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arXiv:2412.11220

Non-Markovian Noise Suppression Simplified through Channel Representation

Zhenhuan Liu, Yunlong Xiao, Zhenyu Cai

Published Dec 15, 2024• 0 citations
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Abstract

Non-Markovian noise, arising from memory effects in the environment, poses substantial challenges to conventional quantum noise suppression protocols, including quantum error correction and mitigation. We introduce a channel representation for arbitrary non-Markovian quantum dynamics, termed the Choi channel, which translates the complex dynamics of non-Markovian noise into the familiar picture of noise channels. It therefore provides a systematic way to design non-Markovian noise suppression protocols: one can apply existing channel-level error-suppression techniques in the Choi-channel picture and then translate them back to the circuit picture. The performance of the resulting non-Markovian protocols, including their noise-suppression effects and complexity, can often be inherited directly from the corresponding Choi-channel protocols without requiring a separate analysis. With this framework, we devise new protocols using Pauli twirling, probabilistic error cancellation, and virtual channel purification. Pauli twirling reduces non-Markovian noise to noise with only classical temporal correlations; probabilistic error cancellation can fully cancel non-Markovian noise; and virtual channel purification can suppress non-Markovian noise without detailed knowledge of its specific form. Through these examples, the Choi channel serves as a foundational bridge for systematically converting existing channel-level techniques into non-Markovian noise suppression protocols.

