Estimating the gradient and higher-order derivatives on quantum hardware

For a large class of variational quantum circuits, we show how arbitrary-order derivatives can be analytically evaluated in terms of simple parameter-shift rules, i.e., by running the same circuit with different shifts of the parameters. As particular cases, we obtain parameter-shift rules for the Hessian of an expectation value and for the metric tensor of a variational state, both of which can be efficiently used to analytically implement second-order optimization algorithms on a quantum computer. We also consider the impact of statistical noise by studying the mean-square error of different derivative estimators. Some of the theoretical techniques for evaluating quantum derivatives are applied to their typical use case: the implementation of quantum optimizers. We find that the performance of different estimators and optimizers is intertwined with the values of different hyperparameters, such as the step size or the number of shots. Our findings are supported by several numerical and hardware experiments, including an experimental estimation of the Hessian of a simple variational circuit and an implementation of the Newton optimizer.

Figure 1

Variational circuit run on a simulator and hardware to investigate the findings in this paper.

Figure 2

Mean-square error for different choices of step size when estimating the gradient using the finite-difference and parameter-shift estimators with ${10}^3$ shots used to evaluate expectation values. The solid lines show the MSE, while the shaded regions illustrate the range of values within one standard deviation. Plots compare the estimators when run on (a) a simulator and (b) ibmq_valencia. Additional gray lines are calculated from theoretical predictions: The dashed lines show the expected behavior of the MSE and the dotted vertical lines show the expected optimal choice of step size: $h^{*}=0.613$ and $s^{*}=\pi /2$.

Figure 3

Relative error for using the parameter-shift estimator to approximate the Hessian matrix on the ibmq_valencia hardware device. Only the upper left $4\times 4$ block is considered, since the exact Hessian is zero in the final row and column.

Figure 4

Comparison of optimizers with circuit evaluation performed on the ibmq_burlington and ibmq_valencia hardware devices. The top row of plots (experiment A) focuses on the gradient descent optimizer for a varying number of shots $N$ per expectation value measurement, while the bottom row of plots (experiment B) compares different optimizers discussed in Sec. 5 when using $N=1000$. (a i) and (b i) Cost function $f\left(\mathit{\theta }\right)$ in terms of the total number of circuit evaluations on the device and (a ii) and (b ii) the path taken by the optimizer through the ${\theta }_1\text{−}{\theta }_2$ space. In (a i) and (b i) the dashed lines give the cost function evaluated on hardware and the solid lines give the cost function evaluated on an exact simulator with the same $\mathit{\theta }$. The dotted line gives the analytic minimum of $-0.874$. In (a ii) and (b ii) the contrasting white and purple circles highlight the maximum and minimum, while the dashed line in (a ii) is the path taken by the GD optimizer with access to exact expectation values.

Figure 5

Optimal step size for the finite-difference and parameter-shift gradient estimators for a range of shot numbers $N$. The shaded regions display the range of step sizes resulting in an MSE that is within $20\%$ of the minimum observed value, while the solid lines are the medians of these regions. The dashed gray lines are the predicted values for the optimal step size.

Figure 6

Mean-square error for different choices of shot number $N$ when using the finite-difference and parameter-shift estimators with corresponding optimal choices of step size to estimate the gradient. The solid lines show the results from simulation, while the dashed gray lines are the predicted scaling. The dotted vertical line indicates the crossover point between the two methods at $N\approx 50$ shots.

Figure 7

Mean-square error for different choices of step size when estimating the Hessian using the finite-difference and parameter-shift estimators with ${10}^3$ shots used to evaluate expectation values on a simulator. The solid lines show the MSE, while the shaded regions illustrate the range of values within one standard deviation.

Figure 8

Mean-square error for different choices of shot number $N$ when using the finite-difference and parameter-shift estimators with optimal choices of step size to estimate the Hessian. A power-law fit to each line gives scalings of $N^{-0.5013}$ and $N^{-1.0008}$ for the finite-difference and parameter-shift methods, respectively.

Figure 9

Comparison of different optimizers discussed in Sec. 5 with circuit evaluation performed on pennylane's noise-free default.qubit simulator. (a) Cost function $f\left(\mathit{\theta }\right)$ in terms of the total number of circuit evaluations and (b) path taken by the optimizer through the ${\theta }_1\text{−}{\theta }_2$ space. In (a) the dotted line gives the analytic minimum of $-0.874$ and in (b) the contrasting white and purple circles highlight the maximum and minimum.

Figure 10

Comparison of different optimizers discussed in Sec. 5 with circuit evaluation performed on pennylane's noise-free default.qubit simulator. (a) Cost function $f\left(\mathit{\theta }\right)$ in terms of the total number of circuit evaluations and (b) path taken by the optimizer through the ${\theta }_1\text{−}{\theta }_2$ space. In (a) the dotted line gives the analytic minimum of $-0.874$ and in (b) the contrasting white and purple circles highlight the maximum and minimum. With respect to the simulation of Fig. 9, in this case a different regularization method is used, as discussed in the text.