Quantum walks with random phase shifts

Quantum Walks on the HypercubeC RISTOPHER M OOREComputer Science DepartmentUniversity of New Mexico,Albuquerque and the Santa Fe Institute,Santa Fe,New Mexicomoore@A LEXANDER R USSELL Department of Computer Science and Engineering University of ConnecticutStorrs,Connecticutacr@November12,2001AbstractRecently,it has been shown that one-dimensional quantum walks can mix more quickly than clas-sical random walks,suggesting that quantum Monte Carlo algorithms can outperform their classicalcounterparts.We study two quantum walks on the n-dimensional hypercube,one in discrete time andone in continuous time.In both cases we show that the instantaneous mixing time isπ4n steps,fasterthan theΘn log n steps required by the classical walk.In the continuous-time case,the probabilitydistribution is exactly uniform at this time.On the other hand,we show that the average mixing timeas defined by Aharonov et al.[AAKV01]isΩn32in the discrete-time case,slower than the classical walk,and nonexistent in the continuous-time case.This suggests that the instantaneous mixing time is amore relevant notion than the average mixing time for quantum walks on large,well-connected graphs.Our analysis treats interference between terms of different phase more carefully than is necessary for thewalk on the cycle;previous general bounds predict an exponential average mixing time when applied tothe hypercube.1IntroductionRandom walks form one of the cornerstones of theoretical computer science.As algorithmic tools,theyhave been applied to a variety of central problems,such as estimating the volume of a convex body[DFK91,LK99],approximating the permanent[JS89,JSV00],andfinding satisfying assignments for Boolean for-mulae[Sch99].Furthermore,the basic technical phenomena appearing in the study of random walks(e.g.,spectral decomposition,couplings,and Fourier analysis)also support several other important areas such aspseudorandomness and derandomization(see,e.g.,[AS92,(9,15)]).The development of efficient quantum algorithms for problems believed to be intractable for classicalrandomized computation,like integer factoring and discrete logarithm[Sho97],has prompted the investi-gation of quantum walks.This is a natural generalization of the traditional notion discussed above where,roughly,the process evolves in a unitary rather than stochastic fashion.The notion of“mixing time,”thefirst time when the distribution induced by a random walk is sufficientlyclose to the stationary distribution,plays a central role in the theory of classical random walks.For a givengraph,then,it is natural to ask if a quantum walk can mix more quickly than its classical counterpart.(Sincea unitary process cannot be mixing,we define a stochastic process from a quantum one by performinga measurement at a given time or a distribution of times.)Several recent articles[AAKV01,ABN01,NV00]have answered this question in the affirmative,showing,for example,that a quantum walk on then-cycle mixes in time O n log n,a substantial improvement over the classical random walk which requires Θn2steps to mix.Quantum walks were also defined in[Wat01],and used to show that undirected graph connectivity is contained in a version of quantum LOGSPACE.These articles raise the exciting possibilitythat quantum Monte Carlo algorithms could form a new family of quantum algorithms that work morequickly than their classical counterparts.Two types of quantum walks exist in the literature.Thefirst,introduced by[AAKV01,ABN01,NV00],studies the behavior of a“directed particle”on the graph;we refer to these as discrete-time quantumwalks.The second,introduced in[FG98,CFG01],defines the dynamics by treating the adjacency matrixof the graph as a Hamiltonian;we refer to these as continuous-time quantum walks.The landscape isfurther complicated by the existence of two distinct notions of mixing time.The“instantaneous”notion[ABN01,NV00]focuses on particular times at which measurement induces a desired distribution,whilethe“average”notion[AAKV01],another natural way to convert a quantum process into a stochastic one,focuses on measurement times selected randomly from some interval.In this article,we analyze both the continuous-time and a discrete-time quantum walk on the hypercube.In both cases,the walk is shown to have an instantaneous mixing time atπ4n.(Of course,in the discretewalk all times are integers.)Recall that the classical walk on the hypercube mixes in timeΘn log n,sothat the quantum walk is faster by a logarithmic factor.Moreover,in the discrete-time case the walk mixesin time less than the diameter of the graph,sinceπ41;astonishingly,in the continuous-time case theprobability distribution at tπ4n is exactly uniform.Both of these things happen due to a marvelousconspiracy of destructive interference between terms of different phase.These walks show i.)a similarity between the two notions of quantum walks,and ii.)a disparity between the two notions of quantum mixing times.As mentioned above,both walks have an instantaneous mixing time at timeπ4n.On the other hand,we show that the average mixing time of the discrete-time walk isΩn32,slower than the classical walk,and that for the continuous-time walk there is no time at which the time-averaged probability distribution is close to uniform in the sense of[AAKV01].Our results suggest that for large graphs(large compared to their mixing time)the instantaneous notion of mixing time is more appropriate than the average one,since the probability distribution is close to uniform only in a narrow window of time.The analysis of the hypercubic quantum walk exhibits a number of features markedly different fromthose appearing in previously studied walks.In particular,the dimension of the relevant Hilbert space is,for the hypercube,exponential in the length of the desired walk,while in the cycle these quantities are roughly equal.This requires that interference be handled in a more delicate way than is required for the walk on the cycle;in particular,the general bound of[AAKV01]yields an exponential upper bound on the mixing time for the discrete-time walk.We begin by defining quantum walks and discussing various notions of mixing time.We then analyze the two quantum walks on the hypercube in Sections2and3.