We explore how a continuous-variable (CV) quantum computer could solve a classic differential equation, making use of its innate capability to represent real numbers in qumodes. Specifically, we construct variational CV quantum circuits [Killoran et al., Phys.~Rev.~Research 1, 033063 (2019)] to approximate the solution of one-dimensional ordinary differential equations (ODEs), with input encoding based on displacement gates and output via measurement averages. Our simulations and parameter optimization using the PennyLane / Strawberry Fields framework demonstrate good convergence for both linear and non-linear ODEs.
It is well known, and appreciated, that quantum computers have the potential to be the most powerful computational devices ever created. This newfound power comes from a quantum parallelism effect that allows the computer to be in multiple states at the same time. This property of quantum parallelism, while suited to handle common problems such as factoring and searching an unorganized database, is extremely well-suited to handle the task of solving a binary maze. I propose an algorithm that can be used to solve a binary maze on a quantum computer, with guaranteed accuracy. While it does work,…
Quantum computer algorithms can exploit the structure of random satisfiability problems. This paper extends a previous empirical evaluation of such an algorithm and gives an approximate asymptotic analysis accounting for both the average and variation of amplitudes among search states with the same costs. The analysis predicts good performance, on average, for a variety of problems including those near a phase transition associated with a high concentration of hard cases. Based on empirical evaluation for small problems, modifying the algorithm in light of this analysis improves its performanc…
We study how to implement quantum stochastic differential equations (QSDEs) on a quantum computer. This is illustrated by an implementation of the QSDE that couples a laser driven two-level atom to the electromagnetic field in the vacuum state on the IBMqx4 Tenerife computer. We compare the resulting master equation and quantum filtering equations to existing theory. In this way we characterize the performance of the computer.
Systems of linear equations are used to model a wide array of problems in all fields of science and engineering. Recently, it has been shown that quantum computers could solve linear systems exponentially faster than classical computers, making for one of the most promising applications of quantum computation. Here, we demonstrate this quantum algorithm by implementing various instances on a photonic quantum computing architecture. Our implementation involves the application of two consecutive entangling gates on the same pair of polarisation-encoded qubits. We realize two separate controlled-…
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 6$^{\mathrm{th…
Solving Differential Equations Using Continuous-Variable Quantum Annealing
Chuo University（中央大学）；The University of Tokyo（东京大学）；Ochanomizu University（御茶水女子大学）；National Institute of Advanced Industrial Science and Technology（产业技术综合研究所）
We present and experimentally realize a quantum algorithm for efficiently solving the following problem: given an $N\times N$ matrix $\mathcal{M}$, an $N$-dimensional vector $\textbf{\emph{b}}$, and an initial vector $\textbf{\emph{x}}(0)$, obtain a target vector $\textbf{\emph{x}}(t)$ as a function of time $t$ according to the constraint $d\textbf{\emph{x}}(t)/dt=\mathcal{M}\textbf{\emph{x}}(t)+\textbf{\emph{b}}$. We show that our algorithm exhibits an exponential speedup over its classical counterpart in certain circumstances. In addition, we demonstrate our quantum algorithm for a $4\times4…
In this work, using a NISQ framework, we obtain the gap of a BCS Hamiltonian. This could lead to interesting implications for superconductivity research. For such task, we choose to use the Variational Quantum Deflation and analyze the hardware restrictions that are needed to find the energy spectra on current quantum hardware. We also compare two different kinds of classical optimizers, Constrained Optimization BY Linear Approximations (COBYLA) and Simultaneous Perturbation Stochastic Approximation (SPSA), and study the effect of decoherence caused by the presence of noise when using simulati…
A quantum algorithm is proposed to solve the Satisfiability problems by the ground-state quantum computer. The scale of the energy gap of the ground-state quantum computer is analyzed for the 3-bit Exact Cover problem. The time cost of this algorithm on the general SAT problems is discussed.
