Quantum Stochastic Walks: A Generalization of Classical Random Walks and Quantum Walks

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1 Quantum Stochastic Walks: A Generalization of Classical Ranom Walks an Quantum Walks The Harvar community has mae this article openly available. Please share how this access benefits you. Your story matters Citation Whitfiel, James D., César A. Roríguez-Rosario, an Alán Aspuru- Guzik Quantum stochastic walks: A generalization of classical ranom walks an quantum walks. Physical Review A 81(2): Publishe Version oi: /PhysRevA Citable link Terms of Use This article was ownloae from Harvar University s DASH repository, an is mae available uner the terms an conitions applicable to Open Access Policy Articles, as set forth at nrs.harvar.eu/urn-3:hul.instrepos:ash.current.terms-ofuse#oap

2 Quantum stochastic walks: A generalization of classical ranom walks an quantum walks César A. Roríguez-Rosario, James D. Whitfiel, an Alán Aspuru-Guzik Department of Chemistry an Chemical Biology, Harvar University, 12 Oxfor St., Cambrige, MA (Date: May 22, 2009) arxiv: v2 [quant-ph] 22 May 2009 We introuce the quantum stochastic walk (QSW), which etermines the evolution of generalize quantum mechanical walk on a graph that obeys a quantum stochastic equation of motion. Using an axiomatic approach, we specify the rules for all possible quantum, classical an quantum-stochastic transitions from a vertex as efine by its connectivity. We show how the family of possible QSWs encompasses both the classical ranom walk (CRW) an the quantum walk (QW) as special cases, but also inclues more general probability istributions. As an example, we stuy the QSW on a line, the QW to CRW transition an transitions to genearlize QSWs that go beyon the CRW an QW. QSWs provie a new framework to the stuy of quantum algorithms as well as of quantum walks with environmental effects. Many classical algorithms, such as most Markov-chain Monte Carlo algorithms, are base on classical ranom walks (CRW), a probabilistic motion through the vertices of a graph. The quantum walk (QW) moel is a unitary analogue of the CRW that is generally use to stuy an evelop quantum algorithms [1, 2, 3]. The quantum mechanical nature of the QW yiels ifferent istributions for the position of the walker, as a QW allows for superposition an interference effects [4]. Algorithms base on QWs exhibit an exponential speeup over their classical counterparts have been evelope [5, 6, 7]. QWs have inspire the evelopment of an intuitive approach to quantum algorithm esign [8], some base on scattering theory [9]. They have recently been shown to be capable of performing universal quantum computation [10]. The transition from the QW into the classical regime has been stuie by introucing ecoherence to specific moels of the iscrete-time QW [11, 12, 13, 14]. Decoherence has also been been stuie as non-unitary effects on continuous-time QW in the context of quantum transport, such as environmentally-assiste energy transfer in photosynthetic complexes [15, 16, 17, 18, 19] an state transfer in superconucting qubits [20, 21]. For the purposes of experimental implementation, the vertices of the graph in a walk can be implemente using a qubit per vertex (an inefficient or unary mapping) or by employing a quantum state per vertex (the binary or efficient mapping). The choice of mapping impacts the simulation efficiency an their robustness uner ecoherence [22, 23, 24]. The previous propose approaches for exploring ecoherence in quantum walks have ae environmental-effects to a QW base on computational or physical moels such as pure ephasing [17] but have not consiere walks where the environmental effects are constructe axiomatically from the unerlying graph. In this work, we efine the quantum stochastic walk (QSW) using a set of axioms that incorporate unitary an non-unitary effects. A CRW is a type of classical stochastic processes. From the point of view of the theory of open quantum systems, the generalization of a classical stochastic process to the quantum