We set the ground for a theory of quantum walks on graphs-the generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.We set the ground for a theory of quantum walks on graphs-the generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.

ACM Symposium on Theory of Computing (STOC)Jul 6, 2001638 citations

Bounds for mixing time of quantum walks on finite graphs

Vladislav Kargin

Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov et al (2002 Proc. 33rd STOC ( 2001 ) (New York: ACM) pp 50–9 arXiv:quant-ph/0012090v2) and it is found that for particular examples of walks on a cycle, a hypercube and a complete graph, quantum walks provide no speedup in mixing over the classical counterparts. In addition, non-unitary quantum walks (i.e. walks with decoherence) are considered and a criterion for their convergence to the unique stationary distribution is derived.Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov et al (2002 Proc. 33rd STOC ( 2001 ) (New York: ACM) pp 50–9 arXiv:quant-ph/0012090v2) and it is found that for particular examples of walks on a cycle, a hypercube and a complete graph, quantum walks provide no speedup in mixing over the classical counterparts. In addition, non-unitary quantum walks (i.e. walks with decoherence) are considered and a criterion for their convergence to the unique stationary distribution is derived.

arXiv:1004.0188Journal of Physics A Mathematical and TheoreticalJul 15, 20108 citations

Relations between quantum walks, open quantum walks, and lifted walks: the cycle graph

Simon Apers, Alain Sarlette

The convergence time of a random walk on a graph towards its stationary distribution is an important indication of the efficiency of random algorithms based on it. Quantum random walks have been shown to allow quadratically accelerated convergence for large graphs, at least in some cases. The famous Grover search algorithm has been shown to actually fit this framework in an abstracted setting. Yet also with classical dynamics, simple mechanisms have been proposed which allow to quadratically accelerate the convergence with respect to a standard random walk. Some basic principles have been conjectured to cause this acceleration, basically transforming a diffusion-like behavior into a more transport-like behavior, but with remaining trail. We have worked out the equivalence of all these accelerating settings for the simplest example: the cycle graph. Quantum coherences turn out to play no major role and a classical feedback structure can be identified.The convergence time of a random walk on a graph towards its stationary distribution is an important indication of the efficiency of random algorithms based on it. Quantum random walks have been shown to allow quadratically accelerated convergence for large graphs, at least in some cases. The famous Grover search algorithm has been shown to actually fit this framework in an abstracted setting. Yet also with classical dynamics, simple mechanisms have been proposed which allow to quadratically accelerate the convergence with respect to a standard random walk. Some basic principles have been conjectured to cause this acceleration, basically transforming a diffusion-like behavior into a more transport-like behavior, but with remaining trail. We have worked out the equivalence of all these accelerating settings for the simplest example: the cycle graph. Quantum coherences turn out to play no major role and a classical feedback structure can be identified.

Nov 16, 20150 citations

Quantum Walks can Unitarily Represent Random Walks on Finite Graphs

Matheus Guedes de Andrade, Franklin de Lima Marquezino, Daniel R. Figueiredo

Quantum and random walks have been shown to be equivalent in the following sense: a time-dependent random walk can be constructed such that its vertex distribution at all time instants is identical to the vertex distribution of any discrete-time coined quantum walk on a finite graph. This equivalence establishes a deep connection between the two processes, far stronger than simply considering quantum walks as quantum analogues of classical random walks. The present work strengthens this connection by providing a construction that establishes this equivalence in the reverse direction: a unitary time-dependent quantum walk can be constructed such that its vertex distribution is identical to the vertex distribution of any random walk on a finite graph at all time instants. The construction shown here describes a quantum walk that matches a random walk without measurements at all time steps (an otherwise trivial statement): measurement is performed in a quantum walk that evolved unitarily until a given time $t$ such that its vertex distribution is identical to the random walk at time $t$ . The construction procedure is general, covering both homogeneous and non-homogeneous random walks. For homogeneous random walks, unitary evolution implies time dependency for the quantum walk, since homogeneous quantum walks do not converge under arbitrary initial conditions, while a broad class of random walks does. Thus, the absence of convergence demonstrated for quantum walks in its debut comes from both time-homogeneity and unitarity, rather than unitarity alone, and our results shed light on the power of quantum walks to generate samples for arbitrary probability distributions. Finally, the construction here proposed is used to simulate quantum walks that match uniform random walks on the cycle and the torus.Quantum and random walks have been shown to be equivalent in the following sense: a time-dependent random walk can be constructed such that its vertex distribution at all time instants is identical to the vertex distribution of any discrete-time coined quantum walk on a finite graph. This equivalence establishes a deep connection between the two processes, far stronger than simply considering quantum walks as quantum analogues of classical random walks. The present work strengthens this connection by providing a construction that establishes this equivalence in the reverse direction: a unitary time-dependent quantum walk can be constructed such that its vertex distribution is identical to the vertex distribution of any random walk on a finite graph at all time instants. The construction shown here describes a quantum walk that matches a random walk without measurements at all time steps (an otherwise trivial statement): measurement is performed in a quantum walk that evolved unitarily until a given time $t$ such that its vertex distribution is identical to the random walk at time $t$. The construction procedure is general, covering both homogeneous and non-homogeneous random walks. For homogeneous random walks, unitary evolution implies time dependency for the quantum walk, since homogeneous quantum walks do not converge under arbitrary initial conditions, while a broad class of random walks does. Thus, the absence of convergence demonstrated for quantum walks in its debut comes from both time-homogeneity and unitarity, rather than unitarity alone, and our results shed light on the power of quantum walks to generate samples for arbitrary probability distributions. Finally, the construction here proposed is used to simulate quantum walks that match uniform random walks on the cycle and the torus.