A hybrid algorithm based on machine learning and quantum ensemble learning is proposed that is capable of finding a solution to a partial differential equation with good precision and favorable scaling in the required number of qubits. The classical part is composed by training several regressors (weak-learners), capable of solving a partial differential equation using machine learning. The quantum part consists of adapting the QBoost algorithm to solve regression problems. We have successfully applied our framework to solve the 1D Burgers' equation with viscosity, showing that the quantum ens…
The quantum approximate optimization algorithm (QAOA) is a hybrid variational quantum-classical algorithm that solves combinatorial optimization problems. While there is evidence suggesting that the fixed form of the standard QAOA ansatz is not optimal, there is no systematic approach for finding better ansätze. We address this problem by developing an iterative version of QAOA that is problem-tailored, and which can also be adapted to specific hardware constraints. We simulate the algorithm on a class of Max-Cut graph problems and show that it converges much faster than the standard QAOA, whi…
Tensor-Programmable Quantum Circuits for Solving Differential Equations
University of Hamburg（汉堡大学）；German Aerospace Center (DLR)（德国航空航天中心）；The Hamburg Centre for Ultrafast Imaging（汉堡超快成像中心）；Clarendon Laboratory, University of Oxford（牛津大学克拉伦登实验室）
Many eigenvalue problems arising in practice are often of the generalized form $A\x=λB\x$. One particularly important case is symmetric, namely $A, B$ are Hermitian and $B$ is positive definite. The standard algorithm for solving this class of eigenvalue problems is to reduce them to Hermitian eigenvalue problems. For a quantum computer, quantum phase estimation is a useful technique to solve Hermitian eigenvalue problems. In this work, we propose a new quantum algorithm for symmetric generalized eigenvalue problems using ordinary differential equations. The algorithm has lower complexity than…
U.S. Army Research Laboratory（美国陆军研究实验室）；U.S. Military Academy（美国军事学院）；Army Cyber Institute（陆军网络研究所）
Classical optimization algorithms in machine learning often take a long time to compute when applied to a multi-dimensional problem and require a huge amount of CPU and GPU resource. Quantum parallelism has a potential to speed up machine learning algorithms. We describe a generic mathematical model to leverage quantum parallelism to speed-up machine learning algorithms. We also apply quantum machine learning and quantum parallelism applied to a $3$-dimensional image that vary with time.
We report an experimental demonstration of a one-way implementation of a quantum algorithm solving Simon's Problem - a black box period-finding problem which has an exponential gap between the classical and quantum runtime. Using an all-optical setup and modifying the bases of single-qubit measurements on a five-qubit cluster state, key representative functions of the logical two-qubit version's black box can be queried and solved. To the best of our knowledge, this work represents the first experimental realization of the quantum algorithm solving Simon's Problem. The experimental results are…
Quantum computing has demonstrated the potential to revolutionize our understanding of nuclear, atomic, and molecular structure by obtaining forefront solutions in non-relativistic quantum many-body theory. In this work, we show that quantum computing can be used to solve for the structure of hadrons, governed by strongly-interacting relativistic quantum field theory. Following our previous work on light unflavored mesons as a relativistic bound-state problem within the nonperturbative Hamiltonian formalism, we present the numerical calculations on simulated quantum devices using the basis lig…
We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how thi…
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 $Δt^2$ in the evolution time $Δt$. As with other algorithms of this type, the solution is encoded in amplitudes of the quantum state, and it is possi…
We present a general method, called Qade, for solving differential equations using a quantum annealer. The solution is obtained as a linear combination of a set of basis functions. On current devices, Qade can solve systems of coupled partial differential equations that depend linearly on the solution and its derivatives, with non-linear variable coefficients and arbitrary inhomogeneous terms. We test the method with several examples and find that state-of-the-art quantum annealers can find the solution accurately for problems requiring a small enough function basis. We provide a Python packag…
We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the boundary (or initial) conditions and contains no adjustable parameters. The second part is constructed so as not to affect the boundary conditions. This part involves a feedforward neural network, containing adjustable parameters (the weights). Hence by construction the boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of t…
Auburn University（奥本大学）；Oak Ridge National Laboratory（橡树岭国家实验室）
Adiabatic quantum computers are a promising platform for approximately solving challenging optimization problems. We present a quantum approach to solving the balanced $k$-means clustering training problem on the D-Wave 2000Q adiabatic quantum computer. Existing classical approaches scale poorly for large datasets and only guarantee a locally optimal solution. We show that our quantum approach better targets the global solution of the training problem, while achieving better theoretic scalability on large datasets. We test our quantum approach on a number of small problems, and observe cluster…