regime is known to be a quantum stochastic process [16, 25, 26, 27, 28, 29] which is the most general type of evolution of a ensity matrix, not simply the Hamiltonian process propose by the QW approach. The main goal of this paper is to introuce a set of axioms that allow for the construction of a quantum stochastic process constraine by a graph. We call all the walks that follow these axioms QSWs. We will show that the family of QSWs inclues both the CRW an the QW as limiting cases. The QSW can yiel new istributions that are not foun either in the CRW or the QW. The connection between the three types of walks iscusse in this manuscript is summarize in Fig. 1. For clarity, we focus on continuous-time walks, but FIG. 1: The quantum stochastic walk (QSW) is a quantum stochastic process efine by a set of axioms that escribe the connectivity of an unerlying graph. The family of possible QSWs encompasses the classical ranom walk (CRW) an the quantum walk (QW) as limiting cases. QSWs can be use to generate walks that continuously interpolate between QW an CRW, an therefore can be employe to analyze algorithmic classical-to-quantum transitions. The family of QSWs also escribes more general quantum stochastic processes with a behavior that cannot be recovere by the QW an CRW approaches. also sketch the corresponing proceure for the iscretetime walks. The QSW provies the funamental tools to stuy the quantum-to-classical transition of walks, as

3 2 well as new methos of control by the application of nonunitary operations on a quantum system. The CRW escribes probabilistic motion over a graph. The ynamics of the probability istribution function is given by the master equation for a classical stochastic process, t p a = b M a b p b, (1) where the vector element p b is the probability of being foun at the vertex b of the graph. The matrix M is the generator for the evolution; its structure is constraine by axioms erive from the connectivity of the graph. For example, if vertices a an b are connecte, Mb a = γ, if they are not, Mb a = 0, an M a a = a γ where a is the egree of vertex a. In analogy to the CRW, the QW [1] has been efine so that the probability vector element p a is replace with a ψ, which evolves accoring to the Schröinger equation, t a ψ = i b a H b b ψ, (2) where H is a Hamiltonian to be efine base on axioms coming from the graph. A choice of this efinition is a H b = Mb a. This unitary evolution effectively rotates populations into coherences an back [32]. The QW fails to completely capture the stochastic nature of CRW. The ranom aspect of the QW comes solely from the probabilistic nature of measurements performe on the wavefunction. Since a classical stochastic process can be generalize to the quantum regime by means of a quantum stochastic process, a CRW shoul be generalize to a QSW erive from the graph [33]. For the generalization, we ientify the probability vector with elements p a with a ensity matrix with elements ρ aα, an generalize the evolution to a quantum stochastic process, t ρ = M[ ρ ], where M is a superoperator [25, 26, 28]. To make this evolution look similar to Eq. (1), we write the ensity matrix in terms of its inices, ρ = a,α ρ aα a α, an the quantum stochastic master equation becomes, t ρ aα = M aα bβ ρ bβ, (3) b,β with the tensor M aα bβ = a M[ b β ] α. This corresponence was pointe out by Mohseni et al. in the context of energy transfer [16]. For a quantum stochastic process to be relate to a walk, the superoperator M must reflect the graph. The connectivity of the vertices will impose conitions on the transition rates of M. Since the quantum stochastic process is more general than both the classical stochastic process an the Schröinger equation, the corresponence of the connectivity of the graph to the rules impose on M shoul inclue an go beyon the connectivity axioms for each of those. For a vertex m connecte to vertices that inclue vertex n, we efine a processes m n which occurs at some rate that can evolve m to an from n. Transition rates for vertices that are not connecte are