The Group Structure of Quantum Cellular Automata
TTHE GROUP STRUCTURE OF QUANTUM CELLULAR AUTOMATA
MICHAEL FREEDMAN, JEONGWAN HAAH, AND MATTHEW B. HASTINGSA
BSTRACT . We consider the group structure of quantum cellular automata (QCA) modulo circuitsand show that it is abelian even without assuming the presence of ancillas, at least for most reason-able choices of control space; this is a corollary of a general method of ancilla removal. Further, weshow how to define a group of QCA that is well-defined without needing to use families, by showinghow to construct a coherent family containing an arbitrary finite QCA; the coherent family consistsof QCA on progressively finer systems of qudits where any two members are related by a shallowquantum circuit. This construction applied to translation invariant QCA shows that all translationinvariant QCA in three dimensions and all translation invariant Clifford QCA in any dimension arecoherent.
The group of quantum cellular automata (QCA) modulo quantum circuits is an abelian group[6,11] (see Appendix). While this group has been fully understood in one dimension using an indextheory[4, 8, 15], and there has been much recent progress in higher dimensions[10, 11], there arestill some unsatisfactory foundational issues. For one issue, the group was only known to beabelian with additional ancillas present and the case without ancillas was not understood. Foranother issue, multiplication of two QCAs will necessarily increase the range of the QCA untileventually (on any finite system) some finite product of QCAs may give a QCA whose range iscomparable to the system size, thus rendering the very idea of locality in that QCA moot. One wayof dealing with this issue is to consider infinite families of QCAs with a fixed control space anddecreasing range of the QCA so that for any finite product of families all but finitely many QCAin the family will have range small compared to the system size. However, this is not completelysatisfactory since an arbitrary family may involve a sequence of QCAs that are “unrelated” to eachother; for example, various indices might fluctuate arbitrarily from one member of the family tothe next.In this paper, we present results to resolve these issues. First, we show how to remove ancillas(for most reasonable choices of control space) so that the group of QCA on those control spacesis abelian even without ancilla. This is a corollary of a theorem that if the action of a quantumcircuit is the identity on ancilla then the action can be realized by a slightly deeper quantum circuitwhose gates are not supported on the ancilla. Second, we show how, given a single fixed QCA,to create a “coherent” family of QCA[6] containing that “mother” QCA. This allows us to definea useful group structure given even a single QCA, where the natural composition of mother QCAis compatible with the elementwise composition of the constructed coherent family. The coherentfamily is parallel to entanglement renormalization groups of topological many-body states, and ourconstruction can be colloquially regarded as a way to construct a “UV theory.” Lastly, we examinetranslation invariant QCA that comes in an obvious family of QCA under periodic boundary con-ditions. Two classes of QCA are shown to be entanglement renormalization group fixed points:Every translation invariant Clifford QCA (one that maps a Pauli operator to a tensor product ofPauli operators) in any dimension and every translation invariant QCA in three dimensions definesa coherent family under periodic boundary conditions. a r X i v : . [ qu a n t - ph ] O c t MICHAEL FREEDMAN, JEONGWAN HAAH, AND MATTHEW B. HASTINGS
1. D
EFINITIONS
Naive Definitions.
Let us first define a QCA, a quantum circuit, and the notion of a controlspace. We will give two definitions in each case, first a simple definition when there is a finitenumber of “sites” and then a more general definition. Here we are defining what we may term a“naive” object, in that we will give a single QCA or circuit, rather than a family.The overall set up is: there is a set {H i } , i ∈ I , of finite dimensional Hilbert spaces indexed by adiscrete set I . The index i labelling the finite dimensional Hilbert spaces is referred to as a “site”index. In turn, each finite dimensional Hilbert space H i is a tensor product of some number ofadditional Hilbert spaces, termed “qudits” associate with that site. This terminology will be usedlater when discussing ancillas.We introduce some metric dist ( · , · ) between sites. This metric may be derived from a controlspace X (where a control space is a fixed smooth manifold with a metric, or alternatively a sim-plicial complex), given a map I : I ! X from sites to points in the control space. The controlspace may have boundary and may be noncompact. We assume the map is locally finite, i.e. for C compact, C ∈ X , I − ( C ) is finite.If the set of sites is finite, then a QCA is a ∗ -automorphism of the algebra of operators on thisHilbert space, subject to a locality condition explained below, and any such QCA can be expressedas conjugation by some unitary, also with a locality condition.More generally, if the set is not finite, we consider the net of operators on ⊗ i ∈ J ⊂ I H i , where J isfinite. Implicitly operators can always be extended to larger finite J (cid:48) ⊂ I by tensoring with id over J (cid:48) \ J . The support supp ( O ) of an operator O is the smallest J over which it may be written. aQCA is a ∗ -automorphism α of the associated net of Endomorphism algebras { ⊗ i ∈ J End ( H i ) } .In both finite and infinite cases, there is a a geometric condition that there is a range R ∈ R + sothat α is R -local, i.e. for all O , supp ( α O ) ⊂ N R ( supp ( O ) , the radius R neighborhood. We will sayeither that the QCA has range R , or that it is an R -QCA.Next we define a circuit more precisely. Informally, a quantum circuit is some sequence of“gates”, each being a unitary acting on a set of bounded range. Formally, a quantum circuit is apair consisting of a unitary and a “circuit decomposition” of that unitary in terms of gates. Such acircuit decomposition of a unitary U consists of writing U in the form(1) U = U D ◦ U D − ◦ . . . ◦ U , where D ≥ U a is written in the form U a = ∏ S ∈ G a U S , a , where G a is a collection of disjoint sets of sites and where U S , a is a unitary supported on S . Dis-jointedness means that the ordering within the product for U a is immaterial. Each U S , a is called agate on set S . We require that the diameter of all S ∈ G a for all a be bounded by a constant calledthe “range” of the gates. The integer a is referred to as labelling the “round” of the circuit. Therange of the circuit is defined to be the range of the gates multiplied by the number of rounds. Wesay that the circuit “implements” the unitary U . Hence such a circuit acts as a QCA by conjugation: O ! U O U † . Informally, then, we may say that some QCA “is equal to a circuit” if the circuit actsas that QCA by conjugation. We emphasize that the range of the resulting QCA is bounded by therange of the circuit, and it may in some cases be significantly smaller; see section 4. HE GROUP STRUCTURE OF QUANTUM CELLULAR AUTOMATA 3
Finally, let us define the notion of an ancilla. As mentioned above, the Hilbert space on each siteis a tensor product of some number of Hilbert spaces termed qudits. Each qudit will be referredto as either “physical” or “ancilla”. We say that a quantum circuit “implements a unitary V usingancillas” if the quantum circuit implements some unitary U = V ⊗ I where V acts on the physicalqudits and I acts on the ancillas.1.2. Families and Coherent Families.
QCA have a natural group structure, in that two QCA canbe multiplied by composing them. However, this composition may increase the range, so that theproduct of an R -QCA with an R -QCA may be an ( R + R ) -QCA. This presents no problem ifthe control space is infinite. For example, Ref. [8] studied an infinite real line as the control space,developing an index theory. However, if the control space has finite diameter, a finite number ofcompositions of QCA may give a QCA whose range is comparable to the injectivity radius of thecontrol space itself, and all locality is lost.One way to resolve this is to consider families of QCA on a fixed control space. A family α isa sequence of QCA, α i , with i = , , , . . . , each with some range R i , with R i ! i ! ∞ . Notethat we will use a bar to denote a family. While all QCA in the family have the same control space,different QCA in the family may have different sets of sites. The product of two families α ◦ β isthe family defined by the sequence of QCA α i ◦ β i . Then, for any finite product of families, for allbut finitely many QCA in the resulting family the range R i will be small compared to the injectivityradius of the control space. This allows the index theory of Refs. [6, 8] to serve as an obstruction towriting a family of QCA as a family of quantum circuits. We define a family of quantum circuitsto be a sequence of quantum circuits, with the depth of all circuits in the family bounded, and withthe range of the gates in the quantum circuit tending to zero. Such a family of quantum circuitswill be called a finite depth quantum circuit (fdqc).Remark: an alternative scaling of distance is to take all QCA in the family to have the samerange R but to rescale the metric on X so that the injectivity radius of X diverges. This differs fromthe definition here simply by an overall scaling of the metric.Remark on notion of bounded range quantum circuit: one might be tempted instead to considerfamilies of bounded range quantum circuits (brqc), defined as a sequence of quantum circuits over X so that the range of the i -th quantum circuit tends to 0 as i ! ∞ (the depth is allowed to divergeso long as the product of range and depth is bounded). More generally, even, one might considercircuits such that the range of the lightcone tends to zero (one may define the range of the lightconein an intuitive way as the maximum range of the QCA implemented by the circuit assuming ageneric choice of gates; we give a more precise definition later). However, assuming the existenceof an appropriate handle decomposition on the control space, all these notions are equivalent. Togive an intuitive idea of the equivalence, consider the case in one-dimension as shown in Fig. 1.We see here a circuit of depth 3 with each gate having range 1. However, it is shown how toregroup it into a circuit of depth 2, with each gate having range O ( ) . In general, assuming anappropriate handle decomposition, we can regroup a circuit to depth d + d -dimensional. We explain this in more detail in section 4.However, there is a problem with the notion of a family. One may imagine a family composedof QCA which are “unrelated to each other” in some sense. For example, take the control space tobe a circle and let QCA α i act on a Hilbert space built from 2 i sites, with one qudit per site and withsites uniformly spaced around the circle so distance 2 − i between neighboring sites. If we choosefor example α i for even i to be the identity QCA and α i for odd i to be a shift right by one, thenthe GNVW index fluctuates between different QCA in the family, which is undesirable. Indeed, MICHAEL FREEDMAN, JEONGWAN HAAH, AND MATTHEW B. HASTINGS F IGURE
1. A quantum circuit. Horizontal axis is position, and vertical axis labelsthe round, increasing upwards. Each rectangle is a gate, and lines represent qudits.The pattern repeats indefinitely. The dashed lines show how to combine a set ofgates into a single unitary: all gates between a pair of neighboring dashed lines canbe combined into a single unitary. This rewrites the circuit as depth two but withgates having diameter 5 in this example.the QCA in the sequence could be even more arbitrary than this example; for example, it is quitepossible to have the GNVW index diverge with i .As a resolution, the following definition of a “coherent” family was proposed in Ref. [6] (anearlier version called this family “uniform”). To define the notion of coherent family, we first needa notion of path equivalence and stable path equivalence as follows: Definition 1.1.
