Preparing Ground and Excited States Using Adiabatic CoVaR

TL;DR

This paper introduces an adiabatic approach to enhance CoVaR, a quantum algorithm for state preparation, enabling it to find ground and excited states without a good initial guess, even with small energy gaps.

Contribution

It extends CoVaR with an adiabatic morphing technique, allowing robust eigenstate preparation without prior warm start, and demonstrates its effectiveness across various applications.

Findings

CoVaR can prepare eigenstates without a good initial state.

Adiabatic CoVaR is effective even with small energy gaps.

It can map out low-lying spectra useful for thermal and high-energy physics applications.

Abstract

CoVarince Root finding with classical shadows (CoVaR) was recently introduced as a new paradigm for training variational quantum circuits. Common approaches, such as variants of the Variational Quantum Eigensolver, aim to optimise a non-linear classical cost function and thus suffer from, e.g., poor local minima, high shot requirements and barren plateaus. In contrast, CoVaR fully exploits powerful classical shadows and finds joint roots of a very large number of covariances using only a logarithmic number of shots and linearly scaling classical computing resources. As a result, CoVaR has been demonstrated to be particularly robust against local traps, however, its main limitation has been that it requires a sufficiently good initial state. We address this limitation by introducing an adiabatic morphing of the target Hamiltonian and demonstrate in a broad range of application examples that CoVaR can successfully prepare eigenstates of the target Hamiltonian when no initial warm start is known. CoVaR succeeds even when Hamiltonian energy gaps are very small -- this is in stark contrast to adiabatic evolution and phase estimation algorithms where circuit depths scale inversely with the Hamiltonian energy gaps. On the other hand, when the energy gaps are relatively small then adiabatic CoVaR may converge to higher excited states as opposed to a targeted specific low-lying state. Nevertheless, we exploit this feature of adiabatic CoVaR and demonstrate that it can be used to map out the low lying spectrum of a Hamiltonian which can be useful in practical applications, such as estimating thermal properties or in high-energy physics.