cid:16) Y ( k ) = y ( k ) obs | X ( j ) = x , { Y ( m ) = y ( m ) obs } m>k (cid:17) (33)To compute p ( j ) ( x ) from ( q ( k ) ( x )) k children of j , we need thefollowing variation of Bayes’ formula: Pr( A | B, C ) Pr( B | C ) = Pr( B | A, C ) Pr( A | C ) (34)Indeed the left hand side of the above equation equals Pr( A ∩ B ∩ C )Pr( B ∩ C ) Pr( B ∩ C )Pr( C ) = Pr( A ∩ B ∩ C )Pr( C ) (35)which is symmetric in ( A, B ) and therefore equals the righthand side. We apply this formula to the events A = (cid:16) X ( j ) = x (cid:17) B = (cid:16) { Y ( k ) = y ( k ) obs } k children of j (cid:17) C = (cid:16) { Y ( m ) = y ( m ) obs } m>j , m not a child of j (cid:17) (36)and define the normalization constant Z = Pr( B | C ) . Weknow that Pr( A ) = p and obtain p ( j ) ( x ) = Pr( A | B, C )= pZ (cid:89) k children of j q ( k ) ( x ) . (37)Since p ( j ) (0) + p ( j ) (1) = 1 , we obtain that Z = p (cid:89) k children of j q ( k ) ( x ) + (1 − p ) (cid:89) k children of j q ( k ) (1 − x ) . (38)We now compute q ( k ) ( x ) from ( p ( l ) ( x )) l children of k . Wehave q ( k ) ( x ) = Pr (cid:16) Y ( k ) = y ( k ) obs | X ( j ) = x , { Y ( m ) = y ( m ) obs } m>k (cid:17) (39)giving − q ( k ) ( x ) = ( − y ( k ) obs + x +1 (cid:89) l children of k (1 − p ( l ) (1)) . (40)We could use Equations (38) and (40) directly to define theiterative Belief Propagation algorithm. However for numericalstability reasons we will follow [33] and use logarithmic ratios: lp ( j ) = log p ( j ) (0) p ( j ) (1) lq ( k ) = log q ( k ) (0) q ( k ) (1) (41) Under this transformation Equation (38) translates into: lp ( j ) = log 1 − pp + (cid:88) k children of j lq ( k ) . (42)Observing that q ( k ) (1) = (exp ( lq ( k ) ) + 1) − , we obtain − q ( k ) (1) = tanh lq ( k ) . (43)Similarly − p ( j ) (1) = tanh ( lp ( j ) / and thereforeEquation (40) translates into: lq ( k ) = ( − y ( k ) obs argtanh (cid:32) (cid:89) l children of k tanh lp ( l ) (cid:33) (44)The BP decoder we use is defined from Equations (42)and (44): the check node k sends the message lq ( k ) to itsparent node. The qubit j sends the message lp ( j ) to its parentnode. The first message is sent by the leaves of the tree,which we assume are qubits. It is initialized to log ( − pp ) . Thelast message is received by the root of the tree, which weassume is a qubit. The value (exp ( lp ( root ) ) + 1) − gives theprobability that the random variable corresponding to the rootis 1 conditioned on the observation of all the check variables.The derivation assumed that the Tanner graph was a tree.However, even for codes for which this is not the case we canstill use the Belief Propagation algorithm as it was describedabove. Although it does not compute exact probabilities anymore: it is a heuristic whose performance we investigatenumerically.
2) Monte Carlo (perfect measurements):
We first considerthe setting where measurements can be performed perfectly,meaning without errors. We apply the Belief Propagationdecoder in parallel. A round of message-passing consists ineach qubit sending a message to each of its neighbor checknode and each check node sending a message to each of itsneighbor qubit. After each round r of Belief Propagation wecompute w r , the weight of the syndrome if we were to flipthe qubits whose belief to have an error is higher than . . Westop as soon as w r ≥ w r − or when w r = 0 . If we stoppedbecause w r = 0 and there is no logical error, we say that thedecoding succeeded. Figure 6a shows the statistical frequencyof unsuccessful decoding as a function of the physical errorrate.The data is consistent with a threshold above 5% physicalerror rate. However, we would like to note that due to thecomplicated dynamics of belief propagation it is generallyhard to prove that a decoding threshold exists. In fact it isknown from classical coding theory that the performance ofBP reaches an “error floor” for low physical error rates whichoccurs due to loops in the Tanner graph [33]. It is possibleto eliminate this problem by postprocessing the output ofBP with the ordered statistics decoder (OSD) which has acomputational complexity of O ( n ) [35]. We have not donethis here and leave it as future work.
