Practical Quantum Error Mitigation for Near-Future Applications

It is vital to minimize the impact of errors for near-future quantum devices that will lack the resources for full fault tolerance. Two quantum error mitigation (QEM) techniques have been introduced recently, namely, error extrapolation [Y. Li and S. C. Benjamin, Phys. Rev. X 7, 021050 (2017); K. Temme et al., Phys. Rev. Lett. 119, 180509 (2017)] and quasiprobability decomposition [K. Temme et al., Phys. Rev. Lett. 119, 180509 (2017)]. To enable practical implementation of these ideas, here we account for the inevitable imperfections in the experimentalist’s knowledge of the error model itself. We describe a protocol for systematically measuring the effect of errors so as to design efficient QEM circuits. We find that the effect of localized Markovian errors can be fully eliminated by inserting or replacing some gates with certain single-qubit Clifford gates and measurements. Finally, having introduced an exponential variant of the extrapolation method we contrast the QEM techniques using exact numerical simulation of up to 19 qubits in the context of a “swap” test circuit. Our optimized methods dramatically reduce the circuit’s output error without increasing the qubit count.

Popular Summary

The first generation of quantum computers with more than 50 quantum bits, or qubits, is expected to emerge in the next 12 months. That is large enough to go beyond the predictive power of conventional supercomputers. However, these early quantum computers will also be prone to errors, potentially preventing them from being useful. We show that one recently proposed solution—writing quantum software in such a way that errors do as little harm as possible—can work even if we have imperfect knowledge of the nature of the errors, as will certainly be the case in reality.

The goal of “quantum error mitigation” is to estimate the value that some observable would take in a circuit free from errors. Focusing on two recent proposals for practically accomplishing this goal, we account for inevitable imperfections in the knowledge of the underlying error model. We describe a protocol for systematically measuring the effect of errors so as to design efficient circuits. We prove that quantum error mitigation can work for quantum computers with up to 20 qubits, and we estimate that it will still work for computers with more than 50 qubits.

This is good news. While quantum computers might start to be useful beyond 50 qubits, the lack of any error mitigation would be a showstopper. Also, this technique is “free,” in the sense that it does not need any extra qubits or other technological features to operate—it is just a smarter way to structure the quantum software.

Figure 1

Quantum computing of the expected value of an observable (a) without quantum error mitigation (QEM) and (b) with QEM. In QEM circuit (b), each operation (including the memory operation) in the original circuit (a) is replaced by an operation depending on the corresponding random numbers [see Fig. 2].

Figure 2

(a) Error mitigation circuits. The choice of a basis operation is determined by the corresponding random number $i$, $j$, or $k$. Original gate that is identity (memory operation) also has to be error mitigated, unless memory error is negligible. In the compensation method, either the original gate or basis operations are applied depending on the random number. (b) The schematic of the linear extrapolation (orange curve) and exponential extrapolation (green curve).

Figure 3

Cost ($C-1$) for correcting errors. We consider a universal set of operations, including the initialization, measurement, single-qubit Clifford gates, a single-qubit non-Clifford gate, and a two-qubit entangling Clifford gate. Sixteen basis operations of each qubit can be generated using these operations. Every operation in the set has error, and the memory error is also included. We assume that qualities of the initialization and single-qubit gates are 10 times better than the measurement and two-qubit gates, and also that the quality of the memory operation is 100 times better. Details of the model are given in Appendix pp8. The cost for correcting error in each operation in the universal set is calculated, and the maximum cost over all operations is plotted. (a) For the depolarizing error model, the cost is lower if we directly use actual operations to correct errors, and the cost is higher if we use gate set tomography (GST) estimations to correct errors. (b) For the overrotation model, the cost is higher without using the Pauli twirling, and the cost is lower when the Pauli twirling is used. In (a) and (b), solid curves correspond to the compensation method (with the optimized $\lambda$), and dashed curves correspond to the inverse method. (c) The cost as a function of the distance between operations with errors and operations without error. For the operation with error $\mathcal{O}$ and the operation without error ${\mathcal{O}}^{(0)}$, the distance is ${\epsilon }_{\mathcal{O}}=\parallel \mathcal{O}-{\mathcal{O}}^{(0)}{\parallel }_{\max }$. The $x$ axis illustrates the maximum distance over all operations in the universal set. In (b) and (c), we always use GST estimations. In (c), Pauli twirling and the inverse method are used for all the data. Pauli twirling is applied to the measurement and two-qubit gate, and the inverse method is applied only to the two-qubit gate, while errors in other operations are corrected using the compensation method. We remark that usually the maximum distance and the maximum cost are given by the two-qubit gate.

Figure 4

swap-test circuit. The first qubit (denoted black) is a probe qubit, and the expected value of $Z$ gives the overlap between states of two groups of qubits (denoted green and orange, respectively). Green qubits are prepared in the Greenberger-Horne-Zeilinger (GHZ) state $(|00\cdots ⟩+|11\cdots ⟩)/\sqrt{2}$, and orange qubits are prepared in $|00\cdots ⟩$. Therefore, the ideal expected value of $Z$ is 0.5.

Figure 5

Histograms of the estimation of $⟨Z⟩$ using quantum computers with inhomogeneous Pauli error and leakage error. For the inhomogeneous Pauli error model, the swap-test circuit involving 19 qubits is simulated: one qubit is the probe qubit, and each group has 9 qubits. For the leakage error model, the swap-test circuit involving fewer qubits (15 qubits) is simulated, because in the numerical simulation we need to use an additional qubit to introduce the leakage process. The ideal value of $⟨Z⟩$ is marked by the red arrow.

Figure 6

Comparison of optimized quantum error mitigation techniques. The green outlines correspond to the quasiprobability technique while solid histograms correspond to the extrapolation technique using a presumption of an underlying linear (blue) or exponential (red) dependence. For the inhomogeneous Pauli error model, the swap-test circuit involving 19 qubits is simulated. For the leakage error model, the swap-test circuit involving fewer qubits (15 qubits) is simulated. The horizontal axis is the estimate of $⟨Z⟩$ that an experimentalist who performs $10^4$ experiments will obtain. Ideally the circuit produces $⟨Z⟩=0.5$. Panel (a) corresponds to physical errors of the inhomogeneous Pauli type, while panel (b) corresponds to physical leakage errors. Note that the horizontal scale differs between the two panels; a gray bar showing the scale from 0.4 to 0.6 appears in both panels to facilitate comparison. For either type of noise, it is clear that exponential extrapolation mitigates noise more than linear extrapolation.

Figure 7

The graphs show the cost of matching the performance of an ideal noiseless circuit with a noisy circuit, using the quasiprobability method. The vertical axis ($C^2$) is a multiplicative factor indicating how many more repetitions of the circuit execution are requited. In each graph the upper pair of lines correspond to error rates achievable in ion trap experiments [28, 29]; i.e., the error rate of two-qubit gate is 0.1% and error rates of single-qubit operations are 0.01%. The lower pair of lines indicate the result of reducing these error rates by a factor of 10. Panel (a) corresponds to the swap-test circuit. Panel (b) corresponds to a circuit where every qubit is actively gated in every time step, and the number of steps equals to the number of qubits.