(Most of the technical details for the discrete-time walk are relegated to an appendix.)1.1Quantum walks and mixing timesAny graph G V E gives rise to a familiar Markov chain by assigning probability1d to all edges leaving each vertex v of degree d.Let P t u v be the probability of visiting a vertex v at step t of the random walk on G starting at u.If G is undirected,connected,and not bipartite,then lim t∞P t u exists1and is independent of u.A variety of well-developed techniques exist for establishing bounds on the rate at which P t u achieves this limit(e.g.,[Vaz92]);if G happens to be the Cayley graph of a group(as are,for example,the cycle and the hypercube),then techniques from Fourier analysis can be applied[Dia88].Below we will use some aspects of this approach,especially the Diaconis-Shahshahani bound on the total variation distance[DS81].For simplicity,we restrict our discussion to quantum walks on Cayley graphs;more general treatments of quantum walks appear in[AAKV01,FG98].Before describing the quantum walk models we set down some notation.For a group G and a set of generatorsΓsuch thatΓΓ1,let X GΓdenote the undirected Cayley graph of G with respect toΓ.For a finite set S,we let L S f:S denote the collection of-valued functions on S with∑s S f s2 1. This is a Hilbert space under the natural inner product f g∑s S f s g s.For a Hilbert space V,a linear operator U:V V is unitary if for all v w V,v w U v U w;if U is represented as a matrix,this is equivalent to the condition that U†U1where†denotes the Hermitian conjugate.There are two natural quantum walks that one can define for such graphs,which we now describe.T HE DISCRETE-TIME WALK:This model,introduced by[AAKV01,ABN01,NV00],augments the graph with a direction space,each basis vector of which corresponds one of the generators inΓ.A step of the walk then consists of the composition of two unitary transformations;a shift operator which leaves the direction unchanged while moving the particle in its current direction,and a local transformation which op-erates on the direction while leaving the position unchanged.To be precise,the quantum walk on X GΓis defined on the space L GΓL G LΓ.LetδγγΓbe the natural basis for LΓ,andδg g G the natural basis for L G.Then the shift operator is S:δgδγδgγδγ,and the local transformation isˇD1D where D is defined on LΓalone and1is the identity on L G.Then one“step”of the walk corresponds to the operator UˇDV.If we measure the position of the particle,but not its direction,at time t,we observe a vertex v with probability P t v∑γΓU tψ0δvδγ2whereψ0L GΓis the initial state.T HE CONTINUOUS-TIME WALK:This model,introduced in[FG98],works directly on L G.The walk evolves by treating the adjacency matrix of the graph as a Hamiltonian and using the Schr¨o dinger equation. Specifically,if H is the adjacency matrix of X GΓ,the evolution of the system at time t is given by U t, where U t eq e iHt(here we use the matrix exponential,and U t is unitary since H is real and symmetric). Then if we measure the position of the particle at time t,we observe a vertex v with probability P t vU tψ0δv2whereψ0is the initial state.Note the analogy to classical Poisson processes:since U t e iHt 1In fact,this limit exists under more general circumstances;see e.g.[MR95].1iHt iHt22,the amplitude of making s steps is the coefficient it s s!of H s,which up to normalization is Poisson-distributed with mean t.Remark.In[CFG01],the authors point out that defining quantum walks in continuous time allows unitarity without having to extend the graph with a direction space and a chosen local operation.On the other hand,it is harder to see how to carry out such a walk in a generically programmable way using only local information about the graph,for instance in a model where we query a graph tofind out who our neighbors are.Instead,continuous-time walks might correspond to special-purpose analog computers,where we build in interactions corresponding to the desired Hamiltonian and allow the system to evolve in continuous time.In both cases we start with an initial wave function concentrated at a single vertex corresponding to the identity u of the group.For the continuous-time walk,this corresponds to a wave functionψ0v ψ0δv(so thatψ0u1andψ0v0for all v u).For the discrete-time walk,we start with a uniform superposition over all possible directions,ψ0vγψ0δvδγ1Γif v u 0otherwise.For the hypercube,u000.In order to define a discrete-time quantum walk,one must select a local operator D on the directionspace.In principle,this introduces some arbitrariness into the definition.However,if we wish D to respectthe permutation symmetry of the n-cube,and if we wish to maximize the operator distance between D andthe identity,we show in Appendix A that we are forced to choose Grover’s diffusion operator[Gro96],which we recall below.We call the resulting walk the“symmetric discrete-time quantum walk”on then-cube.