Experimental error mitigation via symmetry verification in a variational quantum eigensolver

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.

Figure 1

Quantum circuit and energy landscape of the variational eigensolver. (a) Quantum circuit for generating and measuring the variational ansatz state. (b) Coherent excitation exchange, produced as $Q_0$ is fluxed into resonance with $Q_1$ by a square flux pulse of fixed amplitude ($x$ axis) and duration ($y$ axis). The amplitude controls the frequency to which $Q_0$ is pulsed ($\sim 1.428\mathrm{V}$ bringing it on resonance with $Q_1$). (c) Zoom-in of (b) into the region used in the experiment to control the exchange of population between $Q_0$ and $Q_1$. Colored lines illustrate the combinations of square-pulse amplitudes and duration used to achieve fine adjustment of $\tilde{\theta }$. (d) Excitation of $Q_0$ for the combinations of pulse amplitudes and duration marked by colored lines in (c), showing the matching of the experimentally defined $\tilde{\theta }$ to the target $\theta$ defined in Eq. (7) (black dashed curve). Colors [matching (c)] correspond to pulse duration. (e) Landscape of energies $E^{\left(\mathrm{raw}\right)}(\tilde{\theta },R)$ as a function of the experimentally defined $\tilde{\theta }$ angle and the interatomic distance $R$.

Figure 2

Convergence of the VQE algorithm. (a) Experimental VQE estimate of ${\mathrm{H}}_2$ ground-state energy as a function of interatomic distance $R$. At each chosen $R$, we minimize the raw energy $E^{\left(\mathrm{raw}\right)}$ (blue data points) over the variational parameter $\tilde{\theta }$ using the CMA-ES evolutionary algorithm [36]. Applying SV to the converged solution (orange data points) lowers the energy estimate toward the exact solution (dashed curve). Inset: A typical optimization trace for the convergence of the energy estimate. (b)–(d) The reconstructed density matrices of the converged states at (b) $R=0.25$ Å, (c) $R=0.80$ Å, and (d) $R=2.00$ Å, showing that the converged states lie mostly in the single-excitation subspace, and that entanglement increases with the interatomic distance $R$.

Figure 3

Impact of SV in ground-state energy and state fidelity, and dissected error budget. (a) Experimental (solid circles) energy error $\mathrm{\Delta }E$ without and with SV compared to the result (empty circles and dashed line) of a full density-matrix simulation using the full error model. The contributions from optimizer inaccuracy, qubit dephasing, qubit relaxation, residual qubit excitations, and increased $Q_0$ dephasing during the exchange gate are shown as shaded regions for the case of no SV applied. Without SV, $\mathrm{\Delta }E$ is clearly dominated by residual qubit excitation. (b) Zoom-in on experimental and simulated $\mathrm{\Delta }E$ with SV and corresponding error budget. With SV, the effects of residual excitation and qubit relaxation are successfully mitigated, as predicted in Ref. [19]. The remaining energy error is dominated by optimizer inaccuracy. Simulation error bars are obtained by modeling measured fluctuations of $T_1, T_2^{*}$, and residual excitation. (c) Experimental (solid circles with error bars) infidelity to the true ground state without and with SV compared to simulation using the full error model (empty circles and dashed line). Error bars are propagated through the linear inversion procedure for experiment and calculated from sampling noise for simulations. For simulations, error bars are smaller than the markers.

Figure 4

Constraining positivity with symmetry verification to mitigate the effect of sampling noise. The experimental data from Fig. 3 is split into 100 sample simulations for each $R$, increasing the sampling noise by a factor of 10 and making it comparable to other sources of experimental error. For each sample, we plot (red) the relative energy error and infidelity [Eq. (9)]. Values below 1 (dashed lines) indicate that SV has not provided an improvement, as may be the case when the density matrix has negative eigenvalues. We restore the improvement from SV by constraining the positivity of the 2-reduced density matrix [39] (green). Histograms on the top and right axes show the marginal distribution of the two scatter plots. When the density matrices are constrained to be positive, we observe the points fall along the line $y=x$ (blue dashed line), indicating that SV improves both metrics by the same amount.