Error Mitigation via Verified Phase Estimation

The accumulation of noise in quantum computers is the dominant issue stymieing the push of quantum algorithms beyond their classical counterparts. We do not expect to be able to afford the overhead required for quantum error correction in the next decade, so in the meantime we must rely on low-cost, unscalable error mitigation techniques to bring quantum computing to its full potential. In this paper we present a new error mitigation technique based on quantum phase estimation that can also reduce errors in expectation value estimation (e.g., for variational algorithms). The general idea is to apply phase estimation while effectively postselecting for the system register to be in the starting state, which allows us to catch and discard errors that knock us away from there. We refer to this technique as “verified phase estimation” (VPE) and show that it can be adapted to function without the use of control qubits in order to simplify the control circuitry for near-term implementations. Using VPE, we demonstrate the estimation of expectation values on numerical simulations of intermediate-scale quantum circuits with multiple orders of magnitude improvement over unmitigated estimation at near-term error rates (even after accounting for the additional complexity of phase estimation). Our numerical results suggest that VPE can mitigate against any single errors that might occur; i.e., the error in the estimated expectation values often scale as $\mathcal{O}(p^2)$, where $p$ is the probability of an error occurring at any point in the circuit. This property reveals VPE as a practical technique for mitigating errors in near-term quantum experiments.

Popular Summary

Before quantum computers become large enough to pay the overhead required for quantum error correction, low-cost error mitigation techniques are needed for quantum algorithms to push beyond the capabilities of classical computers. A popular class of such error mitigation techniques is based on postselection, where natural or artificial parity checks are used to determine whether errors have occurred, at which point the experiment may be thrown away.

In this work, we find a method to effectively postselect a popular quantum algorithm—quantum phase estimation (QPE)—by determining whether the system returns to its initial state after the QPE routine is complete. This is technically not postselection, as we do not throw experimental data away. Therefore, we call the resulting method “verified phase estimation.” As one may use QPE as a subroutine to estimate expectation values at low cost, we extend this method to a verification protocol for expectation value estimation. This yields a powerful tool, e.g., for the verification of the quantum subroutine in a variational quantum eigensolver, a popular near-term quantum algorithm.

We find our method to be highly effective, demonstrating up to 10 000 times reduction in energy estimation for states generated from different variational circuits, and with a relatively low cost. This makes verified phase estimation and verified expectation value estimation highly useful tools in the belt of near-term quantum algorithmists and experimentalists.

Figure 1

Process diagram of the protocol for verified estimation of the expectation value of a Hamiltonian on a state $|\psi ⟩=U_p|\overrightarrow{0}⟩$. Blue denotes circuits to be executed or data to be extracted from a quantum computer; red denotes signal details to be estimated via classical postprocessing. The protocol proceeds as follows. Top left: a complex Hamiltonian $H$ is split into a number of fast-forwardable summands $H_s$. The spectral function $g(t)$ of $|\mathrm{\Psi }⟩$ under time evolution of each piece is obtained (bottom left) via verified, fast-forwarded phase estimation. In this example, a control qubit is used to extract the phase function via phase kickback. The resulting data form a weighted sum of oscillations with frequencies equal to the eigenvalues $E_j^{(s)}$ of the corresponding factor (bottom middle). This may be decomposed in a variety of classical postprocessing techniques to obtain estimations of the expectation values $⟨H_s⟩$ depending on the type of $H_s$ chosen (bottom right). Regardless of the method used, the expectation values must be normalized to obey Eq. (24), the last step in the verification process. As the expectation value is linear, the verified estimates of $⟨H_s⟩$ obtained may be immediately summed together to give a verified estimate for $⟨H⟩$ (top right).

Figure 2

Quantum circuit for control-free verified phase estimation. The preparation unitary $U_p$ is defined in Eq. (42). The first gate in the circuit is a Hadamard gate (roman H) on the top-most qubit (labeled the target qubit in the text), which should not be confused with the Hamiltonian $H$.

Figure 3

Mitigation of a four-qubit Givens rotation circuit via verified phase estimation. Left: error in estimation of random states in a free-fermion system [Eq. (56)] under a uniform depolarizing channel. Right: error in the same estimation, but this time under an amplitude and phase damping model. In both plots, the RMS error (crosses) is calculated over $50$ different estimations for each error rate using either standard partial state tomography (red) or using verified control-free phase estimation. Individual data points (dashes) are additionally shown. For reference, dashed lines showing linear (red), quadratic (black), and cubic (blue) dependence on the gate error rate are plotted.

Figure 4

Mitigation of a four-qubit VHA circuit via verified phase estimation. Left: error in estimation of the energy of random states generated by the quantum approximate optimization ansatz in the critical phase of the transverse-field Ising model [Eq. (62)] under a uniform depolarizing channel. Right: error in the same estimation, but this time under an amplitude and phase damping model. In both plots, the rms error (crosses) is calculated over $50$ different estimations for each error rate (with randomly chosen ansatz parameters) using either standard partial state tomography (red) or using verified control-free phase estimation. Individual data points (dashes) are additionally shown. For reference, dashed lines showing linear (red) dependence on the gate error rate are plotted.

Figure 5

Error in estimating the ground state energy of a four-site transverse-field Ising model [Eq. (62)] by variational optimization of a VHA ansatz. The resulting expectation values are measured either by verified single-control phase estimation (black) or taken directly from the simulated state (red). We plot the median (crosses) of the absolute energy error over ten optimization attempts, each starting from a different initial point. Individual errors are plotted behind (faint dashes). Guide lines showing a linear dependence are additionally plotted (red dashed lines).

Figure 6

Mitigation of a four-qubit fermionic swap network via verified phase estimation. Three different VPE protocols are explored—a low-rank factorization (top row), a control-free number-conserving Pauli decomposition (middle row), and a single-control Pauli decomposition (bottom row). Details of all decompositions are given in the text. The low-rank factorization is studied for the $H_2$ Hamiltonian at the equilibrium bond distance with a swap network of depth 4, while the other two models are studied at a bond distance of $2$ Å with a swap network of depth 6. All protocols are tested under depolarizing (left column) and amplitude and phase damping (right column) noise models. In all plots, the median error (crosses) is calculated over 50 different estimations for each error rate using either standard partial state tomography (red) or using verified control-free phase estimation. Individual data points (dashes) are additionally shown. For reference, dashed lines showing linear (red), quadratic (black), and cubic (blue) dependence on the gate error rate are plotted.

Figure 7

Convergence of the estimation of a single point in a four-site transverse-field Ising model with the number of samples taken, using verified phase estimation processed either with Prony’s method (blue) or by fitting known phases to the phase function (green), or standard partial state tomography (red) on individual Pauli terms. Left: convergence in the absence of error. Right: convergence in the presence of $1\mathrm{\%}$ depolarizing error per qubit per moment. In each subfigure we plot the median energy error (crosses and lines) over $200$ simulations, which are plotted themselves behind (faint dashes).

Figure 8

Predicted [Eq. (C1)] versus found bias from estimating expectation values using Prony’s method.

Figure 9

Mitigation of the same four-qubit VHA circuit as in Fig. 4 of the main text, but with either depolarizing noise on only the system register (black) or on only the control qubit (blue). This is compared with the error in estimation using partial state tomography instead of VPE (red). For each dataset, the rms error (crosses) is plotted over $50$ different estimations for each error rate (with randomly chosen ansatz parameters), and individual data points are plotted as dashes behind. For reference, dashed lines showing linear (red) dependence on the gate error rate are plotted.

Figure 10

Error in estimating the ground state energy of a free-fermion system [Eq. (58) of the main text] of four fermions (on four qubits), using control-free verified phase estimation and a VQE. Noise model is a mixture of amplitude and phase damping and constant two-qubit control error (details in the text). Median absolute errors for both verified estimation (black crosses) and standard partial state tomography (red crosses) are calculated over ten different optimization attempts. Individual simulations are plotted behind (faint dashes). Each optimization starts from a different parameter set. Linear (red dashed) and cubic (blue dashed) lines are shown as guides.

Figure 11

Expectation value estimation of two individual $H_s$ terms from the control-free number-conserving Pauli operator decomposition of the $H_2$ Hamiltonian studied in Fig. 6 of the main text on states prepared by a fermionic swap network. The two terms here comprise part of the sum [Eq. (6) of the main text] for the expectation value of Fig. 6(b2) of the main text—but are studied here without prefactors (i.e., $\parallel H_s\parallel =1$). Each figure is labeled with the studied term, and guide lines (dashed red and blue) are given to show observed scaling laws. Data presented are the median (crosses) over $50$ individual data points (faint dashes) of the absolute error in estimation using VPE (black) and standard partial state tomography (red).

Figure 12

Comparison between verified phase estimation and symmetry verification. Both techniques are compared to the estimation of the expectation value of the electronic structure Hamiltonian for $H_2$ under a depolarizing noise model. Verified and unverified results are from the same simulated experiment as Fig. 6(b1) of the main text. Although symmetry verification improves the energy error by around a factor of $10$, it still exhibits first-order scaling, as it cannot correct for phase ($Z$) errors during the experiment.