Path-Integral Solution of the One-Dimensional Dirac Quantum Cellular Automaton

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Physics Letters A 378 (2014) 3165–3168

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Physics Letters A

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Path-integral solution of the one-dimensional Dirac quantum cellular automaton

a,b a a,b a, Giacomo Mauro D’Ariano , Nicola Mosco , Paolo Perinotti , Alessandro Tosini ∗ 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 a r t i c l e i n f o a b s t r a c t

Article history: Quantum cellular automata, which describe the discrete and exactly causal unitary evolution of a lattice Received 4 June 2014 of quantum systems, have been recently considered as a fundamental approach to quantum ﬁeld theory Received in revised form 23 August 2014 and a linear automaton for the Dirac equation in one dimension has been derived. In the linear case a Accepted 12 September 2014 quantum cellular automaton is isomorphic to a quantum walk and its evolution is conveniently formu- Available online 16 September 2014 lated in terms of transition matrices. The semigroup structure of the matrices leads to a new kind of Communicated by P.R. Holland discrete path-integral, different from the well known Feynman checkerboard one, that is solved analyti- Keywords: cally in terms of Jacobi polynomials of the arbitrary mass parameter. Quantum cellular automata © 2014 Elsevier B.V. All rights reserved. Quantum walks Dirac quantum cellular automaton Discrete path-integral

