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Quantum EigenGame for excited state calculation

David Quiroga, Jason Han, Anastasios Kyrillidis

TL;DR

This work introduces QuantumGame, a deflation-free, game-theoretic extension of EigenGame tailored for excited-state calculation within the VQE framework. It defines per-player objectives that incorporate an orthogonality penalty and leverages quantum-inner-product estimation (via Hadamard tests) to compute cross terms, enabling a distributed approach to finding the top-$k$ eigenstates of a Hamiltonian without deflation. The authors provide convergence analyses for both a 0th-order, derivative-free variant and a parameterized quantum variant, including error-accumulation results under noisy gradients and finite-difference approximations. Empirically, QuantumGame demonstrates favorable convergence and accuracy on the H$_2$ molecule compared to VQD under both noiseless and shot-noise conditions, and shows ablations validating the scalability and robustness of the approach. The results indicate a practical, deflation-free path to excited-state calculations on NISQ devices with potential for parallel deployment across quantum hardware.

Abstract

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.

Quantum EigenGame for excited state calculation

TL;DR

This work introduces QuantumGame, a deflation-free, game-theoretic extension of EigenGame tailored for excited-state calculation within the VQE framework. It defines per-player objectives that incorporate an orthogonality penalty and leverages quantum-inner-product estimation (via Hadamard tests) to compute cross terms, enabling a distributed approach to finding the top- eigenstates of a Hamiltonian without deflation. The authors provide convergence analyses for both a 0th-order, derivative-free variant and a parameterized quantum variant, including error-accumulation results under noisy gradients and finite-difference approximations. Empirically, QuantumGame demonstrates favorable convergence and accuracy on the H molecule compared to VQD under both noiseless and shot-noise conditions, and shows ablations validating the scalability and robustness of the approach. The results indicate a practical, deflation-free path to excited-state calculations on NISQ devices with potential for parallel deployment across quantum hardware.

Abstract

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 -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.

Paper Structure

This paper contains 12 sections, 24 theorems, 79 equations, 11 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. An EigenGame for VQE
  4. Experiments and discussion
  5. Conclusions
  6. Calculating the expression of the finite differences EigenGame gradient
  7. Swap test and Hadamard test derivations
  8. Classical finite differences convergence analysis
  9. Proof of the main $0^\text{th}$-order convergence theorem
  10. Parameterized convergence analysis
  11. Proof of the main parameterized convergence theorem
  12. Error accumulation

Key Result

Theorem 1

(Convergence of $0^{\text{th}}$-order EigenGame for all players). Consider the Algorithm alg:EigenGame-zeroth with input matrix $M \in \mathbb{R}^{p \times p}$ and learned "parent" eigenvectors $v_{j<i} \in \mathbb{R}^p$ that are accurate enough, i.e., that $|\phi_{j<i}| \leq \frac{c_i g_i}{(i-1)\La where $L_i(\sigma)$ is the Lipschitz continuity assumption of the $0^\text{th}$-order EigenGame bas

Figures (11)

  • Figure 1: Circuit schematic for VQE where $|\psi(\theta)\rangle$ is prepared as $|\psi(\theta)\rangle = \mathbf{U}(\theta) |\psi_0\rangle$ when $|\psi_0\rangle = |+\rangle$. The initial layer of Hadamard gates may be excluded to have $|\psi_0\rangle = |0\rangle$. The circuit is then measured over an observable $M$ to retrieve $\langle \psi(\theta)|M|\psi(\theta)\rangle$.
  • Figure 2: To compute terms of the form $\langle \psi(\theta^{(r)})| M |\psi(\theta^{(j)})\rangle$, we utilize interference between $| \psi(\theta^{(r)}) \rangle$ and $| \psi(\theta^{(j)}) \rangle$.
  • Figure 3: QuantumGame with three players.
  • Figure 4: Error accumulation of a child eigenvector $\hat{v}_i$ from its parent eigenvector $\hat{v}_j$, given $\phi_j \leq \epsilon \ll 1$.
  • Figure 5: Excited state energy levels of the two-qubit $H_2$ molecule mapping calculated with noiseless QuantumGame (blue line), noisy QuantumGame (orange line), noiseless VQD (green line) and noisy VQD (purple line) over accumulative iterations. The noisy setting only uses statistical shot noise with 10k shots. The exact energy level (dashed line) is extracted using Numpy. The inset plot shows the total number of iterations for noiseless VQD and noisy VQD with values of $\beta \in [0.1, 0.5, 1, 2, 3, 4, 5]$.
  • ...and 6 more figures

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Theorem 5: Thm 3 in boumal_global_2019
  • Theorem 6
  • Theorem 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 14 more