# Quantum EigenGame for excited state calculation

arXiv: https://arxiv.org/abs/2503.13644 · HTML: https://mlsanity.com/paper/2503.13644/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 47 references that are in the map.
- First posted on arXiv 17 Mar 2025.

## Builds on (most cited first, 7 of 7)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [A Theoretical and Empirical Comparison of Gradient Approximations in Derivative-Free Optimization](https://mlsanity.com/paper/1905.01332/) (2019)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [The Noisy Power Method: A Meta Algorithm with Applications](https://mlsanity.com/paper/1311.2495/) (2013)
- [EigenGame: PCA as a Nash Equilibrium](https://mlsanity.com/paper/2010.00554/) (2020)
- [Efficient charge-preserving excited state preparation with variational quantum algorithms](https://mlsanity.com/paper/2410.14357/) (2024)

## Abstract (arXiv)

Computing the excited states of a given Hamiltonian is computationally hard for large systems, but methods that do so using quantum computers scale tractably. This problem is equivalent to the PCA problem where we are interested in decomposing a matrix into a collection of principal components. Classically, PCA is a well-studied problem setting, for which both centralized and distributed approaches have been developed. On the distributed side, one recent approach is that of EigenGame, a game-theoretic approach to finding eigenvectors where each eigenvector reaches a Nash equilibrium either sequentially or in parallel. With this work, we extend the EigenGame algorithm for both a 0th-order approach and for quantum computers, and harness the framework that quantum computing provides in computing excited states. Results show that using the Quantum EigenGame allows us to converge to excited states of a given Hamiltonian without the need of a deflation step. We also develop theory on error accumulation for finite-differences and parameterized approaches.