1. Introduction replacing the quantum state with a quantum field on the lattice, a QW describes a QCA that is linear in the field (providing the dis- The simplest example of discrete evolution of physical systems crete evolution of non-interacting particles with a given statistics). is that of a particle moving on a lattice. A (classical) random walk This corresponds to a “second quantization” of the QW and can ul- is exactly the description of a particle that moves in discrete time timately be regarded as a QCA. This is what we call field QCA in steps and with certain probabilities from one lattice position to the present paper. the neighboring positions. This is a special instance of a cellular Both QCAs and QWs have been a subject of investigation in automaton, a more general discrete dynamical model introduced computer-science and quantum information, where the two no- by von Neumann [1]. tions have been extensively studied and mathematically formalized A quantum version of the random walk, called quantum walk (see Refs. [5–7,3,8,9]). The interest in these models was also moti- (QW), was first introduced in [2] where a measurement of the vated by the use of QWs in designing efficient quantum algorithms z-component of a spin-1/2particle decides whether the particle [10–13]. moves to the right or to the left. Later the measurement was re- In Ref. [3] Ambainis et al. provided two general ideas for ana- placed by a unitary operator on the spin-1/2quantum system, also lyzing the evolution of a walk. The first idea consists in studying denoted internal degree of freedom or coin system, with the QW the walk in the momentum space, providing both exact analyti- representing a discrete unitary evolution of a particle with internal cal solutions and approximate solutions in the asymptotic limit of degree of freedom given by the spin [3]. In the most general case very long time. The second idea is to use the discrete path-integral the internal degree of freedom at a site x of the lattice corresponds approach, expressing the QW transition amplitude to a given site to a Hilbert space x, and the total Hilbert space of the system is H as a combinatorial sum over all possible paths leading to that the direct sum x encompassing the Hilbert spaces of all the x site. Ref. [3] provides a path-sum solution of the Hadamard walk sites. As in the classicalH scenario, a QW is a special case of a quan- (the Hadamard unitary is the operator on the coin system), while tum cellular automaton! (QCA) [4], with cells of quantum systems locally interacting with a finite number of neighboring cells via a Ref. [31] gives the solution for the coined QW, with an arbitrary unitary operator. While QWs provide the one-step free evolution unitary acting on the coin space. The same author considered the of one-particle quantum states, QCAs can describe the evolution path-integral formulation for disordered QWs [32] where the coin of an arbitrary number of particles on the same lattice. However, unitary is a varying function of time. The first attempt to mimic the Feynman path-integral in a discrete physical context is the Feynman checkerboard problem [14] * Corresponding author. that consists in finding a simple rule to represent the quantum http://dx.doi.org/10.1016/j.physleta.2014.09.020 0375-9601/© 2014 Elsevier B.V. All rights reserved. 3166 G.M. D’Ariano et al. / Physics Letters A 378 (2014) 3165–3168 dynamics of a Dirac particle in 1 1dimensions as a discrete shown that the usual kinematics of the Dirac equation is recov- + path-integral. In Ref. [15] Kaufmann and Noyes simplify previous ered for small momenta (k 0) and small mass (m 0). The → → approaches [16,17] to the Feynman checkerboard, providing a so- Dirac limit of the automaton is not proved taking a sequence of lution of the finite-difference Dirac equation for a fixed value of automata with smaller and smaller lattice and time spacing, i.e. the mass. However, such finite-difference equations have no corre- the continuum limit given by lattice spacings and the time steps sponding QW or QCA, and generally lead to non-unitary evolutions. sent to 0, but rather fixing the automaton and computing the evo- More recently, following the pioneering papers [18–20], a discrete lution of a class of states with limited band in momentum. For model of dynamics for a relativistic particle has been considered these states the automaton dynamics and the usual dynamics of in a QWs scenario [21–30]. the Dirac equation turn out to be indistinguishable. Here we consider the unique automaton in one space dimen- It is worth noticing that the discrete model of evolution pro- sion that satisfies the following basic principles: unitarity, linearity, vided by the automaton (1) differs with respect to the one at the locality, homogeneity and invariance with respect to the symme- basis of the Feynman checkerboard. Indeed the checkerboard so- tries of the lattice. In