3) Monte Carlo (noisy measurements):
To simulate noisysyndrome extraction we flip the syndrome with probability q .For our simulations we chose q = p . We consider T roundsof error correction. In each round t ∈ { , ..., T } , each qubit independently undergoes a Z error e noiset with probability p .If t (cid:54) = 1 , this error e t is added to e res.t − , the residual error atround t − . The noiseless syndrome is computed: s noiselesst = H ( e res.t − ⊕ e noiset ) . For t ∈ { , ..., T − } , each check node independentlyundergoes an error with probability q . This defines a syndromenoise s noiset . The noisy syndrome is given to the BP decoder: s noisyt = s noiselesst ⊕ s noiset . (45)The BP decoder outputs an inferred error: e inf.t = BP dec. ( s noisyt ) (46)and the residual error is updated: e res.t = e res.t − ⊕ e noiset ⊕ e inf.t (47)For the last round, t = T , we assume perfect measurementsand therefore have s noiseT = 0 . If the weight of the syndromeafter the BP correction of this last round is zero and theresidual error e res.T is not a logical error, we say that thedecoding succeeded and otherwise, that it has failed. Fig-ure 6b shows the statistical frequency of unsuccessful decodingagainst the physical error rate for T = 5 . Note that thenoiseless measurement scenario corresponds to T = 1 . Wesee that increasing the system size decreases the logical errorprobability for a physical/syndrome error probability of p = q up to about .In Figure 6c we show the results of running the BP decoderon the same code ( n = 19 , ) for different number ofiterations T . The performance becomes worse as T = 1 isessentially the noiseless case, however the recession of curvesappears to recede with the number of time steps T .The threshold of the surface code under the same errormodel, but having to repeat the syndrome measurement for d rounds and using a decoder with much less favourablecomputational complexity, is about [37]. Clearly, this is nota fair comparison, as the ckeck weight of the { , , , } -codeis three times higer than the one of the surface code. However,one also needs to factor in the linear encoding rate and hencethe reduction in overhead, i.e. the number of physical qubitsthat need to be spend to obtain a a given number of logicalqubits and for a desired suppression of logical errors [30]. Weleave a detailed analysis as an open problem for future study.V. C ONCLUSION
We have shown how to construct quantum codes with aconstant encoding rate and polynomial scaling distance fromregular tessellations of four-dimensional hyperbolic manifolds.Some of the manifolds we constructed were known, butmany have (to the best of our knowledge) not appeared inprevious literature. We focussed on a particular tessellationof hyperbolic 4-space called the { , , , } tessellation. Theresulting code family has an asymptotic encoding rate of k/n → /
72 = 0 . ... , stabilizer checks of weight 12 anddistance scaling polynomially as Θ( n (cid:15) ) . For the constructionbased on linear representations it can be shown that (cid:15) ≤ . , .
02 0 .
04 0 . . p P n (a) noiseless syndrome ( T = 1 ) .
02 0 .
04 0 . . p P n (b) noisy syndrome with T = 5 .
02 0 .
04 0 . . p P T (c) n = 19 , Fig. 6. (a) Performance of BP decoder for a single time step T = 1 . Wesee that errors are suppressed in the system size for p < . . The curvescross very close to P = 1 . The dashed lines indicate the error probabilityif we were to take k unencoded qubits − (1 − p ) k . Vertical error barscorrespond to the approximate 95 % confidence interval given by p = ˆ p ± . (cid:112) ˆ p (1 − ˆ p ) /n trials where ˆ p is the mean. Here n trials = 1000 for eachphysical error rate and each quantum code. (b) Increasing the number of timesteps to T = 5 lets the curves recede backwards. The pseudo-threshold forthe two largest codes is around . For the codes with more than 10 000qubits, the transition between the successful and the unsuccessful decoding isquite sharp and gives numerical evidence for a noiseless threshold above 5%.