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arXiv:1807.04553 · Physical Review A

Quantum algorithm for solving linear differential equations: Theory and experiment

Tao Xin, Shijie Wei, Jianlian Cui, Junxiang Xiao, Íñigo Arrazola, Lucas Lamata, Xiangyu Kong, Dawei Lu, E. Solano, Guilu Long

Published Mar 6, 2020• 86 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.101.032307

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Many important problems in science and engineering can be reduced to the problem of solving linear equations. The quantum algorithm discovered recently indicates that one can solve an NNN-dimensional linear equation in O(logN)O(logN)O(logN) time, which provides an exponential speedup over the classical counterpart. Here we report an experimental demonstration of the quantum algorithm when the scale of the linear equation is $2\ifmmode\times\else\texttimes\fi{}2$ using a nuclear magnetic resonance quantum information processor. For all sets of experiments, the fidelities of the final four-qubit states are all above 96%. This experiment gives the possibility of solving a series of practical problems related to linear systems of equations and can serve as the basis to realize many potential quantum algorithms.
Abstract Solving linear differential equations is a common problem in almost all fields of science and engineering. Here, we present a variational algorithm with shallow circuits for solving such a problem: given an N×NN \times NN×N N × N matrix $${\varvec{A}}$$ A , an N-dimensional vector $$\varvec{b}$$ b , and an initial vector $$\varvec{x}(0)$$ x ( 0 ) , how to obtain the solution vector $$\varvec{x}(T)$$ x ( T ) at time T according to the constraint $$\textrm{d}\varvec{x}(t)/\textrm{d} t = {\varvec{A}}\varvec{x}(t) + \varvec{b}$$ d x ( t ) / d t = A x ( t ) + b . The core idea of the algorithm is to encode the equations into a ground state problem of the Hamiltonian, which is solved via hybrid quantum-classical methods with high fidelities. Compared with the previous works, our algorithm requires the least qubit resources and can restore the entire evolutionary process. In particular, we show its application in simulating the evolution of harmonic oscillators and dynamics of non-Hermitian systems with PT\mathcal{P}\mathcal{T}PT P T -symmetry. Our algorithm framework provides a key technique for solving so many important problems whose essence is the solution of linear differential equations.
Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time proportional to the number of variables N. A recently proposed quantum algorithm shows that quantum computers could solve linear systems in a time scale of order log(N), giving an exponential speedup over classical computers. Here we realize the simplest instance of this algorithm, solving 2×2 linear equations for various input vectors on a quantum computer. We use four quantum bits and four controlled logic gates to implement every subroutine required, demonstrating the working principle of this algorithm.
We present an efficient, nearly optimal quantum algorithm for solving linear matrix differential equations, with applications to the simulation of open quantum systems and beyond. For unitary or dissipative dynamics, the algorithm computes an entry of the solution matrix with query complexity O~(νLt/ε)\widetilde{\mathcal{O}}(ν\mathcal{L} t/ε)O(νLt/ε), where the constant ννν depends on the problem parameters, L\mathcal{L}L involves a time integral of upper bounds on the norms of evolution operators, and εεε is the error. In particular, νLν\mathcal{L}νL is linear in ttt for unitary dynamics and can be a constant for dissipative dynamics. Our result contrasts prior quantum approaches for differential equations that typically require exponential time for this problem due to the encoding in a quantum state, which can lead to exponentially small amplitudes. We demonstrate the utility of the algorithm through an end-to-end application, namely the simulation of dissipative dynamics for non-interacting fermions, which can be extended to other quantum and classical systems. We compare with classical algorithms and give evidence of polynomial quantum speedups for systems in a lattice, which become more pronounced for systems with long-range interactions and can be shown to be exponential in general. We also provide a lower bound of Ω(νLt/ε)Ω(ν\mathcal{L} t/ε)Ω(νLt/ε) for unitary or dissipative dynamics that proves our algorithm is optimal up to logarithmic factors.
Recently, it has been demonstrated that quantum computers can be used for solving linear systems of algebraic equations with exponential speedup compared with classical computers. Here, we present an efficient and generic quantum circuit design for implementing the algorithm for solving linear systems. In particular, we show the detailed construction of a quantum circuit which solves a 4 × 4 linear system with seven qubits. It consists of only the basic quantum gates that can be realized with present physical devices, implying great possibility for experimental implementation. Furthermore, the performance of the circuit is numerically simulated and its ability to solve the intended linear system is verified.
Adiabatic quantum computation is of fundamental importance in the field of quantum computation as it offers an alternative approach to the gate-based model for the manipulation of quantum systems. Recently, an interesting work [arXiv:1805.10549] indicated that we can solve a linear equation system via an algorithm inspired by adiabatic quantum computing. Here we demonstrate the algorithm in a four-qubit nuclear magnetic resonance system by determining the solution of an eight-dimensional linear equation Ax=bA\mathbf{x}=\mathbf{b}Ax=b. The result is by far the maximum-dimensional linear equation solution with a limited number of qubits in experiments, which include some ingenious simplifications. Our experiment provides the possibility of solving so many practical problems related to linear equations systems and has the potential applications in designing the future quantum algorithms.