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arXiv:2406.18669 · Physical review. D/Physical review. D.

Solving the homogeneous Bethe-Salpeter equation with a quantum annealer

Filippo Fornetti, Alex Gnech, T. Frederico, Francesco Pederiva, Matteo Rinaldi, Alessandro Roggero, Giovanni Salmè, Sergio Scopetta, M. Viviani

Published Sep 6, 2024• 4 citations
arXivHTMLPDF

Abstract

The homogeneous Bethe-Salpeter equation (hBSE), describing a bound system in a genuinely relativistic quantum-field theory framework, was solved for the first time by using a D-Wave quantum annealer. After applying standard techniques of discretization, the hBSE, in ladder approximation, can be formally transformed in a generalized eigenvalue problem (GEVP), with two square matrices: one symmetric and the other nonsymmetric. The latter matrix poses the challenge of obtaining a suitable formal approach for investigating the GEVP by means of a quantum annealer, i.e., to recast it as a quadratic unconstrained binary optimization problem. A broad numerical analysis of the proposed algorithms, applied to matrices of dimension up to 64, was carried out by using both the simulated-annealing package and the D-Wave . The numerical results very nicely compare with those obtained with standard classical algorithms, and also show interesting scalability features. Published by the American Physical Society 2024

DOI
10.1103/physrevd.110.056012

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The homogeneous Bethe-Salpeter equation (hBSE), describing a bound system in a genuinely relativistic quantum-field theory framework, was solved for the first time by using a D-Wave quantum annealer. After applying standard techniques of discretization, the hBSE, in ladder approximation, can be formally transformed in a generalized eigenvalue problem (GEVP), with two square matrices: one symmetric and the other nonsymmetric. The latter matrix poses the challenge of obtaining a suitable formal approach for investigating the GEVP by means of a quantum annealer, i.e., to recast it as a quadratic unconstrained binary optimization problem. A broad numerical analysis of the proposed algorithms, applied to matrices of dimension up to 64, was carried out by using both the simulated-annealing package and the D-Wave . The numerical results very nicely compare with those obtained with standard classical algorithms, and also show interesting scalability features. Published by the American Physical Society 2024
We present a gate-based quantum computing solution of the homogeneous Bethe-Salpeter equation (hBSE) for the bound state of two massive relativistic scalar particles interacting via ladder-approximation scalar exchange. After Wick rotation to Euclidean space and O(4) S-wave partial-wave projection, the hBSE is reduced to a symmetric matrix eigenvalue problem of dimension N = 2^n. We decompose the resulting Hamiltonian into a sum of n-qubit Pauli operators and solve it with the Variational Quantum Eigensolver (VQE) using a Matrix Product State (MPS) tensor-network ansatz. For N=16 (n=4 qubits), the VQE recovers the maximum eigenvalue - which encodes the minimum coupling constant for binding - to better than 1 % mean relative error compared to classical diagonalization. Entanglement analysis of the BSE amplitude shows low, area-law-like entanglement over the tested sizes, which motivates the MPS ansatz. A critical analysis of three independent barriers - exponential Pauli overhead, approximately size-independent low entanglement, and decreasing VQE gradient scales for the tested ansatz, consistent with generic barren-plateau concerns at larger n - reveals that the specific problem studied here lies in a classically tractable regime: over the tested range it is well described by low-bond-dimension tensor networks and efficiently handled by MPS/Lanczos methods. This negative result provides, to our knowledge, the first entanglement quantification of the BSE amplitude in qubit encoding, establishes the Pauli Hamiltonian framework for future BSE variants, and identifies 2D Minkowski-space BSE, N-body bound states, and non-ladder kernels as physically motivated extensions where genuine quantum advantage may become plausible.
