Short-depth circuits for efficient expectation-value estimation

The evaluation of expectation values $\text{Tr}\left[\rho \mathcal{O}\right]$ for some pure state $\rho$ and Hermitian operator $\mathcal{O}$ is of central importance in a variety of quantum algorithms. Near-optimal techniques have been developed in the past and require a number of measurements $N$ approaching the Heisenberg limit $N=O\left(1/\epsilon \right)$ as a function of target accuracy $\epsilon$. The use of quantum phase estimation (QPE) requires, however, long circuit depths $C=O\left(1/\epsilon \right)$ making its implementation difficult on near-term noisy devices. The more direct strategy of operator averaging is usually preferred as it can be performed using $N=O\left(1/{\epsilon }^2\right)$ measurements and no additional gates aside from those needed for the state preparation. In this work we use a simple but realistic model to describe the bound state of a neutron and a proton (the deuteron) to show that the latter strategy can require an overly large number of measurements in order to achieve a prefixed relative target accuracy ${\epsilon }_r$. We propose to overcome this problem using a single step of QPE and classical postprocessing. This approach leads to a circuit depth $C=O\left({\epsilon }^{\mu }\right)$ (with $\mu \ge 0$) and to a number of measurements $N=O\left(1/{\epsilon }^{2+\nu }\right)$ for $0<\nu \le 1$ and a much smaller prefactor. We provide detailed descriptions of two implementations of our strategy for $\nu =1$ and $\nu \approx 0.5$ and derive appropriate conditions that a particular problem instance has to satisfy in order for our method to provide an advantage.

Figure 1

The solid black curves indicate the set of points where Eq. (33) is satisfied with the equality for different choices of the sQPE order $K$ with an ordering from top left to bottom right: the first solid line refers to $K=1$ while the second is for $K=2$ and the shaded region in-between indicates the region where the linear method is no longer sufficient and we need to employ sQPE with $K>1$. The innermost (darkest) region is bordered by the $K=8$ line. With the inset we are zooming in the top-right corner of the main plot.

Figure 2

Results of numerical simulations of the linear algorithm explained in the text together with the operator averaging method of Sec. 1. The gray band indicates the expected variation caused by different choices for the time step $\tau$ in the linear algorithm. The horizontal orange line indicates an indicative ${\epsilon }_r=1\%$ target error threshold.

Figure 3

Central (black solid line) and tensor (green dashed line) contributions to the nuclear potential Argonne $\mathrm{V}{6}^{\prime }$ used to obtain the deuteron Hamiltonian (63). In the inset the range of pion exchange is also shown. The gray dotted line at 0 energy is simply a guide to the eye.

Figure 4

Results of numerical simulations of the algorithm explained in the text. Results correspond to five independent runs with different random number seeds. The blue dotted line corresponds to the expected asymptotic behavior from Eq. (40) while the red dot corresponds to the upper bound from Eq. (64). The inset shows the convergence of the VQE optimization using low-resolution expectation values with errors $\approx 5$ MeV (these were obtained using $N_{\mathrm{tot}}={10}^3$ measurements per function evaluation). The horizontal line indicates the value of the ground-state energy $E_{\text{gs}}=-2.1174$ MeV.

Figure 5

Final error in the estimator for the ground-state energy of the deuteron obtained with the techniques discussed in this work. From top to bottom, they are as follows: operator averaging (black line), linear sQPE with optimal time step (blue line), linear sQPE with half of the optimal time step (green line), and the adaptive cubic sQPE algorithm (purple line). Dotted lines correspond to the analytical results presented in Fig. 2. See text for the meaning of the marked points.

Figure 6

Different estimators of the energy bias for a cubic sQPE calculation of the deuteron ground-state energy, from top to bottom (in all panels): $A1$ approximation $B_{A1}, B_{A2}$ approximation $B_{A2}$, and exact estimator $B_E$ (see text for the definitions). The top panel shows results for an idealized case while panels (b) and (c) are obtained using approximate optimizations employing the estimators $B_{A1}$ and $B_{A2}$ [cf. Eqs. (67) and (69)].

Figure 7

Different estimators for the final error in the deuteron ground-state energy form ideal runs where the time steps are optimized exactly using Eq. (58), from top to bottom: approximate $A1$ results, approximate $A2$ results, and exact MSE result. The cyan up triangle is the same as the blue triangle in Fig. 5. The inset show results obtained using the approximate cost function (61).

Figure 8

Results for the deuteron ground-state energy obtained using the emulated “ibmqx4” quantum computer without error mitigation. The left panel shows results obtained using operator averaging on each qubit while the right panel shows the performance of the linear sQPE algorithm using pairs $[\text{qubit}2,\text{qubit}1]$ (green data points) and $[\text{qubit}3,\text{qubit}2]$ (blue data points) as ancilla and system qubit, respectively.

Figure 9

Same as in Fig. 8 but using the error mitigation strategy described in the text for the measurement noise.

Figure 10

Estimated number of measurements required to reproduce the ground-state energy of the deuteron to ${\epsilon }_r=1\%$ accuracy as a function of the readout error rate. Continuous lines show the total number ($N_{\mathrm{tot}}=2M+N_C$) while the dashed lines show the fraction of the total measurements that has to be dedicated to characterize the readout noise.

Figure 11

Estimated parameter regions for problems expected to be efficiently solvable with the scheme of Sec. 2 as predicted by Eq. (97) (black solid lines) and by the looser condition in Eq. (102) (red contours). See text for more details.