Spatiotemporal Pauli processes: Quantum combs for modelling correlated noise in quantum error correction

Abstract

Correlated noise is a critical failure mode in quantum error correction (QEC), as temporal memory and spatial structure concentrate faults into error bursts that undermine standard threshold assumptions. Yet, a fundamental gap persists between the stochastic Pauli models ubiquitous in QEC and the microscopic, non-Markovian descriptions of physical device dynamics. We close this gap by introducing \emph{Spatiotemporal Pauli Processes} (SPPs). By applying a multi-time Pauli twirl -- operationally realised by Pauli-frame randomisation -- to a general process tensor, we map arbitrary multi-time, non-Markovian dynamics to a multi-time Pauli process. This process is represented by a process-separable comb, or equivalently, a well-defined joint probability distribution over Pauli trajectories in spacetime. We show that SPPs inherit efficient tensor network representations whose bond dimensions are bounded by the environment's Liouville-space dimension. To interpret these structures, we develop transfer operator diagnostics linking spectra to correlation decay, and exact hidden Markov representations for suitable classes of SPPs. We demonstrate the framework via surface code memory and stability simulations of up to distance \(19\) for (i) a temporally correlated ``storm'' model that tunes correlation length at fixed marginal error rates, and (ii) a genuinely spatiotemporal 2D quantum cellular automaton bath that maps exactly to a nonlinear probabilistic cellular automaton under twirling. Tuning coherent bath interactions drives the system into a pseudo-critical regime, exhibiting critical slowing down and macroscopic error avalanches that cause a complete breakdown of surface code distance scaling. Together, these results justify SPPs as an operationally grounded, scalable toolkit for modelling, diagnosing, and benchmarking correlated noise in QEC.

Figures (12)

• Figure 1: Framework overview: Process tensors to spatiotemporal Pauli processes. Starting from microscopic open quantum system dynamics (bottom left), we represent a process tensor description of noise and its tensor network form (top left). Applying the multi-time Pauli twirl (top centre)---as induced operationally by Pauli-frame randomisation---projects $\Upsilon_{0:k}$ onto a spatiotemporal Pauli process (SPP) (top right), i.e., a joint distribution over Pauli trajectories with a classical tensor network representation. The resulting SPP admits scalable analysis and simulation tools (bottom right), including transfer operator correlation diagnostics, hidden Markov model realisations, and Monte Carlo sampling for QEC benchmarking.
• Figure 2: Spatiotemporal tensor network form of a process tensor. Horizontal bonds $\mu_j^i$ encode temporal correlations, bonds $\nu_j^i$ carry spatial correlations, and open indices $\alpha_j^i$, $\beta_j^i$ correspond to system inputs and outputs at each time $j \in [0,k]$ and subsystem $i \in [0,n)$.
• Figure 3: 1D SPP as an MPS over Pauli trajectories. (a) Local tensor obtained by contracting the system--environment unitary tensor $\mathbf{U}_{(j)}$ with the Pauli tensor $\mathbf{P}$, corresponding to a local Pauli twirl on the system. (b) Resulting matrix product state (MPS) representation of the SPP trajectory weights, where environment bonds $\mu_j$ mediate temporal correlations and physical indices $x_j$ label Pauli outcomes at each time step. Sampling this MPS yields Pauli trajectories $\mathbf{x}_{0:k}$ with probability $\Pr(\mathbf{x}_{0:k})$.
• Figure 4: 2D SPP as a PEPS over spatiotemporal Pauli faults. (a) Local tensor transformation for 2D tensor network representation. (b) Projected entangled-pair state (PEPS) representation of the SPP trajectory weights. Spatial correlations are encoded along one axis by bonds $\nu^i_j$, while temporal correlations are encoded along the other axis by bonds $\mu^i_j$.
• Figure 5: Quantum relative entropy diagnostics of SPPs. We plot the generalised quantum mutual information $\mathcal{I}[\Upsilon_{0:1}] = S(\Upsilon_{0:1} \| \Upsilon_{0:1}^{\mathrm{Markov}})$, its twirled counterpart $\mathcal{I}[\Upsilon_{0:1}^{\mathcal{T}_P}]$, and the twirl relative entropy $\mathcal{J}[\Upsilon_{0:1}]$ as functions of interaction strength $\theta\in[0,2\pi]$ for three different Hamiltonian families. Choi operators are normalised such that $\mathrm{Tr}[\Upsilon_{0:1}] = 1$, and the environment is initialised in $\ket{+}$. Panels: (a) Heisenberg exchange interaction, (b) controlled-$X$ rotation, (c) Heisenberg exchange with local field.

Theorems & Definitions (14)

• Definition 3.1: MPO representation of a process tensor
• Definition 4.1: Multi-time Pauli twirl
• Proposition 4.2: Properties
• Theorem 4.3: Twirling yields a process-separable Pauli process
• Definition 4.4: Spatiotemporal Pauli process
• Proposition 4.5: Local tensor network implementation of the multi-time Pauli twirl
• Definition 4.6: Process tensor MPO to SPP MPS
• Lemma 4.7: Bond dimension bound for SPP MPS
• proof
• Corollary 4.8: Environment dimension bound