(c) Performance curves for fixed system size n = 19 , . The recession ofthe curves appears to slow down with the number of time steps T . but we would like to stress that (cid:15) may be higher whenconsidering general { , , , } -codes.Future work – A drawback of our construction is thehigh stabilizer weight. It was shown in [36] that by refiningthe primal and dual tessellation of a homological code in acontrolled way one can reduce the stabilizer weight whilekeeping the asymptotic code parameters invariant. A relatedprocedure is refining the hyperbolic tessellation using a eu-clidean tessellation as done for 2D hyperbolic codes in [30].It has to be determined how to do this in a systematic wayin our construction. Alternatively, it may be possible to definea subsystem version in which stabilizers can be measured bylow-weight gauge operators.We are confident that the decoding performance may beincreased by considering more sophisticated decoding algo-rithms. For example, it would be worthwile to combine theBP decoder with an ordered statistics decoder (OSD) to helpin cases where BP alone would fail [35]. Another interestingavenue of research would be determining the threshold ofthe maximum-likelihood decoder by analyzing the associated4D hyperbolic random-plaquette gauge model, as previouslydone for homological codes in euclidean space [10], [37], [38]. However, it seems to be non-trivial to find a suitableorder parameter, as bulk-boundary scaling works differentlyin hyperbolic geometry.Finally, we hope to perform a detailed comparison tocurrently favoured quantum fault-tolerance schemes such asthe ones based on the surface code. The logical operatorsof hyperbolic codes share their support, making them muchmore efficient in terms of physical qubits for a fixed logicalerror rate [30]. We blieve that the numerical single-shot per-formance, the simplified decoding and the high encoding ratemake this LDPC code family highly competitive in situationswhere planarity is not an issue, such as in modular quantumcomputing architectures.A PPENDIX AT HE I DEALS OF Z [ ξ ] In Section III-E we discuss a construction based on a repre-sentation of the symmetry group of the { , , , } tessellation.The coefficients of the matrices in this representation containlinear combinations of powers of the golden ratio φ . The ringof these elements is denoted Z [ φ ] . To obtain finite exampleswe map this representation to one over finite fields. This isachieved by factoring out maximal ideals of Z [ φ ] . An ideal is maximal if it is a proper subset of the ring and all other idealsare contained in it. It is well-known that the quotient of a ringwith respect to a ideal is a field if and only if that ideal ismaximal. In what follows we discuss a slightly more settingwhere the ideals are prime. An ideal I is prime if for any a and b with ab ∈ I it holds that either a ∈ I or b ∈ I .We characterize the ideals of any ring of integers Z [ ξ ] towhich an element ξ is added. Let h be the minimal polynomialof ξ in Z [ ξ ] . As in Section III-E3 we will directly identify Z [ ξ ] with Z [ x ] / (cid:104) h (cid:105) . Lemma 1.
Let h ∈ Z [ x ] be an irreducible polynomial and I a non-zero prime ideal of Z [ x ] / (cid:104) h (cid:105) . Then I must contain aunique prime number p .Proof. Let ˜ I be the preimage of I under the natural epi-morphism. Since I (cid:54) = (cid:104) (cid:105) by assumption, ˜ I must contain a g ∈ Z [ x ] which is not a multiple of h . Since h is irreduciblewe must have gcd( h, g ) = 1 . By B´ezout’s identity there existpolynomials u, v ∈ Z [ x ] such that uh + vg = p , where p isa positive integer such that the gcd of p and the coefficientsof u and v is 1. Since I is a prime ideal, so is I ∩ Z and since p ∈ I we have that p must be a prime number. Lemma 2.
Let h ∈ Z [ x ] be an irreducible polynomial and I a non-zero prime ideal of Z [ ξ ] = Z [ x ] / (cid:104) h (cid:105) . Then the ring Z [ ξ ] /I is a finite field and I is a maximal ideal.Proof. Since I contains a prime number by Lemma 1 andsince h (the minimal polynomial of ξ ) has finite degree wehave | Z [ ξ ] /I | < ∞ . Furthermore, since I is a prime ideal weknow that Z [ ξ ] /I has no zero divisors.We will now show that every non-zero element a ∈ Z [ ξ ] /I has a multiplicative inverse. Since Z [ ξ ] /I is finite there mustexist positive integers m and n with m > n such that a m = a n , which is equivalent to a n ( a m − n −
1) = 0 . Since Z [ ξ ] /I has no zero divisors this means a m − n = 1 which implies a · a m − n − = 1 . Thus a m − n − is the multiplicative inverseof a .If p is the prime number from Lemma 1 then Z [ ξ ] /I is afinite field of characteristic p . Theorem 2.
