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Europhysics Letters (EPL)

Stochastic simulation of quantum computation

L. R. U. MANSSUR, Renato Portugal

Published Aug 1, 2003• 0 citations
Publisher

Abstract

We discuss a stochastic method to simulate quantum computing algorithms in a classical computer using the Hubbard-Stratonovich decomposition of n -qubit gates into one-qubit gates integrated over auxiliary fields. The problem reduces to finding the fixed points of the associated system of Langevin differential equations. We show that the method can be applied to Grover's algorithm.

DOI
10.1209/epl/i2003-00560-3

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We discuss a stochastic method to simulate quantum computing algorithms in a classical computer using the Hubbard-Stratonovich decomposition of n -qubit gates into one-qubit gates integrated over auxiliary fields. The problem reduces to finding the fixed points of the associated system of Langevin differential equations. We show that the method can be applied to Grover's algorithm.
We simulate Grover's algorithm in a classical computer by means of a stochastic method using the Hubbard-Stratonovich decomposition of n-qubit gates into one-qubit gates integrated over auxiliary fields. The problem reduces to finding the fixed points of the associated system of Langevin differential equations. The equations are obtained automatically for any number of qubits by employing a computer algebra program. We present the numerical results of the simulation for a small search space.
This tutorial paper introduces quantum approaches to Monte Carlo computation with applications in computational finance. We outline the basics of quantum computing using Grover's algorithm for unstructured search to build intuition. We then move slowly to amplitude estimation problems and applications to counting and Monte Carlo integration, again using Grover-type iterations. A hands-on Python/Qiskit implementation illustrates these concepts applied to finance. The paper concludes with a discussion on current challenges in scaling quantum simulation techniques.
Finding the optimal solution to a complex optimisation problem is of great importance in practically all fields of science, technology, technical design and econometrics. We demonstrate that a modified Grover's quantum algorithm can be applied to real problems of finding a global minimum using modest numbers of quantum bits. Calculations of the global minimum of simple test functions and Lennard-Jones clusters have been carried out on a quantum computer simulator using a modified Grover's algorithm. The number of function evaluations N reduced from O(N) in classical simulation to O(N 1/2) in quantum simulation. We also show how the Grover's quantum algorithm can be combined with the classical Pivot method for global optimisation to treat larger systems.
We discuss quantum algorithms that calculate numerical integrals and descriptive statistics of stochastic processes. With either of two distinct approaches, one obtains an exponential speed increase in comparison to the fastest known classical deterministic algorithms and a quadratic speed increase in comparison to classical Monte Carlo (probabilistic) methods. We derive a simpler and slightly faster version of Grover's mean algorithm, demonstrate how to apply quantum counting to the problem, develop some variations of these algorithms, and show how both (apparently quite different) approaches can be understood from the same unified framework. Finally, we discuss how the exponential speed increase appears to (but does not) violate results obtained via the method of polynomials, from which it is known that a bounded-error quantum algorithm for computing a total function can be only polynomially more efficient than the fastest deterministic classical algorithm.
We propose a quantum-classical hybrid scheme for implementing the nonunitary Gutzwiller factor using a discrete Hubbard-Stratonovich transformation, which allows us to express the Gutzwiller factor as a linear combination of unitary operators involving only single-qubit rotations, at the cost of the sum over the auxiliary fields. To perform the sum over the auxiliary fields, we introduce two approaches that have complementary features. The first approach employs a linear-combination-of-unitaries circuit, which enables one to probabilistically prepare the Gutzwiller wave function on a quantum computer, while the second approach uses importance sampling to estimate observables stochastically, similar to a quantum Monte Carlo method in classical computation. The proposed scheme is demonstrated with numerical simulations for the half-filled Fermi-Hubbard model. Furthermore, we perform quantum simulations using a real quantum device, demonstrating that the proposed scheme can reproduce the exact ground-state energy of the two-site Fermi-Hubbard model within error bars.