Quasiparticle Variational Quantum Eigensolver

TL;DR

This work develops a momentum-space, symmetry-preserving VQE framework to simulate quasiparticle excitations in interacting quantum many-body systems, using the XXZ spin chain as a benchmark and exploiting translational symmetry to reconstruct momentum-resolved spectra. The method initializes a free-fermion particle-hole state at fixed momentum $q$, maps it to real space via an inverse FFFT, and variationally evolves it with a Hamiltonian Variational Ansatz (HVA) that preserves translation and parity. Results on 8- and 16-qubit systems show accurate replication of the low-lying quasiparticle dispersion and renormalized velocity, in good agreement with Bethe ansatz, with improved accuracy as the HVA depth increases. These findings validate a targeted, symmetry-aware VQE strategy for excited-state spectroscopy and point to extensions to other interacting models such as the Fermi-Hubbard, highlighting the practical potential of near-term quantum devices for quasiparticle physics.

Abstract

We propose a momentum-space based variational quantum eigensolver (VQE) framework for simulating quasiparticle excitations in interacting quantum many-body systems on near-term quantum devices. Leveraging translational invariance and other symmetries of the Hamiltonian, we reconstruct the momentum-resolved quasiparticle excitation spectrum through targeted simulation of low-lying excited states using VQE. We construct a translationally symmetric variational ansatz designed to evolve a free-fermion particle-hole excited state with definite momentum $q$ to an excited state of the interacting system at the same momentum, employing a fermionic fast Fourier transform (FFFT) circuit coupled to a Hamiltonian Variational Ansatz (HVA) circuit. Even though the particle number is not explicitly conserved in the variational ansatz, the correct quasiparticle state is reached by energetic optimization. We benchmark the performance of the proposed VQE implementation on the XXZ Hamiltonian, which maps onto the Tomonaga-Luttinger liquid in the fermionic representation. Our numerical results show that VQE can capture the low-lying excitation spectrum of the bosonic quasiparticle/two-spinon dispersion of this model at various interaction strengths. We estimate the renormalized velocity of the quasiparticles by calculating the slope of the dispersion near zero momentum using the VQE-optimized energies at different system sizes, and demonstrate that it closely matches theoretical results obtained from Bethe ansatz. Finally, we highlight extensions of our proposed VQE implementation to simulate quasiparticles in other interacting quantum many-body systems.

Figures (10)

• Figure 1: (a) Energy dispersion of the XX model, $\epsilon(k)=4\cos\,k$, with the Fermi points at $k_{F}=\pm\frac{\pi}{2}$. The shaded area indicates the filled negative energy states of the ground state (half-filled sector). (b) Spectrum of particle-hole excitations in the XX model in the half-filled sector. The bounding curves are highlighted in black and red, and the shaded area is the continuum representing the different excitation energies possible, with $q=k_{p}-k_{h}$.
• Figure 2: (a) High-level illustration of the variational ansatz circuit structure, comprised of the free-fermion state preparation circuit $\mathcal{U}_{\text{PH}}$, the inverse fermionic fast Fourier transform (FFFT) circuit $\mathcal{U}_{\text{FFFT}}^{\dagger}$, and the Hamiltonian Variational Ansatz (HVA) circuit $U_{\text{HVA}}(\boldsymbol{\theta})$. (b) Displays the Trotterized structure of the HVA circuit for $p$ layers and parameter vector $\boldsymbol{\theta}=\theta_{1}^{(1)},\theta_{1}^{(2)},\cdots\theta_{p}^{(1)},\theta_{p}^{(2)}$.
• Figure 3: Quantum circuit diagrams of the variational ansatz for quasiparticle simulation for an $N=8$ qubit system; convention for indexing the qubit register is shown in (a). (a) The free-fermion state preparation circuit $\mathcal{U}_{\text{PH}}$ and inverse fermionic fast Fourier transform (FFFT) circuit $\mathcal{U}_{\text{FFFT}}^{\dagger}$ are illustrated for the XX model. $\mathcal{U}_{\text{PH}}$ prepares a particle-hole symmetric state in the half-filled ($\mathcal{N}_{F}=4$) sector with total momentum $q=\frac{\pi}{4}$, and $\mathcal{U}_{\text{FFFT}}^{\dagger}$ is the 8 qubit inverse fermionic fast Fourier transform (FFFT) circuit. fSWAP gates are denoted by the crossed-out square boxes. (b) Illustration of a single layer of the HVA circuit, consisting of parameterized $R_{\text{XX}}$, $R_{\text{YY}}$, and $R_{\text{ZZ}}$ gates.
• Figure 4: Plots of the excitation spectrum for different values of the anisotropy strength $\Delta$ obtained from VQE and exact diagonalization (ED). The quasiparticle excitation spectrum is plotted for the half-filled sector ($S_{z}=0$) for (a) $N=8$ and (b) $N=16$. For reference, the ground state of the XXZ model is located in the half-filled sector at momentum $q=0$. The solid circles are the VQE-optimized energy values, and the open circles are the ED values.
• Figure 5: Semi-log plots of the obtained from VQE for the optimized energies relative errors ((a)-(b)) and the infidelities ((c)-(d)) for $\Delta=0.5$ in the $\mathcal{N}_{F}=N/2$ sector. The errors are plotted for the lowest momentum values ($q\in\left\{0,\frac{\pi}{4},\frac{\pi}{2}\right\}$ for $N=8$ in (a) & (c) and $q\in\left\{0,\frac{\pi}{8},\frac{\pi}{4},\frac{3\pi}{8},\frac{\pi}{2}\right\}$ for $N=16$ in (b) & (d)).