Nash embedding: a road map to realizing quantum hardware

Duong Do, Matthew Aldridge, Duong Tung Nguyen

We propose a projected variational quantum extragradient (VQEG) framework for computing approximate Nash equilibria in two-player zero-sum matrix games. Mixed strategies are parameterized as Born distributions of parameterized quantum circuits (PQCs), transforming the classical bilinear saddle point problem into a smooth but generally minmax optimization in circuit-parameter space. The expected payoff is expressed as the expectation of a diagonal observable, enabling gradient evaluation via the parameter shift rule and compatibility with shot based quantum hardware. To support arbitrary game sizes, we introduce a dominated embedding that maps (m,n) games to qubit-compatible power-of-two dimensions while preserving equilibrium structure. We then develop a projected extragradient method using stochastic gradient estimates derived from finite measurement shots, and establish variance bounds scaling as O(1/S) with respect to the number of measurement shots S, along with convergence to approximate first-order stationarity under standard assumptions. Since stationarity does not guarantee equilibrium optimality, we evaluate performance using the game-space Nash gap. Numerical results demonstrate high-precision solutions on structured instances up to 32x32, while highlighting challenges in unstructured settings.

Bettina Just

Mika A. Sillanpää, Pertti Juhani Hakonen

Faisal Shah Khan, Travis S. Humble

With respect to probabilistic mixtures of the strategies in non-cooperative games, quantum game theory provides guarantee of fixed-point stability, the so-called Nash equilibrium. This permits players to choose mixed quantum strategies that prepare mixed quantum states optimally under constraints. We show here that fixed-point stability of Nash equilibrium can also be guaranteed for pure quantum strategies via an application of the Nash embedding theorem, permitting players to prepare pure quantum states optimally under constraints.

Yann Pouillon, Federico Marchesin, Ivano Tavernelli +1 more

Density Functional Theory (DFT) is the established workhorse of computational materials science, yet it faces fundamental challenges with strongly correlated electrons, such as in transition metal complexes and bond-breaking processes. While quantum computing offers a theoretically exact path to resolve these problems, current hardware limitations prohibit fully quantum simulations of realistic material systems. To bridge this gap, we present SIESTA-QCOMP, a modular suite embedding quantum methodologies within DFT. Harnessing SIESTA's localised basis set, we implement a projection-based embedding scheme that treats the bulk environment classically while mapping the strongly correlated "impurity" subspace to a quantum processor. The architecture uses a socket-based driver to orchestrate workflows between the Fortran-based SIESTA engine and Python-based frameworks like Qiskit, ensuring high interoperability. We detail the hybrid workflow, currently utilising a Variational Quantum Eigensolver (VQE), and discuss the road map toward methods such as Sample-based Quantum Diagonalisation (SQD). We illustrate the framework on an hypothetical iron porphyrin calculation and argue that such hybrid architectures are not merely interim solutions but essential foundations for integrating quantum utility into materials design, as envisioned by our broader QCOMP4DFT initiative.

Chien‐Hung Cho, Chih-Yu Chen, Kuo-Chin Chen +6 more