Path-integral solution of the one-dimensional Dirac quantum cellular automaton
Giacomo Mauro D'Ariano, Nicola Mosco, Paolo Perinotti, Alessandro Tosini
aa r X i v : . [ qu a n t - ph ] J un Path-integral solution of the one-dimensional Dirac quantum cellularautomaton
Giacomo Mauro D’Ariano a,b , Nicola Mosco a , Paolo Perinotti a,b , Alessandro Tosini a a QUIT group, Dipartimento di Fisica, via Bassi 6, Pavia, 27100, Italy b INFN Gruppo IV, Sezione di Pavia, via Bassi 6, Pavia, 27100, Italy
Abstract
Quantum cellular automata have been recently considered as a fundamental approach to quantum field theory,resorting to a precise automaton, linear in the field, for the Dirac equation in one dimension. In such linearcase a quantum automaton is isomorphic to a quantum walk, and a convenient formulation can be given interms of transition matrices, leading to a new kind of discrete path integral that we solve analytically in termsof Jacobi polynomials versus the arbitrary mass parameter.
Keywords:
Quantum cellular automata; Quantum walks; Dirac quantum cellular automaton; Discretepath-integral.
1. Introduction
The simplest example of discrete evolution ofphysical systems is that of a particle moving on a lat-tice. A (classical) random walk is exactly the descrip-tion of a particle that moves in discrete time steps andwith certain probabilities from one lattice position tothe neighboring positions. This is a special instanceof a more general discrete dynamical model knownas cellular automaton introduced by von Neumann[1].A quantum version of the random walk, called quantum walk (QW), was first introduced in [2]where measurements of the z -component of a spin-1 / / coin system,with the QW representing a discrete unitary evolu-tion of a particle state with the internal degree of free-dom given by the spin [3]. In the most general casethe internal degree of freedom at a site x of the latticecan be represented by an Hilbert space H x , and thetotal Hilbert space of the system is the direct sum ofall sites Hilbert spaces. The quantum walk is also aspecial case of a quantum cellular automaton (QCA) [4], with cells of quantum systems locally interact-ing with a finite number of other cells via a unitaryoperator describing the single step evolution. QWsprovide the one-step free evolution of one-particlequantum states, whereas QCAs more generally de-scribe the evolution of an arbitrary number of par-ticles on the same lattice. However, replacing thequantum state with a quantum field on the lattice, aQW describes a QCA linear in the field, correspond-ing to the discrete evolution of non interacting par-ticles with a given statistics–a “second quantization”of the QW–which then can ultimately be regarded asa QCA. This is what we call field QCA in the presentpaper.Both QCAs and QWs have been a subject ofinvestigation in the fields of computer-science andquantum information, where these notions have beenmathematically formalized and studied extensivelyin Refs. [5, 6, 7, 3, 8, 9]. The interest in this modelswas also motivated by the use of QWs in designinge ffi cient quantum algorithms [10, 11, 12, 13].The first attempt to mimic the Feynman path-integral in a discrete physical context is the Feyn-man chessboard problem [14] that consists in find-ing a simple rule to represent the quantum dynam- Preprint submitted to Physics Letters A June 5, 2014 cs of a Dirac particle in 1 + ff erence Dirac equation for a fixed value of themass. However, such finite-di ff erence equation hasno corresponding QW. More recently, following thepioneering papers [18, 19, 20], a discrete model ofdynamics for a relativistic particle has been consid-ered in a QWs scenario [21, 22, 23, 24, 25, 26, 27,28, 29, 30].In Ref. [3] Ambainis et al. gave two generalideas for analyzing the evolution of a walk. Oneidea consists in studying the walk in the momentumspace providing both exact analytical solutions andapproximate solutions in the asymptotic limit of verylong time. The other idea is using the discrete path-integral approach, expressing the QW transition am-plitude to a given site as a combinatorial sum overall possible paths leading to that site. Ref. [3] pro-vides a path-sum solution of the Hadamard walk (theHadamard unitary is the operator on the coin system),while Ref. [31] gives the solution for the coined
QW,with an arbitrary unitary acting on the coin space.The same author considered the path-integral formu-lation for disordered
QWs [32] where the coin uni-tary is a varying function of time.In this work, we consider the Dirac automaton thathas been derived in Refs. [25, 26] from basic prin-ciples about the topology of interactions (unitarity,linearity, locality, homogeneity, isotropy). Such au-tomaton, which is not a coined QW, gives the usualDirac equation in the relativistic limit of small wave-vectors for lattice step at the Planck scale. After re-viewing the one dimensional Dirac automaton, wesolve analytically the automaton in the position spacevia a discrete path-integral. The discrete path cor-responds to a sequence of the automaton transitionmatrices, which are closed under multiplication. Ex-ploting this feature we derive the analytical solutionfor arbitrary initial state.