(Watrous[Wat01]also used Grover’s operator to define quantum walks on undirected graphs.)Since for large n Grover’s operator is close to the identity matrix,one might imagine that it would takeΩn12steps to even change direction,giving the quantum walk a mixing time of n32,slower than the classical random walk.However,like many intuitions about quantum mechanics,this is simply wrong.Since the evolution of the quantum walk is governed by a unitary operator rather than a stochastic one,unless P t is constant for all t,there can be no“stationary distribution”lim t∞P t.In particular,for anyε0,there are infinitely many(positive,integer)times t for which U t1εso that U tψuψuεand P t is close to the initial distribution.However,there may be particular stopping times t which induce distributions close to,say,the uniform distribution,and we call these instantaneous mixing times:Definition1We say that t is anε-instantaneous mixing time for a quantum walk if P t Uε,where A B12∑v A v B v denotes total variation distance and U denotes the uniform distribution.For these walks we show:Theorem1For the symmetric discrete-time quantum walk on the n-cube,t kπ4n is anε-instantaneousmixing time withεO n76for all odd k.and,even more surprisingly,Theorem2For the continuous-time quantum walk on the n-cube,t kπ4n is a0-instantaneous mixingtime for all odd k.Thus in both cases the mixing time isΘn,as opposed toΘn log n as it is in the classical case.Aharonov et al.[AAKV01]define another natural notion of mixing time for quantum walks,in whichthe stopping time t is selected uniformly from the set0T1.They show that the time-averageddistributions¯P T1T∑T1t0P t do converge as T∞and study the rate at which this occurs.For a continuous-time random walk,we analogously define the distribution¯P T v1T T0P t v d t.Then we call a time at which¯P T is close to uniform an average mixing time:Definition2We say that T is anε-average mixing time for a quantum walk if¯P T Uε.In this paper we also calculate theε-average mixing times for the hypercube.For the discrete-time walk,it is even longer than the mixing time of the classical random walk:Theorem3For the discrete-time quantum walk on the n-cube,theε-average mixing time isΩn32ε. This is surprising given that the instantaneous mixing time is only linear in n.However,the probability distribution is close to uniform only in narrow windows around the odd multiples ofπ4n,so¯P T is far from uniform for significantly longer times.We also observe that the general bound given in[AAKV01]yields an exponential upper bound on the average mixing time,showing that it is necessary to handle interference for walks on the hypercube more carefully than for those on the cycle.For the continuous-time walk the situation is even worse:while it possesses0-instantaneous mixing times at all odd multiples ofπ4n,the limiting distribution lim T∞¯P T is not uniform,and we show the following:Theorem4For the continuous-time quantum walk on the n-cube,there existsε0such that no time is an ε-average mixing time.Our results suggest that in both the discrete and continuous-time case,the instantaneous mixing time is a more relevant notion than the average mixing time for large,well-connected graphs.2The symmetric discrete-time walkIn this section we prove Theorem1.We treat the n-cube as the Cayley graph of n2with the regular basis vectors e i010with the1appearing in the i th place.Then the discrete-time walk takes place in the Hilbert space L n2n where n1n.Here thefirst component represents the position x of the particle in the hypercube,and the second component represents the particle’s current“direction”;if this is i,the shift operator willflip the i th bit of x.As in[AAKV01,NV00],we will not impose a group structure on the direction space,and will Fourier transform only over the position space.For this reason,we will express the wave functionψL n2n as a functionΨ:n2n,where the i th coordinate ofΨx is the projection ofψintoδxδi,i.e.the complex amplitude of the particle being at position x with direction i.The Fourier transform of such an elementΨis˜Ψ:n2n,where˜Ψk∑x1k xΨx.Then the shift operator for the hypercube is S:Ψx∑n i1πiΨx e i where e i is the i th basis vector in the n-cube,andπi is the projection operator for the i th direction.The reason for considering the Fourier transform above is that the shift operator isdiagonal in the Fourier basis:specifically it maps˜Ψk Sk˜Ψk whereS k 1k101k2...01k nFor the local transformation,we use Grover’s diffusion operator on n states,D i j2nδi j.The advantage of Grover’s operator is that,like the n-cube itself,it is permutation symmetric.We use thissymmetry to rearrange UkSkD to put the negated rows on the bottom,UkSkD2n12n2n2n12n......2n12n2n2n2n1......where the top and bottom blocks have n k and k rows respectively;here k is the Hamming weight of k.The eigenvalues of Uk then depend only on k.Specifically,Ukhas the eigenvalues1and1withmultiplicity k1and n k1respectively,plus the eigenvaluesλλwhereλ12kn2ink n k e iωkandωk0πis described bycosωk12knsinωk2nk n kIts eigenvectors with eigenvalue1span the k1-dimensional subspace consisting of vectors with support on the k“flipped”directions that sum to zero,and similarly the eigenvectors with eigenvalue1span the n k1-dimensional subspace of vectors on the n k other directions that sum to zero.We call these the trivial eigenvectors.The eigenvectors ofλλe iωk arev k v k1in k 1 kWe call these the non-trivial eigenvectors for a given k.Over the space of positions and directions these eigenvectors are multiplied by the Fourier coefficient1k x,so as a function of x and direction1j n the two non-trivial eigenstates of the entire system,for a given k,arev k x j1k x2n21if kj1i n k if k j0with eigenvalue e iωk,and its conjugate vkwith eigenvalue e iωk.We take for our initial wave function a particle