Refs. [25,26] it has been shown that these lutions to the Dirac equation are valid for discrete physics using constraints lead to a specific automaton model which describes finite differences calculus (where one usually recovers solutions the evolution of a quantum field with two internal degrees of to the infinitesimal Dirac equation in the appropriate continuous freedom. Such a field automaton, which does not correspond to limit). On the other hand the automaton dynamics does not cor- a coined QW, gives the usual Dirac equation in the relativistic responds to a finite difference Hamiltonian or Lagrangian but to limit of small wave-vectors where the lattice step is hypothetically a discrete and exactly causal unitary evolution. The difference is assumed to be the Planck length. After reviewing the one dimen- even more clear observing that it does not exist a QCA whose evo- sional Dirac automaton and the physical significance of its solu- lution exactly corresponds to the finite difference Dirac differential tions, we solve analytically the automaton in the position space equation. For these reasons the path-sum formulation of the Dirac via a discrete path-integral, providing a discrete version of the QCA presented in the following does not coincide with the Feyn- Feynman propagator. In this formulation the discrete paths corre- man checkerboard one (see for example [15]) and it is based on spond to a sequence of the automaton transition matrices, which the algebraic features of the transition matrices in Eq. (1). are proved to be closed under multiplication. Exploiting this feature, and the binary encoding of the admissible paths between two 3. Path-sum formulation of the Dirac QCA causally connected sites, we derive the analytical solution for an arbitrary initial state and mass parameter. Given the field initial condition ψ(0), after t time steps one has ψ(t) At ψ(0), and by linearity the field ψ(x, t) must be a lin- = 2. The one-dimensional Dirac QCA ear combination of the field at the points (y, 0) lying in the past causal cone of (x, t). In general each point (y, 0) is connected to The Dirac QCA of Refs. [25,26] describes the one-step evolution (x, t) in t time steps via a number of different discrete paths. Ac- of a two-component quantum field cording to Eq. (1) at each step of the automaton the local field ψ(y, 0) undergoes a shift T l , l 0, 1, and the internal degree of = ± ψ (x,t) freedom is multiplied by the corresponding transition matrix A , ψ(x,t) R ,(x,t) 2, h ψ (x,t) Z with h R, L, F . A generic path σ connecting x to y in t steps := L ∈ ∈{ } " # is conveniently identified with a string σ htht 1 ...h1 of transi- = − ψR and ψL denoting the right and the left mode of the field. Here tions, corresponding to the overall transition matrix given by the we restrict to one-particle states and the statistics is not relevant, product but the presented solution could be extended to multi-particle state for any statistics consistent with the evolution. In the single- A(σ ) Aht Aht 1 ...Ah1 . (2) 2 = − particle Hilbert space C l2(Z), we will use the factorized basis ⊗ s x , with s R, L. Summing over all admissible path σ and over all points (y, 0) in | ⟩| ⟩ = Here the evolution of the field is restricted to be linear, namely the past causal cone of (x, t), one has there exists a unitary operator A such that the one step evolution of the field is given by ψ(t 1) U ψ(t)U † Aψ(t). In the present ψ(x,t) A(σ )ψ(y, 0). (3) + = = = case the assumption of locality corresponds to writing ψ(x, t 1) y σ + $ $ as linear a combination of ψ(x l, t) with l 0, 1. Homogeneity + = ± We now evaluate analytically the sum over σ in Eq. (3) as a func- (or translation invariance) corresponds to a unitary operator A of tion of the variables x, y, t. the form Upon denoting by r, l, f the numbers of R, L, F transitions in σ , using t r l f and x y r l, one has 1 = + + − = − A A R T AL T − A F I, = ⊗ + ⊗ + ⊗ t f x y t f x y n 0 00 0im r − + − , l − − + . (4) A R , AL , A F , = 2 = 2 = 00 = 0 n = im 0 " # " # " # The overall transition matrix σ A(σ ) in (2) can be efficiently T x 1 x , (1) = | + ⟩⟨ | computed taking the following binary encoding x % $∈Z 2 2 A R nA00, AL nA11, A F im(A10 A10) (5) where A R , AL , A F are called transition matrices, and n m 1, = = = + + = 10 00 n, m R+. ∈ A00 , A11 , (6) The specific form of the transition matrices in Eq. (1) has been = 00 = 01 " # " # derived in Refs. [25,26] as a consequence of the constraints of uni- 01 00 tarity and invariance with respect to the symmetries of the lattice, A01 , A10 , (7) = 00 = 10 and the dynamics of the automaton (1) has been rigorously an- " # " # alyzed in the k wave-vector space. Interpreting the parameter k and observing that the matrices Aab satisfy the simple composition and m of the Dirac automaton as momentum and mass it has been rule G.M. D’Ariano et al. / Physics Letters A 378 (2014) 3165–3168 3167

