Non-Markovianity induced by Pauli-twirling

TL;DR

Pauli twirling converts general noise into Pauli channels but can induce non-Markovianity even when the underlying process is Markovian. The authors establish a bijection between Pauli channels and generalized Pauli-Lindblad channels, showing that Pauli-twirled noise can break, conserve, or instate channel semigroup Markovianity depending on the PL parameters, with negative PL parameters signaling non-Markovianity. They illustrate this with Hadamard dephasing and a noisy sqrt(X) gate, deriving explicit PL-parameter conditions and connecting them to experimental timescales and platform-specific noise biases. The work has direct implications for quantum error mitigation, requiring that negative PL parameters be allowed in noise characterizations to avoid biased error-cancellation results and to correctly interpret Pauli-twirled noise in realistic devices.

Abstract

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 $\sqrt{X}$-gate under standard Markovian noise. As such, our results have direct implications for quantum error mitigation protocols that rely on accurate noise characterization.

Figures (2)

• Figure 1: (a) The relations between the maps in this paper. The Pauli-Lindblad (PL) maps (orange, $P_1$) are those maps of the form of Eq. (\ref{['eq:PL']}) with $\lambda_a\in \mathbb C$. The generalized PL channels are defined as those PL maps that are also quantum channels (olive green, $P_2$). This set is essentially equal to the set of Pauli channels (the latter is the closure of the former, with limit points depicted as the boundary, $P_3$). The channel semigroup Markovian channels (teal, CSM) contain channels that are Pauli channels ($P_4$) and non-Pauli channels ($P_5$). The Pauli-Lindblad channels are essentially the CSM channels that are Pauli channels ($P_4$), forming a strict subset of the generalized PL channels. All possible nontrivial effects of Pauli twirling a quantum channel are indicated by the yellow arrows. Pauli twirling can break (CSMB), conserve (CSMC), or instate (CSMI) channel semigroup Markovianity. It can also conserve the non-CSM property of a channel (nCSMC). (b) The noise channel of a noisy Clifford gate $\mathcal{U}\circ \mathcal{E}$, with $\mathcal{U}(\rho)=U\rho\,U^\dagger$ the noiseless Clifford, can be Pauli twirled by inserting a random Pauli word $P_a$ before, and the conjugated Pauli word $P^U_a=U P_a U^\dagger$ after the noisy gate.
• Figure 2: Transition diagram of channel semigroup Markovianity for the error channel of a $\sqrt{X}$ gate due to Pauli twirling. The two-dimensional parameter space $(\gamma t_g,\gamma_\varphi t_g)$ is divided into four regions corresponding to the effects of Pauli twirling on channel semigroup Markovianity. In the region labeled as CSMB (red) the initial CSM property is broken by Pauli twirling and the resulting channel can only be described by a generalized PL channel with a negative PL parameter. In region CSMC (dark blue) CSM is conserved and the resulting channel has only nonnegative PL parameters. At extreme dephasing rates ($\gamma_\varphi t_g\gtrsim2$) the error channel is non-CSM; Pauli twirling instates Markovianity in region CSMI (purple) and conserves non-Markovianity in region nCSMC (teal). The solid white line is the boundary separating the regions with different channel semigroup Markovianity after Pauli twirling, the dashed white line depicts the implicit curve $\lambda_x(t_g)=0$ [Eq. \ref{['eq:sqrtx_lax']}], the second-order approximation of this boundary. Most importantly, region CSMB contains experimentally relevant scales, shown in the inset, with excellent agreement between the numerical and analytical second-order boundary.

Theorems & Definitions (3)

• Proposition
• Proposition
• proof