at the origin000in an equal superposition of directions.Since its position is aδ-function in real space it is uniform in Fourier space as well as over thedirection space,giving˜Ψ0k2n2n 11This is perpendicular to all the trivial eigenvectors,so theiramplitudes are all zero.The amplitude of its component along the non-trivial eigenvector vkisa k Ψ0vk2n2kni1kn(1)and the amplitude of vk is ak.Note that ak22n2,so a particle is equally likely to appear in eithernon-trivial eigenstate with any given wave vector.At this point,we note that there are an exponential number of eigenvectors in which the initial state has a non-zero amplitude.In Section2.1,we observe that for this reason the general bound of Aharonov et al. [AAKV01]yields an exponential(upper)bound on the mixing time.In general,this bound performs poorly whenever the number of important eigenvalues is greater than the mixing time.Instead,we will use the Diaconis-Shahshahani bound on the total variation distance in terms of the Fourier coefficients of the probability[Dia88].If P t x is the probability of the particle being observed at position x at time t,and U is the uniform distribution,then the total variation distance is bounded byP t U214∑k00k11˜Pt k214n1∑k1nk˜Pt k2(2)Here we exclude both the constant term and the parity term k 11;since our walk changes position at every step,we only visit vertices with odd or even parity at odd or even times respectively.Thus U here means the uniform distribution with probability 2n 1on the vertices of appropriate parity.To find ˜Pt k ,we first need ˜Ψt k .As Nayak and Vishwanath [NV00]did for the walk on the line,we start by calculating the t th matrix power of U k .This isU t k a 1t aa a1t c ......b 1t bc bb 1t ......wherea cos ωk t 1t n k b cos ωk t 1t k and c sin ωk t Starting with the uniform initial state,the wave function after t steps is˜Ψt k 2n 2n cos ωk t n k sin ωk t n kcos ωk t n kksin ωk t k (3)In the next two sections we will use this diagonalization to calculate the average and instantaneous mixing times,which are Ωn 32and Θn respectively.2.1Bounds on the average mixing time of the discrete-time walkIn this section,we prove Theorem 3.To do this,it’s sufficient to calculate the amplitude at the origin.Fouriertransforming Equation 3back to real space at x 00gives ψt 02n 2∑k ˜Ψt k 2n n n ∑k 0n k cos ωk t cos ωk tnThe probability the particle is observed at the origin after t steps is thenP t 0ψt 022nn ∑k 0n k cos ωk t 2Let k 1x n 2.For small x ,k is near the peak of the binomial distribution,and ωk cos 1x π2x O x 3so the angle θbetween ωk for successive values of k is roughly constant,θ2n O x 2leading to constructive interference if θt 2π.Specifically,let t m be the even integer closest to πmn forinteger m .Then cos ωk t mcos 2πkm O x 3mn 1O x 6m 2n 2.By a standard Chernoff bound,2n ∑k 1x n 2n k o 1so long as x ωn 12.Let x νn n 12where νn is a function that goes to infinity slowly as a function of n .We then write P t 0o 12n1x n 2∑k 1x n 2n k 1O x 6m 2n 221O νn 6m 2nwhich is 1o 1as long as m o n 12νn 3,in which case t mo n 32νn 3.For a functionψ:n2n withψ21and a set S n2,we say thatψis c-supported on S if the probability x S is at least c,i.e.∑x S d nψx d2c.The discussion above shows thatψt m is 1o1-supported on0for appropriate t mπmn.Note that ifψis c-supported on0then,as U is local,U kψmust be c1c2c1-supported on W k,the set of vertices of weight k.(The factor of1c is due to potential cancellation with portions of the wave function supported outside0.)Inparticular,at times t m k,for k n2n,ψt m k is1o1-supported on W n2x.If x x nωn,then W12δn2n o1and,evidently,the average1T∑T i P i has total variation distance1o1from the uniform distribution if T o n32.Thus we see that in the sense of[AAKV01],the discrete-time quantum walk is actually slower than the classical walk.In the next section,however,we show that its instantaneous mixing time is only linear in n.We now observe that the general bound of[AAKV01]predicts an average mixing time for the n-cube which is exponential in n.In that article it is shown that the variation distance between¯P T and the uniform distribution(or more generally,the limiting distribution lim T∞¯P T)is bounded by a sum over distinct pairs of eigenvalues,¯PT U 2T∑i j s tλiλja i2λiλj(4)where a iψ0v i is the component of the initial state along the eigenvector v i.(Since this bound includes eigenvaluesλj for which a j0,we note that it also holds when we replace a i2with a i a j,using the same reasoning as in[AAKV01].)For the quantum walk on the cycle of length n,this bound gives an average mixing time of O n log n. For the n-cube,however,there are exponentially many pairs of eigenvectors with distinct eigenvalues,all ofwhich have a non-zero component in the initial state.Specifically,for each Hamming weight k there are nk non-trivial eigenvectors each with eigenvalue e iωk and e iωk.These complex conjugates are distinct from each other for0k n,and eigenvalues with distinct k are also distinct.The number of distinct pairs is thenn1∑k1nk24n∑k k0nknkΩ4nTaking a k2n2from Equation1and the fact thatλiλj2since theλi are on the unit circle, we see that Equation4gives an upper bound on theε-average mixing time of sizeΩ2nε.In general,this bound will give a mixing time of O Mεwhenever the initial state is distributed roughly equally over M eigenvectors,and when these are roughly equally distributed overω1distinct eigenvalues.2.2The instantaneous mixing time of the discrete-time walkTo prove Theorem1we could calculateΨt x by Fourier transforming˜P t k back to real space for all x. However,this calculation turns out to be significantly more awkward than calculating the Fourier