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Rethinking Parity Check Enhanced Symmetry-Preserving Ansatz
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Combines a Hamming-weight-preserving (HWP) ansatz with topology-aware parity checks on physical qubits to mitigate bit-flip errors in VQE and to enforce hard permutation constraints for the Quadratic Assignment Problem.
Problem
VQAs struggle to enforce hard constraints (e.g., symmetry in quantum chemistry, permutation constraints in QAP) and suffer from noise; penalty-based soft constraints are inefficient.
Approach
Use parity checks as projective measurements on ancilla qubits interleaved within HWP ansatz layers so that the state is restricted to the feasible (QAP or number-conserving) subspace; parity checks serve both as error mitigation and as hard-constraint enforcement, using nearest-neighbor CNOT cascades on a 2D-grid qubit topology.
Key results
- On QAP m=6 simulation, NBS-HARD reaches approximation ratio 0.9826 and p_optimal 0.5000 (in-constraint), versus best soft-constraint XYmixer-FC approx 0.9957 but p_optimal 0.4707 at m=3 and far lower at larger m.
- For H2O under noise, SV+PC gives energy error 6.88e-1 Ha vs SV 8.28e-1 Ha, but on LiH SV+PC feasibility drops to 0.073 vs 0.505 for SV alone.
- Noiseless HWP ansatz (NBS) reaches energy errors below 1e-10 for H2, LiH and H2O with fewer gates than UCCSD/UCCSDT (e.g., 441 vs 5871 gates for H2O).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add parity checks between HWP layers (ablation) | HWP ansatz with symmetry verification · H2 noise | Energy (Ha): -0.7599740203 -> -0.7697837117 | Helped |
| add parity checks between HWP layers (ablation) | HWP ansatz with symmetry verification · LiH noise | Energy (Ha): -7.1312360044 -> -7.2627704120 | Helped |
| add parity checks between HWP layers (ablation) | HWP ansatz with symmetry verification · H2O noise | Energy (Ha): -73.9089583257 -> -74.0486388806 | Helped |
| use NBS-HARD instead of XYMIXER-HARD | QAP hard-constraint method · QAP m=4 simulator | poptimal: 0.8498 -> 0.9448 | Helped |
| use NBS-HARD instead of XYMIXER-HARD | QAP hard-constraint method · QAP m=5 simulator | poptimal: 0.7500 -> 0.8500 | Helped |
| use NBS-HARD instead of XYMIXER-HARD | QAP hard-constraint method · QAP m=6 simulator | poptimal: 0.3500 -> 0.5000 | Helped |
| vary the number of parity checks from 6 to 10 (ablation) | HWP ansatz · LiH noisy simulation | energy error: 0.657022752 -> 0.600911213 | Helped |
| vary the number of parity checks from 6 to 10 (ablation) | HWP ansatz · H2O noisy simulation | energy error: 0.748342082 -> 0.688285861 | Helped |
Surprising findings
- Parity check with a deep ansatz on LiH reduced the in-constraint success rate (0.073 for SV+PC vs 0.505 for SV alone), while improving energy error on H2O.
- The expected probability of obtaining feasible states from random sampling decreases exponentially with m (about sqrt(2 pi m)/e^m), so feasibility is only a small fraction for larger QAP sizes.
Limitations
- Results on the superconducting processor are limited to m=3 (12 qubits) with noisy hardware; authors note quantum advantage is not demonstrated.
- Only equality constraints handled; inequality constraints left to future work.
- Feasible-state probability decreases exponentially with m in the QAP setting.
Experimental setup
Datasets: Random QAP instances (100 per m in {3,4,5,6}) OpenFermion molecular Hamiltonians (H2, LiH, H2O)
Benchmarks: H2 LiH H2O ground state energy (STO-3G) QAP instances m=3,4,5,6
Metrics: energy error vs FCI success probability approximation ratio probability of optimal solution probability of feasible solutions
Baselines: HEA QAOA XYmixer-NN XYmixer-FC UCCS UCCSD UCCSDT symmetry verification
Builds on: Hamming weight preserving ansatz XYmixer QAOA UCCSD
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