[Paper Review] Error Mitigation by Symmetry Verification on a Variational Quantum Eigensolver

Ramiro Sagastizabal, Xavier Bonet-Monroig+12 more|arXiv (Cornell University)|Feb 28, 2019

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper demonstrates error mitigation via symmetry verification (SV) in a variational quantum eigensolver (VQE) for the hydrogen molecule using a superconducting qubit processor. By leveraging parity symmetry in the single-excitation subspace and applying SV to detect and discard states violating this symmetry, the authors reduce energy and state estimation errors by an order of magnitude on average across the bond-dissociation curve, with improved fidelity linked to energy accuracy when physicality of the density matrix is enforced.

ABSTRACT

Variational quantum eigensolvers offer a small-scale testbed to demonstrate the performance of error mitigation techniques with low experimental overhead. We present successful error mitigation by applying the recently proposed symmetry verification technique to the experimental estimation of the ground-state energy and ground state of the hydrogen molecule. A finely adjustable exchange interaction between two qubits in a circuit QED processor efficiently prepares variational ansatz states in the single-excitation subspace respecting the parity symmetry of the qubit-mapped Hamiltonian. Symmetry verification improves the energy and state estimates by mitigating the effects of qubit relaxation and residual qubit excitation, which violate the symmetry. A full-density-matrix simulation matching the experiment dissects the contribution of these mechanisms from other calibrated error sources. Enforcing positivity of the measured density matrix via scalable convex optimization correlates the energy and state estimate improvements when using symmetry verification, with interesting implications for determining system properties beyond the ground-state energy.

Motivation & Objective

• To demonstrate practical error mitigation in NISQ-era quantum devices using symmetry verification (SV) with minimal experimental overhead.
• To address the challenge of relaxation and residual excitation errors in variational quantum algorithms that degrade ground-state energy and state estimates.
• To investigate how SV improves both energy and state fidelity, particularly when combined with physicality constraints on the density matrix.
• To establish a link between energy error reduction and ground-state fidelity improvement under SV, especially when sampling noise and non-physical density matrices are corrected.

Proposed method

• Implement a variational ansatz using a tunable exchange interaction between two transmon qubits to prepare states within the single-excitation subspace, preserving the Z2 parity symmetry of the H2 Hamiltonian.
• Apply symmetry verification by measuring the parity operator (Z1Z2) and post-selecting only outcomes with correct symmetry (e.g., odd parity), discarding states that violate it.
• Use full density-matrix simulations matching experimental parameters to isolate and quantify contributions from relaxation, residual excitation, and other error mechanisms.
• Perform quantum state tomography on both raw and SV-processed density matrices to compare energy and fidelity outcomes.
• Enforce physicality of the reconstructed density matrix via convex optimization (L2-norm closest positive semidefinite matrix in Pauli basis) to assess its impact on SV performance.
• Quantify improvements in energy error and state fidelity using metrics η_E and η_F, comparing results with and without physicality enforcement.

Experimental results

Research questions

• RQ1 Can symmetry verification effectively reduce energy and state estimation errors in a VQE implementation on a real quantum processor?
• RQ2 How do relaxation and residual excitation errors affect the accuracy of VQE for the H2 molecule, and to what extent can SV mitigate them?
• RQ3 Does enforcing physicality on the reconstructed density matrix enhance the correlation between energy error reduction and ground-state fidelity improvement under SV?
• RQ4 What is the role of sampling noise in undermining the benefits of SV, and how can it be mitigated?
• RQ5 Is the improvement in energy estimation via SV directly linked to improved ground-state fidelity when physicality constraints are applied?

Key findings

• Symmetry verification reduces the average energy estimation error of the VQE for the H2 molecule by approximately one order of magnitude across the bond-dissociation curve.
• SV significantly mitigates errors arising from qubit relaxation and residual qubit excitation, which break the total excitation number symmetry.
• Without enforcing physicality of the density matrix, SV improves energy error but not state fidelity; however, when physicality is enforced, SV improves both energy and fidelity in a correlated manner.
• The improvement in energy error from SV is directly proportional to the improvement in ground-state fidelity when the initial density matrix is physical, indicating a strong link between the two metrics.
• The maximum benefit of SV occurs when the denominator in the variance formula Tr[ρ^(raw)Ŝ] is small, indicating that SV is most effective when the raw state is already close to the symmetric subspace.
• Enforcing positivity of the fermionic 2-reduced density matrix enables scalable physicality constraints and strengthens the reliability of SV for properties beyond ground-state energy.