arXiv:2103.06463Mar 11, 20210 citations

Quantum walks, the discrete wave equation and Chebyshev polynomials

Simon Apers, Laurent Miclo

A quantum walk is the quantum analogue of a random walk. While it is relatively well understood how quantum walks can speed up random walk hitting times, it is a long-standing open question to what extent quantum walks can speed up the spreading or mixing rate of random walks on graphs. In this expository paper, inspired by a blog post by Terence Tao, we describe a particular perspective on this question that derives quantum walks from the discrete wave equation on graphs. This yields a description of the quantum walk dynamics as simply applying a Chebyshev polynomial to the random walk transition matrix. This perspective decouples the problem from its quantum origin, and highlights connections to earlier (non-quantum) work and the use of Chebyshev polynomials in random walk theory as in the Varopoulos-Carne bound. We illustrate the approach by proving a weak limit of the quantum walk dynamics on the lattice. This gives a different proof of the quadratically improved spreading behavior of quantum walks on lattices.A quantum walk is the quantum analogue of a random walk. While it is relatively well understood how quantum walks can speed up random walk hitting times, it is a long-standing open question to what extent quantum walks can speed up the spreading or mixing rate of random walks on graphs. In this expository paper, inspired by a blog post by Terence Tao, we describe a particular perspective on this question that derives quantum walks from the discrete wave equation on graphs. This yields a description of the quantum walk dynamics as simply applying a Chebyshev polynomial to the random walk transition matrix. This perspective decouples the problem from its quantum origin, and highlights connections to earlier (non-quantum) work and the use of Chebyshev polynomials in random walk theory as in the Varopoulos-Carne bound. We illustrate the approach by proving a weak limit of the quantum walk dynamics on the lattice. This gives a different proof of the quadratically improved spreading behavior of quantum walks on lattices.

arXiv:2402.07809Feb 12, 20240 citations

Quantum Walk on the Line

Nayak Ashwin, Ashvin Vishwanath

Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider {\em quantum\/} walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the ``Hadamard walk''''. In particular, we show that after~ $t$ time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval~ $[-t/\sqrt{2},\;t/\sqrt{2}]$ . This implies that the same walk defined on the circle mixes in {\em linear\/} time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider {\em quantum\/} walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the ``Hadamard walk''''. In particular, we show that after~$t$ time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval~$[-t/\sqrt{2},\;t/\sqrt{2}]$. This implies that the same walk defined on the circle mixes in {\em linear\/} time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.

Dec 13, 2000122 citations

t

such that its vertex distribution is identical to the random walk at time

t

. The construction procedure is general, covering both homogeneous and non-homogeneous random walks. For homogeneous random walks, unitary evolution implies time dependency for the quantum walk, since homogeneous quantum walks do not converge under arbitrary initial conditions, while a broad class of random walks does. Thus, the absence of convergence demonstrated for quantum walks in its debut comes from both time-homogeneity and unitarity, rather than unitarity alone, and our results shed light on the power of quantum walks to generate samples for arbitrary probability distributions. Finally, the construction here proposed is used to simulate quantum walks that match uniform random walks on the cycle and the torus.

t

time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval~

[-t/\sqrt{2},\;t/\sqrt{2}]

. This implies that the same walk defined on the circle mixes in {\em linear\/} time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.