2. The one-dimensional Dirac QCA
The Dirac QCA of Refs. [25, 26] describes the one-step evolution of a two-component quantum field ψ ( x , t ) : = ψ R ( x , t ) ψ L ( x , t ) ! , ( x , t ) ∈ Z ,ψ R and ψ L denoting the right and the left mode ofthe field. Here we restrict to one-particle states andthe statistics is not relevant, but the presented solu-tion could be extended to multi-particle state for anystatistics consistent with the evolution. In the single-particle Hilbert space C ⊗ l ( Z ), we will use the fac-torized basis | s i | x i , with s = R , L .Here the evolution of the field is restricted to belinear, namely there exists a unitary operator A suchthat the one step evolution of the field is given by ψ ( t + = U ψ ( t ) U † = A ψ ( t ). In the present case theassumption of locality corresponds to writing ψ ( x , t +
1) as linear combination of ψ ( x + l , t ) with l = , ± A of the form A = A R ⊗ T + A L ⊗ T − + A F ⊗ I , A R = n
00 0 ! , A L = n ! , A F = m i m ! , (1) T = X x ∈ Z | x + ih x | , where A R , A L , A F are called transition matrices , and n + m = , n , m ∈ R + .
3. Path-sum formulation of the Dirac QCA
After t steps one has ψ ( t ) = A t ψ (0), and due tolinearity the field ψ ( x , t ) is a linear combination ofthe field at the points ( y ,
0) in the past causal cone of( x , t ). Each point ( y ,
0) is connected to ( x , t ) in t timesteps via a number of di ff erent discrete paths. Ac-cording to Eq. (1) at each step of the automaton thelocal field ψ ( y ,
0) undergoes a shift T l , l = , ±
1, withthe internal degree of freedom multiplied by the cor-responding transition matrix A h , with h ∈ { R , L , F } .A generic path σ connecting x to y in t steps is con-veniently identified with the string σ = h t h t − . . . h of transitions, corresponding to the overall transitionmatrix given by the product A ( σ ) = A h t A h t − . . . A h , (2)2nd summing over over all possible paths σ and allpoints ( y ,
0) in the past causal cone of ( x , t ), one has ψ ( x , t ) = X y X σ A ( σ ) ψ ( y , . (3)We now evaluate the sum over σ in Eq. (3) analyti-cally versus x , y , t . Upon denoting by r , l , f the num-bers of R , L , F transitions in σ , respectively, using t = r + l + f and x − y = r − l , one has r = t − f + x − y , l = t − f − x + y . (4)The analytical solution is then evaluated observingthat the overall transition matrix P σ A ( σ ) in (2) canbe e ffi ciently expressed by encoding the transitionmatrices as follows A R = nA , A L = nA , A F = im ( A + A ) (5) A = ! , A = ! , (6) A = ! , A = ! , (7)with matrices A ab satisfying the composition rule A ab A cd = + ( − b ⊕ c A ad , (8)where ⊕ denotes the sum modulo 2. It is now con-venient to denote by σ f = h t , . . . h a generic pathhaving f occurrences of the F -transition, and writeEq. (3) as follows ψ ( x , t ) = X y t −| x − y | X f = X σ f A ( σ ) ψ ( y , . (9)In a path σ f the F transitions identify f + τ F τ F . . . . . . F τ f + , (10)where τ i denotes a (possibly empty) string of R and L . According to Eq. (2) the general σ cannot containsubstrings of the form h i h i − = RL , h i h i − = LR , (11) h i h i − h i − = RFR , h i h i − h i − = LFL , (12)which give null transition amplitude. Therefore, ac-cording to Eq. (11) each τ i in Eq. (10) is made only of equal letters, i.e. τ i = hh . . . h , with h = R , L .On the other hand Eq. (12) shows that two consecu-tive strings τ i and τ i + must be made with di ff erent h .This corresponds to having all τ i = hh . . . h and all τ i + = h ′ h ′ . . . h ′ , with h , h ′ . In the following wewill denote by Ω R and Ω L the sets of strings having