Noise-Agnostic Unbiased Quantum Error Mitigation for Logical Qubits

TL;DR

The paper tackles the challenge of achieving unbiased quantum error mitigation without fully characterizing complex noise models. It proposes spacetime noise inversion (SNI), which uses a single total error rate parameter $P$ together with a sampler of Pauli errors to invert the circuit’s maximum spacetime noise $\mathcal{N}_{max}$ via a Taylor-series expansion, enabling unbiased results even under correlated noise. The authors provide rigorous bias and cost bounds, discuss ideal and practical error samplers, and show robustness to temporal fluctuations, arguing that SNI can be naturally integrated with quantum error correction in the early fault-tolerant era. The work also analyzes resource overheads, outlines sampling strategies for surface codes and qLDPC codes, and demonstrates numerical robustness to non-Pauli and time-varying noise, highlighting the practical potential of combining error mitigation with error correction.

Abstract

Probabilistic error cancellation is a quantum error mitigation technique capable of producing unbiased computation results but requires an accurate error model. Constructing this model involves estimating a set of parameters, which, in the worst case, may scale exponentially with the number of qubits. In this paper, we introduce a method called spacetime noise inversion, revealing that unbiased quantum error mitigation can be achieved with just a single accurately measured error parameter and a sampler of Pauli errors. The error sampler can be efficiently implemented in conjunction with quantum error correction. We provide rigorous analyses of bias and cost, showing that the cost of measuring the parameter and sampling errors is low -- comparable to the cost of the computation itself. Moreover, our method is robust to the fluctuation of error parameters, a limitation of unbiased quantum error mitigation in practice. These findings highlight the potential of integrating quantum error mitigation with error correction as a promising approach to suppress computational errors in the early fault-tolerant era.

Figures (11)

• Figure 1: Workflow of spacetime noise inversion. The error sampler generates spacetime Pauli errors, which are used to estimate the total error rate and modify the quantum circuit: Pauli gates are inserted into the circuit according to the generated errors. The final error-mitigated result is obtained as the expected value over measurement outcomes from the modified circuits, incorporating necessary normalization and phase factors from the Monte Carlo summation. For detailed protocols and pseudocodes, see Appendices \ref{['app:ideal_protocol']}, \ref{['app:practical_protocol']}, and \ref{['app:codes']}.
• Figure 2: (a) Error sampler circuit for a gate. An ancilla qubit is introduced for each qubit the gate $U$ acts on. First, each qubit pair is initialized in the Bell state (BS) $(\vert{ 00 }\rangle+\vert{ 11 }\rangle)/\sqrt{2}$. Then, the inverse gate $U^\dag$ and the gate $U$ are applied sequentially. Finally, the Bell measurement (BM) is applied to each qubit pair, i.e. measuring $XX$ and $ZZ$ operators. Triangles represent encoding (En) and decoding (De) operations, which transfer states between low-distance logical qubits and high-distance super qubits, which are represented by thin black and thick green lines, respectively. (b) Noise-boosted computation circuit. Smooth-boundary squares represent ideal operations, while zigzagged squares denote noise maps (which describe the noises associated with corresponding operations). Encoding and decoding errors are stochastically inserted after state preparation (SP) operations and unitary gates ($U_1$ and $U_2$) but not after measurement (M) operations. Each operation, together with its inherent noise and the inserted encoding/decoding noise, constitutes the effective noisy operation, as indicated by the blue and purple boxes. Note that Pauli twirling is applied to the encoding and decoding operations, such that the associated errors are effectively Pauli, i.e., commutative.
• Figure 3: Bias in error mitigation under error parameter fluctuations. In the numerical simulation, we evaluate the bias of a quantum circuit implementing the transformation $[e^{-i\frac{\pi}{8}(X_1+X_2)}e^{-i\frac{\pi}{8}Z_1Z_2}]^8$ on two qubits initialized in the $\vert{ + }\rangle$ state, with the observable being $X$ on the first qubit. The transformation is decomposed into controlled-NOT, Hadamard, and $T$ gates. The simulation is performed at the logical-qubit level, assuming a model of logical errors: Each non-Pauli operation is subject to a noise map, where all noise maps are parameterized by an error rate $p$ that is randomly drawn from $\{0.001,0.003\}$ with equal probabilities for each individual run of the computation circuit and spacetime error generation. The noise includes both Pauli and coherent errors. SNI with a practical error sampler and the conventional PEC (cPEC) are applied for error mitigation. In both protocols, $M_P$ instances of the spacetime error are used to estimate error parameters. In cPEC, a sparse error model is assumed without accounting for temporal correlations. Error bars represent standard deviations, each estimated from 100 instances. See Appendix \ref{['app:time']} for details.
• Figure 4: (a) Error Sampler. It takes an operation $\alpha\in \mathbb{O}$ as the input, and the output is a Pauli operator $\tau\in\mathbb{P}_n$. (b-e) Circuits for realizing the error sampler. Thin black lines represent logical qubits, while thick green lines represent super qubits. In (b), (c) and (f), each pair of qubits is prepared in the Bell state (BS) $(\vert{ 0 }\rangle\otimes\vert{ 0 }\rangle+\vert{ 1 }\rangle\otimes\vert{ 1 }\rangle)/\sqrt{2}$ and measured in the basis $\{X\otimes X,Z\otimes Z\}$, called Bell measurement (BS). Triangles denote encoding (En.) and decoding (De.) operations that enable transitions between logical qubits and super qubits. $U$ is the unitary operator of a quantum gate, and $\vert{ \psi }\rangle$ is the eigenstate of the operator $\kappa$ with the eigenvalue $+1$. In the circuits, Pauli twirling is applied to encoding, decoding and non-Pauli stabilizer operations $\alpha\in\mathbb{O}_{S}$ on logical qubits. The output Pauli operator $\tau$ is determined from the measurement outcomes according to Tables \ref{['tab:SqErrorsampler']}, \ref{['tab:TqErrorsampler']} and \ref{['tab:SpErrorsampler']}.
• Figure 5: Effective operations after inserting encoding/decoding errors. Triangles represent encoding/decoding noise maps, respectively; see Fig. \ref{['fig:error_sampler']}.

Theorems & Definitions (26)

• Theorem 1
• Example 1
• Definition 1
• Example 2
• Definition 2
• Definition 3
• Definition 4
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• Definition 7