Two QCA α , β are R (cid:48) path equivalent if there exists a continuous path of QCA withrange at most R (cid:48) from α to β . Definition 1.2.
Two QCA α , β are stably R (cid:48) equivalent if one can tensor with additional ancilladegrees of freedom such that α ⊗ Id is R (cid:48) path equivalent to β ⊗ Id , where Id denotes the identityQCA and in this case Id acts on the additional degrees of freedom. Here, α , β may act on differentHilbert spaces with different sites (although they have the same control space), so that one maytensor α with Id acting on one set of ancillas and tensor β with Id acting on a different set ofancillas. There must be a one-to-one correspondence between sites acted on by α ⊗ Id with thoseacted on by β ⊗ Id , with corresponding sites having the same finite dimensional Hilbert space andthe same image under the map to the control space. In particular, sites which are ancillas for α ⊗ Id might not be ancillas for β ⊗ Id . To define a coherent family, we take R i = R · − i for some R and then define: Definition 1.3.
Such a family α of QCA is “path coherent” (or simply “coherent”) if there existsa constant c such that for all i, QCA α i is stably cR i path equivalent to α i + . We emphasize that this definition implies also that for any j > c j suchthat for all i QCA α i is stably c j R i -equivalent to α i + j . Proof: indeed, we can take c j = c + c / + c / + . . . c / j − ≤ c , by composing the circuits that map α k to α k + for k = i , i + , . . . , i + j − R i = − i could be replaced any exponential decay,i.e., with the assumption that R i = x − i for any x >
1. If x >
2, then such a family is also a familyof QCA with range R i = − i . If x <
2, then one can pass to a subfamily and obtain a family withrange R i = − i . So long as R i decays exponentially, then passing to a subfamily does not affect thecoherence of the family, essentially for the reasons explained in the above paragraph, taking now c j = c + c / x + c / x + . . . + c / x j − . HE GROUP STRUCTURE OF QUANTUM CELLULAR AUTOMATA 5
In fact, one may define an even stronger notion of coherence where QCA are related by circuitsrather than by paths, as follows.
Definition 1.4.
Two QCA α , β are R (cid:48) circuit equivalent if there exists a quantum circuit of rangeR (cid:48) such that α is equal to β followed by the quantum circuit, i.e., acting by conjugation by thequantum circuit implements QCA α ◦ β − . Definition 1.5.
Two QCA α , β are stably R (cid:48) circuit equivalent if one can tensor with additionalancilla degrees of freedom such that α ⊗ Id is R (cid:48) circuit equivalent to β ⊗ Id , where Id acts on theadditional degrees of freedom. Definition 1.6.
A family α of QCA is“circuit coherent” if there exists a constant c such that for alli, QCA α i is stably cR i circuit equivalent to α i + . Note that every circuit coherent family is path coherent. Proof: choose a path joining each gatein the circuit to the identity.With this definition, the example above of shifts is not coherent (indeed, it is not even pathcoherent) but if we choose each QCA to shift right by one, then the family is circuit coherent.Given this definition of coherent families, the reader might be tempted to assume that one shouldonly consider coherent families and that “incoherent families” (i.e., those which are not coherent)are somehow artificial. However, consider the following natural way of constructing a family.Consider a translation invariant QCA acting on an infinite lattice of sites in a hypercubic lattice in n spatial dimensions. Then, there is an obvious way to define a family of QCA acting on a finitelattice, with the control space being the n -torus. Assume the translation invariance holds for anytranslation by a single site, in any of the n different spatial directions.For all sufficiently large i , let QCA α i act on a lattice of ( m i ) n sites for some integer m >
1. Thesites will be arranged in a hypercubic lattice within the torus, with distance m − i between neighbor-ing sites. We define α i in the obvious way from the translation invariant QCA. For example, thefamily of shifts where each QCA shifts right by one can be obtained from a translation invariantQCA which shifts right by one on an infinite system. Remark: here we require that i be sufficientlylarge, as to define the QCA we need m i large enough compared to the range of the translationinvariant QCA.Then, it is not clear whether or not this obvious choice of a family is a coherent family. Indeed,one can give explicit examples where the family is not expected to be coherent. For example, inRef. [11] a translation invariant QCA α WW was constructed in three dimensions whose square isa quantum circuit but strong evidence was given that α WW itself could not be implemented by aquantum circuit. A generalization of this was given in Ref. [10] where QCA γ were constructedwhose fourth power was a quantum circuit but it is believed that γ , γ , γ cannot be implementby quantum circuits. Consider a four dimensional translation invariant QCA β given by stackingcopies of γ , so that sites in the fourth dimension are labelled by some integer, and for each choiceof this integer we have an independent copy of γ . Finally, choose m =
3. Then, m i = , , , . . . .Under the assumption that γ , γ , γ are not quantum circuits, this is not a coherent family. Proof:dimensionally reduce, i.e., ignore the requirement of locality in the fourth dimension. Then, forvarying m we have 1 , , , . . . copies of γ . Mod 4 this is 1 , − , , − , . . . copies of γ . However, byassumption, γ is not related to γ − by a quantum circuit.We further discuss families of translation invariant QCA in section 5 and give some sufficientconditions for the family to be coherent.This phenomenon may be reminiscent of something that occurs in Hamiltonians which are sumsof commuting projectors with topologically ordered ground states. If one takes a toric code, then MICHAEL FREEDMAN, JEONGWAN HAAH, AND MATTHEW B. HASTINGS for any fixed manifold, for any cellulation C , one may construct an isometry from the ground statesubspace of the Hamiltonian on that cellulation C to the ground state subspace of the Hamiltonianon a refinement C (cid:48) of that cellulation by tensoring with additional additional degrees of freedomin some product state and applying a local quantum circuit[1]. Thus, the ground states of thisHamiltonian have some similar “coherence” property. However, the ground state structure of thecubic code[9] is much more complicated and does not have this property. This is related to adifference in the renormalization structure of the Hamiltonian, where the toric code Hamiltonianrenormalizes to itself plus topologically trivial terms but the cubic code Hamiltonian renormalizesto itself plus terms describing an additional topological phase.2. A NCILLA R EMOVAL IN Q UANTUM C IRCUITS
Overview.
As remarked before, several authors have noted that the group of quantum cellularautomata modulo quantum circuits is an abelian group in the presence of ancilla. That is, givenany two QCA α , β , both with some bounded range R , the product α ◦ β ◦ α − ◦ β − can be writtenas a circuit with ancillas, i.e., by adding some additional ancilla degrees of freedom, we can find aquantum circuit of bounded depth and range that implements the unitary (cid:16) α ◦ β ◦ α − ◦ β − (cid:17) ⊗ I ,where the tensor product is between the physical and ancilla degrees of freedom, so the circuit actsas α ◦ β ◦ α − ◦ β − on the physical degrees of freedom and as the identity on the ancilla degreesof freedom.This raises the question: if we do not allow ancillas, is the group still abelian? We answerthis question in the affirmative (at least for certain choices of control space) by showing a moregeneral result: for certain choices of control space (defined below), given any circuit that actstrivially on the ancilla degrees of freedom, there is a circuit implements the same unitary on thephysical degrees of freedom without using any ancilla degrees of freedom. The latter circuit hasan increased depth and range, but they are bounded by some function of the depth and range of theoriginal circuit.In this note qudits will be assumed to have the same dimension. Given the construction for an-cilla removal below, the reader will see how to generalize it to cases with varying dimension though(assumption 2.4 below must be generalized so that it holds separately for each prime dividing thedimension of the qudits).The basic idea of the construction involves taking a given decomposition of the circuit andconstructing a new decomposition. A trivial example of this is that given any circuit, we canincrease the depth by 1 by taking one of the U a to be the identity operator, i.e., given a circuit withgiven U a and given depth D , we could define a new circuit U (cid:48) with depth D + U (cid:48) a with U (cid:48) a = U a for a ≤ D and U D + = I . However, more interesting and useful choices ofdecomposition are possible. We will explain how, given a circuit, to change the decomposition sothat in each round, the density of the gates decreases (roughly meaning that only a small fraction ofthe qudits are in the support of any gate; we make this more precise later), at the cost of increasingthe depth of the circuit. We use this modified decomposition to show how it is possible to removeancillas from a quantum circuit, turning a circuit that uses ancillas into one without ancillas, at thecost of increased depth and under certain assumptions on the ancilla density.2.2. Simplest Construction.
