Abstract Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of 10−8.
Abstract Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of 10−8.
Given a matrix A of dimension M×N and a vector b, the quantum linear system (QLS) problem asks for the preparation of a quantum state ∣y⟩ proportional to the solution of Ay=b. Existing QLS algorithms have runtimes that scale linearly with the condition number κ(A), the sparsity of A, and logarithmically with inverse precision, but often overlook structural properties of b, whose alignment with A's eigenspaces can greatly affect performance. In this work, we present a new QLS algorithm that explicitly leverages the structure of the right-hand side vector b. The runtime of our algorithm depends polynomially on the sparsity of the augmented matrix H=[A,−b], the inverse precision, the ℓ2 norm of the solution y=A+b, and a new instance-dependent parameter ET=∑i=1Mpi2⋅di, where p=(AA⊤)+b, and di denotes the squared ℓ2 norm of the i-th row of H. We also introduce a structure-aware rescaling technique tailored to the solution y=A+b. Unlike left preconditioning methods, which transform the linear system to DAy=Db, our approach applies a right rescaling matrix, reformulating the linear system as ADz=b. As an application of our instance-aware QLS algorithm and new rescaling scheme, we develop a quantum algorithm for solving multivariate polynomial systems in regimes where prior QLS-based methods fail. This yields an end-to-end framework applicable to a broad class of problems. In particular, we apply it to the maximum independent set (MIS) problem, formulated as a special case of a polynomial system, and show through detailed analysis that, under certain conditions, our quantum algorithm for MIS runs in polynomial time.
With the increasing popularity of quantum computing and in particular quantum annealing, there has been growing research to evaluate the meta-heuristic for various problems in linear algebra: from linear least squares to matrix and tensor factorization. At the core of this effort is to evaluate quantum annealing for solving linear least squares and linear systems of equations. In this work, we focus on the viability of using quantum annealing for solving these problems. We use simulations based on the adiabatic principle to provide new insights for previously observed phenomena with the D-wave machines, such as quantum annealing being robust against ill-conditioned systems of equations and scaling quite well against the number of rows in a system. We then propose a hybrid approach which uses a quantum annealer to provide a initial guess of the solution x0, which would then be iteratively improved with classical fixed point iteration methods.
This paper explores the application of quantum annealing to the field of symbolic regression, a technique for automatically discovering mathematical equations from data. The core idea is to frame the symbolic regression problem as a quadratic unconstrained binary optimization (QUBO) problem, which can then be solved using the quantum annealing algorithm. This approach aims to identify previously unknown mathematical relationships that accurately represent a given dataset. The paper details the methodology, including the mapping process, the QUBO formulation, and the utilization of the quantum annealing solver. We demonstrate, through theoretical analysis and conceptual design, the potential for quantum annealing to outperform classical methods in certain scenarios, particularly when dealing with complex datasets and high-dimensional search spaces. The resulting equations are presented as a set of symbolic expressions, enabling both quantitative analysis and further investigation. The ultimate goal is to provide a framework for automated equation discovery, potentially leading to advancements in various scientific and engineering domains.
We consider various iterative algorithms for solving the linear equation ax=b using a quantum computer operating on the principle of quantum annealing. Assuming that the computer's output is described by the Boltzmann distribution, it is shown under which conditions the equation-solving algorithms converge, and an estimate of their convergence rate is provided. The application of this approach to algorithms using both an infinite number of qubits and a small number of qubits is discussed.
We introduce a quantum linear system solving algorithm based on the Kaczmarz method, a widely used workhorse for large linear systems and least-squares problems that updates the solution by enforcing one equation at a time. Its simplicity and low memory cost make it a practical choice across data regression, tomographic reconstruction, and optimization. In contrast to many existing quantum linear solvers, our method does not rely on oracle access to query entries, relaxing a key practicality bottleneck. In particular, when the rank of the system of interest is sufficiently small and the rows of the matrix of interest admit an appropriate structure, we achieve circuit complexity O(ε1logm), where m is the number of variables and ε is the target precision, without dependence on the sparsity s, and could possibly be without explicit dependence on condition number κ. This shows a significant improvement over previous quantum linear solvers where the dependence on κ,s is at least linear. At the same time, when the rows have an arbitrary structure and have at most s nonzero entries, we obtain the circuit depth O(ε1logs) using extra O(s) ancilla qubits, so the depth grows only logarithmically with sparsity s. When the sparsity s grows as O(logm), then our method can achieve an exponential improvement with respect to circuit depth compared to existing quantum algorithms, while using (asymptotically) the same amount of qubits.
Various iterative algorithms for solving the linear equation ax=b using a quantum computer operating on the principle of quantum annealing are studied. Assuming that the result produced by the computer is described by the Boltzmann distribution, conditions under which these algorithms converge are obtained and an estimate of their convergence rate is provided. Application of this approach for algorithms that use an infinite number of qubits and a small number of qubits is considered.