τ i + = RR . . . R and τ i + = LL . . . L , respectively,for all i .The above structure for strings σ f can be exploitedto determine the matrix A ( σ f ). We consider sepa-rately the cases of f even and f odd. For f even onehas A ( σ f ) = α ( f ) A + A , f = tA , f < t , σ f ∈ Ω R , A , f < t , σ f ∈ Ω L , (13)while for odd f one has A ( σ f ) = α ( f ) A + A , f = tA , f < t , σ f ∈ Ω R , A , f < t , σ f ∈ Ω L , (14)with the factor α ( f ) given by α ( f ) : = (i m ) f n t − f . (15)According to Eqs. (13) and (14) we can finally re-state Eq. (9) as ψ ( x , t ) = X y X a , b ∈{ , } t −| x − y | X f = c ab ( f ) α ( f ) A ab ψ ( y , , (16)where c aa (2 k + = c (2 k ) = c (2 k ) =
0. The coef-ficients c ab ( f ) count the occurrences of the matrices A ab in the transition matrices of all paths σ f , and aregiven by c ab ( f ) = µ + − ν f − − ν ! µ − + ν f − + ν ! , (17) ν = ab − ¯ a ¯ b , µ ± = t ± ( x − y ) − , where ¯ c : = c ⊕
1, and the binomials are null for non in-teger arguments. The expression of c ab is computedvia combinatorial considerations based on the struc-ture (10) of the paths, and on Eqs. (13) and (14). Let3s start with coe ffi cients c and c . The matrices A and A appear only for f even (see Eq. (13))in which case one has f + odd strings τ i + and f even strings τ i . A appears whenever σ f ∈ Ω R ,namely when the R -transitions are arranged in thestrings τ i + . This means that we have to count in howmany ways the r identical characters R and l identi-cal characters L can be arranged in f + and f strings,respectively. These arrangements can be viewed ascombinations with repetitions which give c ( f ) = f + rr ! f + l − l ! = t + x − y f ! t − x + y − f − ! , where the second equality trivially follows fromEq. (4). Similarly A appears whenever σ f ∈ Ω L which gives c ( f ) = f + ll ! f + r − r ! = t − x + y f ! t + x − y − f − ! . Consider now the other two coe ffi cients c and c counting the occurrences of A and A . The lastones appears only when f is odd (see Eq. (13)) andthen one has the same number f + of odd strings τ i + and even strings τ i . Counting the combinations withrepetitions as in the previous cases we get c ( f ) = c ( f ) = f − + rr ! f − + ll ! = t + x − y − f − ! t − x + y − f − ! , which concludes the derivation of the coe ffi cients c ab ( f ) in Eq. (17).The analytical solution of the Dirac automaton canalso be expressed in terms of Jacobi polynomials P ( ζ,ρ ) k performing the sum over f in Eq. (16) whichfinally gives ψ ( x , t ) = X y X a , b ∈{ , } γ a , b P (1 , − t ) k + (cid:18) mn (cid:19) ! A ab ψ ( y , , k = µ + − a ⊕ b + ,γ a , b = − (i a ⊕ b ) n t (cid:18) mn (cid:19) + a ⊕ b k ! (cid:16) µ ( − ) ab + a ⊕ b (cid:17) (2) k , (18)where γ = γ = γ = γ =
0) for t + x − y odd(even) and ( x ) k = x ( x + · · · ( x + k −
4. Conclusions
We studied the one dimensional Dirac automaton,considering a discrete path integral formulation. Theanalytical solution of the automaton evolution hasbeen derived, adding a relevant case to the set ofquantum automata solved in one space dimension, in-cluding only the coined quantum walk and the disor-dered coined quantum walk. The main novelty of thiswork is the technique used in the derivation of the an-alytical solution, based on the closure under multipli-cation of the automaton transition matrices. This ap-proach can be extended to automata in higher spacedimension. For example the transition matrices ofthe Weyl and Dirac QCAs in 2 + + Acknowledgments
This work has been supported in part by the Tem-pleton Foundation under the project ID
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