efine to be zero. We employ these connectivity rules as the main principle for efining vali QSWs from a given graph. To further explore the connection from the QSW to the CRW an QW as well as more general behaviors, we iscuss the ifferent limiting cases. For the classical case, the allowe transitions come from incoherent population hopping of the form m m n n, an, for completeness, m m m m. These conitions constrain M to operate only on the iagonal elements of ρ, like Eq. (1). In other wors, the QSW transition tensor must have the property α,β δ aα M aα bβ δ bβ = Mb a. The QSW shoul also recover the QW, where evolution among vertices happens through coherences evelope by a Hamiltonian. These transitions inclue terms of the form, m m m n, that exchange populations to coherences with the connecte vertices. For completeness, we also consier transitions of the form m n m n. If m is also connecte to another vertex l, aitional transitions can happen. One transition exchanges populations into coherences among the two vertices connecte to m, m m l n ; the other exchanges coherences between m an a connecte vertex into coherence among two vertices connecte to m, m n l n. These conitions allow for the recovery of Eq. (2). By incluing the conjugates of these, we have now exhauste all the possibilities of the transitions that can happen following the connectivity of vertex m. These rules can be applie to vertices with any number of connections, an serve as the basis for the corresponence of the graph to the QSW. To complete the generalization, we nee an equation of motion for a quantum stochastic process. We choose the Kossakowski-Linbla master equation [26, 27, 28], t ρ = L[ ρ ] = k i [H, ρ] 1 2 L k L kρ 1 2 ρl k L k +L k ρl k, which evolves the ensity matrix as a quantum stochastic process inspire by environmental effects uner the Markov approximation. The Hamiltonian term escribes the coherent evolution (Schröinger equation) while the rest escribe a stochastic evolution. We now set M L in Eq. (3), where, L aα bβ = k δ αβ a ( ih 1 ) 2 L k L k b (4) ( + δ ab β ih 1 ) 2 L k L k α + a L k b β L k α. The connectivity conitions between a vertex m an some connecte vertices n an l an the corresponing

4 3 Axiom Matrix elements connectivity Transition rate 1 m m m m L mm mm = m L k m m L k m m L k L k m 2 m m n n L nn mm = n L k m m L k n 3 m m m n L mn mm = m L k m m L k n + i m H n 1 2 m L k L k n 4 m n m n L mn mn = m L k m n L k n i m H m + i n H n 1 2 m L k L k m 1 2 n L k L k n 5 m n l n L ln mn = l L k m n L k n i l H m 1 2 l L k L k m 6 m m l n L ln mm = l L k m m L k n TABLE I: Axioms for the quantum stochastic walk for processes that connect vertex m to its neighbors. Sum over k, an conjugate elements are implie. Axioms (1) an (2) correspon to the classical ranom walk. Axioms (3), (4) an (5) contain the quantum walk using H an aitional evolution ue to {L k } terms. Axiom (6) comes only from the quantum stochastic walk, having no equivalent in the classical ranom walk or quantum walk. non-zero transition rates accoring to Eq. (4) can be summarize in Table I [34]. The Axioms from Table I capture the behavior sketche in Fig. (1). To recover the CRW, it suffices to consier Axioms (1) an (2). These Axioms are a classical subset of the transition rates of the QSW. On the other han, the QW is obtaine by making all the rates of each element of {L k } zero. In this case, only the subset of Axioms (3), (4) an (5) from Table I are relevant, corresponing to the Hamiltonian, an the QW is recovere. If the rates of {L k } are nonzero, these Axioms contain behavior beyon the QW. Finally, Axiom (6) has properties that have no equivalent in either the CRW or the QW; it is a type of transition that appears exclusively in the QSW an leas to ifferent istributions. The choice of H or {L k } is not uniquely etermine by the Axioms. For example, a CRW is equivalent to a QSW with the choice of each element of {L k } to be associate to M in the following manner. First, since connections are efine between two vertices, it is easier to write