Abstract We investigate the possibility and current limitations of flow computations using quantum annealers by solving a fundamental flow problem on Ising machines. As a fundamental problem, we consider the one-dimensional advection–diffusion equation. We formulate it in a form suited to Ising machines (i.e. both classical and quantum annealers), perform extensive numerical tests on a classical annealer, and finally test it on an actual quantum annealer. To make it possible to process with an Ising machine, the problem is formulated as a minimization problem of the residual of the governing equation discretized using either the spectral method or the finite difference method. The resulting system equation is then converted to the quadratic unconstrained binary optimization (QUBO) form through the quantization of variables. It is found in numerical tests using a classical annealer that the spectral method requiring a smaller number of variables has a particular merit over the finite difference method because the accuracy deteriorates with the increase of the number of variables. We also found that the computational error varies depending on the condition number of the coefficient matrix. In addition, we extended it to a two-dimensional problem and confirmed its fundamental applicability. From the numerical test using a quantum annealer, however, it turns out that the computation using a quantum annealer is still challenging due largely to the structural difference from the classical annealer, which leaves a number of issues toward its practical use.
Abstract We investigate the possibility and current limitations of flow computations using quantum annealers by solving a fundamental flow problem on Ising machines. As a fundamental problem, we consider the one-dimensional advection–diffusion equation. We formulate it in a form suited to Ising machines (i.e. both classical and quantum annealers), perform extensive numerical tests on a classical annealer, and finally test it on an actual quantum annealer. To make it possible to process with an Ising machine, the problem is formulated as a minimization problem of the residual of the governing equation discretized using either the spectral method or the finite difference method. The resulting system equation is then converted to the quadratic unconstrained binary optimization (QUBO) form through the quantization of variables. It is found in numerical tests using a classical annealer that the spectral method requiring a smaller number of variables has a particular merit over the finite difference method because the accuracy deteriorates with the increase of the number of variables. We also found that the computational error varies depending on the condition number of the coefficient matrix. In addition, we extended it to a two-dimensional problem and confirmed its fundamental applicability. From the numerical test using a quantum annealer, however, it turns out that the computation using a quantum annealer is still challenging due largely to the structural difference from the classical annealer, which leaves a number of issues toward its practical use.
This paper explores the integration of Carleman linearization and quantum annealing to solve nonlinear ordinary differential equations (ODEs) under the Chebyshev spectral method. Using Carleman linearization, the computation of optimal Chebyshev coefficients is reformulated as a quadratic program and further transformed into a Quadratic Unconstrained Binary Optimization (QUBO) problem via binary expansion. This QUBO formulation is well-suited for quantum annealers, where solutions are obtained by minimizing the Hamiltonian of the corresponding Ising model. To address hardware limitations, we employ an iterative quantum annealing strategy with dynamically updated parameters. Case studies on the D-Wave platform demonstrate the potential of our approach to harness the strengths of Carleman linearization and Chebyshev approximation theory, offering continuous and differentiable solutions to nonlinear ODEs, especially promising as quantum resources become more economically viable.
Most quantum annealers, or Ising machines have the hardware limitations in its qubit size and thus the computable problem size is much limited. This paper proposes a novel classical-Ising hybrid annealing method which virtually extends the computable size of Ising machines based on a theoretical analysis. Given a quadratic unconstrained binary optimization (QUBO) model, the proposed method selects the cutting binary variable that is most likely to have the same value as the ground-state solution. Then, we recursively cut the QUBO model based on the cutting binary variable until we obtain sufficiently small-sized subQUBO models. Experimental evaluations demonstrate that the proposed hybrid annealing method can give much better quasi-ground-state solutions than state-of-the-art existing methods for large-sized QUBO models.