trans-form of the probability distribution,˜P t k,which we need to apply the Diaconis-Shahshahani bound.Since P t xΨt xΨt x,and since multiplications in real space are convolutions in Fourier space,we perform a convolution over n2:˜Pt k∑k ˜Ψt k˜Ψt k kwhere the inner product is defined on the direction space,u v∑n i1u i v i.We write this as a sum over j, the number of bits of overlap between k and k,and l,the number of bits of k outside the bits of k(and so overlapping with k k).Thus k has weight j l,and k k has weight k j l.Calculating the dot product˜Ψt k˜Ψt k k explicitly from Equation3as a function of these weights and overlaps gives˜P t k 12k∑j0n k∑l0kjn klcosωj l t cosωk j l t A sinωj l t sinωk j l t(5)whereA cosωk cosωj l cosωk j l sinωj l sinωk j lThe reader can check that this gives˜P t01for the trivial Fourier component where k0,and˜P t n 1t for the parity term where k n.Using the identities cos a cos b12cos a b cos a b and sin a sin b12cos a b cos a b we can re-write Equation5as˜P t k 12k∑j0n k∑l0kjn kl1A2cosωt1A2cosωt12k∑j0n k∑l0kjn klY(6)whereωωj lωk j l.The terms cosωt in Y are rapidly oscillating with a frequency that increases with t.Thus,unlike the walk on the cycle,the phase is rapidly oscillating everywhere,as a function of either l or j.This will make the dominant contribution to˜P t k exponentially small when t nπ4,giving us a small variation distance when we sum over all k.To give some intuition for the remainder of the proof,we pause here to note that if Equation6were an integral rather than a sum,we could immediately approximate the rate of oscillation of Y tofirst order at the peaks of the binomials,where j k2and l n k 2.One can check that dωk d k2n and hence dωd l dωd j4n.Since A1,we would then write˜Pt k O 12k∑j0n k∑l0kjn kle4i jt n e4ilt nwhich,using the binomial theorem,would give˜Pt k O 1e4it n2k1e4it n2n kcos k2tncos n k2tn(7)In this case the Diaconis-Shahshahani bound and the binomial theorem giveP t U214∑0k nnkcos k2tncos n k2tn2122cos22tnn1cos22tnn1If we could take t to be the non-integer valueπ4n,these cosines would be zero.This will,in fact,turn out to be the right answer.But since Equation6is a sum,not an integral,we have to be wary of resonances where the oscillations are such that the phase changes by a multiple of2πbetween adjacent terms,in which case these terms will interfere constructively rather than destructively.Thus to show that thefirst-order oscillation indeed dominates,we have a significant amount of work left to do.The details of managing these resonances can be found in Appendix B.The process can be summarized as follows:i.)we compute the Fourier transform of the quantity Y in Equation6,since the sum of Equation6 can be calculated for a single Fourier basis function using the binomial theorem;ii.)the Fourier transform0.20.40.60.810.20.40.60.81(a)Variation distance at time t as a function of t n .(b)Log 2Probability as a function of Hamming weight.Figure 1:Graph (a)plots an exact calculation of the total variation distance after t steps of the quantum walk for hypercubes of dimension 50,100,and 200,as a function of t n .At t n π4the variation distance is small even though the walk has not had time to cross the entire graph.This happens because the distribution is roughly uniform across the equator of the n -cube where the vast majority of the points are located.Note that the window in which the variation distance is small gets narrower as n increases.Graph (b)shows the log 2probability distribution on the 200-dimensional hypercube as a function of Hamming distance from the starting point after 157π4n steps.The probability distribution has a plateau of 2199at the equator,matching the uniform distribution up to parity.of Y can be asymptotically bounded by the method of stationary phase.The dominant stationary point corresponds to the first-order oscillation,but there are also lower-order stationary points corresponding to faster oscillations;so iii.)we use an entropy bound to show that the contribution of the other stationary points is exponentially small.To illustrate our result,we have calculated the probability distribution,and the total variation distance from the uniform distribution (up to parity),as a function of time for hypercubes of dimension 50,100,and 200.In order to do this exactly,we use the walk’s permutation symmetry to collapse its dynamics to a function only of Hamming distance.In Figure 1(a)we see that the total variation distance becomes small when t n π4,and in Figure 1(b)we see how the probability distribution is close to uniform on a “plateau”across the hypercube’s equator.Since this is where the vast majority of the points are located,the total variation distance is small even though the walk has not yet had time to cross the entire graph.3The continuous-time walkIn the case of the hypercube,the continuous-time walk turns out to be particularly easy to analyze.Theadjacency matrix,normalized by the degree,is H x y1n if x and y are adjacent,and 0otherwise.Interpreting H as the Hamiltonian treats it as the energy operator,and of course increasing the energy makes the system run faster;we normalize by the degree n in order to keep the maximum energy of the system,and so the rate at which transitions occur,constant as a function of n .The eigenvectors of H and U t are simply the Fourier basis functions:if v k x1k x then Hv k 12k n v k and U t v k e it 12k n v k where we again use k to denote the Hamming weight of k .If our initial wave vector has a particle at 0,then its initial Fourier spectrum is uniform,and at time t we have˜Ψt k 2n 2e it 12k n Again writing the probability P as the convolution of Ψwith Ψin Fourier space,。

