Spectral Gap Estimation via Adiabatic Preparation

TL;DR

This work tackles the challenge of estimating spectral gaps in quantum systems using a hardware-friendly approach that relies on Adiabatic Preparation to generate a targeted superposition of eigenstates. By monitoring and fitting the time-dependent fluctuations of a chosen observable on the evolved state, the gap between the corresponding eigenstates is extracted without computing the individual eigenvalues. The method is validated on Ising models (1D and 2D) and small molecules (H2, He2), showing accurate gap estimates with shallow circuits on simulators and real devices, including IonQ Aria up to 20 qubits. The results highlight a practical NISQ-era pathway for energy-gap calculations across complex Hamiltonians, with potential extensions to larger systems in the fault-tolerant era.

Abstract

Estimating energy gaps, i.e. the energy difference between two different states, in quantum systems is crucial for understanding their properties. Conventionally, spectral gap estimation relies on independently computing the ground-state and first-excited-state energies and then taking their difference. This work introduces an alternative procedure for estimating spectral gaps on digital quantum devices using the Adiabatic Preparation technique to create a specific superposition state. The expectation values of observables measured on such a state exhibit time-dependent fluctuations which, through a fitting process, can be used to estimate the energy gap. We successfully test our method on the 1D and 2D Ising models, and H2 and He2 molecules, implementing relatively shallow circuits both on noiseless and noisy simulators. The robustness of the approach is corroborated by additional experiments on the real IonQ Aria device for the 1D Ising model up to 20 qubits, demonstrating the applicability of the proposed method for currently available digital quantum devices and paving the way for more complex energy gap calculation requiring deeper circuits in the fault-tolerant era to come.

Figures (9)

• Figure 1: Schematic representation of the proposed methodology. Initially, a quantum circuit generates the superposition state $\ket{\Psi_0}$ of two eigenstates of the auxiliary Hamiltonian $H_0$. Through Adiabatic Preparation this state is evolved to the $\ket{\Psi}$ state of the system Hamiltonian $H$. Sequential measurements of the chosen observable $O(t)$ at different time steps are recorded. Finally a fitting process is used to determine the spectral gap. The obtained value is compared to the benchmark spectral gap.
• Figure 2: Spectral gap as a function of $h_3/J_1$ for a (a) $L=4$ and (b) $L=20$ Ising chain with PBC. The plots include simulations on a classical computer, showing both noiseless results and experiments on IonQ Aria 1. The adiabatic and infrared limits are regions where the circuit depth used is insufficient. Classical simulations of the noisy hardware are also shown in panel (a). Error bars, shown but not always visible, represent a single standard deviation computed from the fit.
• Figure 3: Spectral gap for (a) H2 and (b) He2 molecules as a function of the bond length. Noiseless simulations show excellent agreement with benchmark values. Error bars represent a single standard deviation computed from the fit.
• Figure 4: Numerical estimations of the expectation value $\mathcal{A} = \abs{\bra{\Phi^+}\sigma^1\otimes \left(\mathds{1}^{\otimes L-1}\right)\ket{\Phi^-}}$ as a function of the Ising's coupling constants ratio $J_1/h_3$. For all chain lengths considered, $L = \{4,6,8,10,12\}$, $\mathcal{A}$ remains larger than $L^{-1/2}$ for any value of $h_3/J_1$, ensuring that the amplitude is strictly positive across the entire range of coupling ratios.
• Figure 5: Fidelity of the approximate superposition state $\ket{\Psi(0,1)}$, in the noiseless and noisy case ($\ket{\nu}$, ${\rho_\nu}$), with the ground and first excited state ($\ket{\psi_0}$, $\ket{\psi_1}$) for an Ising chain of $L=8$ qubits. The fidelity is found to be sub-optimal, decaying and oscillating for increasing $h_3$. The discontinuities are given by the change in $\delta \tau$ (at $h_3=\{3.6,5\}$) imposed in the adiabatic process.