Robustness of variational quantum algorithms against stochastic parameter perturbation

Variational quantum algorithms are tailored to perform within the constraints of current quantum devices, yet they are limited by performance-degrading errors. In this study we consider a noise model that reflects realistic gate errors inherent to variational quantum algorithms. We investigate the decoherence of a variationally prepared quantum state due to this noise model, which causes a deviation from the energy estimation in the variational approach. By performing a perturbative analysis of optimized circuits, we determine the noise threshold at which the criterion set by the stability lemma is met. We assess our findings against the variational quantum eigensolver and quantum approximate optimization algorithm for various problems with up to 14 qubits. Moreover, we show that certain gate errors have a significantly smaller impact on the coherence of the state, allowing us to reduce the execution time without compromising performance.

Figure 1

Average energy increase of the perturbed optimal VQE circuit of $n=6,8,10$ qubits for circuit depths $p=8,10,12$, obtained by the perturbation of ${\mathit{\theta }}^{*}$ by $\delta$ uniformly sampled from the range $(-\sigma ,\sigma )$. Error bars depict standard errors. Polynomial fits of data points confirm that $\delta E\propto {\sigma }^2$.

Figure 2

Average energy shift of 100 uniformly generated 3-SAT instances of 6, 8 and 10 qubits obtained from perturbed optimal circuits of depths 15, 25, and 30, respectively. All instances were designed to have a clause-to-variable ratio of 4.2 and a unique satisfying assignment. Error bars depict standard errors. Polynomial fits of data indicate $\delta E\propto {\sigma }^2$ up to $q{\sigma }^2\gtrsim 5$.

Figure 3

Average energy shift of 100 uniformly generated 3-SAT instances of 6, 8, and 10 qubits obtained from perturbed optimal circuits of depth 10. All instances were designed to have a clause-to-variable ratio of 4.2 and a unique satisfying assignment. Error bars depict standard errors. Polynomial fits of data indicate $\delta E\propto {\sigma }^2$ up to $q{\sigma }^2\gtrsim 5$.

Figure 4

Energy $\langle H\rangle =\langle \psi ({\mathit{\theta }}^{*}+\mathit{\delta }\mathit{\theta })|H|\psi ({\mathit{\theta }}^{*}+\mathit{\delta }\mathit{\theta })\rangle$ obtained for an unstructured search with $n=10$ and $p=25$ when only (a) ${\beta }_k$ or (b) ${\gamma }_k$ is perturbed by $\delta$.

Figure 5

Average energy $\langle H\rangle =\langle \psi ({\mathit{\theta }}^{*}+\mathit{\delta }\mathit{\theta })|H|\psi ({\mathit{\theta }}^{*}+\mathit{\delta }\mathit{\theta })\rangle$ of 100 uniformly generated 3-SAT instances solved with $p=30$ layers, where (a) ${\beta }_k$ or (b) ${\gamma }_k$, from the $k\mathrm{th}$ layer, is perturbed by $\delta$. The instances are of $n=10$ qubits with a clause-to-variable ratio of 4.2 and a unique satisfying assignment.

Figure 6

Each cell in the figure represents the optimized energy for a given depth and a fixed maximal execution time. The coloring scheme is indicated on the right. Green and orange rectangles depict the two branches of angles that minimize the expectation value for a given depth.

Figure 7

Average energy shift of 100 instances of Max-Cut on 6, 8, and 10 qubits obtained from perturbed optimal circuits of depths of 10, 13, and 15, respectively. The instances were designed to have a graph density of approximately 0.5. Error bars depict standard errors. Polynomial fits of data indicate $\delta E\propto {\sigma }^2$ up to $q{\sigma }^2\gtrsim 1$.

Figure 8

Average energy for the problem of an unstructured search obtained by the perturbation of ${\mathit{\gamma }}^{*}$ and ${\mathit{\beta }}^{*}$ by $\delta$ uniformly sampled from the range $(-\sigma ,\sigma )$. Error bars depict standard errors. Polynomial fits of data points of 6, 8, and 10 qubits in the ranges $\sigma \in [0,0.1]$, [0,0.07], and [0,0.05], respectively, confirm that $\delta E\propto {\sigma }^2$.