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A quantum algorithm for linear differential equations with layerwise parameterized quantum circuits

Junxiang Xiao, Jingwei Wen, Zengrong Zhou, Ling Qian, Zhiguo Huang, Shijie Wei, Guilu Long

Published Mar 4, 2024• 3 citations
Publisher

Abstract

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 $$N \times N$$ 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 $$\mathcal{P}\mathcal{T}$$ 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.

DOI
10.1007/s43673-023-00115-1

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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 M\mathcal{M}M, an NNN-dimensional vector b\mathbf{b}b, and an initial vector x(0)\mathbf{x}(0)x(0), we obtain a target vector x(t)\mathbf{x}(t)x(t) as a function of time ttt according to the constraint dx(t)/dt=Mx(t)+bd\mathbf{x}(t)/dt=\mathcal{M}\mathbf{x}(t)+\mathbf{b}dx(t)/dt=Mx(t)+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.
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Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms to general inhomogeneous sparse linear differential equations, which describe many classical physical systems. We examine the use of high-order methods to improve the efficiency. These provide scaling close to Δt2Δt^2Δt2 in the evolution time ΔtΔtΔt. As with other algorithms of this type, the solution is encoded in amplitudes of the quantum state, and it is possible to extract global features of the solution.
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We propose a distinct approach to solving linear and nonlinear differential equations (DEs) on quantum computers by encoding the problem into ground states of effective Hamiltonian operators. Our algorithm relies on constructing such operators in the Chebyshev space, where an effective Hamiltonian is a sum of global differential and data constraints. Once the effective Hamiltonian is formed, solutions of differential equations can be obtained using the ground state preparation techniques (e.g. imaginary-time evolution and quantum singular value transformation), bypassing variational search. Unlike approaches based on discrete grids, the algorithm enables evaluation of solutions beyond fixed grid points and implements constraints in the physics-informed way. Our proposal inherits the best traits from quantum machine learning-based DE solving (compact basis representation, automatic differentiation, nonlinearity) and quantum linear algebra-based approaches (fine-grid encoding, provable speed-up for state preparation), offering a robust strategy for quantum scientific computing in the early fault-tolerant era.
Quantum algorithms offer an exponential advantage with respect to the number of dependent variables for solving certain nonlinear ordinary differential equations (ODEs). These algorithms typically begin by transforming the original nonlinear ODE into a higher-dimensional linear ODE using a linearization technique, most commonly Carleman linearization. Existing works restrict their analysis to ODEs where the nonlinearities are polynomial functions of the dependent variables, significantly limiting their applicability. In this work we construct an efficient quantum algorithm for solving ODEs with `Fourier' nonlinear terms expressible as du/dt=G0+G1eiud{\bf u}/dt = G_0 + G_1 e^{i{\bf u}}du/dt=G0​+G1​eiu, where u{\bf u}u denotes a vector of nnn complex variables evolving with ttt, G0G_0G0​ is an nnn-dimensional complex vector, G1G_1G1​ is an n×nn \times nn×n complex matrix and eiue^{i{\bf u}}eiu denotes the vector with entries {eiuj}\{e^{iu_j}\}{eiuj​}. To tackle the Fourier nonlinear term, which is not expressible as a finite sum of polynomials of u{\bf u}u, our algorithm employs a generalization of the Carleman linearization technique known as Koopman linearization. We also make other methodological advances towards relaxing the stringent dissipativity condition required for efficient solution extraction and towards integrated readout of classical quantities from the solution state. Our results open avenues to the development of efficient quantum algorithms for a significantly wider class of high-dimensional nonlinear ODEs, thereby broadening the scope of their applications.