A unified quantum computing quantum Monte Carlo framework through structured state preparation

Abstract

We extend Quantum Computing Quantum Monte Carlo (QCQMC) beyond ground-state energy estimation by systematically constructing the quantum circuits used for state preparation. Replacing the original Variational Quantum Eigensolver (VQE) prescription with task-adapted unitaries, we show that QCQMC can address excited-state spectra via Variational Fast Forwarding and the Variational Unitary Matrix Product Operator (VUMPO), combinatorial optimization via a symmetry-preserving VQE ansatz, and finite-temperature observables via Haar-random unitaries. Benchmarks on molecular, condensed-matter, nuclear-structure, and graph-optimization problems demostrate that the QMC diffusion step consistently improves the energy accuracy of the underlying state-preparation method across all tested domains. For weakly correlated systems, VUMPO achieves near-exact energies with significantly shallower circuits by offloading optimization to a classical tensor-network pre-training step, while for strongly correlated systems, the QMC correction becomes essential. We further provide a proof-of-concept demonstration that Haar-random basis state preparation within QCQMC yields finite-temperature estimates from pure-state dynamics.

Figures (26)

• Figure 1: A schematic workflow of the Cross-domain QCQMC.
• Figure 2: Modified Hadamard test to compute the absolute value of all the transition matrix elements $|\widetilde{P}_{ij}| = |\langle b_j|U^{\dagger}PU|b_i \rangle|$ for a fixed input state $|b_i\rangle$.
• Figure 3: Modified Hadamard test to compute the transition matrix element $\widetilde{P}_{ij} = \langle b_j|U^{\dagger}PU|b_i \rangle$ and thus estimate the sign, where $X_i|0\rangle = |b_i\rangle$.
• Figure 4: Overview of the VUMPO ansatz. (a) An $L$ layer brickwork network of two-qubit unitaries $\{u^m_\ell\}$ is equivalent to an MPO with bond dimension $D \leq 2^{2L}$. (b) The unitaries are classically optimized using DMRG-like sweeps to approximately diagonalize the Hamiltonian $H$ (written as a bond dimension $\chi$ MPO with tensors $W$) by minimizing the total energy variance; looped legs denote contraction over all indices, $\boldsymbol{\tau}$. (c) The optimized brickwork tensor network with parameters $\boldsymbol{\theta}^*$ maps directly to a quantum circuit $V(\boldsymbol{\theta}^*)$ for state preparation targeting ground and/or excited states.
• Figure 5: Phase diagram of the $2\times 2$ Fermi-Hubbard model with $\mu=0$ showing the free-Fermion ($U=0$), Fermi liquid ($U/t \lesssim 4$), Non-Fermi liquid ($4\lesssim U/t \lesssim 8$) and strong coupling ($U/t \gg 8$) phases. For half filling, $\mu = \frac{U}{2}$, the strong coupling phase becomes a Mott insulating phase.