Low-Resource Quantum Energy Gap Estimation via Randomization

TL;DR

A hybrid quantum-classical protocol that integrates Time Evolution via Probabilistic Angle Interpolation (TE-PAI) into the shadow spectroscopy framework that enables the simulation of time evolution using shallow stochastic circuits while preserving unbiased estimates through quasiprobability sampling is proposed.

Abstract

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 standard 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.

Figures (8)

• Figure 1: Energy spectra of the 10-qubit Heisenberg model from our TE-PAI-based method (solid lines) and the Trotter-based method (dotted line), with the total number of circuit executions fixed at 1000. The legend details the combinations of TE-PAI samples ($M_{\text{TE-PAI}}$) and shadow snapshots ($N_s$). For the same execution cost, the Trotter-based method shows a higher peak intensity, while the TE-PAI results are consistent across different configurations.
• Figure 2: Effect of depolarizing noise on the energy spectra obtained from TE-PAI (red) and Trotter-based (blue) methods. Dotted lines represent the noise-free benchmarks, while solid lines show the results under noise. The TE-PAI spectrum demonstrates strong robustness, with its peak remaining clearly visible. In contrast, the Trotter-based spectrum collapses, losing nearly all spectral information. This highlights the practical advantage of TE-PAI's shallower circuits in noisy environments.
• Figure 3: Spectra of a 20-qubit Transverse Ising Hamiltonian measured on quantum hardware. The red and green line show a TE-PAI based spectrum performed respectively on ibm_kobe and ibm_kingston. The blue dotted line shows a Trotter based spectrum performed on ibm_kingston. The grey dotted vertical line indicates the exact energy gap.
• Figure 4: Comparison of TE-PAI spectra computed with different shadow sizes. The spectra were generated using a fixed number of TE-PAI samples while varying the shadow size $N_s \in \{1, 10, 100\}$, respectively blue, orange and green spectrum. While all cases resolve the theoretical energy gap, increasing $N_s$ significantly enhances the peak intensity. Simulation parameters match those detailed in Sec. \ref{['sec:classical_simulation']}.
• Figure 5: Architecture and error landscape of the ibm_kobe quantum processor. Each node represents a qubit, with its color indicating the average readout assignment error. Each edge represents a physical connection between two qubits, and its color encodes the average controlled-Z (CZ) gate error. Grey nodes or edges indicate that the corresponding error exceeds a predefined threshold, set to 0.05 for the readout error and 0.02 for the CZ error. The qubits and connections highlighted by the red outline denote the specific subset of the device that is utilized in our experiments.

Theorems & Definitions (6)

• Theorem 1: Unbiasedness
• Theorem 2: Variance of the overall estimator
• Lemma 1
• proof
• proof
• proof