The basic idea of the decomposition that we will use can be seenin the following example that we present pictorially. After giving this example, we give a moregeneral construction with formal definitions. We consider a one-dimensional system. Sites arelabelled by integers, and the distance between sites is simply the absolute value of the difference
HE GROUP STRUCTURE OF QUANTUM CELLULAR AUTOMATA 7
21 2 3 4 3 2 1
A B C D C B A B F IGURE
2. Horizontal axis is position. Vertical axis labels the round, increasingupwards. The bottom line labelled by numbers represents the qudits, with eachnumber 1 , , , , , , A , B , C , D contains many gates horizontally and vertically. Each gate is entirely contained within asingle block of gates; the angled line represents a “lightcone”.Let V a for a ∈ { A , B , C , D } be the product of the unitaries in all blocks labelled a . Then, thecircuit is equal to the product V D V C V B V A . This product gives a different decomposition of thecircuit, in which gates in blocks labelled D are done in a later round than they would be in theoriginal decomposition. The depth of the circuit is increased by at most a factor of 4.This new decomposition gives rise to what one may pictorially describe as a staircase or a zig-zag: if, as a function of the coordinate of sites, one considers those rounds over which some gate isacting on that site, these will increase and decrease periodically in a zig-zag, with sites in a blocklabelled 4 having gates at the latest rounds (i.e., they are in gates in blocks D ) and those in a blocklabelled 1 having gates in the earliest rounds (i.e., in gates in blocks A , B ).We then use an idea of “borrowing” to remove ancillas: for each gate acting on an ancilla quditin block 1, we replace its action on that ancilla with the same action on a physical qudit in block3. We similarly replace the action on ancillas in block 2 with physical qudits in block 4, ancillasin block 3 with physical qudits in block 1, and ancillas in block 4 with physical qudits in block 2.Precisely, we construct a one-to-one pairing between ancilla qudits in block 1 with physical quditsin block 3, and similarly between ancillas in block 2 with physical qudits in block 4, and so on.Such a pairing exists because we will take all blocks the same size. We pair each ancilla qudit inblock 1 with a physical qudit in a neighboring block 3 (for example, the block 3 immediately tothe right). Then, we conjugate V A by a swap gate which swaps the ancillas in blocks 1 with thecorresponding physical qudits in blocks 3. Similarly, we conjugate V B by a swap which swaps the MICHAEL FREEDMAN, JEONGWAN HAAH, AND MATTHEW B. HASTINGS ancillas in block 1 with physical qudits in block 3 and swaps the ancillas in block 2 with physicalqudits in blocks 4; note crucially that V B does not act on the physical qudits in blocks 3 or 4. Weconjugate V C by a swap which swaps the ancillas in block 2 with physical qudits in block 4 andswaps the ancillas in block 3 with physical qudits in block 1. We conjugate V D by a swap whichswaps the ancillas in block 3 with physical qudits in block 1 and swaps the ancillas in block 4 withphysical qudits in block 2.Then one may show under the assumption that the original circuit acted trivially on the ancillas,that this replacement gives a new circuit that implements the same unitary. We call this “borrow-ing” because one “borrows” the physical qudit in block 3 to act as an ancilla for block 1, and thenreturns it unchanged so later it can be used as a physical qudit. The qudit is returned unchangedbecause the original circuit acted as the identity on the ancilla qudits. Since the blocks have widththat is proportional to the range of the quantum circuit, this requires increasing the range of thegates in the quantum circuit only by a factor proportional to the range of the whole circuit.A similar kind of decomposition can be used when the control space is more than one-dimensional,so long as it has the form of a one-dimensional line R times some arbitrary space X ⊥ and so longthere is a certain symmetry under translation along the one-dimensional line. Label coordinates inthe control space by a pair ( x , y ) with x ∈ R and y ∈ X ⊥ . Then, we require that for all x , y , there bea site at ( x , y ) if and only if there is one at ( x + , y ) , and that both such sites (if they exist) have thesame Hilbert space dimension. If these conditions hold, one simply uses the coordinate along theone-dimensional line to define the block decomposition. One picks the swap gates to swap quditsat the same value of coordinate in the space X ⊥ but shifted coordinate along the one-dimensionalline.If we consider the more general setting in which only some of the sites have physical qudits, wecan still make such a borrowing construction, given an assumption on the density of the physicalqudits. For example, suppose that sites labelled by integers equal to 0 mod K for some integer K > / K of the qudits are physical. One way to proceed here is to iterate the above construction, removinghalf of the ancilla qudits at each step, i.e., suppose for simplicity that K is a power of 2 (if it is not,one may tensor with the identity on additional ancillas so that it is). Consider a new system withone physical qudit and one ancilla qudit on each site, where the physical qudit is the tensor productof the original physical qudit and K / − K / General Construction.
We now give a more general construction for other control spaces.Given a quantum circuit U and some decomposition of that circuit, and some pair ( S , a ) where S is a set and a is an integer, define the “forward lightcone” of ( S , a ) inductively as follows. It is thesmallest set of pairs which contains ( S , a ) and such that if U T , b and U T (cid:48) , b (cid:48) are in the decompositionof the circuit and b (cid:48) > b and T (cid:48) ∩ T (cid:54) = /0 then ( T (cid:48) , b (cid:48) ) is in the forward lightcone whenever ( T , b ) is.Given some decomposition of a quantum circuit, and given some function τ ( · ) from pairs ( S , a ) to integers, we say that this function τ ( · ) (which we call a “time function”) is “causal” if givenany pair of gates U S , a and U T , b with b > a and S ∩ T (cid:54) = /0, then τ ( T , b ) > τ ( S , a ) . Note that for anycausal time function, it follows that given any pair of gates U S , a and U T , b with ( T , b ) in the forward HE GROUP STRUCTURE OF QUANTUM CELLULAR AUTOMATA 9 lightcone of ( S , a ) we have that τ ( T , b ) > τ ( S , a ) . Proof: if ( T , b ) is in the forward lightcone of ( S , a ) then there is some sequence ( S , a ) , ( S , a ) , . . . , ( S k , a k ) with b > a k > . . . > a > a and T ∩ a k (cid:54) = /0 and a k ∩ a k − (cid:54) = /0 and a ∩ S (cid:54) = /0, and then the claim follows inductively.We will say that a gate U S , a “acts at time t ” if τ ( S , a ) = t .We now show that Lemma 2.1.
Given some decomposition D of a unitary U and given some causal function τ withthe range of τ being , . . . , τ max for some arbitrary integer τ max ≥ , then U = V τ max ◦ . . . ◦ V , where (2) V b = ∏ ( S , a ) s . t . τ (( S , a ))= b U S , a . That is, V b is the product of U S , a over all gates S , a in the decomposition D such that τ ( S , a ) = b.Proof. The proof is inductive in τ max . The base case τ max = U is the identityand has no gates in its circuit decomposition.Let U have depth D . Define ˜ U by ˜ U = ˜ U D ◦ . . . ◦ ˜ U , where ˜ U a = ∏ S ∈ G a , τ (( S , a )) < τ max U S , a . Inwords, ˜ U has the same circuit decomposition as U , except that we remove all gates U S , a with τ (( S , a )) = τ max , and ˜ U a is the same as U a except that we remove all gates U S , a with τ (( S , a )) = τ max .By assumption, there is no gate U T , b that is in the forward lightcone of any of the gates in V τ max . Hence, in the decomposition U = U D ◦ . . . ◦ U , the gates U S , a with τ (( S , a )) = τ max can becommuted to the left of all other gates, which shows that U = V τ max ˜ U . By the inductive assumption,˜ U = V τ max − ◦ . . . ◦ V since τ ( · ) is also a causal time function for ˜ U . Hence, U = V τ max ◦ . . . ◦ V . (cid:3) We claim that
Lemma 2.2.
