Error Analysis for the Estimated Energy from a [[4,2,2]] Encoded Variational Quantum Eigensolver 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 provide 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 that show that the [[4,2,2]] error detection code improves the accuracy of computational chemistry calculations 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 unitary coupled-cluster doubles ansatz 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 energy, logical fidelity and probabilities for error-free calculations. We analyze the precision and accuracy of the estimated energy against the benchmark of chemical accuracy of 1mHa. We find that post-selection improves accuracy of the energy estimate over the unencoded ansatz and decreases precision due to the loss of samples. The estimated energy is found to be within chemical accuracy at 0.08% noise from 49.6% samples retained after post-selection from 200000 samples, and it is nearly 2 mHa lower than the corresponding unencoded simulation.