Quantum error mitigation for error-resilient quantum computation

Quantum computers have the potential of providing unprecedented computational power for solving problems that are known to be difficult for classical computers, including integer factoring, database searching, molecular simulation and combinatorial optimization. At the time of writing, the largest real-world quantum computers are only handling 100–200 noisy qubits, which are also known as “noisy intermediate-scale quantum computers”. Quantum error mitigation (QEM) constitutes a class of promising techniques, which are capable of reducing the computational error caused by the decoherence-induced impairments in the qubits. However, this is achieved at the cost of an undesired computational overhead termed as the sampling overhead. In this thesis, we aim for striking a beneficial and flexible computational accuracy vs. sampling overhead trade-off, for both circuit-level and algorithm-level QEM. We commence by presenting an overview of existing QEM techniques, highlighting their main sources of computational overhead and their requirements concerning the prior knowledge about the computational task or the noise model. Specifically, for channel-inversion based QEM, we present a comprehensive sampling overhead analysis, and propose “quantum channel precoders” that reduce the sampling overhead. We show that Pauli channels have the lowest sampling overhead in a large class of practical quantum channels, and that the depolarizing channels have the lowest sampling overhead among Pauli channels. Furthermore, we conceive a beneficial amalgam of channel-inversion based QEM and quantum error-correction codes (QECCs). Regarding the specific implementation strategy of channel-inversion based QEM, we analyze the error scaling versus sampling overhead trade off of the Monte Carlo based channel inversion, which drastically reduces the complexity of candidate circuit generation, at the cost of a moderate accuracy degradation. In particular, we show that the computational error of the Monte Carlo based channel inversion is on the order of O (√ NG), where NG is the number of gates. This is similar to that of the exact channel inversion. By contrast, the computational error is on the order of O(NG) when no QEM is applied. However, the candidate circuit generation complexity of the Monte Carlo based strategy is exponentially lower than that of the exact channel inversion, implying that the Monte Carlo based channel inversion strategy has a favorable accuracy vs. overhead trade-off. Next, we design symmetry-based QEM methods having high sample-efficiency. In general, the intrinsic symmetry conditions regarding the computational tasks may be exploited for the mitigation of the computational error using the method of symmetry verification. However, it is limited to state symmetries, hence it has a restricted scope of application. We extend the symmetry verification method to circuit symmetries by proposing the technique of spatio-temporal stabilizers. Specifically, as a natural generalization of the conventional stabilizers, spatio-temporal stabilizers are capable of detecting whether a quantum circuit commutes with certain operators. This enables us to mitigate the errors violating the commutativity conditions. We then discuss the detailed design strategy of the spatio-temporal stabilizer method for mitigating the errors of practical quantum algorithms, including the quantum Fourier transform and the quantum approximate optimization algorithm. Regarding a specific kind of symmetry conditions, namely the permutation symmetry across different activations of quantum circuits, we propose the method of permutation filtering, inspired by the philosophy of finite impulse response (FIR) filters of classical signal processing theory. Remarkably, the existing virtual distillation method may be viewed as a special case of permutation filters. We show that the proposed design of these filters converges to the global optimum, and that the error reduction performance of the optimal filter is particularly good for narrowband noises, corresponding to the scenario of deep quantum circuits. Concerning the spectral leakage issue of the quantum phase estimation algorithm, we propose an algorithm-level error mitigation method, namely the dual-frequency estimator. This is potentially useful in the context of noisy intermediate-scale quantum computing, since the maximum achievable recording length N is ultimately restricted by the coherence time, hence we have to rely on multiple samples when a high phase estimation accuracy is required. In particular, we show that when the number of samples is sufficiently large, the dual-frequency estimator outperforms the cosine window, which is shown to be optimal for single-sample estimation. Furthermore, we also show that the dual-frequency estimator achieves the Cramer-Rao bound when the number of samples is large. This implies that the dual-frequency estimator has a beneficial accuracy vs. sampling overhead trade-off in the asymptotic regime.