the inex k in terms of two inices k (κ, κ ). Using this notation, we enforce the graph by choosing L (κ,κ ) = Mκ κ κ κ. The Hamiltonian rates of H are set to zero. This choice ensures all the transition rates to be zero except Axiom (2) from Table I, thereby recovering the CRW. An interpolation between the CRW an the QW conitions can be use to obtain the classical-to-quantum transition of any graph by changing the relative rates of the unitary an non-unitary processes. Other sets of {L k } can yiel behavior ifferent from both the classical an quantum walks. To illustrate the ifference between these choices, we stuy a simple example: the walk on an infinite line, where a vertex j is only connecte to its nearest neighbors. The conitions for the ynamics of the walk on the line can be obtaine from Table I by making j = m, j 1 = l an j + 1 = n. To obtain behavior that is ifferent from both the CRW an the QW, we specify a set {L k } with only one member that is L = κ,κ M κ κ κ κ. This choice makes Axiom (6) nonzero, a type of transition that cannot be interprete either as the CRW or the QW. We illustrate the resulting istribution, an compare it to CRW an the QW, in Fig. 2. The QSW provies a general framework to stuy the transition between the CRW an the QW. In Fig. 2a, we use the parameter ω = [0, 1] to interpolate between the QW an the CRW: L ω [ρ] = (1 ω) i [H, ρ] + ω ( ) k 1 2 L k L kρ 1 2 ρl k L k + L k ρl k. By combining the unitary evolution an the evolution ue to the {L (κ,κ )} terms, the transition from the QW to the CRW can be stuie as one quantum stochastic walk. This proceure can be one in general to stuy the transition from any QW to a CRW for any graph [35]. To highlight the ifference between the QSW an the QW, we show the transition between them with L = κ,κ M κ κ κ κ on Fig. 2b. Although in this paper we focuse on continuous-time walks, a parallel argument hols for iscrete-time walks. A iscrete-time CRW evolves each time step by a Markov chain with a stochastic matrix S following p a = b Sa b p b. The quantum analogue is to use a quantum stochastic map B [25] by means of ρ aα = bβ Baα bβ ρ bβ, or equivalently as a superoperator, ρ = B [ρ] k C kρc k. A similar set of Axioms to Table I can be compute by using Bbβ aα = k a C k b β C k α [36]. The connection between the iscrete-time QW an the continuous-time QW has been stuie by Strauch [30]. In conclusion, we introuce an axiomatic approach to efine the quantum stochastic walk, which behaves accoring to a quantum stochastic process as constraine by a particular graph. This walk recovers the classical ranom walk an the quantum walk as special cases, as well as the transition between them. The quantum stochastic walk allows for the construction of new types of walks that result in ifferent probability istributions. As an example, we stuie the walk on a line. The quantum stochastic walk provies a framework to stuy quantum walks uner ecoherence. Reexamination of previous work that consiere environmental effects from physical motivations might suggest that, if interprete

5 4 FIG. 2: Probability istributions of a walker starting at position 0 of an infinite line efine by M are shown at time t = 5. The heavy line represents the quantum walk, the thin line represents the classical ranom walk an the otte line represents a choice of a quantum stochastic walk ue to a single operator L = P κ,κ M κ κ κ κ whose istribution cannot be obtaine from the classical or quantum walks. a) The transition of the quantum walk to the classical ranom walk is shown. The iagonal elements of the ensity matrix (Population) are plotte as a function of the vertex of the walker (Position) an a coupling parameter ω. When ω = 0 the evolution is ue only to the Hamiltonian, an when ω = 1 the Hamiltonian ynamics are not present with the environmental ynamics taking over. b) The transition of the quantum walk to the quantum stochastic walk is shown. as a quantum stochastic walk, the environment is changing the effective graph. Since the quantum stochastic walk is more general than the