Resource Estimation for Quantum Variational Simulations of the Hubbard Model

As the advances in quantum hardware bring us into the noisy intermediate-scale quantum (NISQ) era, one possible task we can perform without quantum error correction using NISQ machines is the variational quantum eigensolver (VQE) due to its shallow depth. A specific problem that we can tackle is the strongly interacting Fermi-Hubbard model, which is classically intractable and has practical implications in areas like superconductivity. In this Article, we outline the details about the gate sequence, the measurement scheme and the relevant error mitigation techniques for the implementation of the Hubbard VQE on a NISQ platform. We perform resource estimation for both silicon spin qubits and superconducting qubits for a 50-qubit simulation, which cannot be solved exactly via classical means, and find similar results. The number of two-qubit gates required is on the order of 20000. Hence, to suppress the mean circuit error count to a level such that we can obtain meaningful results with the aid of error mitigation, we need to achieve a two-qubit gate error rate of $\sim 10^{-4}$. When searching for the ground state, we need a few days for one gradient-descent iteration, which is impractical. This can be reduced to around $10$ minutes if we distribute our task among hundreds of quantum processing units. Hence, implementing a 50-qubit Hubbard model VQE on a NISQ machine can be on the brink of being feasible in near term, but further optimisation of our simulation scheme, improvements in the gate fidelity, improvements in the optimisation scheme and advances in the error mitigation techniques are needed to overcome the remaining obstacles. The scalability of the hardware platform is also essential to overcome the runtime issue via parallelisation, which can be done on one single silicon multi-core processor or across multiple superconducting processors.