Let h : X ! R be a Lipschitz function with Lipschitz constant K. Given such afunction, define h ( S ) for any set S to equal min x ∈ S h ( x ) .Then, given a quantum circuit where the gates have range at most r, and given any set of sitesB, the time function τ ( S , a ) = (cid:98) h ( S ) (cid:99) + ca is causal for any c ≥ rK + .Proof. Consider any pair of gates U S , a and U T , b with b > a and S ∩ T (cid:54) = /0. Then, by the Lipschitzcondition, h ( S ) ≤ h ( T ) + rK . Hence, (cid:98) h ( S ) (cid:99) < (cid:98) h ( T ) (cid:99) + rK +
1. Hence for c ≥ rK + (cid:98) h ( S ) (cid:99) + ca < (cid:98) h ( T ) (cid:99) + cb . (cid:3) We will call such a function h ( · ) a “height” function. Remark: an example of such a heightfunction that we will use later is h ( x ) = dist ( x , B ) , where B is any set of points in the control space.This function has Lipschitz constant 1.We make one heuristic remark on this construction. This time function (cid:98) h ( S ) (cid:99) + ca can be under-stood as defining a new time depending both on the original “time”, i.e. a , as well as the positionin “space”, i.e., the function h ( S ) . This is somewhat reminiscent of coordinate transformations inspecial relativity, in which the time in a moving reference frame depends linearly on time and spacein another reference frame. Here, we find some range of possible a for which the time functionis causal, reminiscent of the fact that in special relativity, only certain linear transformations willpreserve timelike intervals. In the same spirit we remark that the computer science tradition oftreating space and time complexities separately must at some point encounter special relativity.Now we describe a more general form of “borrowing”. Given a circuit C implementing unitary U , with a causal time function τ , let a “borrowing function” f ( · ) be a function from sites to siteswith the following properties. First, the range of the borrowing function is a subset of the set ofsites with a physical qudit. Next, for any site i , for each site j such that f ( j ) = i , let S j ( i ) be the set of ( S , a ) such that there is a gate U S , a with j ∈ S . Let also S i ( i ) be the set of ( S , a ) such that there is agate U S , a with i ∈ S . Let T j ( i ) be the image S j ( i ) under τ , i.e., in words T j ( i ) for is the sets of timesat which some gate acts on j . Then, for a borrowing function we require that for all i , for any pair j , k (cid:54) = i with f ( j ) = f ( k ) = i with j (cid:54) = k , we have that either T j ( i ) < T k ( i ) or T k ( i ) < T j ( i ) where theinequality T j ( i ) < T k ( i ) means that every element of T j ( i ) is smaller than every element of T k ( i ) .In words, we require that either all gates with support on j act before all gates with support on k orvice-versa. Further we require that for any j (cid:54) = i that either T j ( i ) < T i ( i ) or T j ( i ) > T i ( i ) . In words,we require that all gates with support on j act before all gates with support on i , or vice-versa.From such a borrowing function, we can construct a new circuit C (cid:48) as follows: for every gate U S , a in C , we interchange every ancilla qubit on site j in its support with the physical qubit on site f ( j ) . That is, define σ i , j to swap the physical degree of freedom on site i with the ancilla degree offreedom on site j and define(3) U (cid:48) S , a ! (cid:16) ∏ j ∈ S σ f ( j ) , j (cid:17) U S , a (cid:16) ∏ j ∈ S σ f ( j ) , j (cid:17) , with U (cid:48) S , a the gates in circuit C (cid:48) . The gates U (cid:48) S , a do not act on the ancilla qubits, so we can thendefine in the obvious way a circuit C (cid:48)(cid:48) that acts on a system that includes only physical degrees offreedom, implementing unitary U (cid:48)(cid:48) with U (cid:48) = U (cid:48)(cid:48) ⊗ I .We now show that Lemma 2.3.
If U acts trivially on the ancilla degrees of freedom, then U = U (cid:48) .Proof. Let there be N sites. Let Λ , . . . , Λ N be a sequence of subsets of sites, with | Λ m | = m andeach Λ i obtained from Λ i − by a adding a single site. We prove the lemma by considering asequence of circuits C , C , C , . . . , C N with C = C and C N = C (cid:48) and showing that C m and C m − implement the same unitary for each m > C m to have gates U ( m ) S , a with(4) U ( m ) S , a = (cid:16) ∏ j ∈ S ∩ Λ m σ f ( j ) , j (cid:17) U S , a (cid:16) ∏ j ∈ S ∩ Λ m σ f ( j ) , j (cid:17) . In words, in each circuit C m we make the swap of ancilla qubit on site j with physical qubit on site f ( j ) only for those j ∈ Λ m .Let C m implement unitary U m . We will show that U m = U m − . Let Λ m \ Λ m − = j with i = f ( j ) .Let t min be the smallest element of T j ( i ) and t max be the largest element.Define V ( m − ) b = ∏ ( S , a ) s . t . τ (( S , a ))= b U ( m − ) S , a , as in lemma 2.1. By assumption, the product V ( m − ) t max ◦ . . . ◦ V ( m − ) t min acts trivally on the ancilla qudit on site j , since all gates that act on this ancilla are inthis product. Further, by assumption, the product V ( m − ) t max ◦ . . . ◦ V ( m − ) t min acts trivally on the physicalqudit on site i since either T i ( i ) < T j ( i ) or T i ( i ) > T j ( i ) . Hence, V ( m − ) t max ◦ . . . ◦ V ( m − ) t min = σ f ( j ) , j ) V ( m − ) t max ◦ . . . ◦ V ( m − ) t min σ f ( j ) , j (5) = V ( m ) t max ◦ . . . ◦ V ( m ) t min . Hence U m = U m − . (cid:3) Thus, we have shown that the existence of a causal time function and a borrowing functionenables us to remove the ancillas. Now we give some conditions under which, given some circuit,
HE GROUP STRUCTURE OF QUANTUM CELLULAR AUTOMATA 11 we can construct a causal time function and a borrowing function. The goal is to remove ancillaswithout increasing the circuit depth or range of the gates by an excessive amount.We make the following assumption on the control space and sites. The assumption is phrased interms of the number of physical and ancilla qudits, so they may be applied to the setting in whicheach site has one physical and one ancilla qudit or to a more general setting with a lower densityof physical qudits.
Assumption 2.4.
There are constants < c ≤ c (cid:48) and d > and there is some (cid:96) which dependsonly on the range and depth of the quantum circuit and on c , c (cid:48) , d, such that: for any (cid:96) ≤ (cid:96) , thenumber of physical qudits within distance (cid:96) of x is at least (cid:98) c (cid:96) d (cid:99) and further the number of ancillaqudits with distance from x in the interval [ (cid:96) − , (cid:96) ] is at most (cid:98) c (cid:48) (cid:96) d − (cid:99) . Remark: note that the lower bound on the number of physical qudits is on the number containedin some ball of radius (cid:96) while the upper bound on the number of ancilla is on some “shell” ofthickness 1.It is not clear to us how much this assumption can be generalized. While the assumption holdsfor many reasonable choices of control space and control map, it is likely that for other controlspaces borrowing functions can still be constructed. A particularly interesting case to considerwould be a hyperbolic space with a uniform density of qudits (uniform on some scale) and withthe range of the circuit large compared to the curvature radius. Our construction given Assumption2.4 will use a particularly simple choice of function h to construct a causal time function. Aftergiving this construction, we will discuss other possible choices of function h and explain how toconstruct a borrowing function on the hyperbolic plane.We consider a quantum circuit whose gates have range r and with the quantum circuit havingdepth D . Recall that an ε -net is a set of points x , x , . . . , such that every point in the space is withindistance ε of some point in the ε -net and no two distinct points in the sequence are distance lessthan ε of each other. We construct a (cid:96) -net for some (cid:96) that we pick later. We choose the timefunction τ ( S , a ) = (cid:98) dist ( S , B ) (cid:99) + ca with c = r + B being the set of points in the net. Bylemma 2.2, this time function is causal.We now construct a borrowing function. Recall that for a borrowing function, we require thatif f ( j ) = f ( k ) with j (cid:54) = k , then either all gates with support on j act before all gates with supporton k or vice-versa. Consider a site j with (cid:98) dist ( j , B ) (cid:99) = m . Then, gates acting on j act at timesin the interval [ m , m + ( r + ) D ] , where we use the time function τ . We will decompose the set ofancilla qudits into several disjoint sets, called S , S , . . . , where S a is the set of ancilla qudits j with ( r + ) Da ≤ (cid:98) dist ( j , B ) (cid:99) < ( r + ) D ( a + ) . Then, gates acting on qudits in in S a will all act before(or will all act after) those acting on qudits in S b for any a , b with | a − b | >
1. So, for qudit j ∈ S a and k ∈ S b with | a − b | >
1, we do not have a constraint that f ( j ) (cid:54) = f ( k ) . We now make a choiceto ensure that for j ∈ S a and k ∈ S a + that f ( j ) (cid:54) = f ( k ) .Let us first introduce some notation. For each site j , let b ( j ) be a closest point in the net B andfor any x ∈ T let T ( x ) be the set of j with b ( j ) = x . We will decompose T ( x ) into two disjointsets, T ( x ) , T ( x ) , each of size at least (cid:98) T ( x ) / (cid:99) . We will then make an arbitrary choice: we willchoose that for j ∈ T ( x ) , if j ∈ S a for a even (or odd), we will pick f ( j ) in T ( x ) (respectively, f ( j ) ∈ T ( x )) . This guarantees the needed property to get a borrowing function. Further, byconstruction dist ( j , f ( j )) ≤ (cid:96) .So, it remains to show that we can indeed make such a choice of f ( j ) . However, this is possibleby the density assumption: for each point x ∈ B , T ( x ) ∩ S a is bounded by c (cid:48) (( r + ) d ) d ( a + ) d − .However, the number of physical qudits closer to x than to any other point in the net is at least c (cid:96) d .For sufficiently large (cid:96) , the second number is larger than the first and so the matching exists. Thus we have
Theorem 2.5.