量子随机行走的理论模型与实验操作方法量子随机行走（Quantum Random Walk, QRW）是量子力学中的一种基本运动模型，它描述的是粒子在格子上的随机行走过程。

相较于经典随机行走，量子随机行走具有更为复杂的行为，对于探索新的量子计算和量子信息领域具有重要的意义。

本文将介绍量子随机行走的理论模型，并探讨实验操作方法。

量子随机行走的理论模型可以分为离散和连续两类。

离散型量子随机行走是最早被研究的模型之一，它由Aharonov等人于1993年提出。

在离散型量子随机行走模型中，粒子在一个二维的格子上进行运动，每个格点上有两个状态，通常记作|0>和|1>。

在每一个时间步长，粒子根据一组量子门的演化规律进行状态转移。

这组量子门通常由Hadamard门和位移门组成，它们将粒子的行走规则由经典的概率性转化为量子的幺正动力学演化。

连续型量子随机行走是离散型量子随机行走的推广，它在时间和空间上都是连续的。

连续型量子随机行走模型于1996年由Farhi和Gutmann提出，其理论基础是量子速度速率方程。

在这种模型中，粒子在连续时间和连续空间上运动，其位置和动量可以通过量子力学中的坐标和动量算符来描述。

通过改变不同的哈密顿量，可以得到不同的连续型量子随机行走模型。

实验操作方法是研究量子随机行走的重要手段。

目前已经有多种方法用于实现量子随机行走的实验。

其中一种方法是使用光子作为量子比特，通过操纵光子的偏振态来模拟量子随机行走。

这种方法的优点是实验操作简单，可扩展性好。

另一种方法是使用超冷原子气体，通过调控超冷原子的内部自旋态来实现量子随机行走。

这种方法的优点是可以实现精密控制，粒子之间的相互作用比较强。

同时，还有其他一些实现量子随机行走的方法，如使用量子电路、核磁共振等。

实验操作方法的选择取决于具体的研究需求。

如果研究的是量子随机行走的基本行为和特性，那么光子或超冷原子的实验方法是比较合适的选择；如果研究的是量子随机行走在实际应用中的潜力，那么可能需要更加复杂的实验设置和技术手段。

