Efficient Calculation of Equilibrium Correlation Functions

Abstract:A majority of measurable dynamic quantities of a physical system is described as an equilibrium correlation function of the form $G_{AB}(t-t') = \left<A(t)B(t')\right>$, where $t'$ represents the impact time and $t$ represents the response time. Conventional quantum algorithms to compute such quantities via Hamiltonian simulation such as the Hadamard test, variational methods, or linear-response-based algorithms share one feature in common: each time point $t-t'$ is calculated by a different quantum circuit, leading to at least $O(N_t)$ total shots, and at least $O(N_t^2)$ total quantum gates. Here, we provide a different approach: utilizing $O(\log N_t)$ ancilla qubits to store time as a quantum variable, and effectively parallelize the computation of an equilibrium correlation function by requiring only a single quantum circuit for all time points. We show that this circuit needs to be run only $O(\log N_t)$ times, which in total requires $O(N_t \log N_t)$ quantum resources in the large system size limits. We prove that this scaling is optimal, and demonstrate our algorithm calculating the Green's function of a one-dimensional spinless Hubbard model.

Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Computational Physics (physics.comp-ph)

arXiv:2609.40349 [quant-ph]

(or arXiv:2609.40349v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.40349 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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