Noise tailoring for error mitigation and for diagnozing digital quantum computers

TL;DR

This work introduces Noise Tailoring (NT), a strategy to reshape two-qubit gate noise into a target Pauli/depolarizing channel to enhance error mitigation on NISQ devices. By combining Randomized Compiling (RC), Pauli Noise Tomography (PNT), NT, and NEC, the authors demonstrate significant potential gains in classical simulations (up to ~5x improvements) and show how NT can diagnose otherwise hidden noise sources in hardware. On real IBMQ devices, finite-sampling NT can underperform due to residual non-Pauli, coherent, and time-varying noise, though infinite-sampling NT indicates potential improvements and provides rich diagnostics for hardware development. The work highlights both the practical limitations and the diagnostic value of shaping noise, with implications for improved EM, hardware benchmarking, and future open-system quantum simulations on noisy devices.

Abstract

Error mitigation (EM) methods are crucial for obtaining reliable results in the realm of noisy intermediate-scale quantum (NISQ) computers, where noise significantly impacts output accuracy. Some EM protocols are particularly efficient for specific types of noise. Yet the noise in the actual hardware may not align with that. In this article, we introduce Noise Tailoring (NT) -- an innovative strategy designed to modify the structure of the noise associated with two-qubit gates through statistical sampling. We perform classical emulation of the protocol behavior and find that the NT+EM results can be up to 5 times more accurate than the results of EM alone for realistic Pauli noise acting on two-qubit gates. At the same time, on actual IBM quantum computers, the NT method falls victim to various small error sources beyond Markovian Pauli noise. We propose to use the NT method for characterizing such error sources on quantum computers in order to inform hardware development.

Figures (16)

• Figure 1: Workflow diagram describing the implementation of Noise Tailoring (NT) and how we combine it with other techniques in our protocol. (a) Randomized Compiling (RC) is performed in order to convert the error channel of CNOT gates to Pauli error channel. Pauli Noise Tomography (PNT) is then performed in order to characterize this Pauli error channel. (b) Due to RC, the average outputs of the circuits correspond to an effective Pauli noise acting on CNOT gates. This is the noise assumed by the rest of the protocol. (c) The knowledge of the Pauli noise coefficients is used to determine the target noise channel to be aimed for by NT. (Here for illustration, the target channel is the depolarizing noise whose strength is the average of all the Pauli coefficients). (d) RC and NT are applied simultaneously to the circuit of interest. The probability of applying various single-qubit Pauli gates for the NT protocol is computed based on the Pauli noise characterized in the previous steps and the target noise chosen. (e) As a result of the RC+NT combination, the effective noise acting on the CNOT gates of the circuit is the target noise (which is chosen to be depolarizing here). (f) The obesrvables of interest are measured. The efficiency of the error mitigation scheme (here, Noise Estimation Circuits --- NEC) is enhanced by the noise structure adjustment due to the use of NT. The NT technique used in the protocol constitutes the main novelty of the present work.
• Figure 2: Illustration of different sampling approaches to control noise of 2-qubit gates. While PEC attempts to completely cancel the noise (right), PER partially reduces it, while keeping the structure (center). Both techniques typically come with a very large circuit overhead. Our novel NT approach aims to map a given Pauli noise channel to another one, selected by the user. We illustrate this in the figure by targeting the depolarizing noise channel, whose Pauli error rates are all equal (for the purpose of illustration, their value is set to the average of the original Pauli rates---dashed horizontal line on all plots). Depending on the chosen target channel, the sampling overhead might be moderate.
• Figure 3: Classical emulation of various error mitigation protocols, as defined in Sec. \ref{['subsec:protocol_classical_sims']}. The emulations include the noise that corresponds to the Pauli noise on ibm_hanoi, cf. Sec. \ref{['subsec:noise_in_classical_emulation']}. We show the evolution of $X_0$, $Y_1$, and $Z_2$ observables over $15$ Trotter steps (the circuits for the longest simulation time contain 135 CNOT gates). The red dots show the perfect result in the noiseless case. The other curves show the results that correspond to trial protocols defined in Sec. \ref{['subsec:protocol_classical_sims']}. The orange dots show the result obtained by using NEC directly on a Pauli noise channel $\mathcal{N}_p^\infty$ (T1). The light green dots show the results obtained by using NEC with an optimally selected purely depolarizing channel $\mathcal{N}_d^\infty$ (T2); these results show the potential improvement due to NT in the infinite sampling limit. The dark green dots show the results of the NT protocol using a finite sample size of $N_{\mathrm{NT}} = 10^4$ circuits, which can realistically be run on a NISQ device (T3); error bars indicate the uncertainty associated with both the NT finite sampling and shot noise.
• Figure 4: Average weighted absolute error (AWAE) for our classical emulations of the BCS model simulation for the four trial protocols T1--T4 defined in Sec. \ref{['subsec:protocol_classical_sims']}. The average is performed over the expectation values of 7 different observables, cf. Eq. \ref{['eq:set_of_observables']}. The results involve averaging over all time points (left side) and over the last two time points (right side). The bars are labeled with trial numbers T1-T4, and the corresponding noise channels are marked on the $x$-axis. In all cases, the expectation values are error-mitigated using NEC. The contribution to the AWAE due to the finite-sampling of the NT protocol, $\Delta \zeta_{\text{NT}}$, is indicated on the plot by black double arrows. One sees that the use of NT (even in the realistic finite-sampling version, see T3) allows for a substantial reduction of the errors (compare to T1). One further sees that part of the improvement comes purely from changing the noise structure (see T4).
• Figure 5: Comparison of the average weighted absolute error $\zeta$ for the last two time points in classical emulations (CE) and in actual quantum computer runs (QC). Purple bars show the QC results obtained on ibm_hanoi, and the green and the orange bars show the classical emulation results from Fig. \ref{['fig:error_class_sim']}. The purple bar in the center shows $\zeta$ obtained in the quantum computer runs with RC+NT+NEC protocol using $N_{\text{NT}} = 10^4$ sampling circuits. Comparing it to the dark-green bar in the center, one identifies the contributions of the residual coherent and unknown noise channels in the QC runs. The dashed purple blocks on top of the orange bar show the estimated contribution of these noise sources in the RC+NEC protocol, see Sec. \ref{['sec:accuracy_non-improvement_NT+NEC']} for details. The purple bar on the right shows an estimate for AWAE in the limit of $N_{\text{NT}} = \infty$, which eliminates the residual coherent noise completely; the extrapolation procedure producing this estimate is described in Sec. \ref{['sec:extrapolation']}. Overall, the application of NT does not produce an accuracy improvement when using $10^4$ sampling circuits due to the amplification of non-Pauli noise sources by the NT protocol. However, eliminating the residual coherent noise by increasing the number of sampling circuits can provide improvement compared to the bare RC+NEC protocol. Lastly, comparing the two protocols provides a diagnostic for various sources of noise present on a quantum computer, as discussed in Sec. \ref{['sec:NT_for_diagnostics']}.