We investigate the use of discrete-time quantum walks to sample from an almost-uniform distribution, in the absence of any external source of randomness. Integers are encoded on the vertices of a cycle graph, and a quantum walker evolves for a fixed number of steps before its position is measured and recorded. The walker is then reset to the measured site, and the procedure is iterated to produce the sequence of random numbers. We show that when the quantum walk parameters, such as the coin operator and initial state, satisfy the conditions of the ergodic theorem for random walks on finite groups, the resulting sequence converges asymptotically to the uniform distribution. Although correlations between successive outcomes are unavoidable, they can be significantly reduced by a suitable choice of the evolution time. By analyzing the iterated convolution of the quantum walk transition probability and exploiting the ergodic theorem, we demonstrate convergence of the marginal distributions toward the uniform distribution in the asymptotic limit.
We investigate the use of discrete-time quantum walks to sample from an almost-uniform distribution, in the absence of any external source of randomness. Integers are encoded on the vertices of a cycle graph, and a quantum walker evolves for a fixed number of steps before its position is measured and recorded. The walker is then reset to the measured site, and the procedure is iterated to produce the sequence of random numbers. We show that when the quantum walk parameters, such as the coin operator and initial state, satisfy the conditions of the ergodic theorem for random walks on finite groups, the resulting sequence converges asymptotically to the uniform distribution. Although correlations between successive outcomes are unavoidable, they can be significantly reduced by a suitable choice of the evolution time. By analyzing the iterated convolution of the quantum walk transition probability and exploiting the ergodic theorem, we demonstrate convergence of the marginal distributions toward the uniform distribution in the asymptotic limit.
Abstract We investigate the use of discrete-time quantum walks to sample from an almost-uniform distribution, in the absence of any external source of randomness. Integers are encoded on the vertices of a cycle graph, and a quantum walker evolves for a fixed number of steps before its position is measured and recorded. The walker is then reset to the measured site, and the procedure is iterated to generate the sequence of outcomes. We show that when the number of nodes of the graph is odd, i.e. the condition of the ergodic theorem for classical random walks on finite groups are satisfied, the marginal distributions converges to the uniform distribution. Although correlations between successive outcomes are unavoidable, they can be significantly reduced by a suitable choice of the evolution time.
Quantum walks on graphs are ubiquitous in quantum computing finding a myriad of applications. Likewise, random walks on graphs are a fundamental building block for a large number of algorithms with diverse applications. While the relationship between quantum and random walks has been recently discussed in specific scenarios, this work establishes a formal equivalence between the two processes on arbitrary finite graphs and general conditions for shift and coin operators. It requires empowering random walks with time heterogeneity, where the transition probability of the walker is non-uniform and time dependent. The equivalence is obtained by equating the probability of measuring the quantum walk on a given node of the graph and the probability that the random walk is at that same node, for all nodes and time steps. The first result establishes procedure for a stochastic matrix sequence to induce a random walk that yields the exact same vertex probability distribution sequence of any given quantum walk, including the scenario with multiple interfering walkers. The second result establishes a similar procedure in the opposite direction. Given any random walk, a time-dependent quantum walk with the exact same vertex probability distribution is constructed. Interestingly, the matrices constructed by the first procedure allows for a different simulation approach for quantum walks where node samples respect neighbor locality and convergence is guaranteed by the law of large numbers, enabling efficient (polynomial-time) sampling of quantum graph trajectories. Furthermore, the complexity of constructing this sequence of matrices is discussed in the general case.
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 analogs of classical random walks. The present paper 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 nonhomogeneous 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 a quantum walk 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.
With unitary coin operator that is a random matrix drawn from a uniform distribution with respect to the Haar measure, we construct a variant of a discrete quantum random walk using a d-regular undirected connected simple graph. With each step of the walk, the coin operator random matrix is drawn independently from the distribution. The expectation value over the distribution of random unitaries for the associated quantum channel gives a depolarization channel for the coin subspace and resembles the associated classical random walk where the direction a walker steps depends upon the outcome of a fair coin or balanced d-sided dice. Remarkably, despite the fact that the averaged channel depolarizes the coin subspace, measurements in the vertex subspace can be designed that would reveal information about the initial quantum state, even after many iterations of the channel. We illustrate with examples of quantum walks on Cayley graphs of Abelian groups, such as the cycle and hypercube graphs. For Cayley graphs of Abelian groups, the averaged channel is dephasing in the Fourier basis. These quantum walks are examples of bipartite strongly interacting systems, where one subsystem is strongly perturbed, yet information about the initial state in the other subsystem is potentially measurable forever. Due to its decoherence, a quantum random walk with a Haar-random coin would not be useful in search algorithms but could aid in understanding quantum systems with strongly perturbed subsystems.
We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., nonrandom) case, we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other. For example, a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense, with any lattice dimension and coin dimension. We find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time, namely as the distribution of the discrete time analog of the group velocity. In the random case, we let a Markov chain (control process) pick in each step one of finitely many unitary walks, in the sense described above. In ballistic order, we find a nonrandom drift which depends only on the mean of the control process and not on the initial state. In diffusive scaling, the limiting distribution is asymptotically Gaussian, with a covariance matrix (diffusion matrix) depending on momentum. The diffusion matrix depends not only on the mean but also on the transition rates of the control process. In the nonrandom limit, i.e., when the coins chosen are all very close or the transition rates of the control process are small, leading to long intervals of ballistic evolution, the diffusion matrix diverges. Our method is based on spatial Fourier transforms, and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum.