Algorithmic Shadow Spectroscopy
We present shadow spectroscopy as a simulator-agnostic quantum algorithm for
estimating energy gaps using very few 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
can reveal them by post-processing classical shadows of time-evolved quantum
states to extract a large number of time-periodic signals $N_o\propto 10^8$,
whose frequencies correspond to 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 processed signals as $\propto \sqrt{N_o}$, and (c) spectral peak
positions are immune to reasonable levels of noise. We demonstrate our approach
on model spin systems and the excited state conical intersection of molecular
CH$_2$ and verify that our method is indeed intuitively easy to use in
practice, 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. Finally, we measured a high-quality, experimental shadow spectrum
of a spin chain on readily-available IBM quantum computers, achieving the same
precision as in noise-free simulations without using any advanced error
mitigation, and verified scalability in tensor-network simulations of up to
100-qubit systems.
Hans Hon Sang Chan Richard Meister Matthew L. Goh Bálint Koczor