An algorithm to factorize quantum walks into shift and coin operations
AAn algorithm to factorize quantum walks into shift and coin operations
C. Cedzich, T. Geib, and R. F. Werner Quantum Technology Group, Heinrich Heine Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstr. 2, 30167 Hannover, Germany
We provide an algorithm that factorizes one-dimensional quantum walks into a protocol of twobasic operations: A fixed conditional shift that transports particles between cells and suitable coinoperators that act locally in each cell. This allows to tailor quantum walk protocols to any experi-mental setup by rephrasing it on the cell structure determined by the experimental limitations. Wegive the example of a walk defined on a qutrit chain compiled to run an a qubit chain.
In this paper we demonstrate an algorithm that fac-torizes any one-dimensional quantum walk into a finite“protocol” of shift and coin operations within a given cellstructure.Such a factorization is of fundamental importance forthe understanding of single-particle dynamics in discretetime and space, and closes an important gap betweentwo different perspectives. Only the two together givea complete understanding, and allow to decide whethera given task be achieved with available building blocks.The description of the tasks is by conditions “from with-out”. The corresponding definition of a quantum walkin this perspective is that of a one-step unitary oper-ator on a lattice system satisfying a locality condition.Many overarching results, like the topological classifica-tion of walks with symmetries [7, 8, 10, 11] are based onjust such an axiomatic characterization. On the otherhand, one may take a constructive approach to quantumwalks, defining the class of quantum walks “from within”in terms of a few available operations. This is the natu-ral approach for experimental implementation. Also theconstruction of explicit models as analogues of condensedmatter systems [4, 6, 15, 22, 25, 28, 36] and the design ofquantum walk based algorithms [1, 2, 30, 35, 39] followsthis approach.Only when the two approaches are demonstrablyequivalent one is sure that no road blocks to implementa-tion have been overlooked. The strongest way of showingthis equivalence is a compilation algorithm, which pro-duces from any abstractly given walk a factorization intoa sequence of operations. In this paper we provide sucha compilation method, breaking a general walk into twokinds of operations: Coins, rotating each cell separatelyand the conditional shift. The shift is specified by sin-gling out one basis vector in each cell and then shiftingonly these components. The shift operation and the cellstructure, with cells of dimension ≤ d < ∞ , will bearbitrary but fixed throughout.Clearly, every shift-coin protocol is a quantum walkaccording to the axiomatic definition. For the converse,only partial results exist. In the translation invariant set-ting, Fourier transformation maps any quantum walk toa finite dimensional unitary, which then can be factorizedinto shifts and coins via techniques that were originally developed for filter banks [16, 23, 40]. Another knowntechnique provides a factorization using a grouping of thesystem into sufficiently large cells [18]. We actually usethat technique below as one step of our algorithm, butadditional work is required, when the shift capabilitiesand with it the cells are fixed.A shift-coin decomposition is required for a number oftheoretical tasks. Firstly, to couple walks to gauge fields:Such fields are implemented as commutation phases ofthe shift operators [9, 12, 13, 17]. Secondly, one wouldlike to consider a walk as the one-particle sector of aninteracting system, known abstractly as a quantum cel-lular automaton (QCA) [3, 14, 34]. The best constructionfor this [40] uses exactly a decomposition as provided inour paper. The analogy between walks and QCAs was aguiding idea in [18], so that one can hope for an extensionof our results to the QCA setting.Being able to factorize any local unitary is highlydesirable also from an experimenter’s perspective sinceshift and coin operations are implementable in variousplatforms such as neutral atoms in optical lattices [20],trapped ions [31, 41], light-pulses in optical fibres [32, 33]and photonic waveguide arrays [26, 29]. On the otherhand, our algorithm provides a concrete method to adapta given quantum walk to the experimental set-up at hand.We demonstrate this by compiling a quantum walk withthree-dimensional coins [5, 19, 38] for systems where theexperimental setup provides only qubit cells. The systems.
We consider the discrete time dynamics of single par-ticles, so-called quantum walks , on the 1D lattice Z “fromwithout”. These systems are described by a unitary op-erator W subject to a locality condition on an arbitrarybut fixed cell structure H = (cid:77) x ∈ Z H x , (1)with uniformly bounded cell dimensions ≤ d x < N < ∞ . We denote the basis of such H by | x, i (cid:105) with x ∈ Z and i = 1 , . . . , d x . The locality condition is expressed asa finite upper bound L < ∞ on the interaction length, a r X i v : . [ qu a n t - ph ] F e b i.e. | x − y | > L ⇒ (cid:104) x, i | W | y, j (cid:105) = 0 . (2)Clearly, this abstract point of view is independent ofthe given cell structure, and reorganizing the cells leadsmerely to a different but still finite interaction length.Below we provide a factorization algorithm that allowsus to compile any such banded unitary W as a quantumwalk “from within”, i.e. as a sequence W = C S n C · · · S n i C i (3)of two basic operations that are determined by the cellstructure (1): The conditional shift operator S transportsthe first basis vector of each cell one cell to the right, i.e. S = , (4)and coin operators C that act locally as a d x -dimensionalunitary C ( x ) . The Algorithm.
