Quantum algorithms for solving linear differential equations

Solving linear differential equations (LDEs) is a hard problem for classical computers, while quantum algorithms have been proposed to be capable of speeding up the calculation. However, they are yet to be realized in experiment as it cannot be easily converted into an implementable quantum circuit. Here, we present and experimentally realize an implementable gate-based quantum algorithm for efficiently solving the LDE problem: given an $N\ifmmode\times\else\texttimes\fi{}N$ matrix $\mathcal{M}$ , an $N$ -dimensional vector $\mathbf{b}$ , and an initial vector $\mathbf{x}(0)$ , we obtain a target vector $\mathbf{x}(t)$ as a function of time $t$ according to the constraint $d\mathbf{x}(t)/dt=\mathcal{M}\mathbf{x}(t)+\mathbf{b}$ . We show that our algorithm exhibits an exponential speedup over its classical counterpart in certain circumstances, and a gate-based quantum circuit is produced which is friendly to the experimentalists and implementable in current quantum techniques. In addition, we experimentally solve a $4\ifmmode\times\else\texttimes\fi{}4$ linear differential equation using our quantum algorithm in a four-qubit nuclear magnetic resonance quantum information processor. Our algorithm provides a key technique for solving many important problems which rely on the solutions to linear differential equations.Solving linear differential equations (LDEs) is a hard problem for classical computers, while quantum algorithms have been proposed to be capable of speeding up the calculation. However, they are yet to be realized in experiment as it cannot be easily converted into an implementable quantum circuit. Here, we present and experimentally realize an implementable gate-based quantum algorithm for efficiently solving the LDE problem: given an $N\ifmmode\times\else\texttimes\fi{}N$ matrix $\mathcal{M}$, an $N$-dimensional vector $\mathbf{b}$, and an initial vector $\mathbf{x}(0)$, we obtain a target vector $\mathbf{x}(t)$ as a function of time $t$ according to the constraint $d\mathbf{x}(t)/dt=\mathcal{M}\mathbf{x}(t)+\mathbf{b}$. We show that our algorithm exhibits an exponential speedup over its classical counterpart in certain circumstances, and a gate-based quantum circuit is produced which is friendly to the experimentalists and implementable in current quantum techniques. In addition, we experimentally solve a $4\ifmmode\times\else\texttimes\fi{}4$ linear differential equation using our quantum algorithm in a four-qubit nuclear magnetic resonance quantum information processor. Our algorithm provides a key technique for solving many important problems which rely on the solutions to linear differential equations.

arXiv:1807.04553Physical Review AMar 6, 202086 citations

Quantum algorithm for time-dependent differential equations using Dyson series

Dominic William Berry, Pedro C. S. Costa

Time-dependent linear differential equations are a common type of problem that needs to be solved in classical physics. Here we provide a quantum algorithm for solving time-dependent linear differential equations with logarithmic dependence of the complexity on the error and derivative. As usual, there is an exponential improvement over classical approaches in the scaling of the complexity with the dimension, with the caveat that the solution is encoded in the amplitudes of a quantum state. Our method is to encode the Dyson series in a system of linear equations, then solve via the optimal quantum linear equation solver. Our method also provides a simplified approach in the case of time-independent differential equations.Time-dependent linear differential equations are a common type of problem that needs to be solved in classical physics. Here we provide a quantum algorithm for solving time-dependent linear differential equations with logarithmic dependence of the complexity on the error and derivative. As usual, there is an exponential improvement over classical approaches in the scaling of the complexity with the dimension, with the caveat that the solution is encoded in the amplitudes of a quantum state. Our method is to encode the Dyson series in a system of linear equations, then solve via the optimal quantum linear equation solver. Our method also provides a simplified approach in the case of time-independent differential equations.

arXiv:2212.03544QuantumJun 13, 202431 citations

Quantum Algorithm for Linear Differential Equations with Exponentially Improved Dependence on Precision

Dominic William Berry, Andrew M. Childs, Aaron Ostrander +1 more

arXiv:1701.03684Communications in Mathematical PhysicsOct 7, 2017219 citations

Quantum algorithm for linear systems

Yunah Choi, Jeihee Cho, Shiho Kim

Advances in computersNov 29, 20250 citations

Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

Cheng Xue, Yu-Chun Wu, Guo‐Ping Guo

Abstract While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation method, we propose a quantum algorithm for solving n -dimensional nonlinear dissipative ordinary differential equations (ODEs). Our algorithm first converts the original nonlinear ODEs into the other nonlinear ODEs which can be embedded into finite-dimensional linear ODEs. Then we solve the embedded linear ODEs with quantum linear ODEs algorithm and obtain a state ϵ -close to the normalized exact solution of the original nonlinear ODEs with success probability Ω(1). The complexity of our algorithm is O ( gηT poly(log( nT / ϵ ))), where η , g measure the decay of the solution. Our algorithm provides exponential improvement over the best classical algorithms or previous quantum algorithms in n or ϵ .Abstract While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation method, we propose a quantum algorithm for solving n -dimensional nonlinear dissipative ordinary differential equations (ODEs). Our algorithm first converts the original nonlinear ODEs into the other nonlinear ODEs which can be embedded into finite-dimensional linear ODEs. Then we solve the embedded linear ODEs with quantum linear ODEs algorithm and obtain a state ϵ -close to the normalized exact solution of the original nonlinear ODEs with success probability Ω(1). The complexity of our algorithm is O ( gηT poly(log( nT / ϵ ))), where η , g measure the decay of the solution. Our algorithm provides exponential improvement over the best classical algorithms or previous quantum algorithms in n or ϵ .

arXiv:2111.07486New Journal of PhysicsDec 1, 202133 citations

\mathcal{M}

, an

N

-dimensional vector

\mathbf{b}

, and an initial vector

\mathbf{x}(0)

, we obtain a target vector

\mathbf{x}(t)

as a function of time

t

according to the constraint

d\mathbf{x}(t)/dt=\mathcal{M}\mathbf{x}(t)+\mathbf{b}

. We show that our algorithm exhibits an exponential speedup over its classical counterpart in certain circumstances, and a gate-based quantum circuit is produced which is friendly to the experimentalists and implementable in current quantum techniques. In addition, we experimentally solve a $4\ifmmode\times\else\texttimes\fi{}4$ linear differential equation using our quantum algorithm in a four-qubit nuclear magnetic resonance quantum information processor. Our algorithm provides a key technique for solving many important problems which rely on the solutions to linear differential equations.