# Algorithmic Shadow Spectroscopy

arXiv (Cornell University)（2023）

Abstract

We present shadow spectroscopy as a simulator-agnostic quantum algorithm for estimating energy gaps using an extremely low number of circuit repetitions (shots) and no extra resources (ancilla qubits) beyond performing time evolution and measurements. The approach builds on the fundamental feature that every observable property of a quantum system must evolve according to the same harmonic components: We post-process classical shadows of time-evolved quantum states to extract a large number of time-periodic signals $N_o\propto 10^8$, whose frequencies directly reveal Hamiltonian energy differences with Heisenberg-limited precision. We provide strong analytical guarantees that (a) quantum resources scale as $O(\log N_o)$, while the classical computational complexity is linear $O(N_o)$, (b) the signal-to-noise ratio increases with the number of analysed signals as $\propto \sqrt{N_o}$, and (c) peak frequencies are immune to reasonable levels of noise. Moreover, applying shadow spectroscopy numerically to probe model systems and the excited state conical intersection of molecular CH$_2$ verifies that the approach is intuitively easy to use in practice, very robust against gate noise, amiable to a new type of algorithmic-error mitigation technique, and uses orders of magnitude fewer number of shots than typical near-term quantum algorithms -- as low as 10 shots per timestep is sufficient.

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Key words

spectroscopy

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