Given some banded unitary W , fix some cell struc-ture according to (1), thereby determining the interactionlength L in (2). Our factorization algorithm consists offive steps: Step 0 - deal with non-vanishing indices.
In this preparatory step we bring the walk into a stan-dard form with zero net flow of information. This ismeasured by an integer-valued index ind ( W ) [18, 21].Important for our purpose is that the index is addi-tive, i.e. ind ( W W ) = ind ( W ) + ind ( W ) . Moreover, ind ( S ) = 1 so that the walk S − n W , with n = ind ( W ) has vanishing index. This allows us to henceforth assumethat the walks we consider have vanishing net informa-tion flow, i.e. ind ( W ) = 0 . Step 1 - decouple periodically.
Any walk W on the integers with vanishing index canbe decoupled into two half-line walks via a local decou-pling [18] , i.e. we can write W as =
11 11 , (5)where the first operator on the right acts non-triviallyonly on a block of L cells and the second one is decou-pled into a left and a right part. Due to the locality of the decoupling we can repeatthis periodically every L cells which gives = , (6)where the blocks on the right side are of different sizesince the local dimensions of the cells they entail varies.Moreover, it is important to note that these blocks acton overlapping subspaces.If arbitrary shift operations were available, this wouldessentially solve our problem. The remaining steps arenecessary to make do with just the given shift and cellstructure. Step 2 - parametrize blocks by elementary unitaries.
Each block in (6) is a D × D -dimensional unitary with D depending on the dimensions of the L cells in theblock. Every such block can be parametrized as a prod-uct of D ( D − / “elementary unitaries” [24, 27, 37] ofthe form M nm = a b c d ≡ (cid:20)(cid:18) a bc d (cid:19)(cid:21) nm , (7)which differ from the identity only in the four matrixelements at ( n, n ) , ( n, m ) , ( m, n ) and ( m, m ) . These arereplaced by the entries a , b , c and d of a unitary × matrix. We indicate the positions of these elements bythe superscript in the matrix M . With this notationevery block in (6) can be written as D − (cid:89) n =1 (cid:32) D (cid:89) m = n +1 M nm (cid:33) . (8)Importantly, the algorithm of [24, 27, 37] affects only theblock itself, and therefore can be applied simultaneouslyin each block in (6). Step 3 - factorize elementary unitaries into shifts andcoins.
The previous step reduces our task to writing each M nm as a shift-coin sequence on the given cell struc-ture. To this end, we embed M nm into H by padding itwith identities on both sides, and take the indices n, m as basis labels in the given cell structure, i.e. n = | x, i (cid:105) and m = | y, j (cid:105) .We distinguish two cases: If y = x , M nm is already acoin. Otherwise, if k = y − x (cid:54) = 0 , the factorization of M nm requires the use of the conditional shift. First, weinitialize the matrix elements of M nm in a coin C M = a bc d . (9)at y . Then, we conjugate C M with S k , which translatesthe basis element | y, (cid:105) to the correct cell at x . Last, weconjugate with a coin C that swaps basis elements in thecells at x and y according to C | x, i (cid:105) = | x, (cid:105) and C | y, j (cid:105) = | y, (cid:105) . (10)Combining these steps amounts to M nm = C † S − k C M S k C, (11)i.e. a sequence of shift and coin-operations specified by k , the swapping coin C and the coin C M at y .Let us illustrate the above by applying it to an elemen-tary unitary M on C ⊕ C . Initializing C M , shiftingby k = 1 , and swapping with C = 1I ⊕ σ gives indeed S −→ C −→ . Step 4 - assemble.
