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arXiv:2607.01364

Decohered toric code under quantum damping noise and its mapping to a classical spin model

Nihar Ranjan Dash, Sanjoy Dutta, R. Srikanth, Subhashish Banerjee

quant-phSubmitted Jul 1, 2026• Announced Jul 3, 2026• 0 citations
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Abstract

We investigate properties of toric codes under realistic damping error channels, which include squeezing, thermal and non-Markovian effects. First, we map the decohered toric code under the generalized amplitude-damping (GAD) and the squeezed generalized amplitude-damping (SGAD) channels to the statistical-mechanical models using the double Hilbert-space formalism. Second, we map the action of the GAD and SGAD channels on the toric code to stochastic Pauli-type errors via Pauli twirling, yielding asymmetric depolarizing channels, and obtain the logical failure probabilities as a function of temperature and squeezing. In both cases, we relate the channel parameters of the GAD and SGAD channels to the spin-coupling constants of the statistical-mechanical model.

Comments
11 pages, 6 figures

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We investigate properties of toric codes under realistic damping error channels, which include squeezing, thermal and non-Markovian effects. First, we map the decohered toric code under the generalized amplitude-damping (GAD) and the squeezed generalized amplitude-damping (SGAD) channels to the statistical-mechanical models using the double Hilbert-space formalism. Second, we map the action of the GAD and SGAD channels on the toric code to stochastic Pauli-type errors via Pauli twirling, yielding asymmetric depolarizing channels, and obtain the logical failure probabilities as a function of temperature and squeezing. In both cases, we relate the channel parameters of the GAD and SGAD channels to the spin-coupling constants of the statistical-mechanical model.
A squeezed generalized amplitude damping (SGAD) channel is a quantum channel that models a general noise process incorporating the effects of bath squeezing, dissipation, and decoherence. In this paper, we analyze the dynamics of quantum entanglement and discord in the SGAD channel with memory. By obtaining a stochastic map defining this noisy quantum channel, we derive the concurrence and discord of Werner-like mixed states sent by successive uses of the channel. It is shown that these quantum correlations can be preserved or even generated depending on the initial channel input states, channel parameters, and the degree of channel memory. In particular, the squeezing effect does not contribute to the dynamics of quantum correlation for singlet-like states under correlated noise.
We present analytic estimates of the performances of various approximate quantum error-correction schemes for the generalized amplitude-damping (GAD) qubit channel. Specifically, we consider both stabilizer and nonadditive quantum codes. The performance of such error-correcting schemes is quantified by means of the entanglement fidelity as a function of the damping probability and the nonzero environmental temperature. The recovery scheme employed throughout our work applies, in principle, to arbitrary quantum codes and is the analog of the perfect Knill-Laflamme recovery scheme adapted to the approximate quantum error-correction framework for the GAD error model. We also analytically recover and/or clarify some previously known numerical results in the limiting case of vanishing temperature of the environment, the well-known traditional amplitude-damping channel. In addition, our study suggests that degenerate stabilizer codes and self-complementary nonadditive codes are especially suitable for the error correction of the GAD noise model. Finally, comparing the properly normalized entanglement fidelities of the best performant stabilizer and nonadditive codes characterized by the same length, we show that nonadditive codes outperform stabilizer codes not only in terms of encoded dimension but also in terms of entanglement fidelity.
Stabilizer-based simulation of quantum error-correcting codes typically relies on the Pauli-twirling approximation (PTA) to render non-Clifford noise classically tractable, but PTA can distort the behavior of physically relevant channels such as thermal relaxation. Physically accurate noise simulation is needed to train decoders and understand the noise suppression capabilities of quantum error correction codes. In this work, we develop an exact and stabilizer-compatible model of qubit thermal relaxation noise and show that the combined amplitude damping and dephasing channel admits a fully positive probability decomposition into Clifford operations and reset whenever T2⩽T1T_2 \leqslant T_1T2​⩽T1​. For T2>T1T_2 > T_1T2​>T1​, the resulting decomposition is negative, but allows a smaller sampling overhead versus independent channels. We further introduce an approximated error channel with reset that removes the negativity of the decomposition while achieving higher channel fidelity to the true thermal relaxation than PTA, and extend our construction to finite temperature relaxation. We apply the exact combined model to investigate large surface codes and bivariate bicycle codes on superconducting platforms with realistic thermal relaxation error. The differing logical performances across code states further indicate that noise-model-informed decoders will be essential for accurately capturing thermal-noise structure in future fault-tolerant architectures.
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Traditional quantum error-correcting codes are designed for the depolarizing channel modeled by generalized Pauli errors occurring with equal probability. Amplitude damping channels, in general, model the decay process of a multilevel atom or energy dissipation of a bosonic system at zero temperature. We discuss quantum error-correcting codes adapted to amplitude damping channels for higher dimensional systems (qudits). For multi-level atoms, we consider a natural kind of decay process, and for bosonic systems, we consider the qudit amplitude damping channel obtained by truncating the Fock basis of the bosonic modes to a certain maximum occupation number. We construct families of single-error-correcting quantum codes that can be used for both cases. Our codes have larger code dimensions than the previously known single-error-correcting codes of the same lengths.
A powerful method for analyzing quantum error-correcting codes is to map them onto classical statistical mechanics models. Such mappings have thus far mostly focused on static codes, possibly subject to repeated syndrome measurements. Recent progress in quantum error correction, however, has prompted new paradigms where codes emerge from stabilizer circuits in spacetime -- a unifying perspective encompassing syndrome extraction circuits of static codes, dynamically generated codes, and logical operations. We show how to construct statistical mechanical models for stabilizer circuits subject to independent Pauli errors, by mapping logical equivalence class probabilities of errors to partition functions using the spacetime subsystem code formalism. We also introduce a modular language of spin diagrams for constructing the spin Hamiltonians, which we describe in detail focusing on independent circuit-level X-Z error channels. With the repetition and toric codes as examples, we use our approach to analytically rank logical error rates and thresholds between code implementations with standard and dynamic syndrome extraction circuits, describe the effect of transversal logical Clifford gates on logical error rates, and perform Monte Carlo simulations to estimate maximum likelihood thresholds. Our framework offers a universal prescription to analyze, simulate, and compare the decoding properties of any stabilizer circuit, while revealing the innate connections between dynamical quantum systems and noise-resilient phases of matter.