Mitigating Errors in Variational Quantum Eigensolver Calculations through Extended Random Phase Approximation

Abstract

Quantum hardware noise remains a major obstacle to accurate molecular simulations using the variational quantum eigensolver (VQE). We introduce a novel error-mitigation strategy based on the extended random phase approximation (ERPA), which reconstructs the two-particle reduced density matrix (2-RDM) from approximate transition density matrices, effectively filtering errors in quantum measurements. Unlike quantum subspace expansion, the approach requires only the one- and two-particle RDMs. Since ERPA preserves the input 1-RDM, we further propose a hybrid quantum-classical scheme employing an inexpensive density matrix renormalization group (DMRG) calculation to provide a noise-free approximation to this quantity. Benchmarks on water double-bond stretching and cyclobutadiene automerization, performed on IBM quantum hardware, demonstrate reductions in energy errors by orders of magnitude. Combined with multireference adiabatic connection (AC0) corrections, the approach additionally recovers dynamical correlation beyond the active space, yielding accurate molecular potential energy curves and reaction barriers. Our results establish ERPA as a promising, measurement-efficient route toward reliable quantum-classical electronic structure calculations.

The paper investigates whether extended random phase approximation (ERPA) can mitigate noise in VQE molecular energies using only measured one- and two-particle reduced density matrices (RDMs). ERPA transition densities reconstruct the ground-state 2-RDM through the McLachlan–Ball relation, avoiding the higher-order RDMs generally required by singles quantum subspace expansion. The reconstruction preserves the input 1-RDM, motivating a hybrid variant that supplies an approximate, noise-free 1-RDM from low-bond-dimension DMRG. Tests cover symmetric water stretching and cyclobutadiene automerization, both in CAS(4,4), using four or five tapered qubits on IBM Heron hardware. ADAPT-VQE circuits were constructed and optimized classically, then held fixed during hardware measurements. Results use 1,000 shots and ten independent runs. ERPA alone leaves water energy errors reaching approximately $400\,\mathrm{m}E_h$; supplying a DMRG 1-RDM with bond dimension $M=4$ reduces maximum active-space errors to approximately $15\,\mathrm{m}E_h$ for water and below $10\,\mathrm{m}E_h$ for cyclobutadiene. Adding an AC0 correction incorporates correlation outside the active space. The cyclobutadiene barrier is 6.56 kcal/mol before AC0 and 8.99 kcal/mol afterward, versus corresponding classical references of 7.38 and 10.91 kcal/mol. The reconstruction is neither exact nor guaranteed to be $N$-representable: it can yield energies below the ground-state energy even with exact input RDMs. Finite-shot simulations show variance suppression accompanied by systematic bias reaching roughly $15\,\mathrm{m}E_h$. Hardware accuracy partly reflects cancellation of this negative bias against positive noise-induced errors. The demonstrated protocol does not mitigate errors during VQE optimization.