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arXiv:2608.04601

Solving Differential Equations Using Continuous-Variable Quantum Annealing

Kazuki Miyanishi, Soshun Naito, Asuka Koura, Kazue Kudo, Toshiki Yamaji, Yuichiro Matsuzaki

quant-phSubmitted Aug 5, 2026• Announced Aug 6, 2026• 0 citations
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Abstract

Most existing quantum annealing approaches are formulated for qubit-based architectures. Consequently, applying them to continuous-variable optimization problems requires discretizing the variables, which can incur substantial qubit overhead. Continuous-variable quantum annealing based on bosonic systems has recently been proposed as an alternative framework, in which each optimization variable is directly encoded in a bosonic mode, such as a cavity mode. In this work, we develop a continuous-variable quantum annealing formulation for solving linear differential equations. By recasting the determination of the solution as a continuous-variable optimization problem, the differential equation can be mapped onto an objective function compatible with bosonic quantum annealing. Numerical simulations of second-order linear differential equations demonstrate that, under the conditions considered, the proposed formulation reproduces the corresponding analytical solutions. These results establish a potential route toward solving differential equations without the discretization overhead inherent in qubit-based implementations.

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Most existing quantum annealing approaches are formulated for qubit-based architectures. Consequently, applying them to continuous-variable optimization problems requires discretizing the variables, which can incur substantial qubit overhead. Continuous-variable quantum annealing based on bosonic systems has recently been proposed as an alternative framework, in which each optimization variable is directly encoded in a bosonic mode, such as a cavity mode. In this work, we develop a continuous-variable quantum annealing formulation for solving linear differential equations. By recasting the determination of the solution as a continuous-variable optimization problem, the differential equation can be mapped onto an objective function compatible with bosonic quantum annealing. Numerical simulations of second-order linear differential equations demonstrate that, under the conditions considered, the proposed formulation reproduces the corresponding analytical solutions. These results establish a potential route toward solving differential equations without the discretization overhead inherent in qubit-based implementations.
Abstract Quantum annealing is a method developed to solve combinatorial optimization problems by utilizing quantum bits. Solving such problems corresponds to minimizing a cost function defined over binary variables. However, in many practical cases, the cost function may also involve continuous variables. Representing continuous variables using quantum bits requires binary encoding, which demands a large number of qubits. To overcome this limitation, an approach using quantum resonators has been proposed, enabling the direct handling of continuous variables within the quantum annealing framework. On the other hand, certain optimization problems involve both binary and continuous variables simultaneously, and a quantum annealing method capable of efficiently solving such hybrid problems has not been established. Here, we propose a quantum annealing method based on a hybrid system composed of qubits and resonators, aiming to minimize cost functions that contain both binary and continuous variables. We present a general framework for hybrid quantum annealing using such systems, and investigate its feasibility and effectiveness through numerical simulations.
Abstract The quantum annealing framework provides a general technique for the solution of a wide variety of optimization problems. Recent technological developments have made it possible to build quantum annealing devices with thousands of qubits, bringing them within range to tackle real-world problems in the near future. In this paper, I review this framework, and present a method for its use in the minimization of functions of continuous variables. Then, I discuss how this method can be employed to solve optimization problems with applications in high-energy physics, including training neural networks for classification, solving partial differential equations, and fitting effective field theories to experimental data.
Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6th^{\mathrm{th}}th order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an oracle within quantum search algorithms for inverse problems.
Abstract Linear regression is a data analysis technique that is categorized as supervised learning. By utilizing known data, we can predict unknown data. Recently, researchers have explored the use of quantum annealing (QA) to perform linear regression, where parameters are approximated to discrete values using binary numbers. However, this approach has a limitation: we need to increase the number of qubits to improve the accuracy. Here, we propose a novel linear regression method using QA that leverages continuous variables. In particular, the boson system facilitates the optimization of linear regression without resorting to discrete approximations, as it directly manages continuous variables while engaging in QA. The major benefit of our new approach is that it can ensure accuracy without increasing the number of qubits as long as the adiabatic condition is satisfied.
Linear regression is a data analysis technique, which is categorized as supervised learning. By utilizing known data, we can predict unknown data. Recently, researchers have explored the use of quantum annealing (QA) to perform linear regression where parameters are approximated to discrete values using binary numbers. However, this approach has a limitation: we need to increase the number of qubits to improve the accuracy. Here, we propose a novel linear regression method using QA that leverages continuous variables. In particular, the boson system facilitates the optimization of linear regression without resorting to discrete approximations, as it directly manages continuous variables while engaging in QA. The major benefit of our new approach is that it can ensure accuracy without increasing the number of qubits as long as the adiabatic condition is satisfied.
Quantum computers have long been expected to efficiently solve complex classical differential equations. Most digital, fault-tolerant approaches use Carleman linearization to map nonlinear systems to linear ones and then apply quantum linear-system solvers. However, provable speedups typically require digital truncation and full fault tolerance, rendering such linearization approaches challenging to implement on current hardware. Here we present an analog, continuous-variable algorithm based on coupled bosonic modes with qubit-based adaptive measurements that avoids Hilbert-space digitization. This method encodes classical fields as coherent states and, via Kraus-channel composition derived from the Koopman-von Neumann (KvN) formalism, maps nonlinear evolution to linear dynamics. Unlike many analog schemes, the algorithm is provably efficient: advancing a first-order, LLL-grid point, ddd-dimensional, order-KKK spatial-derivative, degree-rrr polynomial-nonlinearity, strongly dissipative partial differential equations (PDEs) for TTT time steps costs O(T(log⁡L+drlog⁡K))\mathcal{O}\left(T(\log L + d r \log K)\right)O(T(logL+drlogK)). The capability of the scheme is demonstrated by using it to simulate the one-dimensional Burgers' equation and two-dimensional Fisher-KPP equation. The resilience of the method to photon loss is shown under strong-dissipation conditions and an analytic counterterm is derived that systematically cancels the dominant, experimentally calibrated noise. This work establishes a continuous-variable framework for simulating nonlinear systems and identifies a viable pathway toward practical quantum speedup on near-term analog hardware.