Logical Error Rates for the Variational Quantum Eigensolver using a [[4,2,2]] Encoded Ansatz

Application benchmarks that run on noisy intermediate scale quantum computing (NISQ) devices require techniques for detecting and mitigating errors to assess accuracy and performance. Quantum error detection codes offer a framework in which to encode these computations and track the presence of errors, but the subsequent logical error rate depends on the application circuit as well as the underlying hardware noise. Here we extend recent results using the [[4,2,2]] error detection code to improve the accuracy of computational chemistry calculations by calculating the logical error rate of an encoded variational ansatz. Within the context of the variational quantum eigensolver (VQE), we numerically simulate the mixed state generated by noisy execution of a UCC ansatz circuit for the case of the hydrogen molecule, accounting for variations in circuit parameters due to noise and the underlying noise models. Simulations of the unencoded, encoded, and post-selected states lead to estimates of logical error rate and probabilities for error-free calculations. For the case of one- and two-qubit depolarizing gate noise, we find that the error detection code reduces the logical infidelity by 4% relative to the unencoded physical rate when the noise parameter is <5%. This yields a corresponding decrease in the estimated energy by 6% (0.07 Ha). We also evaluate the change in the logical state fidelity with circuits modified to account for hardware connectivity constraints for comparison with simulations on hardware.