Under Assumption 2.4 above, for sufficiently large (cid:96) depending only on c , c (cid:48) , d andon the range and depth of the quantum circuit, there is a borrowing function with dist ( j , f ( j )) ≤ (cid:96) and so ancillas may be removed. As a corollary
Corollary 2.6.
Consider any pair of QCA α , β with range R. Let the Assumption 2.4 hold. Further,we assume that there are no ancilla qudits. Then α ◦ β ◦ α − ◦ β − is a quantum circuit with depthand range depending only on R , c , c (cid:48) , d.Proof. Using ancillas, α ◦ β ◦ α − ◦ β − is a quantum circuit. Since assumption 2.4 held withoutancillas, when we add one extra ancilla for each physical qudit it still holds and so we may removethe ancillas. (cid:3) Now let us consider to what extent we can generalize Assumption 2.4. First, we emphasize that some kind of lower bound on the density of physical qudits is necessary. To see this, consider thefollowing simple example. Imagine a large square lattice, size (cid:96) × (cid:96) for some (cid:96) (cid:29)
1, with periodicboundary conditions in both directions and with one qudit per site. Let the qudits with horizontalcoordinate equal to 0 or to (cid:96)/ O ( ) which implements a vertical shift by + − (cid:96)/
2. This circuit isimplemented by a “swindle” (see for example discussion of such a circuit in Ref. [6]). On the otherhand, no such circuit exists which acts just on the physical qudits.However, we still expect that some form of ancilla removal can be implemented on more generalcontrol spaces so long as the coarse density of physical qudits is comparable to the coarse densityof ancilla qudits. Certainly, if there is one physical and one ancilla qudit per site, one expectsthat ancilla removal can be implemented more generally. It seems that to do this, one way is togeneralize the choice of function h in lemma 2.2. For example, : Lemma 2.7.
Let h : X ! [ , T ] be -Lipschitz where T is O ( ) . Assume that there is one physicaland one ancilla qudit per site. Consider a quantum circuit of depth D with gates of range r. Assumethat there is an involution f on the set of sites such that for any site x, | h ( x ) − h ( f ( x )) | > D ( r + ) and dist ( x − f ( x )) ≤ r · O ( ) . In words, this involution maps each site to a nearby site with asufficiently different value of h.Then, τ ( S , a ) = (cid:98) h ( S ) (cid:99) + ( r + ) a is a causal time function and f ( · ) defines a borrowing functionfor this time function.Removing ancillas using this time function and borrowing function increaes the depth and rangeof the quantum circuit by an O ( ) multiplicative factor.Proof. Consider a pair x , f ( x ) . Assume without loss of generality h ( x ) > h ( f ( x )) + d ( r + ) . Sincethe quantum circuit C has depth D , all gates in circuit C (cid:48) acting on the qudits on site f ( x ) areexecuted at time at most (cid:98) h ( f ( x )) (cid:99) + D ( r + ) ≤ h ( f ( x )) + D ( r + ) < h ( x ) . All gates in C (cid:48) acting on the qudits on site x are in a gate which is supported on some set S withindistance r of x so they are executed at time at least executed at time at least (cid:98) h ( S ) (cid:99) + ( r + ) ≥ h ( S ) + r ≥ h ( x ) . HE GROUP STRUCTURE OF QUANTUM CELLULAR AUTOMATA 13
So, the borrowing condition is satisfied. (cid:3)
Note that we want h to have a bounded variation, so that the depth of the circuit does not increasetoo much. However, to find the involution f , it is important that the density (on some coarse scale)of sites in the inverse image under h of various intervals [ i , i + ] is roughly constant. That is, if wetry the choice h ( x ) = dist ( X , B ) for a hyperbolic space, we find that most of the density is in theinverse image of the interval [ (cid:96) − , (cid:96) ] , i.e., most points are far from the net, so we cannot find aphysical qudit to borrow to remove those ancillas.For the two dimensional hyperbolic plane, however, we can construct borrowing function asfollows. Choose a periodic tiling by large (compared to the circuit range) tiles. Choose the heightfunction to be 0 at the center of the tiles and increase radially outward. For example, far enoughfrom the boundary of a tile, choose the height function to simply be the distance from the centerof a tile. Assuming a site density which is uniform on this scale with one physical and one ancillaqudit per site, we can remove the ancillas far enough from the boundary of the tiles simply becausemost of the volume is not near the center of the tiles. What is left is then some quantum circuitsupported near the 1-skeleton of this tiling. We can then find some borrowing to remove ancillas forthis circuit, choosing the height function to increase from the center of the edges to their boundary.We leave a precise construction and a generalization to higher dimensions for future work.A regular tiling of hyperbolic space is generally constructed by selecting a co-compact reflectiongroup Γ and then each tile is a convex fundamental domain. Surprisingly there is Vinberg’s theo-rem[16] which states that above dimension 30 there are no cocompact reflection groups acting onhyperbolic space ( 30 may not be a sharp bound). This rather surprising theorem is number theo-retic in nature and the does not yet seem to have a geometric explanation. But it serves as a warningthat it may not be straight forward to generalize the sketch we presented for H to hyperbolic spacein all higher dimensions.3. C OHERENT F AMILIES AND THE OFFICIAL DEFINITION OF
QCAThis section completes the circle. We began with naive definitions of QCA and quantum circuitsand then enhanced them to sequence-based definitions. In this section, we prove coherence theo-rems, Theorem 3.1 and 3.3, to show that an initial QCA α (of sufficiently small range R ) in factcontains all the necessary information in the first place. The “mother” α ∈ QCA of a coherent se-quence α ∈ QCA effectively determines α which obeys a strong uniqueness property. This allowsus to dispense with sequences and return, now with greater confidence, to the naive definition.This section will focus on path coherent families and equivalence up to paths. The next section4 will discuss circuit equivalance of families.Consider a coherent family α = { α , α , . . . } . Each α i is of range R i , with α = α called the mother . It is required in definition 1.3 that α i + be obtained from α i , i ≥ α i + to be obtained from α i by what we will call a deformation . We will see later that this in fact is nota change in the definition: all of the allowed deformations can in fact be described by stabilizationand quantum circuit and hence our construction will give a circuit coherent family, not just a pathcoherent family.We will see that the pleating construction will give a circuit coherent family. The uniquenessresults in this section will apply however to path coherent families more generally and will dealwith a more general notion of equivalence under paths. In section 4 we will deal with circuitcoherent families and equivalence using circuits. A deformation is a composition of c local deformations, c a constant independent of i depend-ing only on the control space X . (We may take c = dim ( X ) + local deformation consistsof a composition of four operations all supported on the set h j ⊂ X where h j is the disjoint unionof well seperated (contractible) balls. In our application X is given a handle decomposition h con-sisting of 0-handles h , 1-handles h , . . . , d -handles h d . The well-separated condition can be takento be that the distance between any two disjoint handles in X is > · d R i , d the dimension ofthe control space X , and each handle will be assumed to have diameter bounded by a constanttimes R i ; a precise sufficient condition is given below (If X is P.L. triangulated, a well-separatedhandle decomposition can be obtained from a refinement of the original triangulation). The fouroperations, which together constitute what is allowed by a quantum in a stable setting, are:(1) Stabilization: the index set I is enlarged to I (cid:48) with new finite Hilbert spaces H i , i ∈ I (cid:48) \ I introduced and the control map I : I ! X extended by a locallyfinite mapping of I (cid:48) \ I to X .(2) Acting by arbitrary quantum circuits over h i .independent of i .(3) Destabilization: the inverse operation to (1)(4) Composing the control map I : I ! X by a diffeomorphism g : X ! X ,isotopic to the identity.(6)Note: It is possible for a deformation to implement all stabilizations at one time at the beginningand all destabilizations at one time at the end. Then since steps (2) and (4) can be done adiabati-cally, it is legitimate to think of “deformation” indeed as a continuous process.Further, clearly, (1-3) are all examples of quantum circuits or stabilization (the destabilizationoperation (3) is equivalent to stabilizing α i + ), while (4) can be described by circuits and stabiliza-tion by tensoring with ancillas and using swap gates. For example, to a site from some point x tosome point x (cid:48) , we tensor with an additional ancilla at x (cid:48) (stabilization), conjugate by a swap, andthen remove the the original degree of freedom at x (destabilization). Since the handle diameteris bounded by R i , (2) can be implemented by a single gate of range bounded by R i and hence is aquantum circuit. Similarly, the range bound on the swaps in (4) is obeyed.Now we define the group QCA ( X ) to be the group of coherent families over X subject to theequivalence relation α ≡ β if there exists a k so that for k ≥ k , α k = β k i.e. sequences areequivalent if they agree on a terminal segment. Notice that all finite compositions are defined and(up to equivalence) act with arbitrarily small range.We also consider finite depth quantum circuits fdqc ( X ) on X whose depth is O ( ) and not allowed to diverge as the degrees of freedom become more closely packed in X and the sequenceindex i approaches infinity. When fdqc ( X ) is similarly defined as terminal equivalence classes offamilies of circuits, with each family having range of the gates in the circuit bounded by R / i forsome i , we have(7) fdqc ( X ) (cid:67) QCA ( X ) and as proved in [6, 11] the quotient is again abelian:(8) Q ( X ) = QCA ( X ) / fdqc ( X ) Theorem 3.3 will allow us to complete the circle and remove the “bars” which we added formathematical precision. It says that if the initial R is small w.r.t. the handle structure on X , then HE GROUP STRUCTURE OF QUANTUM CELLULAR AUTOMATA 15 R -sequences exist and are unique in a very strong sense. But first we state a purely topologicalresult.Given a d -dimensional Riemannian manifold ( X , g ) , possibly with boundary and possibly non-compact, define a map p : X ! X to be a pleating if it is a finite (at most d + basic pleatings which are defined to be smooth maps p i : X ! X so that for every point in thesource smooth local coordinates in source and target can be found so that the map takes on one ofthree forms:(1) rank = d , in local coordinates about (cid:126) x = ( , . . . , ) , p ( x , . . . , x d ) = ( x , . . . , x d ) (2) fold, p ( x , . . . , x d ) = ( x , x , . . . , x d ) , rank = d − p ( x , . . . , x d ) = ( x + x x , x , . . . , x d ) , rank = d − p = p n ◦ · · · ◦ p .Given p , let X p be the graph of p ; X p = { ( x , p ( x ) } ⊂ X × X . Of course π : X p ! X is a diffeo-morphism. Our interest in X p comes from the fact that it inherits a piecewise-smooth Riemannianmetric g p from the second projection: g p = π ∗ ( g ) . The next theorem says that we may choose p so that the first projection π : X p ! X is coarsely compressive. Theorem 3.1 (Manifold pleating) . Let ( X g ) be a Riemannian manifold of dimension d ≥ , possiblywith boundary and possibly non-compact. For every ε , ε (cid:48) > there is a pleating p : X ! X so that dist g ( x , x ) ≤ ε dist g p (( x , p ( x )) , ( x , p ( x ))) , whenever dist g ( x , x ) ≥ ε (cid:48) .The next theorem expresses the fact that the set of pleatings is a very well-behaved functionspace, and well-adapted for application to parameter families. Theorem 3.2.