quantum walk, it is therefore universal for quantum computation. Its quantum to classical transiton can also be use to examine ecoherence phenomena in quantum computation an classicalto-quantum phase transitions. New quantum stochastic istributions might suggest new kins of quantum algorithms or even new classes of quantum algorithms base on this moel. Acknowlegments: We thank F. Strauch for many insightful iscussions an comments. We also thank I. Kassal, L. Vogt, S. Venegas-Anraca, P. Rebentrost, K. Moi, S. Joran an E.C.G. Suarshan for their comments on the manuscript. A.A.G. an J.D.W. thank the Army Research Office uner contract W911NF an C.A.R. thanks the Mary-Fieser Postoctoral Fellowship program. roriguez@chemistry.harvar.eu whitfiel@chemistry.harvar.eu aspuru@chemistry.harvar.eu [1] E. Farhi an S. Gutmann, Phys. Rev. A 58, 915 (1998). [2] S. Venegas-Anraca, Quantum Walks for Computer Scientists (Morgan an Claypool, 2008). [3] A. Ambainis, Lecture Notes in Computer Science 4910, 1 (2008). [4] Y. Aharonov, L. Daviovich, an N. Zagury, Phys. Rev. A 48, 1687 (1993). [5] J. Watrous, J. Comput. Syst. Sci 62, 376 (2001). [6] A. Chils, E. Farhi, an S. Gutmann, Quantum Information Processing 1, 35 (2002). [7] A. Chils, R. Cleve, E. Deotto, an E. Farhi, in Proc. 36th ACM Symposium on Theory of Computing (2003), p. 59. [8] N. Shenvi, J. Kempe, an K. Whaley, Phys. Rev. A 67, (2003). [9] E. Farhi, J. Golstone, an S. Gutmann, Theory of Computing 4, 169 (2008). [10] A. Chils, Phys. Rev. Lett. 102, (2009). [11] V. Kenon, Mathematical Structures in Computer Science 17, 1169 (2007). [12] P. Love an B. Boghosian, Quantum Information Processing 4, 335 (2005). [13] T. Brun, H. Carteret, an A. Ambainis, Phys. Rev. Lett. 91, (2003). [14] A. Romanelli, R. Siri, G. Abal, A. Auyuanet, an R. Donangelo, Physica A 347, 137 (2005). [15] P. Rebentrost, M. Mohseni, an A. Aspuru-Guzik, J. Phys. Chem. C. In press. Also at arxiv: (2009). [16] M. Mohseni, P. Rebentrost, S. Lloy, an A. Aspuru- Guzik, Journal of Chemical Physics 129 (2008). [17] P. Rebentrost, M. Mohseni, I. Kassal, S. Lloy, an A. Aspuru-Guzik, New Journal of Physics 11, (2009). [18] F. Caruso, A. Chin, A. Datta, S. Huelga, an M. Plenio, arxiv: (2009). [19] M. Plenio an S. Huelga, New J. Phys. 10, (2008). [20] F. Strauch an C. Williams, Phys. Rev. B 78, (2008). [21] F. Strauch, Phys. Rev. A 79, (2009). [22] A. Hines an P. Stamp, arxiv: (2007). [23] M. Drezgic, A. Hines, M. Sarovar, an S. Sastry, arxiv: (2008). [24] F. Strauch, Private Communication (2009). [25] E. C. G. Suarshan, P. M. Mathews, an J. Rau, Phys. Rev. 121, 920 (1961). [26] A. Kossakowski, Rep. Math. Phys 3, 247 (1972). [27] G. Linbla, Commun. Math. Phys 48, 119 (1975). [28] V. Gorini, A. Kossakokowski, an E. C. G. Suarshan, J. Math. Phys 17, 821 (1976). [29] C. A. Roriguez-Rosario an E. C. G. Suarshan, arxiv: (2008). [30] F. Strauch, Phys. Rev. A 74, (2006). [31] S. Salimi an A. Sorouri, arxiv: (2009). [32] The Hermitian form of the Hamiltonian eliminates irecte graphs from consieration, motivating work on pseuo-hermitian Hamiltonians [31]. [33] Just like a classical stochastic process inclues, but is not limite, to bistochastic processes, a quantum stochastic processes inclues but is not limite to unitary processes [25]. [34] For compactness, we assume that all connections are equally weighte; the general case can be obtaine by having coefficients that epen on the vertices. [35] Our ecoherence moel comes from the graph, while the

6 ones iscusse in Ref. [11] were inspire by a physical moel that localizes the position an thus give ifferent istributions. We propose stuies like Ref. [11] coul be reinterprete as a QSW, where the environmental part, as interprete from Table I, affects the connectivity of the graph an thus generates ifferent istributions. [36] There is no nee for an explicit coin, as the stochastic map is the most general evolution of a ensity matrix, incluing ranom processes. However, if esire, a coin coul be implemente as an environment from the open quantum system interpretation of the map. 5