Finding the ground state of the Hubbard model by variational methods on a quantum computer with gate errors

Abstract

A key goal of digital quantum computing is the simulation of fermionic systems such as molecules or the Hubbard model. Unfortunately, for present and near-future quantum computers the use of quantum error correction schemes is still out of reach. Hence, the finite error rate limits the use of quantum computers to algorithms with a low number of gates. The variational Hamiltonian ansatz (VHA) has been shown to produce the ground state in good approximation in a manageable number of steps. Here we study explicitly the effect of gate errors on its performance. The VHA is inspired by the adiabatic quantum evolution under the influence of a time-dependent Hamiltonian, where the -ideally short -fixed Trotter time steps are replaced by variational parameters. The method profits substantially from quantum variational error suppression, e.g., unitary quasi-static errors are mitigated within the algorithm. We test the performance of the VHA when applied to the Hubbard model in the presence of unitary control errors on quantum computers with realistic gate fidelities.

Resource Estimation for Quantum Variational Simulations of the Hubbard Model

“…Hence, if N blk scale better than O( √ V), then the Slater determinant preparation dominates the gate count at large V, otherwise the Hamiltonian ansatz dominates the gate count at large V. In the previous simulation of the Hubbard model in a ladder grid structure with periodic boundaries [3], N blk scale superlinear to V, while Refs. [15,16] also show similar results for the open-boundary Hubbard model. Hence, we expect the gate cost due to the Slaterdeterminant preparation part of the circuit to be negligible at large V.…”

“…There can be variability in the estimates when we choose a different set of hyperparameters. One important assumption we make is the number of ansatz blocks in the circuit is equal to the number of sites: N blk = V, which is an optimistic assumption given what we observe in the numerical simulations for a small size system [3,15,16]. An increase in N blk leads to a linear increase in the gate counts and a quadratic increase in the runtime (due to the increase in both the gate counts and the parameter counts).…”

“…To achieve results of sufficient precision, we likely need to go up to double excitation for UCCA, which means a gate scaling of N gates ∼ O(N 5 ) (assuming Jordan-Wigner encoding, see Ref [7]). To achieve similar precision, N blk for HA is likely to scale better than O(N 7/2 ) based on the previous Hubbard-model numerical-simulation results [3,15,16], thus HA should have a scaling advantage over UCCA in simulating the Hubbard model.…”

“…The sampling cost of applying both direct symmetry verification and linear error extrapolation is C SE ∼ 25 as shown in Eq. (15). Thus the number of circuit runs needed to estimate the energy and the energy-gradient vector with error mitigation is…”

Mapping the metal-insulator phase diagram by algebraically fast-forwarding dynamics on a cloud quantum computer

“…Here, ⟨𝑖, 𝑗⟩ refers to indices of nearest-neighbor sites 𝑖 and 𝑗 [27]. The Hubbard model has been recently investigated in the context of quantum computing with the applications of VQE algorithms [28,29] and as a benchmark for for quantum simulations [4,30,31].…”