A momentum-space reference goes onto a quantum processor: correlator optimization moves after measurement, reaching 4.40% error on 32 qubits

Explicitly correlated wave functions can now be variationally optimized on a quantum processor without feedback: the quantum side only prepares a compact momentum-space reference state and rotates it to real space, leaving the correlator to the classical side. Previously, variational Monte Carlo needed repeated access to real-space reference amplitudes, and overlaps of a momentum-space matrix product state with arbitrary Slater determinants are #P-hard; the transcorrelated method avoided that conflict but restricted the correlator and gave a nonvariational energy. On 32 qubits of Quantinuum's H2-2, a bond-dimension-4 spin-blocked matrix product state gives a 4.40% relative ground-state energy error after postselection, reduced further by calibration; on the spin-interleaved matrix product state, a 103-parameter correlator gives 0.139% on the original samples, and four independent validation sample sets give 95% relative-error bounds from 0.20% to 0.25%.

The quantum circuits measure only occupation and hopping bases fixed by the Hamiltonian, so the correlator's form and parameters can be chosen, changed and re-optimized after the experiment, with the same measurement record reused for every variational evaluation. Conventional variational quantum algorithms iterate between the quantum processor and a classical optimizer, rerunning circuits as parameters update; here every quantum circuit is fixed before execution. The compiled diagonal-measurement circuit contains 555 two-qubit gates and 1,095 single-qubit gates, each hopping-measurement circuit adds one to three compiled two-qubit gates, and preparation and rotation act only on nearest-neighbor qubits in a linear chain, so no two-dimensional connectivity is needed.

Distribution errors from finite sampling and hardware noise can be corrected by reweighting against reference-state expectation values known from the classical matrix product state, pulling low-order moments of the sampled distributions back to their correct values. Without calibration, the postselected samples still give an energy well above the full-sample noise-free benchmark; calibration reduces both the bootstrap standard error and the deviation from that benchmark. Partitioning 157 million spin-interleaved samples into experiments of increasing size, calibration narrows the central 68% interval by factors of 3.3 to 4.4, and among successful experiments no calibration-induced bias is resolved.

The demonstrated scope is the half-filled two-dimensional Fermi–Hubbard model: the 32-qubit hardware experiment uses a bond-dimension-4 spin-blocked matrix product state, numerical simulations extend to bond dimension 1024, and resource comparisons are given against quantum-selected configuration interaction and quantum Krylov methods. The authors explicitly call it a proof of principle and argue that quantum advantage would require larger lattices away from half filling.

How sampling cost grows with lattice size is only a hypothesis at half filling, where the authors propose a polynomial-factor growth in sample count and expect slower growth in dilute systems; calibration becomes less effective at larger sizes, and the hardware estimate exceeds 95.2% of matched noise-free experiments, indicating a residual hardware-dependent upward shift; correlator optimization is nonconvex, and under finite sampling the objective is no longer a Rayleigh quotient, so runs can diverge or fall slightly below the exact ground-state energy.