The space of pleatings homotopic to the identity id X (the topology is either thecompact open topology w.r.t. C or any C k ) is both contractible and locally contractible. Pleatingsmay be constructed pointwise near the identity and also relative to the identity on any open subsetof X .Proof. The allowed singularity types 1) immersion 2) fold and 3) cusp are sufficiently flexiblethat the space of these maps satisfies Gromov’s h -principle [3, 7] without dimension restriction(compare this to the case of immersions alone in which according to Smale-Hirsch theory [12]the index of handles must be strictly less than the ambient dimension of the immersion.) Asa result the space of such maps has the same weak homotopy type as the corresponding func-tion space: { maps X ! X pointwise close to the identity } which is clearly contractible and locallycontractible. (cid:3) Before giving the proof of Theorem 3.1, which inducts on a handle decomposition, it is good tosee how it works for X an interval, where no handle decomposition is needed. In this case p canbe taken to have the graph indicated in Fig. 3.Remark: In applications it is useful that the pleats can be chosen large, (cid:29) R , so when a QCAis pulled back under ( p i ) − we see an isomorphic copy restricted to a large open set of X p . Wewill not make this completely explicit (it is more tedious than illuminating) but merely note, aswe set up the handle by handle pleating, that the size of the handles should be large relative to theQCA radius R . The constants we give are comfortable overestimates. We choose a fine handledecomposition of X , sufficiently coarse w.r.t. the QCA radius R so that the distance betweendisjoint handles ≥ · d R and handles are metrically convex with diameters in the range 100 · d R to 1000 · d R .The reason for the 2 d factors is that while we increase control in the co-core directions of ahandle, h i of index i , by compressing these directions, this comes at the cost of stretching, say by x = [ , ] x = [ , ] ε ≈ maximum slope ε (cid:48) ≈ radius of curvaturenear critical pointsF IGURE
3. Graph p , a basic 1D pleating.a factor of 2, parts of the higher index j , j > i , handles which attach to h i . It is important that thecumulative effect of these stretchings does not increase the range R of the QCA so much that thehandle structure fails to be (approximately) preserved. The key is point is that at any time duringthe handle by handle modifications of the QCA its action on operators supported near the centralregion of any handle core be mapped to operators supported well within that handle. Proof sketch of Theorem 3.1.
The proof is visual and many pictures are now supplied. The firststep is to shrink distances with h , the zero handles. This is done by radially compressing each0-handle toward its origin, see Fig. 4. 2-handle1-handle2-handle0-handle 0-handleF IGURE ( d − ) -cocore direction. In co-core directions where pleating does not shrink distances, we relyon radial compression as in the 0-handle case. Pleating means the differential becomes rank d − S d − embedded transverse to the core of the one handle. There arecusp singularities along an equatorial S d − subspheres. In the d = ( S , S ) pairs;the map is indicated in Fig. 5, and Fig. 3 illustrates the pleating along the core in the 1D case, e.g. X ∼ = S . HE GROUP STRUCTURE OF QUANTUM CELLULAR AUTOMATA 17 core 1-handleattaching regions(a) arcs are folds, dots are cusps pre-image1-handle (b) This projection indicates p restricted tothe core, the folds are gradually reducedto nothing as one moves parallel to thecore on either sideF IGURE d = d -dimensional k -handle, i.e. the pair ( D k × D d − k , ∂ D k × D d − k ) isto pleat in the core ( D k ) direction with (fold locus, cusp locus) a sequence of embedded pairs ( S d − , S d − ) , for k = ( S k − × S d − k , S k − × S d − k − ) , for k ≥ ( D d − k ) direction. (cid:3) Theorems 3.1 and 3.2 guide the stabilizations necessary to prove:
Theorem 3.3.
Given α = α , an R-QCA on a d-dimensional Riemannian manifold X , for suffi-ciently small R there is a deformation of α through QCA of range ≤ R to α with range ≤ R (orany other specific fraction such as Rn of R). Similarly, if α t is a k-parameter family, t ∈ D k , of QCAwith range ≤ R for all t and range ≤ Rn for t ∈ ∂ D k , then there is a deformation continuous in theentire t-family and the identity near ∂ D k which reduces the range to Rn over the entire family. Ifwe do not require the deformation to be the identity near ∂ D k the range can be reduced to Rm overthe entire family for any m.Proof sketch. The key idea is that all pleats discussed in Theorem 3.1 and its proof can be simulatedby introducing, via stabilization, an even 2 j number of parallel sheets of X , with a copy of α on odd pairs of fold circles(no cusps when d =
2) attaching region2-handle(a) Arotate(b) Illustration should be rotated around the z -axisand then vertically projected to pleat a 2-handleF IGURE S (cid:48) × I region F IGURE α − on even numbered copies. This is accomplished by writing α ◦ α − = id on the odd sheets and simply id on the even sheets (stabilization) and then using thetechnique of Fig. 8 to shift α to the even sheets. Finally the precise combinatorics of any pleatingcan then be simulated by judiciously returning bits of α to the odd sheets where there should not be pleating. Finally cancel α with α − and destabilize.Cancellation relies on the fact that α ⊗ α − is a quantum circuit[2, 14] of depth O ( ) and gaterange bounded by R . This circuit can be truncated at the folds of the pleats, giving α ⊗ α − on apair of pleats and Id outside the pleats (and some circuit near the fold; the details do not matter).This point is explained by Theorem 2.3 and its proof in [6], and is illustrated in Fig. 8 in the caseof a single 1D zigzag.The parametric statements in Theorem 3.3 follow from the parametric properties of pleatingdescribed in Theorem 3.2. (cid:3) Theorem 3.3 allows us both to implement the sequence definitions of QCA and also to see thatit is redundant. First given α , a R -QCA on X , we can use Theorem 3.3 to construct a sequence HE GROUP STRUCTURE OF QUANTUM CELLULAR AUTOMATA 19 α X = R in illustration α ◦ α − α α − α α X X αα − α g F IGURE { α = α , α , . . . } ∈ QCA ( X ) . But now suppose α is the mother of another sequence { α , α (cid:48) , . . . } ∈ QCA ( X ) . There is a continuous path of R n -QCA, α tn , joining α n to α (cid:48) n , α n = α n and α n = α (cid:48) n obtained by concatenating the deformation of α to α n with the inverse of the deformation from α to α (cid:48) n . Theorem 3 . α n to α (cid:48) n whichnever increases the range R beyond R n − . Similarly, given two mother QCA α , β connected by acontinuous path of R -QCA, there is a continuous path of R n -QCA from α n to β n .Thus the different terminal sequences with mother α and range R n are all equivalent: the se-quence, in fact, provides no new information and the naive definitions based on individual QCAand bounded depth quantum circuits are essentially correct (given the existence of descendant se-quences and their canonical nature).The official definition of Q ( X ) = QCA ( X ) / fdqc ( X ) goes through sequences, hence the bars, butwe have seen the bars offer no new information when compared with the naive definitions. Thenaive definition may well be criticized on the ground that long composition eventually span the“separate scales” and are not well defined. But they become well defined when the existence anduniqueness of coherent sequences is added to the discussion.In this discussion we have used 1-parameter families of coherent refinements to achieve well-defined group compositions of arbitrary length. Higher k -parameter families of refinements showthe group law is also continuous when applied to ( k − ) parameter families of group elements. Wedo not explore the implications of higher families here. Remark on Pleating Depth:
The pleating trick is an example of the Eilenberg Swindle. Aswe have explained, each step of the allowed deformations (1-4) is allowed under the definition ofcoherence. However, it worth emphasizing that one further application of the Swindle shows that,so long as we impose no a priori bound on the density of ancillas, the depth of the resulting circuitis also O ( ) . Suppose that α and α n are connected by a long sequence of QCA { α , . . . , α n } witheach α i + derived from α i by a fdqc. While at first glance the path from α to α n seems to requiredepth O ( n ) , using the swindle α and α n in fact can be connected by a depth O ( ) circuit. Aninitial circuit of depth O ( ) can be created from the identity as ( α − ⊗ α ) ⊗ ( α − ⊗ α ) ⊗ · · · ⊗ ( α − n − ⊗ α n ) Regrouping and canceling using a second depth O ( ) circuit yields α − ⊗ ( α ⊗ α − ) ⊗ ( α ⊗ α − ) ⊗ · · · ⊗ α n α − ⊗ α n which can then be used to pass from α to α ⊗ ( α − ⊗ α n ) to α n , all with depth O ( ) .Thus the 4 operations (lines 6) suffice to define a single-shot equivalent relation on QCA ad-equate for all our theorems, provided an unbounded ancilla density is permitted. If it is not, thealternative definition via deformation is to be performed.We conclude this section with a comment on normality. When deforming a quantum circuit oreven a QCA there is no difference between allowing deformation at the beginning, middle or endof the circuit (or QCA). For example, suppose we wish to insert a swap s (itself deformable fromidentity) between α and α − we can do this in three equivalent ways:(10) ( α s α − ) αα − = α s α − = αα − ( α s α − ) For this reason we have not been explicit in the main text as to where deformation occurs.
