The accurate computation of ground and excited states of many-fermion quantum systems is one of the most consequential, contemporary challenges in the physical and computational sciences whose solution stands to benefit significantly from the advent of quantum computing devices. Existing methodologies using phase estimation or variational algorithms have potential drawbacks such as deep circuits requiring substantial error correction or non-trivial high-dimensional classical optimization. Here we introduce a quantum solver of contracted eigenvalue equations, the quantum analogue of classical methods for the energies and reduced density matrices of ground and excited states. The solver does not require deep circuits or difficult classical optimization and achieves an exponential speed-up of the exact classical algorithms. We demonstrate the algorithm though computations on both a quantum simulator and two IBM quantum processing units.
The simulation of strongly correlated many-electron systems is one of the most promising applications for near-term quantum devices. Here we use a class of eigenvalue solvers (presented in Phys. Rev. Lett. 126, 070504 (2021)) in which a contraction of the Schrödinger equation is solved for the two-electron reduced density matrix (2-RDM) to resolve the energy splittings of ortho-, meta-, and para-isomers of benzyne ${\textrm C_6} {\textrm H_4}$. In contrast to the traditional variational quantum eigensolver, the contracted quantum eigensolver solves an integration (or contraction) of the many-e…
A quantum eigenvalue solver based on tensor networks
University of California, Berkeley（加州大学伯克利分校）；Berkeley Quantum Information and Computation Center, University of California, Berkeley（加州大学伯克利分校伯克利量子信息与计算中心）
Implicit methods are attractive for hybrid quantum-classical CFD solvers as the flow equations are combined into a single coupled matrix that is solved on the quantum device, leaving only the CFD discretisation and matrix assembly on the classical device. In this paper, an implicit hybrid solver is investigated using emulated HHL circuits. The hybrid solutions are compared with classical solutions including full eigen-system decompositions. A thorough analysis is made of how the number of qubits in the HHL eigenvalue inversion circuit affect the CFD solver's convergence rates. Loss of precisio…
The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far, either suffer from lack of accuracy and/or are limited to very small sizes of the problem, and thus have no practical usage. Here we present an advanced quantum algorithm for solving the Poisson equation with high accuracy and dynamically tunable problem size. After converting the Poisson equation to the linear systems through the finite difference method, we adopt the Harrow-Hassidim-Lloyd (HHL) algorithm as the basic framework. Particu…
The Poisson equation has applications across many areas of physics and engineering, such as the dynamic process simulation of ocean current. Here we present a quantum Fast Poisson Solver, including the algorithm and the complete and modular circuit design. The algorithm takes the HHL algorithm as the template. The controlled rotation is performed based on the arc cotangent function which is evaluated by the Plouffe's binary expansion method. And the same method is used to compute the cosine function for the eigenvalue approximation in phase estimation. Quantum algorithms for solving square roo…
The most advanced D-Wave Advantage quantum annealer has 5000+ qubits, however, every qubit is connected to a small number of neighbors. As such, implementation of a fully-connected graph results in an order of magnitude reduction in qubit count. To compensate for the reduced number of qubits, one has to rely on special heuristic software such as qbsolv, the purpose of which is to decompose a large problem into smaller pieces that fit onto a quantum annealer. In this work, we compare the performance of two implementations of such software: the original open-source qbsolv which is a part of the …
Quantum linear solvers for quantum chemistry: prospects of exponential quantum advantage
Centre for Quantum Engineering, Research and Education, TCG CREST（TCG CREST 量子工程、研究与教育中心）；Department of Physics, IIT Tirupati（蒂鲁帕提印度理工学院物理系）；Deloitte Tohmatsu LLC（德勤审计法人有限公司）；Indian Association for the Cultivation of Science（印度科学培育协会）；Donostia international physics centre（圣塞巴斯蒂安国际物理中心）；Academy of Scientific and Innovative Research (AcSIR)（科学与创新研究学院）
The eigenvalue density of a matrix plays an important role in various types of scientific computing such as electronic-structure calculations. In this paper, we propose a quantum algorithm for computing the eigenvalue density in a given interval. Our quantum algorithm is based on a method that approximates the eigenvalue counts by applying the numerical contour integral and the stochastic trace estimator applied to a matrix involving resolvent matrices. As components of our algorithm, the HHL solver is applied to an augmented linear system of the resolvent matrices, and the quantum Fourier tra…
College of Science, China University of Petroleum（中国石油大学理学院）
In this work, we present a quantum neighborhood preserving embedding and a quantum local discriminant embedding for dimensionality reduction and classification. We demonstrate that these two algorithms have an exponential speedup over their respectively classical counterparts. Along the way, we propose a variational quantum generalized eigenvalue solver that finds the generalized eigenvalues and eigenstates of a matrix pencil $(\mathcal{G},\mathcal{S})$. As a proof-of-principle, we implement our algorithm to solve $2^5\times2^5$ generalized eigenvalue problems. Finally, our results offer two o…
Preconditioning is the most widely used and effective way for treating ill-conditioned linear systems in the context of classical iterative linear system solvers. We introduce a quantum primitive called fast inversion, which can be used as a preconditioner for solving quantum linear systems. The key idea of fast inversion is to directly block-encode a matrix inverse through a quantum circuit implementing the inversion of eigenvalues via classical arithmetics. We demonstrate the application of preconditioned linear system solvers for computing single-particle Green's functions of quantum many-b…
