This paper proposes a quantum algorithm for Markov chain spectral gap estimation that is quasi-optimal (i.e., optimal up to a polylogarithmic factor) in the number of vertices for all parameters, and additionally quasi-optimal in the reciprocal of the spectral gap itself, if the permitted relative error is above some critical value. In particular, these results constitute an almost quadratic advantage over the best-possible classical algorithm. Our algorithm also improves on the quantum state of the art, and we contend that this is not just theoretically interesting but also potentially practically impactful in real-world applications: knowing a Markov chain's spectral gap can speed-up sampling in Markov chain Monte Carlo.
Our approach uses the quantum singular value transformation, and as a result we also develop some theory around block-encoding Markov chain transition matrices, which is potentially of independent interest. In particular, we introduce explicit block-encoding methods for the transition matrices of two algebraically-defined classes of Markov chains.
This paper proposes a quantum algorithm for Markov chain spectral gap estimation that is quasi-optimal (i.e., optimal up to a polylogarithmic factor) in the number of vertices for all parameters, and additionally quasi-optimal in the reciprocal of the spectral gap itself, if the permitted relative error is above some critical value. In particular, these results constitute an almost quadratic advantage over the best-possible classical algorithm. Our algorithm also improves on the quantum state of the art, and we contend that this is not just theoretically interesting but also potentially practically impactful in real-world applications: knowing a Markov chain's spectral gap can speed-up sampling in Markov chain Monte Carlo.
Our approach uses the quantum singular value transformation, and as a result we also develop some theory around block-encoding Markov chain transition matrices, which is potentially of independent interest. In particular, we introduce explicit block-encoding methods for the transition matrices of two algebraically-defined classes of Markov chains.
Quantum-enhanced Markov chain Monte Carlo, an algorithm in which configurations are proposed through a measured quantum quench and accepted or rejected by a classical algorithm, has been proposed as a possible method for robust quantum speedup on imperfect quantum devices. While this procedure is resilient to noise and control imperfections, the potential for quantum advantage is unclear. By upper-bounding the gap of the Markov chain, we identify competing factors that limit the algorithm's performance. One needs the quantum dynamics to efficiently delocalize the system over a range of classical states, but it is also detrimental to introduce too much entropy through the quench. Specifically, we show that in the long-time limit, the gap of the Markov chain is bounded by the inverse participation ratio of the classical states in the eigenstate basis, showing there is no advantage when quenching to an ergodic system. For the paradigmatic Sherrington-Kirkpatrick and three-spin model, we identify the regime of optimal spectral gap scaling and link it to the system's eigenstate properties.
We present a quantum algorithm based on classical fully polynomial randomized approximation schemes (FPRASs) for estimating partition functions that combine simulated annealing with the Monte Carlo Markov chain method and use nonadaptive cooling schedules. We achieve a twofold polynomial improvement in time complexity: a quadratic reduction with respect to the spectral gap of the underlying Markov chains and a quadratic reduction with respect to the parameter characterizing the desired accuracy of the estimate output by the FPRAS. Both reductions are intimately related and cannot be achieved separately. First, we use Grover's fixed-point search, quantum walks, and phase estimation to efficiently prepare approximate coherent encodings of stationary distributions of the Markov chains. The speed up we obtain in this way is due to the quadratic relation between the spectral and phase gaps of classical and quantum walks. The second speed up with respect to accuracy comes from generalized quantum counting used instead of classical sampling to estimate expected values of quantum observables.
Markov chain methods are remarkably successful in computational physics, machine learning, and combinatorial optimization. The cost of such methods often reduces to the mixing time, i.e., the time required to reach the steady state of the Markov chain, which scales as δ−1 , the inverse of the spectral gap. It has long been conjectured that quantum computers offer nearly generic quadratic improvements for mixing problems. However, except in special cases, quantum algorithms achieve a run-time of O(δ−1N) , which introduces a costly dependence on the Markov chain size N, not present in the classical case. Here, we re-address the problem of mixing of Markov chains when these form a slowly evolving sequence. This setting is akin to the simulated annealing setting and is commonly encountered in physics, material sciences and machine learning. We provide a quantum memory-efficient algorithm with a run-time of O(δ−1N4) , neglecting logarithmic terms, which is an important improvement for large state spaces. Moreover, our algorithms output quantum encodings of distributions, which has advantages over classical outputs. Finally, we discuss the run-time bounds of mixing algorithms and show that, under certain assumptions, our algorithms are optimal.
