Advantage of Warm Starts for Electron-Phonon Systems on Quantum Computers

TL;DR

The paper tackles the bottleneck in quantum simulations of electron–phonon systems at strong coupling by introducing a Lang–Firsov–inspired initial state that captures polaronic dressing. It provides an efficient quantum circuit to prepare this state and demonstrates, via overlap benchmarks, that the ground-state fidelity remains high across couplings, enabling a substantial reduction in QPE iterations. Resource analysis shows the preparation cost is modest, while the improved overlap yields exponential savings in overall QPE cost, especially as the electron–phonon coupling strengthens. This work highlights the practical value of incorporating physical intuition into initial-state design for scalable quantum simulations of correlated electron–phonon systems and points to broader applicability to polarons and related bosonic environments.

Abstract

Simulating electron-phonon interactions on quantum computers remains challenging, with most algorithmic effort focused on Hamiltonian simulation and circuit optimization. In this work, we study the single-electron Holstein model and propose an initial-state ansatz that substantially enhances ground state overlap in the strong coupling regime, thereby reducing the number of iterations required in standard quantum phase estimation. We further show that this ansatz can be implemented efficiently and yields an exponential reduction in overall circuit costs relative to conventional initial guesses. Our results highlight the practical value of incorporating physical intuition into initial state preparation for electron-phonon coupled systems.

Figures (11)

• Figure 1: Overlap $\Omega_{\rm gs}=|\langle {\rm GS}|\Psi_{\text{LF}}(\alpha)\rangle|^2$ versus $\lambda$ for $t=\omega_0=1$ and (a) the phonon-free initial state with $\alpha = 0$; (b) the strong-coupling value $\alpha = g/\omega_0$; and (c) the variational solution of Eq. (10). The exact GS was calculated with DMRG for chains of various lengths $N$. We only show results for $N=4$ and $N=8$ because values for $N>8$ are indistinguishable from those with $N=8$, on this scale.
• Figure 2: Circuit for the Lang–Firsov ansatz on a chain of length $N=4$. The first two electron registers specify the electron’s position. Each of the four remaining registers, with $m \approx 6\!-\!8$ qubits per register, encodes the local phonon mode at its site. $U_{0}$ prepares the harmonic–oscillator ground state on every phonon register. The controlled–$\tilde{U}(\alpha)$ applies the site-dependent phonon displacement conditioned on the electron registers.
• Figure 3: $T$-gate cost, on a Logarithmic scale against the number of sites, $N$. We show the costs of preparing either the vacuum state with $U_0$ and the Lang-Firsov state with the added cost of implementing controlled $\tilde{U}_{\alpha}$'s. In both cases, we set $m=6$ (encoding phononic degrees of freedom at a site). We see that the difference in the cost of state preparation circuits are barely visible in this scale.
• Figure 4: We compare the T-count cost of QPE-based ground-state preparation using two different initial states. For fixed $\omega_0 = 1$, the figure shows the cost ratio \ref{['eq:cost_ration']} as a function of the coupling constant dependent parameter $\lambda$. As $\lambda$ increases, the relative cost of the LF-based protocol decreases rapidly, with only weak dependence on the system size $N$.
• Figure 5: Function norm error between $e^{x^2/2}$ and different $h_0(z)$ approximations for degrees $\{20, 22, 24, 26, 28\}$ shown on a log scale. We observe that the error spikes close to $x=0$, and that there is not much benefit to slightly increasing the degree after $d={22}$.