Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity

TL;DR

This paper introduces quantum circuits that efficiently encode the spectra of correlated electron models, enabling faster quantum phase estimation with fewer T gates, which could surpass classical methods in electronic structure calculations.

Contribution

The authors develop circuits with linear T complexity for encoding spectra, improving quantum phase estimation efficiency for electronic Hamiltonians compared to prior approaches.

Findings

T gate complexity is $O(N + ext{log}(1/\epsilon))$ for electronic Hamiltonians.

Sampling in the eigenbasis can be achieved with asymptotic gate efficiency.

Potential to outperform classical methods with about a million qubits in hours.

Abstract

We construct quantum circuits which exactly encode the spectra of correlated electron models up to errors from rotation synthesis. By invoking these circuits as oracles within the recently introduced "qubitization" framework, one can use quantum phase estimation to sample states in the Hamiltonian eigenbasis with optimal query complexity $O(\lambda / \epsilon)$ where $\lambda$ is an absolute sum of Hamiltonian coefficients and $\epsilon$ is target precision. For both the Hubbard model and electronic structure Hamiltonian in a second quantized basis diagonalizing the Coulomb operator, our circuits have T gate complexity $O({N + \log (1/\epsilon}))$ where $N$ is number of orbitals in the basis. This enables sampling in the eigenbasis of electronic structure Hamiltonians with T complexity $O(N^3 /\epsilon + N^2 \log(1/\epsilon)/\epsilon)$. Compared to prior approaches, our algorithms are asymptotically more efficient in gate complexity and require fewer T gates near the classically intractable regime. Compiling to surface code fault-tolerant gates and assuming per gate error rates of one part in a thousand reveals that one can error correct phase estimation on interesting instances of these problems beyond the current capabilities of classical methods using only about a million superconducting qubits in a matter of hours.