Efficient Quantum Simulation of Electron-Phonon Systems by Variational Basis State Encoder

Digital quantum simulation of electron-phonon systems requires truncating infinite phonon levels into $N$ basis states and then encoding them with qubit computational basis. Unary encoding and the more compact binary/Gray encoding are the two most representative encoding schemes, which demand $\mathcal{O}(N)$ and $\mathcal{O}(\log{N})$ qubits as well as $\mathcal{O}(N)$ and $\mathcal{O}(N\log{N})$ quantum gates respectively. In this work, we propose a variational basis state encoding algorithm that reduces the scaling of the number of qubits and quantum gates to both $\mathcal{O}(1)$. The cost for the scaling reduction is a constant amount of additional measurement. The accuracy and efficiency of the approach are verified by both numerical simulation and realistic quantum hardware experiments. In particular, we find using 1 or 2 qubits for each phonon mode is sufficient to produce quantitatively correct results across weak and strong coupling regimes. Our approach paves the way for practical quantum simulation of electron-phonon systems on both near-term hardware and error-corrected quantum computers.