Random matrix perspective on probabilistic error cancellation

TL;DR

This work addresses the challenge of probabilistic error cancellation in quantum computing by modeling noisy circuits as random Haar-unitary layers interleaved with Lindblad noise, and studying the resulting denoisers $D = \mathcal{U} \Lambda_U^{-1}$. By analyzing ensembles of random Lindblad generators and applying a Baker-Campbell-Hausdorff reformulation, it shows that denoiser spectra inherit the universal shape of single random Lindblad spectra, while locality induces a hierarchical set of decay timescales. A linear approximation -- valid for small noise strength $t$ -- connects the denoiser spectrum to the sum of Lindbladians, predicting both the central location and the contour of the spectrum, which agrees well with numerics. Local noise models reveal multiple spectral timescales, reinforcing the view that locality survives scrambling by random circuits and suggesting the potential for shallow, few-body denoisers in near-term devices. These findings provide a spectral-theoretic framework to understand and design probabilistic error cancellation strategies in realistic quantum hardware.

Abstract

Probabilistic error cancellation is an attempt to reverse the effect of dissipative noise channels on quantum computers by applying unphysical channels after the execution of a quantum algorithm on noisy hardware. We investigate on general grounds the properties of such unphysical quantum channels by considering a random matrix ensemble modeling noisy quantum algorithms. We show that the complex spectra of denoiser channels inherit their structure from random Lindbladians. Additional structure imposed by the locality of noise channels of the quantum computer emerges in terms of a hierarchy of timescales.

Figures (12)

• Figure 1: Diagrammatic representation of eq. (\ref{['eq:denoiser_def']}) defining the denoiser $D$. Each layer of the noisy circuit $\Lambda_U$ consists of a unitary operation $\mathcal{U}_i$ followed by a noise channel $\mathcal{N}_i$. The denoiser $D$ recovers the target unitary circuit $\mathcal{U}$ from the noisy channel $\Lambda_U$.
• Figure 2: Spectra of $\Lambda_U$ and the denoiser $D$ for different noise parameters $t= [0.1, 0.2, 0.3, 0.4, 0.5]$, with $N=32$ and $m=2$. The spectra of the noisy channel lie inside the unit circle, while the denoiser spectra lie outside, centered around the real axis. For small noise $t$ the spectrum of the noisy channel is close to the unit circle, while the denoiser is close to the identity with eigenvalue 1. With increasing $t$ the noisy channel spectrum moves towards the center of the unit circle, while the denoiser spectrum increases in modulus and moves away form 1. We draw the predicted centers of the denoiser for different $t$ as vertical lines at $\exp(mt)$, according to Eq. (\ref{['eq:denoiser_spectrum_prediction']}).
• Figure 3: denoiser spectra for different system sizes $N=[8,16,24,32]$. Here $t=0.1$ and $m = 2$. For readability we exclude the stationary point at 1 and we include the prediction of the contour discussed in the next section.
• Figure 4: denoiser spectra for different $m=[2,3,4,5]$. Here $t=0.1$ and $N = 32$. For readability we exclude the stationary point at 1 and we include the prediction of the contour discussed in the next section.
• Figure 5: Minimal distance between eigenvalues of the denoiser and the spectrum of $\exp\left(-t\sum^m_{i}\tilde{\mathcal{L}}_{i}\right)$, for $N=32$ and different combinations of $t$ and $m$ . For each eigenvalue we search for the minimal distance to an eigenvalue in the other spectrum. For $t=0.1, m=2$ the largest deviation is at the order of $10^{-6}$. With $t$ and $m$ increasing the differences of the spectra is also increasing.