原子物理学中的量子随机行走量子随机行走（Quantum Random Walk），是原子物理学中研究的重要课题之一。

它通过模拟单个粒子在离散空间中随机移动的过程，揭示了微观世界中非经典的行为特征。

量子随机行走的基本概念可以从经典随机行走开始理解。

在经典随机行走中，一个粒子在平面上进行二维随机移动，每一步只能向上、下、左或右移动。

经过多次随机选择方向并移动一步，粒子的位置会随着时间的推移逐渐扩散，形成一个随机分布。

而在量子随机行走中，粒子不再是经典粒子，而是由量子态描述的，具有波粒二象性。

它既可以表现为粒子的离散位置，也可以表现为波函数的连续分布。

在量子随机行走中，粒子的行为会受到“硬币”操作的影响。

这里的“硬币”是指量子系统的一个自由度，它的状态可以是0或1，分别代表右移或左移的方向。

在每一步移动之前，粒子都会与这个“硬币”进行相互作用，并根据测量结果决定下一步移动的方向。

而这个“硬币”操作通常由一系列基本的量子逻辑门实现，比如Hadamard门或CNOT门等。

值得注意的是，量子随机行走和经典随机行走之间存在很大差异。

经典随机行走是完全随机的，而量子随机行走则会在某些情况下表现出非经典的行为。

例如，当“硬币”处于纠缠态时，粒子的位置分布会出现干涉现象，形成一种周期性的特征。

这种干涉效应与经典随机行走的扩散过程形成鲜明对比，展现了量子行走的独特之处。

量子随机行走不仅在理论物理中具有重要意义，而且在实验方面也有广泛的应用。

通过实验可以观测到量子随机行走的多种行为特征，如扩散速率、干涉现象等。

这些实验结果不仅丰富了对量子行为的理解，也有助于开发新的量子技术。

量子随机行走的研究还涉及到许多其他领域，比如量子算法、量子通信以及量子传感等。

在量子算法中，量子随机行走可以用于设计一些特定问题的高效算法，如搜索和优化问题。

而在量子通信和传感中，量子随机行走可以用于实现更安全和高效的通信和传感方案。

实际上，量子随机行走已经成为量子信息领域的一个重要研究方向，吸引了越来越多的科学家的关注和研究。

量子随机行走与优化算法引言：量子计算作为一种新兴的计算模型，吸引了广泛的研究兴趣。

量子随机行走是量子计算中的重要概念之一，被广泛应用于优化算法中。

本文将介绍量子随机行走的基本概念和原理，并探讨其在优化算法中的应用。

一、量子随机行走的基本概念量子随机行走是一种模拟随机游走的量子算法，它利用量子叠加态和量子干涉效应，在搜索和优化问题中展现出独特的优势。

在经典随机行走中，一个粒子在一个图上随机游走，而在量子随机行走中，粒子的行为由量子态决定。

量子随机行走的基本概念可以通过一个简单的例子来说明。

假设有一个无限长的一维线性图，粒子从原点出发，每一步都有50%的概率向左或向右移动。

在经典随机行走中，粒子的位置服从高斯分布，而在量子随机行走中，粒子的位置服从波函数的模方分布。

这种分布的特点是波峰和波谷的位置在不断变化，从而提供了更多的搜索空间。

二、量子随机行走的原理量子随机行走的原理可以通过量子叠加态和量子干涉效应来解释。

在经典随机行走中，粒子在每一步都有50%的概率向左或向右移动，而在量子随机行走中，粒子的行为由量子态决定。

量子随机行走的量子态可以表示为一个张量积的形式，即粒子的位置和粒子的自旋态的张量积。

通过对粒子的位置和自旋态进行叠加，可以得到量子随机行走的量子态。

在量子随机行走的过程中，粒子的位置和自旋态会发生相互作用，从而产生量子干涉效应。

量子干涉效应是量子随机行走的核心原理之一。

在经典随机行走中，粒子的位置服从高斯分布，而在量子随机行走中，粒子的位置服从波函数的模方分布。

这种分布的特点是波峰和波谷的位置在不断变化，从而提供了更多的搜索空间。

三、量子随机行走在优化算法中的应用量子随机行走在优化算法中有着广泛的应用。

其中一个重要的应用是在图论中的最短路径问题。

最短路径问题是一个经典的优化问题，通过量子随机行走可以得到更快速的解决方法。

在经典的最短路径算法中，通常使用Dijkstra算法或者Bellman-Ford算法来求解最短路径。

量子随机游走的应用与优化随机游走是一种模拟随机行走过程的数学模型，其在许多领域中都有重要应用。

而随着量子计算的发展，量子随机游走（Quantum Random Walk，以下简称QRW）作为一种量子模拟方法，引起了研究人员的广泛兴趣。

本文将探讨量子随机游走在应用和优化方面的潜力。

首先，QRW在搜索算法中有着广泛的应用。

传统的搜索问题需要通过遍历所有可能的解空间来找到最优解，而这一过程的时间复杂度通常很高。

然而，在量子计算中，QRW可以通过一种"均匀性加速"的方式实现更高效的搜索。

具体来说，量子随机游走通过构建一个幺正变换操作来描述搜索问题，然后通过量子叠加和干涉来加速搜索的过程，并在一定概率下找到最优解。

这一技术在优化问题求解中具有潜在的巨大优势。

其次，QRW也可以应用于图论问题的求解。

图论是计算机科学中的一个重要领域，涉及到网络、路由等方面的优化问题。

传统的图论算法在处理规模较大的图时经常面临着复杂度过高的问题。

而QRW可以通过一种基于概率的扩散方式来解决这一问题。

具体来说，QRW将图的顶点映射到量子比特上，通过量子叠加和干涉的方式实现更高效的图搜索。

这种逼近图搜索问题的方法，有望解决许多实际问题中的优化难题。

另外，QRW还可以应用于优化传染病模型。

对于传染病的传播过程，人们通常使用传染病模型来研究和预测。

而QRW可以通过对传染病模型中的节点和边进行量子映射，从而实现更高效的传染病传播模拟。

这种基于量子随机游走的优化传染病模型，不仅可以提高计算效率，还可以更准确地预测传染病的传播路径和策略。

此外，QRW还有其他一些潜在的应用。

例如，在供应链管理中，通过量子随机游走可以更高效地实现供应链的优化调度；在社交网络分析中，QRW可以应用于社交关系的发现和分析等。

这些应用领域的研究仍处于初级阶段，但已经展示出了QRW在优化问题中的潜力。

总而言之，量子随机游走作为一种量子模拟方法，在应用和优化方面具有广泛的潜力。

量子随机行走及其应用量子随机行走（Quantum Random Walk）是一种基于量子力学原理的随机过程，近年来在量子计算、量子信息和量子模拟等领域引起了广泛的关注。