Let h ∈ Z [ x ] be an irreducible polynomial and I a non-zero prime ideal of Z [ ξ ] = Z [ x ] / (cid:104) h (cid:105) . Any prime ideal I of Z [ ξ ] = Z [ x ] / (cid:104) h (cid:105) is equal to (cid:104) p (cid:105) , if h is irreducible in F p [ x ] or (cid:104) p, g ( ξ ) (cid:105) , if h is not irreducible in F p [ x ] ,where p is a prime number and g is an irreducible factor of h mod p .Proof. Let π I : Z [ ξ ] → Z [ ξ ] /I be the quotient map. FromLemma 2 we know that Z [ ξ ] /I is a finite field and we notethat it is generated (as a ring) by π I ( ξ ) . By Lemma 1 we havethat I must contain a unique prime number p . Hence, I mustbe generated by p and the minimal polynomial g of π I ( ξ ) .As ξ is a root of h in Z [ ξ ] by construction, we must havethat g divides h mod p . The two cases in the statement of thetheorem follow, depending on whether g ≡ h mod p .A PPENDIX BQ UOTIENT C ONDITION
We use the notation of [24]: let G be a symmetry group gen-erated by ( r i ) i ∈{ ,, } . For i ∈ { , , } , let S i be the subgroupof G generated by ( r j ) j ∈{ ,, }\{ i } . Let P be the polytopeassociated with G and ( S i ) i ∈{ ,, } . Let H be a subgroup of G .We are interested in a condition sufficient to define a quantumcode associated with the quotient polytope P /H .The orbit of an i-face F a = g a S i under the action of His { hg a S i | h ∈ H } . By definition this orbit is a face of thequotient abstract polytope. We denote it by HF a . In terms ofelements of Γ , It corresponds to the double coset Hg a S i .We use the same incidence definition: HF a and HF b areincident if Hg a S i ∩ Hg b S j (cid:54) = ∅ .We want to find a condition under which quotienting on theright by S i , i ∈ { , ..., } “does not interact” with quotientingon the left by H . More formally the following so-callednon-local subgroup condition is sufficient to prove the liftingLemma 3. Definition 1 (The non-local subgroup condition) . We say thata subgroup of the symmetry group of the tessellation G fullfillsthe non-local subgroup condition if for any i, j ∈ { , ..., } and for all g ∈ G we have gHg − ∩ S i S j = { id } . (48)The term non-local refers to the subgroup H: since thesubgroups S i , i ∈ { , ..., } are “local” (with respect forinstance to the distance in the Caley graph (cid:0) Γ , ( r i ) i ∈{ ,..., } (cid:1) ,the subgroup H has to be non-local in order to not interact withthe S i , i ∈ { , ..., } . Lemma 3 (Lifting of S i cosets) . Let H be a subgroup of astring C-group G of rank n satisfying the non-local subgroupcondition (Equation (48) ). For i, j ∈ { , ..., n } , let HF i be an i -face of H \P G and let HF j be a j -face of H \P G incidentto HF j .For any i -face K i of P such that HK i = HF i , there existsa unique face K j of P such that HK j = HF j and K j isincident to K i .Proof. There exists g i ∈ G such that K i = g i S i . Then HK i = HF i = Hg i S i . There exists g j ∈ G such that HF j = Hg j S j . There exist h i , h j ∈ H , s i ∈ S i and s j ∈ S j such that h i g i s i = h j g j s j . Therefore g i s i = h − i h j g j s j (49)We can define K j = h − i h j g j S j . Clearly HK j = HF j and K j is incident to K i .To prove uniqueness suppose that a face L j of P satisfies HL j = HF j and L j is incident to K i . There exists g ∈ G such that L j = gS j . Since HgS j = Hg j S j , there exists h ∈ H and s (cid:48)(cid:48) j ∈ S j such that g = hg j s (cid:48)(cid:48) j (50)Since L j is incident to K i , there exists s (cid:48) i ∈ S i and s (cid:48) j ∈ S j such that gs (cid:48) j = g i s (cid:48) i . Using Equations (49) and (50), weobtain hg j s (cid:48)(cid:48) j s (cid:48) j = h − i h j g j s j s − i s (cid:48) i . We can rewrite this as s − j g − j h − j h i hg j s j = s − i s (cid:48) i ( s (cid:48) j ) − ( s (cid:48)(cid:48) j ) − s j .Defining ¯ g = g j s j , ¯ h = h − j h i h, ¯ s i = s − i s (cid:48) i and ¯ s j = ( s (cid:48) j ) − ( s (cid:48)(cid:48) j ) − s j , we have ¯ g − ¯ h ¯ g = ¯ s i ¯ s j . Using the non-local subgroup condition Equation (48), itimplies that ¯ g − ¯ h ¯ g = id and therefore that ¯ h = id. We haveproven that h = h − i h j , which means that L j = K j .Thus under the non-local subgroup condition Equation (48)we can use the lifting Lemma 3 to prove that the orthogonalityof the parity-check matrices is preserved by such quotients:Let HF i − be an ( i − -face and HF i +1 be an ( i + 1) -faceof the quotient polytope. Let { ˜ F i , ..., ˜ F i n } be the collectionof i -faces incident to both HF i − and HF i +1 . Under the non-local subgroup condition Equation (48), the Lemma 3 showsthat there exist faces of the covering polytope K i − covering HF i − , K i +1 covering HF i +1 and ∀ k ∈ { , ..., n } , K i k covering HF i k such that { K i , ..., K i n } is the collection of i -faces incident to both K i − and K i +1 .The preservation of the orthogonality of parity check matriceswould follow immediately from this.A CKNOWLEDGMENT
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Nikolas P. Breuckmann
Nikolas P. Breuckmann holds a UCLQ ResearchFellowship at University College London. He is interested in quantum infor-mation and related fields. He obtained his PhD at RWTH Aachen Universityworking with Prof. Barbara Terhal on quantum fault-tolerance and quantumcomplexity theory. He has worked in industry at PsiQuantum, a Bay Areabased start-up building a silicon-photonics based quantum computer.