Product of Families Equals Family of Product:
Theorem 3.3 also implies another result: if α , β , γ are coherent families, with α ◦ β = γ , then, using ancillas, there is a continuous path from α n ◦ β n to γ n . To prove this, note that there is a continuous path of R n -QCA from ( α n ◦ β n ) ⊗ Id to α n ⊗ β n , where we have added one ancilla degree of freedom for each physical degree of freedom.Then, there is a deformation from α n ⊗ β n to α ⊗ β , a deformation from α ⊗ β to γ ⊗ Id, andfinally a deformation from γ ⊗ Id to γ n ⊗ Id. Composing these formations and applying theorem3.3 gives the desired result.4. C
OHERENT F AMILIES U SING C IRCUITS
The previous section showed how to construct a coherent family given a mother QCA. Theorem3.3 then showed that given two path coherent familes, α , β with α , β , connected by a path of R -QCA, then α n , β n were connected by a path of R n -QCA. In this section we show a similar resultfor circuit-coherent families.First let us show the claim that a circuit with range bounded by R (or more generally the forwardlightcone of every qudit having diameter bounded by R ) can be written in depth d + O ( R ) , assuming an appropriate handle decomposition on a d -dimensional control space.To show this, decompose the qudits into disjoint sets S , S , . . . , S d (in this case, each set S j willrepresent the qudits in the union of j -handles). Let V j be the unitary implemented by the product ofgates in the quantum circuit which are supported on S j and which are not in the forward lightconeof any S k for k > j ; here we take the product of gates in the obvious order. Then, if the circuitimplements unitary U we have U = V d V d − . . . V . Assuming each S j is a union of sets of diameter O ( R ) , with distance between the sets larger than 10 R , then each V j can be regarded as a one-roundquantum circuit using gates of diameter O ( R ) . In this subsection, then, we will assume such ahandle decomposition exists and treat all notions of range for a quantum circuit equivalently.First, we emphasize that the construction of coherent families in the previous section gives acircuit-coherent family since each deformation can be implemented by circuits or stabilization.Let us say that two QCAs α , β are “equal up to a circuit of depth D and range R ” if α ◦ β − isequal to such a circuit. This definition raises the following question: suppose α , β are two circuitcoherent families, with α , β equal up to a circuit of depth O ( ) and range O ( R ) . Is it the case that HE GROUP STRUCTURE OF QUANTUM CELLULAR AUTOMATA 21 β n , α n are equal to a circuit of depth O ( ) and range O ( R n ) ? A second question involves products:suppose α , β , γ are three circuit coherent families, with α ◦ β equal to γ up to a circuit of depth O ( ) and range O ( R ) . Is it the case that α n ◦ β n is equal to γ n up to a circuit of depth O ( ) andrange O ( R n ) ?In this section we answer both questions affirmatively. Note first that in both cases it is straight-forward to show a slightly weaker equivalence up to a circuit of depth O ( ) and range O ( R ) , as wesketch in the next two paragraphs. We then prove a lemma 4.1 which shows that this equivalenceimplies equivalence up to a circuit of depth O ( ) and range O ( R n ) . The key point which allows usto apply lemma 4.1 is that even though the circuit may use gates of range O ( R ) , the circuit acts asa QCA of much smaller range, O ( R n ) . For example, if α n , β n are equal up to a circuit and α n , β n both have range R n , then the circuit acts as a QCA with range at most 2 R n .Consider first the first question above, equivalence of β n and α n . Note first that α is O ( R ) -circuit equivalent to α n by definition of a circuit equivalent family: α i and α i + are O ( R i ) -circuitequivalent, and summing over i this gives an equivalence by a circuit with lightcone bounded by O ( R ) . Similarly, β is O ( R ) -circuit equivalent to β n and α is O ( R ) -circuit equivalent to β byassumption. Combining these equivalences gives the equivalence between α n and β n . Of course,the equivalence between α and α n may involve stabilization or destabilization, and similarly forthe equivalence between β , β n ; if the resulting circuit from α n to β n uses ancillas, these may beremoved using the techniques of section 2 given Assumption 2.4.For the second question, α n is O ( R ) -circuit equivalent to α ; let µ be the QCA correspondingto this circuit. Similarly, β n is O ( R ) -circuit equivalent to β ; call the corresponding QCA ν . Then, α n ◦ β n = ( µ ◦ α ) ◦ ( ν ◦ β ) . However, α ◦ ν = ν (cid:48) ◦ α where ν (cid:48) = α ◦ ν ◦ α − is a quantumcircuit of range O ( R ) . So, α n ◦ β n is equivalent to γ up to a circuit of range O ( R ) ; combining withthe equivalence between γ and γ n gives the desired result.Finally we prove the lemma. Lemma 4.1.
Let C be some circuit, with the range of the lightcone of the circuit bounded by R, forsome sufficiently small R. Assume that the circuit acts as a QCA with range r (cid:28)
R. Then, thereexists a circuit C (cid:48) that implements the same unitary as C, with the range of the lightcone of C (cid:48) bounded by r.Proof.