The last step is to apply step 3 in parallel in eachblock in (6). In the special case of equal cells, each blockis parametrized by elementary unitaries in the same way.Thus, we can use a common “shift-skeleton” for all blocksand we only have to adjust the coins C M and swaps C in (11) in parallel in each block. In total, this gives asequence of at most D ( D − ∼ O ( d L ) coin and shiftoperations.In the general case, the cell configuration varies fromblock to block, and with it the parametrization of eachblock into elementary unitaries. Thus, the above does notapply directly. What helps us out in this case is the uni-form bound N on the local cell dimension which impliesthat there is only a finite number of different cell config-urations of the blocks and thus only a finite number ofdifferent parametrizations (8). We further factorize theblock diagonal matrices in (6) such that each factor con-tains all blocks with the same block configuration. Forexample, if there are two cell configurations that alter-nate we write the block diagonals in (6) as = W - W W - W . (12)To each such sparse factor, we can apply Step 3 with acommon “shift-skeleton”, each contributing O ( D ) shiftand coin operations to the overall sequence. This step is crucial: without it we could still find ashift-coin factorization for every block in (6), but withouta common “shift-skeleton” we could not apply Step 3 in parallel for all blocks which would lead to an infiniteproduct.This concludes our description of the algorithm.
Example.
An important application of the above algorithm is totailor a given quantum walk to another architecture thatis determined by experimental constraints. As an ex-ample, we show how the so-called “three-state” quantumwalk discussed in [5, 19, 38] can be realized in a setupwith qubit cells. As the name suggests, this walk is de-fined on a Hilbert space with three-dimensional cells asthe shift-coin protocol W = S S † C (13)where S = S as in (4) and S shifts the third basisvector in each cell. As coin we choose for all x the three-dimensional Grover matrix C ( x ) = (cid:16) − − − (cid:17) , butthe following analysis applies with appropriate changesfor arbitrary choices of the local coins. The index of W vanishes by ind ( W ) = ind ( S ) + ind ( S † ) = ind ( S ) − ind ( S ) = 0 , such that we can directly start with Step1 . To decouple W we capitalize on the hemiolic rela-tion between the old and the new cell structure, wheretwo three-dimensional cells are interpreted as three two-dimensional cells, i.e.On the new C -cells W has interaction length L = 2 such that Step 1 gives a decoupling every L = 4 new cells. However, by the hemiolic relation between the oldand the new cell structure we can also decouple every L cells in the old cell structure where L = 1 . It thus sufficesto consider blocks that contain only 3 instead of 4 newcells. One possible periodic decoupling according to (6)results in the blocks σ σ = = [ σ ] [ σ ] (14)acting on H x − ⊕ H x ⊕ H x +1 ( x ∈ Z ) with respect tothe C -cells, and − − − − − − = , (15)acting H x ⊕ H x +1 ⊕ H x +2 .The parametrization by elementary unitaries of thefirst block can be read off directly. Applying the algo-rithm of [24, 27, 37] the second block is parametrizedby the elementary unitaries M = [ H ] , M = [ A ] and M = [ σ H ] for the first × block, and anal-ogously M = [ H ] , M = [ A ] and M = [ σ H ] for the second × block, where H = ( σ + σ ) / √ isthe Hadamard coin and A = 13 (cid:18) √ − √ (cid:19) . (16)Hence, the blocks (15) are parametrized as = H A σ H H A σ H . (17)Thus, in Step 3 we only need to consider [ σ ] on H x − ⊕ H x ⊕ H x +1 and [ A ] , [ σ H ] , [ H ] and [ A ] on H x ⊕ H x +1 ⊕ H x +2 , since the remaining elementaryunitaries are already coins. [ σ ] is realized by C M = σ at x , k = 1 and swapping coins at x − and x . The shift-coin factorizations of the elementary unitaries in (17),with k = y − x (cid:54) = 0 are parametrized by the followingdata: C M ( x ) k C ( x )[ A ] A ( x + 1) 1 σ ( x + 1)[ A ] A ( x + 2) 1 σ ( x + 1)[ σ H ] Hσ ( x ) − H ] σ Hσ ( x + 1) − Since the cells have constant dimension, in
Step 4 these block-factorisations can be performed in parallelwith a common “shift-skeleton”.
Summary and Outlook.
We provided a concrete algorithm to factorize any one-dimensional quantum walk into a finite product of shiftand coin operations on any given cell structure. Thiscloses a long-standing gap in the understanding of suchsystems, but also has practical implications: on the onehand, it allows to adapt a given walk to any experimentalset-up and, on the other, to either optimize with respectto the cell dimensions or the interaction length.An interesting direction for future work is to optimizethe length of the shift-coin protocols. One option thatjumps to the eye is to homogenize a given cell structureby “filling up” each cell by locally adding innocent by-standers until all cells have the same dimension. This,however, would violate the assumption of a given fixedcell structure.
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