Variational quantum simulation of general processes

Variational quantum algorithms have been proposed to solve static and dynamic problems of closed many-body quantum systems. Here we investigate variational quantum simulation of three general types of tasks---generalised time evolution with a non-Hermitian Hamiltonian, linear algebra problems, and open quantum system dynamics. The algorithm for generalised time evolution provides a unified framework for variational quantum simulation. In particular, we show its application in solving linear systems of equations and matrix-vector multiplications by converting these algebraic problems into generalised time evolution. Meanwhile, assuming a tensor product structure of the matrices, we also propose another variational approach for these two tasks by combining variational real and imaginary time evolution. Finally, we introduce variational quantum simulation for open system dynamics. We variationally implement the stochastic Schr\"odinger equation, which consists of dissipative evolution and stochastic jump processes. We numerically test the algorithm with a six-qubit 2D transverse field Ising model under dissipation.Comment: 18 page

Theory of variational quantum simulation

The variational method is a versatile tool for classical simulation of a variety of quantum systems. Great efforts have recently been devoted to its extension to quantum computing for efficiently solving static many-body problems and simulating real and imaginary time dynamics. In this work, we first review the conventional variational principles, including the Rayleigh-Ritz method for solving static problems, and the Dirac and Frenkel variational principle, the McLachlan's variational principle, and the time-dependent variational principle, for simulating real time dynamics. We focus on the simulation of dynamics and discuss the connections of the three variational principles. Previous works mainly focus on the unitary evolution of pure states. In this work, we introduce variational quantum simulation of mixed states under general stochastic evolution. We show how the results can be reduced to the pure state case with a correction term that takes accounts of global phase alignment. For variational simulation of imaginary time evolution, we also extend it to the mixed state scenario and discuss variational Gibbs state preparation. We further elaborate on the design of ansatz that is compatible with post-selection measurement and the implementation of the generalised variational algorithms with quantum circuits. Our work completes the theory of variational quantum simulation of general real and imaginary time evolution and it is applicable to near-term quantum hardware.Comment: 41 pages, accepted by Quantu

Variational ansatz-based quantum simulation of imaginary time evolution

Imaginary time evolution is a powerful tool for studying quantum systems. While it is possible to simulate with a classical computer, the time and memory requirements generally scale exponentially with the system size. Conversely, quantum computers can efficiently simulate quantum systems, but not non-unitary imaginary time evolution. We propose a variational algorithm for simulating imaginary time evolution on a hybrid quantum computer. We use this algorithm to find the ground-state energy of many-particle systems; specifically molecular hydrogen and lithium hydride, finding the ground state with high probability. Our method can also be applied to general optimisation problems and quantum machine learning. As our algorithm is hybrid, suitable for error mitigation and can exploit shallow quantum circuits, it can be implemented with current quantum computers.Comment: 14 page

Resource estimations for the Hamiltonian simulation in correlated electron materials

Correlated electron materials, such as superconductors and magnetic materials, are regarded as fascinating targets in quantum computing. However, the quantitative resources, specifically the number of quantum gates and qubits, required to perform a quantum algorithm to simulate correlated electron materials remain unclear. In this study, we estimate the resources required for the Hamiltonian simulation algorithm for correlated electron materials, specifically for organic superconductors, iron-based superconductors, binary transition metal oxides, and perovskite oxides, using the fermionic swap network. The effective Hamiltonian derived using the $ab~initio$ downfolding method is adopted for the Hamiltonian simulation, and a procedure for the resource estimation by using the fermionic swap network for the effective Hamiltonians including the exchange interactions is proposed. For example, in the system for the $10^2$ unit cells, the estimated number of gates per Trotter step and qubits are approximately $10^7$ and $10^3$, respectively, on average for the correlated electron materials. Furthermore, our results show that the number of interaction terms in the effective Hamiltonian, especially for the Coulomb interaction terms, is dominant in the gate resources when the number of unit cells constituting the whole system is up to $10^2$, whereas the number of fermionic swap operations is dominant when the number of unit cells is more than $10^3$.Comment: 10 pages, 4 figures, 3 table