# Efficient Quantum Trace Estimation with Reconfigurable Real-Time Circuits

arxiv（2024）

Abstract

Recently, quantum algorithms that leverage real-time evolution under a
many-body Hamiltonian have proven to be exceptionally effective in estimating
individual eigenvalues near the edge of the Hamiltonian spectrum, such as the
ground state energy. By contrast, evaluating the trace of an operator requires
the aggregation of eigenvalues across the entire spectrum. In this work, we
introduce an efficient near-term quantum algorithm for computing the trace of a
broad class of operators, including matrix functions of the target Hamiltonian.
Our trace estimator is similar to the classical Girard-Hutchinson estimator in
that it involves the preparation of many random states. Although the exact
Girard-Hutchinson estimator is not tractably realizable on a quantum computer,
we can construct random states that match the variance of the Girard-Hutchinson
estimator through only real-time evolution. Importantly, our random states are
all generated using the same Hamiltonians for real-time evolution, with
randomness owing only to stochastic variations in the duration of the
evolutions. In this sense, the circuit is reconfigurable and suitable for
realization on both digital and analog platforms. For numerical illustration,
we highlight important applications in the physical, chemical, and materials
sciences, such as calculations of density of states and free energy.

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