Constant-depth circuits for dynamic simulations of materials on quantum computers

Abstract

Dynamic simulation of materials is a promising application for near-term quantum computers. Current algorithms for Hamiltonian simulation, however, produce circuits that grow in depth with increasing simulation time, limiting feasible simulations to short-time dynamics. Here, we present a method for generating circuits that are constant in depth with increasing simulation time for a specific subset of one-dimensional (1D) materials Hamiltonians, thereby enabling simulations out to arbitrarily long times. Furthermore, by removing the effective limit on the number of feasibly simulatable time-steps, the constant-depth circuits enable Trotter error to be made negligibly small by allowing simulations to be broken into arbitrarily many time-steps. For an N-spin system, the constant-depth circuit contains only $\mathcal {O}(N^{2})$ O ( N 2 ) CNOT gates. Such compact circuits enable us to successfully execute long-time dynamic simulation of ubiquitous models, such as the transverse field Ising and XY models, on current quantum hardware for systems of up to 5 qubits without the need for complex error mitigation techniques. Aside from enabling long-time dynamic simulations with minimal Trotter error for a specific subset of 1D Hamiltonians, our constant-depth circuits can advance materials simulations on quantum computers more broadly in a number of indirect ways.

An Algebraic Quantum Circuit Compression Algorithm for Hamiltonian Simulation

“…All of these are subclasses of TFXY Hamiltonians obtained by restricting some parameters of the full TFXY model. [22], this paper is closely related to two earlier papers [7,23] written by some of us. The results in our current paper were first conjectured in [7].…”

“…[22], this paper is closely related to two earlier papers [7,23] written by some of us. The results in our current paper were first conjectured in [7]. There, the fixed-depth circuit property was identified by using QFAST [42], an optimization-based numerical circuit compiler.…”

“…This compilation method becomes challenging for problems larger than 7 qubits because of the exponential dimensionality of the state space and this prompted us to analyze the problem in more detail. Our analysis presented in the current paper resulted in a constructive proof of the fixed-depth property (Section 5) for all cases conjectured in [7] and a scalable and accurate circuit compression algorithm that easily handles systems with O(10 3 ) qubits. Furthermore, we improve the circuit depth of the classical Ising model to O(1) compared to [7].…”

“…Our analysis presented in the current paper resulted in a constructive proof of the fixed-depth property (Section 5) for all cases conjectured in [7] and a scalable and accurate circuit compression algorithm that easily handles systems with O(10 3 ) qubits. Furthermore, we improve the circuit depth of the classical Ising model to O(1) compared to [7]. In [23] the existence of fixed depth circuits for Hamiltonian simulation is proven through a Cartan decomposition of the Lie algebra generated by the Hamiltonian.…”

“…Depending on the details of the quantum hardware, it is important to remark that the cost of a single two-spin Pauli-X rotation can sometimes be considered approximately the same as the cost of a two-spin XY rotation since both require the same number of two qubit CNOT gates [7]. These are the main source of errors on many devices and in that case we expect the TFXY mapping to perform better.…”