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Landscape Analysis of Excited States Calculation over Quantum Computers

Hengzhun Chen, Yingzhou Li, Bichen Lu, Jianfeng Lu

TL;DR

The paper analyzes three excited-state VQE formulations that embed orthogonality constraints into the objective: qOMM, qTPM, and qL1M. It provides rigorous landscape analyses showing that every local minimum is a global minimum, derives precise forms for stationary points and local minimizers, and demonstrates how orthogonality can emerge without explicit constraints. Numerical results on a small H2 system compare quantum-resource costs and optimization behavior, revealing trade-offs between quantum overhead and classical optimization complexity. The work offers theoretical guarantees and practical guidelines for selecting among these approaches and lays groundwork for broader landscape analyses in VQE methods. Overall, it advances robust, unconstrained strategies for excited-state calculations on NISQ devices through principled orthogonality embedding and rigorous optimization theory.

Abstract

The variational quantum eigensolver (VQE) is one of the most promising algorithms for low-lying eigenstates calculation on Noisy Intermediate-Scale Quantum (NISQ) computers. Specifically, VQE has achieved great success for ground state calculations of a Hamiltonian. However, excited state calculations arising in quantum chemistry and condensed matter often requires solving more challenging problems than the ground state as these states are generally further away from a mean-field description, and involve less straightforward optimization to avoid the variational collapse to the ground state. Maintaining orthogonality between low-lying eigenstates is a key algorithmic hurdle. In this work, we analyze three VQE models that embed orthogonality constraints through specially designed cost functions, avoiding the need for external enforcement of orthogonality between states. Notably, these formulations possess the desirable property that any local minimum is also a global minimum, helping address optimization difficulties. We conduct rigorous landscape analyses of the models' stationary points and local minimizers, theoretically guaranteeing their favorable properties and providing analytical tools applicable to broader VQE methods. A comprehensive comparison between the three models is also provided, considering their quantum resource requirements and classical optimization complexity.

Landscape Analysis of Excited States Calculation over Quantum Computers

TL;DR

The paper analyzes three excited-state VQE formulations that embed orthogonality constraints into the objective: qOMM, qTPM, and qL1M. It provides rigorous landscape analyses showing that every local minimum is a global minimum, derives precise forms for stationary points and local minimizers, and demonstrates how orthogonality can emerge without explicit constraints. Numerical results on a small H2 system compare quantum-resource costs and optimization behavior, revealing trade-offs between quantum overhead and classical optimization complexity. The work offers theoretical guarantees and practical guidelines for selecting among these approaches and lays groundwork for broader landscape analyses in VQE methods. Overall, it advances robust, unconstrained strategies for excited-state calculations on NISQ devices through principled orthogonality embedding and rigorous optimization theory.

Abstract

The variational quantum eigensolver (VQE) is one of the most promising algorithms for low-lying eigenstates calculation on Noisy Intermediate-Scale Quantum (NISQ) computers. Specifically, VQE has achieved great success for ground state calculations of a Hamiltonian. However, excited state calculations arising in quantum chemistry and condensed matter often requires solving more challenging problems than the ground state as these states are generally further away from a mean-field description, and involve less straightforward optimization to avoid the variational collapse to the ground state. Maintaining orthogonality between low-lying eigenstates is a key algorithmic hurdle. In this work, we analyze three VQE models that embed orthogonality constraints through specially designed cost functions, avoiding the need for external enforcement of orthogonality between states. Notably, these formulations possess the desirable property that any local minimum is also a global minimum, helping address optimization difficulties. We conduct rigorous landscape analyses of the models' stationary points and local minimizers, theoretically guaranteeing their favorable properties and providing analytical tools applicable to broader VQE methods. A comprehensive comparison between the three models is also provided, considering their quantum resource requirements and classical optimization complexity.

Paper Structure

This paper contains 12 sections, 9 theorems, 135 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Optimization Landscape and Algorithms in VQE
  3. qOMM
  4. qTPM
  5. qL1M
  6. Numerical Results
  7. Quantum resource estimate
  8. Optimization behavior
  9. Conclusion
  10. Proofs of qOMM
  11. Proofs of qTPM
  12. Proofs of qL1M

Key Result

Theorem 2.1

\newlabelthm:stationary_qomm0 Given a negative definite Hermitian matrix $A$, stationary points of qOMM eq:qomm take the form $XP$, where $P \in \mathbb{R}^{p \times p}$ is an arbitrary permutation matrix, $X \in \mathbb{C}^{n \times p}$ admits a block structure, $X = \left( X_1, \cdots, X_k \righ In addition, any $X\in \mathbb{C}^{n\times p}$ satisfying the conditions above is a stationary point

Figures (2)

  • Figure 1: Relative error of the objective function versus optimization iteration of the three VQE models for the lowest three eigenvalues of the $\mathrm{H}_2$ molecule with two derivative-free optimizers: COBYLA and BOBYQA. The stop criteria is set to a minimum trust region radius of rhoend=1e-7 or a maximum of 600 iterations.
  • Figure 2: Convergence of relative errors of eigenvalues and trajectories of orthogonality constraint under COBYLA optimizer for three VQE models. (Left) The convergence behavior of relative error of eigenvalues in $l_2$ norm for all three models. (Right) The value of $\left\|X^*X - I\right\|_\mathrm{F}$ versus optimization iteration for three models.

Theorems & Definitions (23)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Remark 2.9
  • Definition A.1: Strict Majorization
  • ...and 13 more