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arXiv:2212.11036 · PRX Quantum

Algorithmic Shadow Spectroscopy

Hans Hon Sang Chan, Richard Meister, Matthew L. Goh, Bálint Koczor

Published Mar 17, 2025• 19 citations
arXivHTMLPDF

Abstract

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 postprocessing classical shadows of time-evolved quantum states to extract a large number of time-periodic signals N o ∝ 10 8 , whose frequencies correspond to Hamiltonian energy differences with precision limited as ϵ ∝ 1 / T for simulation time T . 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 ∝ 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 relatively few shots given a reasonable initial state is supplied—we demonstrate that even 10 shots per time step can be 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.

DOI
10.1103/prxquantum.6.010352

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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 postprocessing classical shadows of time-evolved quantum states to extract a large number of time-periodic signals N o ∝ 10 8 , whose frequencies correspond to Hamiltonian energy differences with precision limited as ϵ ∝ 1 / T for simulation time T . 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 ∝ 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 relatively few shots given a reasonable initial state is supplied—we demonstrate that even 10 shots per time step can be 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.
Estimating the energy spectra of quantum many-body systems is a fundamental task in quantum physics, with applications ranging from chemistry to condensed matter. Algorithmic shadow spectroscopy is a recent method that leverages randomized measurements on time-evolved quantum states to extract spectral information. However, implementing accurate time evolution with low-depth circuits remains a key challenge for near-term quantum hardware. In this work, we propose a hybrid quantum-classical protocol that integrates Time Evolution via Probabilistic Angle Interpolation (TE-PAI) into the shadow spectroscopy framework. TE-PAI enables the simulation of time evolution using shallow stochastic circuits while preserving unbiased estimates through quasiprobability sampling. We construct the combined estimator and derive its theoretical properties. Through numerical simulations, we demonstrate that our method accurately resolves energy gaps and exhibits enhanced robustness to gate noise compared to first-order Trotter-based shadow spectroscopy. We further validate the protocol experimentally on up to 20 qubits using IBM quantum hardware. This makes TE-PAI shadow spectroscopy a promising tool for spectral analysis on noisy intermediate-scale quantum (NISQ) devices.
Extracting energy spectra from quantum Hamiltonians is a fundamental task for quantum simulation, yet remains challenging on noisy intermediate-scale quantum (NISQ) devices. We propose Measured Quantum Time Evolution (MQTE), an ancilla-free algorithm that estimates energy gaps by applying real-time evolution to a reference state and measuring time-resolved probabilities via repeated projective measurements. Spectral analysis of these signals reveals oscillation frequencies corresponding to eigenvalue differences. Crucially, MQTE exhibits inherent robustness to quantum hardware noise and sampling errors: these disturbances manifest as a white-noise background, which does not distort the underlying spectral structure but rather obscures the frequency information. By increasing the number of measurement samples, the intensity of the background white noise can be suppressed, thereby recovering the original spectral content. We validate the algorithm's performance via numerical simulations on one- and two-dimensional Heisenberg models, demonstrating accurate extraction of energy gaps and resilience against both sampling and circuit-level noise. Experimental implementation on the superconducting quantum processor Tianyan-176-II further confirms the practical feasibility and noise tolerance of MQTE under real hardware conditions. This work provides a robust and scalable framework for quantum spectral estimation in the NISQ era.
We introduce a postprocessing technique for classical shadow measurement data that enhances the precision of ground state estimation through high-dimensional subspace expansion; the dimensionality is only limited by the amount of classical postprocessing resources rather than by quantum resources. Crucial steps of our approach are the efficient identification of useful observables from shadow data, followed by our regularized subspace expansion that is designed to be numerically stable even when using noisy data. We analytically investigate noise propagation within our method, and upper bound the statistical fluctuations due to the limited number of snapshots in classical shadows. In numerical simulations, our method can achieve a reduction in the energy estimation errors in many cases, sometimes by more than an order of magnitude. We also demonstrate that our performance improvements are robust against both coherent errors (bad initial state) and gate noise in the state-preparation circuits. Furthermore, performance is guaranteed to be at least as good—and in many cases better—than direct energy estimation without using additional quantum resources, and the approach is thus a very natural alternative for estimating ground state energies directly from classical shadow data.
The efficient probing of spectral features is important for characterising and understanding the structure and dynamics of quantum materials. In this work, we establish a framework for probing the excitation spectrum of quantum many-body systems with quantum simulators. Our approach effectively realises a spectral detector by processing the dynamics of observables with time intervals drawn from a defined probability distribution, which only requires native time evolution governed by the Hamiltonian without ancilla. The critical element of our method is the engineered emergence of frequency resonance such that the excitation spectrum can be probed. We show that the time complexity for transition energy estimation has a logarithmic dependence on simulation accuracy and how such observation can be guaranteed in certain many-body systems. We discuss the noise robustness of our spectroscopic method and show that the total running time maintains polynomial dependence on accuracy in the presence of device noise. We further numerically test the error dependence and the scalability of our method for lattice models. We present simulation results for the spectral features of typical quantum systems, either gapped or gapless, including quantum spins, fermions and bosons. We demonstrate how excitation spectra of spin-lattice models can be probed experimentally with IBM quantum devices.
Measuring global quantum properties-such as the fidelity to complex multipartite states-is both an essential and experimentally challenging task. Classical shadow estimation offers favorable sample complexity, but typically relies on many-qubit circuits that are difficult to realize on current platforms. We propose the robust phase shadow scheme, a measurement framework based on random circuits with controlled-ZZZ as the unique entangling gate type, tailored to architectures such as trapped ions and neutral atoms. Leveraging tensor diagrammatic reasoning, we rigorously analyze the induced circuit ensemble and show that phase shadows match the performance of full Clifford-based ones. Importantly, our approach supports a noise-robust extension via purely classical post-processing, enabling reliable estimation under gate-dependent noise where existing techniques often fail. Additionally, by exploiting structural properties of random stabilizer states, we design an efficient post-processing algorithm that resolves a key computational bottleneck in previous shadow protocols. Our results enhance the practicality of shadow-based techniques, providing a robust and scalable route for estimating global properties in noisy quantum systems.
Simulating quantum dynamics is one of the most important applications of quantum computers. Traditional approaches for quantum simulation involve preparing the full evolved state of the system and then measuring some physical quantity. Here, we present a different and novel approach to quantum simulation that uses a compressed quantum state that we call the “shadow state”. The amplitudes of this shadow state are proportional to the time-dependent expectations of a specific set of operators of interest, and it evolves according to its own Schrödinger equation. This evolution can be simulated on a quantum computer efficiently under broad conditions. Applications of this approach to quantum simulation problems include simulating the dynamics of exponentially large systems of free fermions or free bosons, the latter example recovering a recent algorithm for simulating exponentially many classical harmonic oscillators. These simulations are hard for classical methods and also for traditional quantum approaches, as preparing the full states would require exponential resources. Shadow Hamiltonian simulation can also be extended to simulate expectations of more complex operators such as two-time correlators or Green’s functions, and to study the evolution of operators themselves in the Heisenberg picture. When performing quantum simulation, oftentimes the properties of interest are only a subset of the information contained in the entire state. Here, the authors devise a different type of quantum simulation, where they work with a compressed quantum state whose amplitudes are proportional to expected values of some specific observables of interest.