101 research outputs found
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
Calculation of the Green's function on near-term quantum computers
The Green's function plays a crucial role when studying the nature of quantum
many-body systems, especially strongly-correlated systems. Although the
development of quantum computers in the near future may enable us to compute
energy spectra of classically-intractable systems, methods to simulate the
Green's function with near-term quantum algorithms have not been proposed yet.
Here, we propose two methods to calculate the Green's function of a given
Hamiltonian on near-term quantum computers. The first one makes use of a
variational dynamics simulation of quantum systems and computes the dynamics of
the Green's function in real time directly. The second one utilizes the Lehmann
representation of the Green's function and a method which calculates excited
states of the Hamiltonian. Both methods require shallow quantum circuits and
are compatible with near-term quantum computers. We numerically simulated the
Green's function of the Fermi-Hubbard model and demonstrated the validity of
our proposals
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 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 unit cells, the estimated number of gates per Trotter
step and qubits are approximately and , 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 , whereas the number of
fermionic swap operations is dominant when the number of unit cells is more
than .Comment: 10 pages, 4 figures, 3 table
- …