The Signal
ERPA Cuts VQE Energy Errors by Orders of Magnitude
A new study applies extended random phase approximation to noisy VQE measurements of water and cyclobutadiene on IBM quantum hardware. The method sharply reduces energy errors, but depends on a classical estimate of the one-particle density
Quantum.world Team · Written with AI · fact-checked ·
Extended random phase approximation (ERPA) sharply reduces energy errors in two molecular simulations on IBM quantum hardware, according to a paper posted October 28. But the method does not correct errors in one of its key inputs, and its reconstructed energies can fall below the exact ground-state value. The results show both the promise of using physical constraints to clean up noisy measurements and the limits researchers still need to address.
Michael Wagner, Pavel Begy, Jiří Brabec, Alexandra Tucholská, Mikuláš Matoušek, Katarzyna Pernal and Libor Veis tested ERPA on water bond stretching and cyclobutadiene automerization. Their approach reconstructs the two-particle reduced density matrix, or 2-RDM, from measured density-matrix data. That reconstruction filters noise in the 2-RDM, but preserves the supplied one-particle reduced density matrix, or 1-RDM; errors in that input therefore remain.
ERPA relies on a cleaner 1-RDM
The team addresses that limitation by supplying ERPA with a noise-free approximation to the 1-RDM from a low-bond-dimension density matrix renormalization group (DMRG) calculation. DMRG is classical, so the method is hybrid: quantum hardware supplies measured data, while classical computation provides the cleaner one-particle input and performs the reconstruction. The authors say ERPA requires measurements only through the 1-RDM and 2-RDM, rather than the higher-order information needed by the quantum subspace expansion approach they compare it with.
“We have demonstrated that a noise-free approximate 1-RDM obtained from a low-bond-dimension DMRG calculation, combined with the ERPA reconstruction of the noisy QPU-measured 2-RDM, can reduce errors in the active-space energies by orders of magnitude.”
Michael Wagner, Pavel Begy, Jiří Brabec, Alexandra Tucholská, Mikuláš Matoušek, Katarzyna Pernal and Libor Veis, authors of the paper
The hardware tests used IBM’s 156-qubit ibm_pittsburgh processor, based on the Heron r3 architecture. The molecular calculations required five qubits for water and four for cyclobutadiene after symmetry-based reductions. The researchers optimized their quantum circuits using a state-vector simulator, then held the circuit parameters fixed when measuring on the processor; the experiment tests noisy energy evaluation, not a VQE optimization loop running on the device.
Two molecules, 1,000 shots per measurement
For water, the team studied symmetric stretching of the double bond, using 1,000 measurement shots and averaging over 10 independent runs. With the DMRG-derived 1-RDM, ERPA brought the active-space energies to within about 15 millihartree of the CASSCF reference across the potential-energy curve. Adding a multireference adiabatic connection correction, called AC0, recovered correlation beyond the active space and brought the resulting curve close to the paper’s essentially exact reference.
The second test followed cyclobutadiene as it changes from a rectangular structure to a square transition state. ERPA-corrected energies stayed within 10 millihartree of the CASSCF reference along the path. The calculated reaction barrier was 6.56 kilocalories per mole, compared with 7.38 for CASSCF; adding AC0 gave 8.99, while CASSCF with AC0 yielded 10.91. The paper also reports a 8.4-kilocalorie-per-mole result from a multireference average-quadratic coupled-cluster benchmark, while experimental barrier estimates span a much wider range.
The correction can introduce its own bias
ERPA’s improved estimates are not uniformly reliable by construction. The authors report that the reconstructed 2-RDM is not guaranteed to be N-representable—a condition required for a density matrix to correspond to a valid physical state—and that this can produce energies below the exact ground-state energy. For water, they observed a negative deviation of about 15 millihartree at one stretched geometry. The DMRG input also remains an approximation, even though it supplies a less noisy 1-RDM.
The authors say their protocol adds no quantum resources beyond measuring the 1-RDM and 2-RDM already used in the energy calculation and AC0 correction, provided the molecular orbitals stay fixed. The tests cover two strongly correlated molecules and preoptimized VQE states, not a broad set of chemical systems or noisy quantum optimization from start to finish. Their next research direction is to incorporate ERPA into the VQE optimization itself.