Let α C denote the QCA corresponding to circuit C . We will apply the pleating constructionabove to construct a pleated QCA α C P . Previously, we applied the pleating construction to reducethe range of a QCA. However, the pleating construction also reduces the range of the gates in thecircuit: each occurrence of α C or α − C in the pleating construction away from the folds can bedecomposed into gates of C or C − , and the deformation of the pleat reduces the range of thesegates. With this decomposition there is a “pleated circuit” C P which corresponds to QCA α C P .We construct C P as follows. For each occurrence of α or α − in the pleating construction (i.e.,for each sheet), we decompose the corresponding circuit C or C − as a product C near ◦ C f ar where C near includes gates near the folds, i.e., within distance O ( R ) of the folds, and C f ar includes theremaining gates far from the folds which are not in the forward lightcone of U near . Then we write C P = C P , near ◦ C P , f ar where C P , f ar is obtained in the obvious way from the gates in each C f ar , i.e.,for each sheet in the pleat, we insert the gates of C f ar for that sheet. To construct C P , near , recallthe construction near the folds for an arbitrary QCA α . The product α ⊗ α − near the fold isa quantum circuit, as we write α ⊗ α − = (cid:16) ( α ⊗ Id ) ◦ Swap ◦ ( α − ⊗ Id ) (cid:17) ◦ Swap, where Swap swaps degrees of freedom in the two tensor factors. The operator (cid:16) ( α ⊗ Id ) ◦ Swap ◦ ( α − ⊗ Id ) (cid:17) is a circuit for any α , as is Swap, so this gives the circuit near the folds.We will choose the pleats so that the range of the lightcone in C P is bounded by O ( R n ) for n large enough that R n < r . We can regard α C as the mother of a coherent family with α C P being its n -th member. Hence, since α C is a QCA of range r , by definition of a coherent family there is acircuit D of lightcone range O ( r ) from α C P to α C , i.e., D acts as α C ◦ α − C P . Finally, let C (cid:48) = D ◦ C P . (11) (cid:3)
5. F
AMILIES OF T RANSLATION I NVARIANT
QCAIn this section the control space is a torus unless noted otherwise. The set of sites will forma hypercubic lattice with L × L × · · · × L d sites. The distance between neighboring sites will be L − j along j -th direction. A translation invariant (TI) QCA is completely specified by the images ofsingle qudit operators at one site and the numbers L j ; the TI QCA is invariant under any translationby one site along any of d directions. Thus, given a TI QCA α of range R ≤ /
100 on, say, 100 d sites, the “obvious” family of a TI QCA is determined by a sequence of numbers of sites.In Section 1 we pointed out that the obvious family of a TI QCA on d -torus may not be coherent.Given a 3-dimensional TI nontrivial QCA γ of order 4, we considered an example that is a 4-dimensional TI QCA ¯ β constructed by stacking γ . We tuned the boundary conditions ( L j ) suchthat the number of γ in β i does not conform with the order of γ , rendering ¯ β incoherent. However,in this example a sub sequence of ¯ β is coherent as one can insert multiples of four layers of γ by aquantum circuit. So, if we are to make a general claim on the coherence of a TI QCA, the best weshould aim for is to show that some subsequence of the obvious family is coherent. Here we givea result in that direction.Note that there is an open possibility that there exists a TI QCA η of infinite order, i.e., η n isa quantum circuit if and only if n =
0. If such η exists, then the stack of η , which is TI, willnot contain any coherent subsequence. In addition, the hypothetical η of infinite order makes twonotions of equivalence of QCA distinct: One notion of equivalence is by quantum circuits that weare considering in this paper, and the other is by “blending” [6] — two QCAs α and β of range R blend between disjoint regions A and B if there is an interpolating R -QCA that agrees with α on A and β on B . The circuit equivalence implies the blending equivalence. A stack of someQCA always blends with the identity QCA by terminating the stack, but the full stack of η thathas infinite order cannot be a circuit. The potential distinction between blending equivalence andcircuit equivalence has a footprint in the coherence of TI QCA as we will see now. Definition 5.1.
Let I = [ − , ] and S = { ( x , y ) ∈ R : x + y = } . They admit an orientation-reversing diffeomorphism R by interchanging x with − x. If α is a QCA on X = X (cid:48) × I or X = X (cid:48) × S where the geometry of sites is invariant under R acting on I or S , we define α ∗ to be R ◦ α ◦ R , called one-coordinate inversion of α . Note that for any TI QCA α , the product α ⊗ α ∗ blends into the identity: Just project S in T d down to I by ( x , y ) x . If the blending equivalence implied circuit equivalence, then we wouldalways have α ∗ ∼ = α − .In the pleating construction for the coherent family of QCA starting with a mother TI QCA,specialized to the situation where the pleats are all parallel to a coordinate axis, one half of the HE GROUP STRUCTURE OF QUANTUM CELLULAR AUTOMATA 23 (a) 𝛽 (b) 𝛼(ℓ)𝛼(ℓ) 𝛼 2ℓ −1 𝛼(ℓ)𝛼 𝐿 𝛼 𝐿+𝑘ℓ−1 (c) F IGURE
9. Construction of translation invariant pleats. Assuming that the dimen-sionally reduced QCA has finite order in the group of QCA modulo fdqc, the k -thpower of (a) is trivial for some k < ∞ . This implies that k -th power of (b) is trivial.In the specific pleating of (c), β appears exactly k times, implying the coherence ofthe obvious family of translation invariant QCA.added degrees of freedom is acted on by ( α − ) ∗ = : α −∗ and the other half by α , except for “cusps”of the pleats. If the obvious family of some TI QCA is coherent, then, intuitively, α −∗ has tomatch α up to a finite depth quantum circuit. Since α −∗ blends with α , if the blending equivalenceimplied circuit equivalence, then we may expect that the obvious family of TI QCA would alwaysbe coherent. Not knowing this, we instead prove the following theorem, which still bears aninteresting corollary. Theorem 5.2.
Let ¯ α = { α L } on L d sites (L = L , L + , L + , . . . ) be the obvious family of a TIQCA on an d-torus. Here, L is so large that the range of α is at most / of the linear size ofthe torus. Using the same local action as ¯ α , for j = , , . . . , d we define α ( (cid:96), j ) = { α L ( (cid:96), j ) } onL j − × (cid:96) × L d − j sites to be another obvious family of TI QCA on an ( d − ) -torus, obtained bydimensionally reducing in the j-th coordinate. If for any j and for any sufficiently large (cid:96) , α ( (cid:96), j ) has finite order in the group of QCA modulo finite depth quantum circuits, then a subsequence of ¯ α is coherent.Proof. We claim that there is a deformation α L into α L (cid:48) by stabilization and finite depth quantumcircuits where L (cid:48) − L is any multiple of some fixed (cid:96) . We do this by increasing the number of sitesalong one coordinate after another. This is enough for the theorem.Choose (cid:96) such that it is at least 10 times larger than the range R of α L . Let α ( (cid:96), j ) have order k (cid:48) < ∞ and let α ( (cid:96), j ) have order k (cid:48)(cid:48) < ∞ . Then, both α ( (cid:96), j ) and α ( (cid:96), j ) become trivial at the k -th power where k ≤ k (cid:48) k (cid:48)(cid:48) . We will prove the claim with (cid:96) = k (cid:96) .Consider a control space X = “8” × T d − . The figure “8” is the j -th direction along which thereare exactly 2 (cid:96) sites, and T d − is a torus. For clarity of presentation, let us distinguish the one circlethat goes around the outer rim of the figure “8” from the two disjoint circles in the inner rims. Let α ( (cid:96), j ) − be defined along the outer rim, and two α ( (cid:96), j ) along the inner rims. See Figure 9(a). Then, except for the O ( R ) -neighborhood of the middle crossing point of the figure “8”, thecomposed action of the three QCA’s is the identity. Hence, we are left with a TI QCA ¯ β on the ( d − ) -torus at the middle point of the figure “8”. See Figure 9(b). Since α ( (cid:96), j ) k ∼ = Id and α ( (cid:96), j ) k ∼ = Id, we know ¯ β k ∼ = Id.Now, bring one α L and k copies of α L ( (cid:96), j ) that are laid adjacent to α L . See Figure 9(c). Wrapthe arrangement with α − on L j − × ( L + k (cid:96) ) × L d − j sites. We see that there is β L at each of the k singular points where ( d − ) -torus is, which are jointly trivial. (cid:3) Corollary 5.3.
For any d = , , , . . . , every obvious family of d-dimensional TI Clifford QCAwith prime dimensional qudits contains an infinite coherent subsequence. (A Clifford QCA is onethat maps every Pauli matrix to a tensor product of Pauli matrices.) Every obvious family of -dimensional TI QCA contains an infinite coherent subsequence.Proof. In any dimension, every TI Clifford QCA with prime dimensional qudits has order 1, 2, or4 [10]. Every 2-dimensional QCA has order 1 [6]. (cid:3)
Remark: The above argument applies verbatim to any “invertible states” [5, 13] that appear inmathematical many-body physics. One can consider an equivalence relation of the states by finitedepth quantum circuits where each gate is G -symmetric for some (possibly trivial) group G . Then,any translation invariant invertible state is coherent if it becomes finite order upon dimensionalreductions (compactification along any one direction). In more physical terms, such a state isautomatically an entanglement renormalization group fixed point. Similarly, the construction ofsection 3 can be applied to any such invertible state.A PPENDIX
A. QCA A
RE AN A BELIAN G ROUP M ODULO C IRCUITS
Here we recall the proof that QCA are an abelian group modulo circuits[6, 11] . The argumentis summarized in Fig. 10 below. ≡ = = α α α α α α α α F IGURE H = ⊗ i H over { x i } ⊂ X (that is, X is imagined normal to thefigure) and the stacking is composition of QCA, α ◦ α . The line to its right represents doubling thedegrees of freedom by introducing for every H i at x i an identical ancilla at the same location. The ≡ sign represents equality in the quotient group Q . The crossings (under/over has no significancehere) are our notation for the swap operators which at x i swaps the local Hilbert space H i withits ancillary partner. Notice that a swap is a fdqc of depth one, so may be added at will withoutchanging the element in Q . At the far right we have the manifestly abelian group structure. HE GROUP STRUCTURE OF QUANTUM CELLULAR AUTOMATA 25 R EFERENCES [1] Miguel Aguado and Guifre Vidal,
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ICHAEL F REEDMAN , M
ICROSOFT R ESEARCH , S
TATION Q, AND D EPARTMENT OF M ATHEMATICS , U
NIVER - SITY OF C ALIFORNIA , S
ANTA B ARBARA , S
ANTA B ARBARA , CA 93106,
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ICROSOFT Q UANTUM AND M ICROSOFT R ESEARCH , R
EDMOND , WA 98052, USA
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