b c 1 ( 1) ⊕ are arranged in the strings τ2i 1. This means that we have to count Aab Acd + − Aad, (8) + = 2 in how many ways the r identical characters R and l identical f 2 f denoting the sum modulo 2. It is then convenient to denote characters L can be arranged in +2 and 2 strings, respectively. ⊕ by σ h , ...h the generic path having f occurrences of the These arrangements can be viewed as combinations with repeti- f = t 1 F -transition, and write Eq. (3) as follows tions which give f f t x y t x y t x y + − − + 1 −| − | 2 r 2 l 1 2 2 c00( f ) + + − − , ψ(x,t) A(σ )ψ(y, 0). (9) = r l = f f = 2 2 1 y f 0 σ f " #" # " #" − # $ $= $ where the second equality trivially follows from Eq. (4). Simi- In a path σ the F transitions identify f 1slots f larly A appears whenever σ Ω which gives + 11 f ∈ L τ F τ F ...... F τ , (10) f f t x y t x y 1 2 f 1 − + + − 1 + 2 l 2 r 1 2 2 c11( f ) + + − − . where τi denotes a (possibly empty) string of R and L. According = l r = f f 1 to Eq. (8) the generic path σ cannot contain substrings of the form " #" # " 2 #" 2 − # Consider now the other two coefficients c10 and c01 counting the hihi 1 RL, hihi 1 LR, (11) occurrences of A10 and A01. The last ones appears only when f is − = − = f 1 odd (see Eq. (13)) and then one has the same number + of odd hihi 1hi 2 RFR, hihi 1hi 2 LFL, (12) 2 − − = − − = strings τ2i 1 and even strings τ2i . Counting the combinations with + since they give null transition amplitude. Therefore, according to repetitions as in the previous cases we get Eq. (11) each τi in (10) is a string of equal letters, i.e. τi hh ...h, = f 1 f 1 with h R, L. On the other hand Eq. (12) shows that two con- −2 r −2 l = c10( f ) c01( f ) + + secutive strings τi and τi 1 must be made of different h. This = = r l + " #" # corresponds to have all τ2i hh ...h and all τ2i 1 h′h′ ...h′, with t x y 1 t x y 1 = + = + − − − + − h h′. In the following we will denote by ΩR and ΩL the sets 2 2 ≠ f 1 f 1 , of strings having τ2i 1 RR ...R and τ2i 1 LL ...L, respectively, = − − + = + = 2 2 for all i. " #" # The above structure for strings σ f can be exploited to deter- which concludes the derivation of the coefficients cab( f ) in mine the matrix A(σ f ). We consider separately the cases of f Eq. (17). even and f odd. For f even one has The analytical solution of the Dirac automaton can also be ex- (ζ,ρ) pressed in terms of Jacobi polynomials P computing the sum A A , f t k 00 + 11 = over f in Eq. (16) A(σ f ) α( f ) A00, f < t, σ f ΩR , (13) = ⎧ A , f < t, ∈ Ω , 2 ⎨ 11 σ f L (1, t) m ∈ ψ(x,t) γa,b P − 1 2 Aabψ(y, 0), = k + n while for f odd one has y a,b 0,1 " " # # ⎩ $ $∈{ } A A , f t a b 1 10 + 01 = k µ ⊕ + , A(σ ) α( f ) A10, f < t, σ f ΩR , (14) = + − 2 f = ⎧ ∈ A01, f < t, σ ΩL , 2 a b a b f k ( ab ) ⎨ ∈ a b t m + ⊕ µ( ) ⊕2 γa,b i ⊕ n ! − + , (18) with the factor α( f ) given by =− n (2) ⎩ " # k ) * f t f where γ γ 0(γ γ 0) for t x y odd (even) and α( f ) (im) n − . (15) 00 = 11 = 10 = 01 = + − := (x)k x(x 1) (x k 1). According to Eqs. (13) and (14) we can finally restate Eq. (9) as = + ··· + − 4. Conclusions t x y −| − | ψ(x,t) c ( f )α( f )A ψ(y, 0), (16) = ab ab We studied the one dimensional Dirac automaton, considering y a,b 0,1 f 0 $ $∈{ } $= a discrete path-integral formulation. The analytical solution of the where c (2k 1) c (2k) c (2k) 0. The coefficients c ( f ) automaton evolution has been derived, adding a relevant case to aa + = 01 = 10 = ab count the number of paths σ f which give Aab as total transition the set of quantum automata solved in one space dimension, in- matrix, and are given by the following product of binomial coeffi- cluding only the coined QW and the disordered coined QW. The cients main novelty of this work is the technique used in the derivation of the analytical solution, based on the closure under multiplica- µ ν µ ν + − tion of the automaton transition matrices. This approach can be cab( f ) f 1 − f 1 + , = − ν − ν extended to automata in higher space dimension. For example the " 2 − #" 2 + # transition matrices of the Weyl and Dirac QCAs in 2 1and 3 1 ab ab t (x y) 1 + + ν − ¯ ¯ , µ ± − − , (17) dimensions recently derived in Ref. [26] enjoy the closure feature = 2 ± = 2 and their path-sum formulation could lead to the first analytically where c c 1, and the binomials are null for non-integer ar- solved example in dimension higher than one. ¯ := ⊕ guments. The expression of cab is computed via combinatorial considerations based on the structure (10) of the paths, and on Acknowledgements Eqs. (13) and (14). Let us start with the coefficients c00 and c11. The matrices A00 and A11 appear only for f even (see Eq. (13)) in This work has been supported in part by the John Templeton f 2 f which case one has + odd strings τ2i 1 and even strings τ2i . Foundation under the project ID# 43796 A Quantum-Digital Uni- 2 + 2 A appears whenever σ Ω , namely when the R-transitions verse. 00 f ∈ R 3168 G.M. D’Ariano et al. / Physics Letters A 378 (2014) 3165–3168

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