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Register a new userAn HHL-Based Algorithm for Computing Hitting Probabilities of Quantum Random Walks
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We present a novel application of the HHL (Harrow-Hassidim-Lloyd) algorithm -- a quantum algorithm solving systems of linear equations -- in solving an open problem about quantum random walks, namely computing hitting (or absorption) probabilities of a general (not only Hadamard) one-dimensional quantum random walks with two absorbing boundaries. This is achieved by a simple observation that the problem of computing hitting probabilities of quantum random walks can be reduced to inverting a matrix. Then a quantum algorithm with the HHL algorithm as a subroutine is developed for solving the problem, which is faster than the known classical algorithms by numerical experiments.
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Let $X$ be the constrained random walk on $mathbb{Z}_+^d$ $d >2$, having increments $e_1$, $-e_i+e_{i+1}$ $i=1,2,3,...,d-1$ and $-e_d$ with probabilities $lambda$, $mu_1$, $mu_2$,...,$mu_d$, where ${e_1,e_2,..,e_d}$ are the standard basis vectors. The process $X$ is assumed stable, i.e., $lambda < mu_i$ for all $i=1,2,3,...,d.$ Let $tau_n$ be the first time the sum of the components of $X$ equals $n$. We derive approximation formulas for the probability ${mathbb P}_x(tau_n < tau_0)$. For $x in bigcup_{i=1}^d Big{x in {mathbb R}^d_+: sum_{j=1}^{i} x(j)$ $> left(1 - frac{log lambda/min mu_i}{log lambda/mu_i}right) Big}$ and a sequence of initial points $x_n/n rightarrow x$ we show that the relative error of the approximation decays exponentially in $n$. The approximation formula is of the form ${mathbb P}_y(tau < infty)$ where $tau$ is the first time the sum of the components of a limit process $Y$ is $0$; $Y$ is the process $X$ as observed from a point on the exit boundary except that it is unconstrained in its first component (in particular $Y$ is an unstable process); $Y$ and ${mathbb P}_y(tau< infty)$ arise naturally as the limit of an affine transformation of $X$ and the probability ${mathbb P}_x(tau_n < tau_0).$ The analysis of the relative error is based on a new construction of supermartingales. We derive an explicit formula for ${mathbb P}_y(tau < infty)$ in terms of the ratios $lambda/mu_i$ which is based on the concepts of harmonic systems and their solutions and conjugate points on a characteristic surface associated with the process $Y$; the derivation of the formula assumes $mu_i eq mu_j$ for $i eq j.$
In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case. Further, we prove that for any reversible ergodic Markov chain $P$, the quantum hitting time of the quantum analogue of $P$ has the same order as the square root of the classical hitting time of $P$. We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. Finally, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem for the 2D grid. Extending his result, we show that for any state-transitive Markov chain with unique marked state, the quantum hitting time is of the same order for both the detection and finding problems.
We propose an iterative improvement method for the Harrow-Hassidim-Lloyd (HHL) algorithm to solve a linear system of equations. This is a quantum-classical hybrid algorithm. The accuracy is essential to solve the linear system of equations. However, the accuracy of the HHL algorithm is limited by the number of quantum bits used to express the eigenvalues of the matrix. Our iterative method improves the accuracy of the HHL solutions, and gives higher accuracy which surpasses the accuracy limited by the number of quantum bits. In practical HHL algorithm, a huge number of measurements is required to obtain good accuracy, even if we provide a sufficient number of quantum bits for the eigenvalue expression, since the solution is statistically processed from the measurements. Our improved iterative method can reduce the number of measurements. Moreover, the sign information for each eigenstate of the solution is lost once the measurement is made, although the sign is significant. Therefore, the naive iterative method of the HHL algorithm may slow down, especially, when the solution includes wrong signs. In this paper, we propose and evaluate an improved iterative method for the HHL algorithm that is robust against the sign information loss, in terms of the number of iterations and the computational accuracy.
We present a stochastic quantum computing algorithm that can prepare any eigenvector of a quantum Hamiltonian within a selected energy interval $[E-epsilon, E+epsilon]$. In order to reduce the spectral weight of all other eigenvectors by a suppression factor $delta$, the required computational effort scales as $O[|log delta|/(p epsilon)]$, where $p$ is the squared overlap of the initial state with the target eigenvector. The method, which we call the rodeo algorithm, uses auxiliary qubits to control the time evolution of the Hamiltonian minus some tunable parameter $E$. With each auxiliary qubit measurement, the amplitudes of the eigenvectors are multiplied by a stochastic factor that depends on the proximity of their energy to $E$. In this manner, we converge to the target eigenvector with exponential accuracy in the number of measurements. In addition to preparing eigenvectors, the method can also compute the full spectrum of the Hamiltonian. We illustrate the performance with several examples. For energy eigenvalue determination with error $epsilon$, the computational scaling is $O[(log epsilon)^2/(p epsilon)]$. For eigenstate preparation, the computational scaling is $O(log Delta/p)$, where $Delta$ is the magnitude of the orthogonal component of the residual vector. The speed for eigenstate preparation is exponentially faster than that for phase estimation or adiabatic evolution.
Classical random walks on finite graphs have an underrated property: a walk from any vertex can reach every other vertex in finite time, provided they are connected. Discrete-time quantum walks on finite connected graphs however, can have infinite hitting times. This phenomenon is related to graph symmetry, as previously characterized by the group of direction-preserving graph automorphisms that trivially affect the coin Hilbert space. If a graph is symmetric enough (in a particular sense) then the associated quantum walk unitary will contain eigenvectors that do not overlap a set of target vertices, for any coin flip operator. These eigenvectors span the Infinite Hitting Time (IHT) subspace. Quantum states in the IHT subspace never reach the target vertices, leading to infinite hitting times. However, this is not the whole story: the graph of the 3D cube does not satisfy this symmetry constraint, yet quantum walks on this graph with certain symmetric coins can exhibit infinite hitting times. We study the effect of coin symmetry by analyzing the group of coin-permutation symmetries (CPS): graph automorphisms that act nontrivially on the coin Hilbert space but leave the coin operator invariant. Unitaries using highly symmetric coins with large CPS groups, such as the permutation-invariant Grover coin, are associated with higher probabilities of never arriving, as a result of their larger IHT subspaces.
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