Quantum annealing machines are next-generation computers for solving combinatorial optimization problems. Although physical simulations are one of the most promising applications of quantum annealing machines, a method how to embed the target problem into the machines has not been developed except for certain simple examples. In this study, we focus on a method of representing real numbers using binary variables, or quantum bits. One of the most important problems for conducting physical simulation by quantum annealing machines is how to represent the real number with quantum bits. The variables in physical simulations are often represented by real numbers but real numbers must be represented by a combination of binary variables in quantum annealing, such as quadratic unconstrained binary optimization (QUBO). Conventionally, real numbers have been represented by assigning each digit of their binary number representation to a binary variable. Considering the classical annealing point of view, we noticed that when real numbers are represented in binary numbers, there are numbers that can only be reached by inverting several bits simultaneously under the restriction of not increasing a given Hamiltonian, which makes the optimization very difficult. In this work, we propose three new types of real number representation and compared these representations under the problem of solving linear equations. As a result, we found experimentally that the accuracy of the solution varies significantly depending on how the real numbers are represented. We also found that the most appropriate representation depends on the size and difficulty of the problem to be solved and that these differences show a consistent trend for two annealing solvers. Finally, we explain the reasons for these differences using simple models, the minimum required number of simultaneous bit flips, one-way probabilistic bit-flip energy minimization, and simulation of ideal quantum annealing machine.
Quantum annealing is a metaheuristic method aimed at finding the global minimum of a pseudo-Boolean quadratic function. To solve a problem by a quantum annealer, it must be restated as a QUBO (quadratic unconstrained binary optimization) problem, the standard format acceptable to an annealer. However, there are many natural problems that turn out be of cubic or higher order, yet promising solutions can be obtained for many instances using quantum annealers, after converting to the QUBO format, using order reduction criteria. Accordingly, quadratization becomes a crucial preprocessing step in order to make use of present-day quantum annealing hardware that relies on coupling technology, which can only process interactions of at most degree two. A critical challenge is that quadratization methods often require many additional variables, degrading the performance of the annealers. A number of algorithmic and heuristic quadratization techniques have been proposed and used for annealing in the literature, affecting the performance of the annealers differently. We examine a number of such quadratization techniques, focusing on their efficiency, compactness, and applicability. We analyze the approaches motivated by diverse goals and highlight the transition from foundational approaches to more recent compact approaches. An evaluation of the current usage of different quadratization methods in the literature is carried out at the end of the survey. Further, we comment on the limitations on usage and show possible opportunities to enhance the performance of annealers.
Recently, several powerful machines dedicated to solving combinatorial optimization problems through the Ising-model formulation have appeared. The trigger for the paradigm shift to a specialized machine for solving optimization problems was the D-Wave machine, which implements quantum annealing. Quantum annealing employs quantum fluctuations to find an optimal solution to an optimization problem with discrete variables. In particular, we input the optimization problem in the form of the Ising Hamiltonian, which is a specialized form of the quadratic unconstrained binary optimization problem. However, when we employ quantum annealing for a practical optimization problem, there are several issues. One typical issue is absence of the detailed form of the cost function, which characterizes the optimization problem. To input problems into specialized machines for solving an optimization problem, it is necessary to determine the unknown parameters within the Ising model. We propose a method to estimate the unknown parameters in the Ising Hamiltonian using compressed sensing. Furthermore, we analyze the theoretical limitations of our proposed method by employing the replica method, which is a sophisticated tool in statistical mechanics.
Solving Helmholtz problems using finite elements leads to the resolution of a linear system which is challenging to solve for classical computers. In this paper, we investigate how quantum annealers could address this challenge. We first express the linear system arising from the Helmholtz problem as a generalized eigenvalue problem (gEVP). The obtained gEVP is mapped into quadratic unconstrained binary optimization problems (QUBOs) which we solve using an adaptive quantum annealing eigensolver (AQAE) and its classical equivalent. We identify two key parameters in the success of AQAE for solving Helmholtz problems: the system condition number and the integrated control errors (ICE) in the quantum hardware. Our results show that a large system condition number implies a finer discretization grid for AQAE to converge, leading to a variable overhead, and that AQAE is either tolerant or not with respect to ICE depending on the gEVP. Finally, we establish lower bounds on the annealing time, narrowing the possibility of a quantum advantage for solving Helmholtz problems.