Large-scale eigenvalue problems pose a significant challenge to classical computers. While there are efficient quantum algorithms for unitary or Hermitian matrices, eigenvalue problems for non-normal matrices remain open in quantum computing. In this work, we propose a quantum algorithm that given a non-normal matrix, outputs an estimate of an eigenvalue to within additive error ε with probability at least 1−pfail. Our estimation strategy is to sample points on the complex plane and examine the distance between the sampled point and the eigenvalues. We show that the distance is related to the smallest singular value of the shifted matrix, hence reducing the problem to ground state energy estimation via Hermitianization. With the knowledge of an eigenvalue, we are able to prepare the associated eigenvector using ground state preparation. Our estimating scheme can also be modified to approximate the extreme eigenvalue, and in particular the spectral gap. The algorithm is implemented based on the block encoding input model and requires O(κ2ε−(2m−1)log(1/pfail)) queries to the block encoding oracle. Our algorithm is the first general eigenvalue algorithm that achieves this scaling. We also perform numerical simulation to validate the algorithms.
This dissertation is concerned with a variety of problems in quantum computing and the spectral theory of quantum-mechanical channels and Hamiltonians. We begin with the problem of quantum gate synthesis. The challenge is to find a fixed, finite set S of quantum gates such that, given a quantum circuit component U acting on n qubits and target precision ε, there is a short, efficiently computable composition of gates from S that approximates U to precision ε. This problem has already led to a rich literature in the case n=1 with deep connections to modern number theory. In this dissertation we present the state-of-the-art algorithm for a single qubit, achieving length 7 log (1/ε). The techniques used include the lattice structure of certain dense subgroups of PU(2) and Lenstra's algorithm for integer programming in convex domains. The next topic considered is scientific computing algorithms intended to run on a quantum computer. Specifically, we study a framework for computing functions of normal matrices using a quantum computer, with the objective of controlling the ancilla overhead, probability of success, and end precision. This arises as a special case of the multivariate quantum eigenvalue transform, for computing function of multiple commuting Hermitian matrices. The techniques applied include approximation of regular functions by low-degree polynomials and modern quantum primitives like quantum signal processing and linear combinations of unitaries. The third component is a spectral gap comparison result for Davies generators, a class of Lindbladians used for Gibbs state preparation and which are a well-studied example of quantum Markov dynamics, and which contain a canonical classical Markov chain. The main result shows that if the spectrum of the Hamiltonian driving the dynamics is free of long arithmetic progressions, then the spectral gaps of the quantum and classical Markov chains are comparable. Thus, classical techniques may be leveraged to control the behavior of the quantum dynamics. The proof is based on transferring any "off-diagonal" eigenvector of the Davies generator to an observable which commutes with the Hamiltonian and has a comparable Lindbladian Rayleigh quotient. Finally, we study the infimum of the spectrum, or ground state energy, of a discrete Schrödinger operator on the scaled integer grid in n dimensions, parameterized by a nonnegative potential V and a small frequency parameter θ. We relate this ground state energy to that of a corresponding continuous semiclassical Schrödinger operator on the full n-dimensional space with parameter θ, arising from the same choice of potential, and show that these quantities are equal up to vanishing factors depending on the regularity of V. The proofs are elementary and rely on sampling and interpolation to map low-energy functions for the discrete operator on the scaled grid to low-energy functions for the continuous operator on the full space, and vice versa. Coauthors on these works were J. Basso, T. R. Blackman, Y. Borns-Weil, I. Detherage, S. Ganguly, N. Srivastava, T. Saffat, A. Sinclair, and T.-D. Vuong.
Many problems in linear algebra-such as those arising from non-Hermitian physics, transcorrelated quantum chemistry and differential equations-can be solved on a quan-tum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited to this task, as eigenvalues and singular values are different in general. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators, and a related Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra. QEVT has query complexity to the block encoding nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the Heisenberg-limited scaling for diagonalizable input matrices. As applications, we develop a linear differential equation solver with strictly linear time query complexity for average-case diagonalizable operators, as well as a ground state preparation algorithm that upgrades previous nearly optimal results for Hermitian Hamiltonians to diagonalizable matrices with real spectra. Underpinning our algorithms is an efficient method to prepare a quantum super-position of Faber polynomials, which generalize the nearly-best uniform approximation properties of Chebyshev polynomials to the complex plane. Of independent interest, we also develop techniques to generatenFourier coefficients with 0 (poly log (n)) gates compared to prior approaches with linear cost.