Quantum EigenGame for excited state calculation
This work addresses a computational bottleneck in quantum chemistry and physics for researchers using quantum computers, but it is incremental as it adapts an existing classical method to a quantum setting.
The authors tackled the problem of computing excited states of a Hamiltonian, which is computationally hard for large systems, by extending the EigenGame algorithm to quantum computers, resulting in convergence to excited states without needing a deflation step.
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 $0^\text{th}$-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.