Infinite Distance Extrapolation: How error mitigation can enhance quantum error correction

Abstract

Quantum error mitigation (QEM) and quantum error correction (QEC) are two research areas that are often considered as distinct entities, and the problem of combining the two approaches in a non-trivial way has only recently started to be explored. In this paper, we explore a paradigm at the intersection of the two, based on the error mitigation technique of Zero-Noise Extrapolation (ZNE), that uses the distance of an error correcting code as a noise parameter. This is distinct from some alternative approaches, as QEC is here used as a subroutine inside the QEM framework, while other proposals use QEM as a subroutine inside QEC experiments. Intuitively, we exploit the fact that a reduction in the physical noise level is analogous to an increase in the code distance, as both of them result in a decrease in the logical error rate. As such, the extrapolation to zero noise in the case of ZNE becomes comparable to the extrapolation to infinite distance in the case of this method. We describe how to calculate expectation values from a fault-tolerant computation, and we gain some analytical intuition for our ansatz choice. We explore the performance of the considered method to reduce the errors in a range of expectation values for a realistic circuit-level noise model and realistic device imperfections on the rotated surface code, and we particularly show that the performance of the method holds even in the case of non-stabiliser input states.

Figures (8)

• Figure 1: Rotated surface code for $d = 5$.
• Figure 2: Illustration of how rotated surface code patches of different distances can be arranged in parallel on a device with fixed qubit resources. For example, the physical qubits can be used for either one copy of the $d=7$ code, or one copy of the $d=5$ code and three copies of the $d=3$ code.
• Figure 3: Expectation values as a function of surface code distance for different physical error rates $p$. The grey vertical line separates fitting data (coloured points, left) from reference data (black points, right), where the reference values are the true expectation values at those distances. The solid lines are obtained by fitting the left points to the extrapolating function ansatz. Values closer to the horizontal black line, which represents the noiseless expectation value, are better.
• Figure 4: IDE performance for $\braket{T|X|T}$. (a): Absolute errors in expectation values for several values of the physical error rate $p$ and different distances $d$ of the surface code. The grey vertical line separates fitting data (coloured points, left) from reference data (black points, right). The solid lines are obtained by fitting the left points to the extrapolating function ansatz, and the shaded regions around them represent the error bars. The dotted, horizontal lines are the errors in the extrapolated expectation values at infinite distance. The dashed lines represent the linear fit to the exact data points left of the grey line. The $d_\text{eff}$ values, indicated by black arrows and listed in the legend, are calculated over multiple bootstrap resampling trials as the effective distance at which the IDE error matches the linear fit error. Lower $(\downarrow)$ is better.(b): Effective distance of the IDE value as a function of the cutoff fitting distance for different values of the physical error rate $p$. Points above the dashed $y = x$ line indicate that IDE yields an improvement over a single experiment at the cutoff distance. Higher $(\uparrow)$ is better.
• Figure 5: IDE performance for the $X$ observable evaluated on non-stabiliser states in the $XY$-plane for several physical error rates, as a function of the $R_z$ rotation angle preparing the non-stabiliser state. For both subplots, the exact point at $\theta = 0$ is omitted because of numerical instability, while its label is retained on the axis for reference. The cutoff distance for fitting is $d = 13$.