Solving Helmholtz problems using finite elements leads to the resolution of a linear system which is challenging to solve for classical computers. In this paper, we investigate how quantum annealers could address this challenge. We first express the linear system arising from the Helmholtz problem as a generalized eigenvalue problem (gEVP). The obtained gEVP is mapped into quadratic unconstrained binary optimization problems (QUBOs) which we solve using an adaptive quantum annealing eigensolver (AQAE) and its classical equivalent. We identify two key parameters in the success of AQAE for solving Helmholtz problems: the system condition number and the integrated control errors (ICE) in the quantum hardware. Our results show that a large system condition number implies a finer discretization grid for AQAE to converge, leading to a variable overhead, and that AQAE is either tolerant or not with respect to ICE depending on the gEVP. Finally, we establish lower bounds on the annealing time, narrowing the possibility of a quantum advantage for solving Helmholtz problems.
Self-consistent Hartree-Fock approximation combined with solutions of the Bethe-Salpeter equation offers a powerful tool for studies of strong correlation effects arising in condensed matter models, nuclear physics, quantum field theories, and real materials. The standard finite-temperature approach would be to first solve the problem in the Matsubara representation and then apply numerical analytic continuation to the real-frequency axis to link theoretical results with experimental probes, but this ill-conditioned procedure often distorts important spectral features even for very accurate imaginary-frequency data. We demonstrate that the ladder-type finite-temperature Bethe-Salpeter equation in the Hartree-Fock basis for the three-point vertex function and, ultimately, system's polarization can be accurately solved directly on the real frequency axis using the diagrammatic Monte Carlo technique and series resummation. We illustrate the method by applying it to the homogeneous electron gas model and demonstrate how multiple scattering events renormalize Landau damping.
The first-principles quantum simulation of the full electronic Hamiltonian of a material model of realistic size appears daunting even in a future era of fully fault-tolerant quantum computers. In contrast, classical simulations of material excited states rely on heuristic quasiparticle frameworks, such as the GW approximation and Bethe-Salpeter equation, where only a few quasiparticles are explicitly considered. Here we give a detailed description of quasiparticle quantum simulation for multi-excitonic states within the BSE. We develop an explicit first-quantized, orbital-based block encoding of the multi-exciton BSE Hamiltonian for qubitization, and reduce its cost using integral factorization and crystal symmetries. For a materials model of size NNN, and a fixed number of quasiparticles, the resulting algorithms require O~(N)\widetilde{O}(N)O(N) Toffoli gates and O(log⁡N)O(\log N)O(logN) logical qubits, or, using space-time tradeoffs, O~(N)\widetilde{O}(\sqrt{N})O(N​) Toffoli gates and O~(N)\widetilde{O}(\sqrt{N})O(N​) logical qubits, achieving up to a sixth power reduction in Toffoli count, or up to exponential reduction in qubit count, versus standard full Hamiltonian simulation techniques that have been explored in materials simulation. Resource estimates for the classically challenging problem of triexcitonic eigenvalue estimation across a range of semiconductor models of a realistic physical size of O(103)O(10^3)O(103) atoms, show that our Toffoli-optimized implementation reduces the Toffoli count by 106−10710^6-10^7106−107 relative to full Hamiltonian simulation estimates, while a qubit-optimized implementation requires only a few hundred logical qubits. More broadly, these results show that classical heuristic frameworks will be essential in the design of future fault-tolerant quantum algorithms for materials.
Within the Equation of Motion Phonon Method (EMPM) framework, we address the computation of the ground-state eigenpair of nuclear Hamiltonians by reformulating the eigenvalue problem as a Quadratic Unconstrained Binary Optimization (QUBO). To this end, we adopt a QUBO-based iterative steepest-descent algorithm and validate its performance using Simulated An­nealing, demonstrating its feasibility for nuclear many-body calculations. This work represents a first step toward implementations on quantum annealers and extensions to the full eigenspectrum.
Within the Equation of Motion Phonon Method (EMPM) framework, we address the computation of the ground-state eigenpair of nuclear Hamiltonians by reformulating the eigenvalue problem as a Quadratic Unconstrained Binary Optimization (QUBO). To this end, we adopt a QUBO-based iterative steepest-descent algorithm and validate its performance using Simulated An­nealing, demonstrating its feasibility for nuclear many-body calculations. This work represents a first step toward implementations on quantum annealers and extensions to the full eigenspectrum.