DOI
10.48550/arxiv.2412.11220

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Non-Markovian noise, arising from memory effects in the environment, poses substantial challenges to conventional quantum noise suppression protocols, including quantum error correction and mitigation. We introduce a channel representation for arbitrary non-Markovian quantum dynamics, termed the Choi channel, which translates the complex dynamics of non-Markovian noise into the familiar picture of noise channels. It therefore provides a systematic way to design non-Markovian noise suppression protocols: one can apply existing channel-level error-suppression techniques in the Choi-channel picture and then translate them back to the circuit picture. The performance of the resulting non-Markovian protocols, including their noise-suppression effects and complexity, can often be inherited directly from the corresponding Choi-channel protocols without requiring a separate analysis. With this framework, we devise new protocols using Pauli twirling, probabilistic error cancellation, and virtual channel purification. Pauli twirling reduces non-Markovian noise to noise with only classical temporal correlations; probabilistic error cancellation can fully cancel non-Markovian noise; and virtual channel purification can suppress non-Markovian noise without detailed knowledge of its specific form. Through these examples, the Choi channel serves as a foundational bridge for systematically converting existing channel-level techniques into non-Markovian noise suppression protocols.
Noise forms a central obstacle to effective quantum information processing. Recent experimental advances have enabled the tailoring of noise properties through Pauli twirling, transforming arbitrary noise channels into Pauli channels. This underpins theoretical descriptions of fault-tolerant quantum computation and forms an essential tool in noise characterization and error mitigation. Pauli-Lindblad channels have been introduced to aptly parameterize quasi-local Pauli errors across a quantum register, excluding negative Pauli-Lindblad parameters relying on the Markovianity of the underlying noise processes. We point out that caution is required when parameterizing channels as Pauli-Lindblad channels with nonnegative parameters. For this, we study the effects of Pauli twirling on Markovianity. We use the notion of Markovianity of a channel (rather than that of an entire semigroup) and prove a general Pauli channel is non-Markovian if and only if at least one of its Pauli-Lindblad parameters is negative. Using this, we show that Markovian quantum channels often become non-Markovian after Pauli twirling. The Pauli-twirling induced non-Markovianity necessitates the use of negative Pauli-Lindblad parameters for a correct noise description in experimentally realistic scenarios. An important example is the implementation of the X\sqrt{X}X​-gate under standard Markovian noise. As such, our results have direct implications for quantum error mitigation protocols that rely on accurate noise characterization.
Non-Markovian noise, arising from environmental memory effects, is the most general and challenging form of noise in quantum computing, and is typically difficult to characterize and suppress. Here, we analyze and experimentally demonstrate a non-Markovian noise suppression scheme inspired by quantum purification protocols. We theoretically prove that, even without noise calibration and assumptions on specific noise models, the scheme can exponentially reduce non-Markovian error rates with respect to the ancillary system size. We implement the protocol using nuclear spins, demonstrating that non-Markovian noise can be suppressed for both unitary operations and non-unitary channels. The observed fidelities and process tomography show close agreement with theoretical predictions, confirming the practicality and effectiveness of the scheme.
Pauli channels are widely used to describe errors in quantum computers, particularly when noise is shaped via Pauli twirling. A common assumption is that such channels admit a Markovian generator, namely a Pauli-Lindblad model with non-negative rates, but the validity of this assumption has not been systematically examined. Here, using CP-indivisibility as our criterion for non-Markovianity, we study multi-qubit Pauli channels from a single snapshot of the dynamics. We find that while the generator always has the same structure as the standard Pauli-Lindblad model, the rates may be negative or complex. We show that random Pauli channels are almost always non-Markovian, with the probability of encountering a negative rate converging doubly exponentially to unity with the number of qubits. For physically motivated noise models shaped by Pauli twirling, including single-qubit over-rotations and two-qubit amplitude damping errors, we find that negative rates are generic, even when the underlying physical noise is Markovian. We generalize probabilistic error amplification and cancellation to non-Markovian generators, and quantify the sampling overhead introduced by negative and complex rates. Experiments on superconducting qubits confirm that allowing negative rates in the learned noise model yields more accurate predictions than restricting to non-negative rates.
Correlated noise is a critical failure mode in quantum error correction (QEC), yet a gap remains between the stochastic Pauli models used for scalable QEC analyses and the microscopic, non-Markovian descriptions of noise in physical devices. We bridge this gap by introducing Spatiotemporal Pauli Processes (SPPs): the natural multi-time generalization of Pauli channels, joint probability distributions over Pauli faults across space and time, obtained exactly from any noise process, however non-Markovian, under standard Pauli-frame randomization. SPPs thereby provide a single language in which correlated noise can be recast, compared, and extended, compatible with both microscopic open systems modeling and the stabilizer workflow of QEC design and analysis. Our central result is constructive: the multi-time Pauli twirl acts by local contractions on a process tensor network, yielding an explicit classical tensor network whose virtual bonds encode memory, bounded by the environment's Liouville-space dimension. Transfer operator diagnostics link memory spectra to correlation decay, and hidden Markov representations enable efficient sampling of correlated Pauli fault trajectories, mapping microscopically derived noise directly into circuit-level QEC simulation. We demonstrate this with surface code memory and stability benchmarks up to distance 191919. A temporal "storm" model with tunable correlation time shows that memory alone, at strictly fixed marginal error rates, systematically erodes the exponential error suppression expected from code distance. A genuinely spatiotemporal quantum cellular automaton bath maps exactly, under system twirling, to a nonlinear probabilistic cellular automaton; tuning its coherent interactions drives the noise into a pseudo-critical regime with critical slowing down and macroscopic error avalanches that reverse surface code distance scaling.
Non-Markovian noise, arising from environmental memory effects, is the most general and challenging form of noise in quantum computing, and is typically difficult to characterize and suppress. Here, we analyze and experimentally demonstrate a non-Markovian noise suppression scheme inspired by quantum purification protocols. We theoretically prove that, even without noise calibration and assumptions on specific noise models, the scheme can exponentially reduce non-Markovian error rates with respect to the ancillary system size. We implement the protocol using nuclear spins, demonstrating that non-Markovian noise can be suppressed for both unitary operations and nonunitary channels. The observed fidelities and process tomography show close agreement with theoretical predictions, confirming the practicality and effectiveness of the scheme.
Quantum error mitigation(QEM), an error suppression strategy without the need for additional ancilla qubits for noisy intermediate-scale quantum~(NISQ) devices, presents a promising avenue for realizing quantum speedups of quantum computing algorithms on current quantum devices. However, prior investigations have predominantly been focused on Markovian noise. In this paper, we propose a non-Markovian Noise Mitigation(NMNM) method by extending the probabilistic error cancellation (PEC) method in the QEM framework to treat non-Markovian noise. We present the derivation of a time-local quantum master equation where the decoherence coefficients are directly obtained from bath correlation functions(BCFs), key properties of a non-Markovian environment that will make the error mitigation algorithms environment-aware. We further establish a direct connection between the overall approximation error and sampling overhead of QEM and the spectral property of the environment. Numerical simulations performed on a spin-boson model further validate the efficacy of our approach.