本文将介绍量子随机行走的基本概念，并探讨其在信息搜索、优化问题和生物化学等领域的应用。

一、量子随机行走的基本概念量子随机行走是一种泛化的随机游走模型，将经典的随机行走模型与量子力学中的量子叠加效应相结合。

在经典随机游走中，我们将粒子在一维空间上的运动建模为左右两种可能的移动方式。

而在量子随机行走中，粒子可以处于动态的叠加态，同时进行左右两种移动。

这种叠加态的存在使得量子随机行走在某些情况下表现出与经典随机行走截然不同的行为。

二、量子随机行走的应用1. 信息搜索量子随机行走被广泛应用于信息搜索问题。

经典的搜索算法，如随机游走和Grover搜索算法，在某些情况下需要大量的计算时间和资源。

而量子随机行走利用了量子叠加态的特性，可以加速搜索过程。

通过调整初始态和演化操作，可以使得粒子在搜索空间中快速找到目标。

这为高效的信息搜索提供了一种新的思路。

2. 优化问题量子随机行走也可以应用于解决优化问题。

例如，某些组合优化问题需要在庞大的搜索空间中找到全局最优解。

经典的优化算法，如遗传算法和模拟退火算法，通常需要大量的迭代和计算。

而量子随机行走结合了量子计算的优势，可以在较少的迭代次数内找到较好的解。

这对于求解复杂的优化问题具有重要意义。

3. 生物化学量子随机行走在生物化学中也有一定的应用。

生物分子在细胞内的传递过程中常常涉及到一系列的随机行走。

在经典随机行走模型中，这些传递过程可以通过概率转移矩阵描述。

而引入量子效应后，我们可以使用量子随机行走模型更加准确地描述生物分子的传递行为。

这有助于理解生物体内信息传递的机制，以及设计新的药物传递方式。

三、发展前景和挑战量子随机行走作为一种新的随机过程模型，具有广阔的应用前景。

随着量子计算技术的发展，量子随机行走将有更多的机会在不同领域展示其优势。

量子随机行走粒子的奇特漫游量子随机行走是一种具有奇特特性的粒子运动现象。

在经典物理学中，粒子的行为受到牛顿力学的限制，而在量子物理学中，粒子的运动方式则更为复杂和随机。

量子随机行走的概念由英国物理学家斯蒂芬·韦斯特提出，对于这一现象的探索有助于我们对量子世界的理解。

量子随机行走的粒子并不沿着传统的轨迹移动，而是以一种随机的方式在空间中进行移动。

这种移动方式有时会让人感到迷惑，因为它违背了经典物理学中预期的运动规律。

在量子随机行走中，粒子可以同时处于多个位置，以概率的方式进行跳跃，而不是像经典粒子一样按照确定的路径行进。

量子随机行走的粒子在一个由节点组成的结构上进行移动。

这个结构可以是一维线性结构，也可以是二维平面或者更高维度的结构。

在每个节点上，粒子可以按照一定的概率进行向前或向后的跳跃，同时还可以进行左右或者其他方向上的跳跃。

这种随机的移动方式使得粒子在空间中进行漫游，不断地改变位置。

量子随机行走的粒子具有一些奇特的性质。

首先，它可以达到的位置数目比经典粒子要多得多。

在经典物理学中，粒子只能在确定的轨迹上移动，而在量子随机行走中，粒子可以同时出现在几个节点上，这使得它能够到达更多的位置。

其次，量子随机行走的粒子具有波动性。

波动性是量子理论中非常重要的一个概念，它意味着物质的粒子可以表现出波动的特性。

在量子随机行走中，粒子同时处于多个位置，表现出波动的性质。

有趣的是，量子随机行走的粒子还可以通过后续的测量和干涉来影响它的运动。

通过对粒子的位置进行测量，可以观察到它的运动路径，并且可以通过在结构上施加干涉来改变粒子的行为。

这种干涉效应是量子物理学中一项重要的研究内容，也是实现量子随机行走的关键因素之一。

量子随机行走在理论和实验研究中都具有广泛的应用。

在理论研究方面，科学家们通过模拟和计算，探索量子随机行走在不同结构和条件下的行为规律。

在实验研究方面，科学家们使用量子系统，如离子阱、光学系统等，来模拟和观测粒子的随机行走现象。

量子计算技术中的量子随机游走随着科技的不断发展，量子计算技术逐渐成为了一个备受关注的领域。

其中，量子随机游走技术备受瞩目，被认为是未来量子计算机的重要组成部分之一。

那么，什么是量子随机游走技术？它背后的原理又是什么呢？量子随机游走是指，由量子系统进行的一种特殊的计算过程。

在量子随机游走中，一个量子粒子将被放置在一个走廊中，而该走廊由多个节点组成。

通过“量子掷币”，量子粒子将在不同节点中进行随机跳跃。

在这个过程中，粒子将会与其他量子粒子进行干涉，从而得到一些与经典物理难以比拟的结果。

这些结果将被采用来完成一些复杂的计算任务，例如图像处理等。

要理解量子随机游走的原理，首先必须了解量子叠加原理。

量子叠加原理指的是，当一个量子系统处于两个或多个可能状态的叠加态时，该系统将具有这些状态的所有特性。

例如，在一个叠加态粒子系统中，粒子将同时具有“自旋上”和“自旋下”的特性，而这些特性都可以被同时使用。

基于量子叠加原理，量子随机游走利用量子粒子的叠加状态完成随机跳跃。

在量子掷币时，量子系统将同时处于“正面”和“反面”的叠加态中。

这意味着，量子粒子将以等概率的方式跳到任何一个节点中。

然而，在跳跃时，量子粒子也将和其他量子粒子进行干涉，从而会生成一些不同于经典物理的结果。

这些结果将被采用来计算复杂的问题，例如搜索、排序和图像处理等。

当前，量子随机游走技术尚处于研究阶段。

尽管有一些实验已经证实了这种技术的巨大潜力，但要将它转化为实用技术仍然需要时间和资金的大量投入。

在未来，量子随机游走技术有望成为一种重要的量子计算机技术，带来革命性的改变。

在结论中，量子随机游走技术作为量子计算技术中的一项重要内容，它背后的原理是量子叠加原理。

这种技术可以用来计算复杂问题，例如搜索、排序和图像处理等。

虽然目前尚处于研究阶段，但它有着巨大的潜力，是未来量子计算机的一个重要组成部分。

未来，量子随机游走技术有望为我们带来巨大的改变。

量子计算中的量子随机游走算法随着计算机技术的飞速发展，人们对量子计算的研究也越来越深入。

量子计算具有高效性和破解某些加密算法的能力，备受学术界和工业界的关注，其中量子随机游走算法是一种具有潜力的算法。

量子随机游走算法是一种基于量子机制的算法，其基本原理是模拟量子随机游走的过程来解决问题。

量子随机游走算法具有很高的概率收敛性和高效性，可以解决一些经典算法难以处理的问题，例如快速搜索和图形分解问题等。

在量子随机游走算法中，随机漫步是其核心部分，其过程可以概括为：在某个初始状态下，随机漫步选择当前状态的其中一个子状态进行求解，然后以某种概率转移到下一个状态并继续选择子状态，直至随机漫步过程收敛为止。

量子随机游走算法则是将随机漫步的过程转化为量子态演化的过程，使用量子态演化算子实现随机漫步，从而进行问题求解。

举例来说，对于图形分解问题，即对于所给定的一个大整数N，求其质因数分解。

经典算法中，求解N的质因数分解（也称“整数分解”问题）是困难的。

但是，利用量子随机游走算法，可以在多项式时间内求解。

量子随机游走算法首先将整数分解问题转化为有向图问题，其中每个有向边上的权重是一个映射函数，类似于角色“马尔科夫过程” 。

接着，将该有向图转换为一个量子态，用量子态演化算子实现随机漫步过程，在多项式的时间内得到解决方案。

量子随机游走算法的核心之一是量子态的构建。

量子态的构建可以利用量子门操作来实现，通常是通过量子傅里叶变换实现，而量子傅里叶变换又是基于哈达玛变换和位移变换构建的。

因此，量子随机游走算法的实现需要优化量子门和量子傅里叶变换等操作的效率和精度，这需要对量子计算硬件和量子算法进行深入研究，并开发出一些新的量子计算技术和算法来加速量子随机游走算法的实现。

总体来说，量子随机游走算法是一种具有前途的算法，其广泛应用领域包括数学、计算机科学、物理学、化学、统计学等各个领域。

虽然量子计算目前在实现上面临着很多挑战，但是量子随机游走算法的研究和应用将会推动量子计算技术的发展，对未来的科技发展也将会产生深远的影响。