Noise-Resilient Quantum Evolution in Open Systems through Error-Correcting Frameworks

TL;DR

The paper addresses the challenge of preserving quantum information in open systems by embedding quantum error-correcting codes into microscopic system–bath models and solving a second-order master equation. It benchmarks the five-qubit, Steane, and toric codes under local and collective bosonic baths, using state fidelity as the primary metric and exploring both low and high temperature regimes along with single- and two-qubit logical encodings. The key findings are that, in the low-temperature regime, the five-qubit code provides the strongest protection and benefits from multiple correction cycles, while at high temperature thermal noise diminishes the advantage of all codes; for two-qubit Werner states there exists a critical evolution time tc before QEC helps, which increases with entanglement. Collectively, the results establish a quantitative framework for evaluating QEC in realistic open-system environments and guide the design of noise-resilient quantum architectures for near-term devices, highlighting the efficiency of compact codes like the five-qubit code over topological or CSS variants in these settings.

Abstract

We analyze quantum state preservation in open quantum systems using quantum error-correcting (QEC) codes that are explicitly embedded into microscopic system-bath models. Instead of abstract quantum channels, we consider multi-qubit registers coupled to bosonic thermal environments, derive a second-order master equation for the reduced dynamics, and use it to benchmark the five-qubit, Steane, and toric codes under local and collective noise. We compute state fidelities for logical qubits as functions of coupling strength, bath temperature, and the number of correction cycles. In the low-temperature regime, we find that repeated error-correction with the five-qubit code strongly suppresses decoherence and relaxation, while in the high-temperature regime, thermal excitations dominate the dynamics and reduce the benefit of all codes, though the five-qubit code still outperforms the Steane and toric codes. For two-qubit Werner states, we identify a critical evolution time before which QEC does not improve fidelity, and this time increases as entanglement grows. After this critical time, QEC does improve fidelity. Comparative analysis further reveals that the five-qubit code (the smallest perfect code) offers consistently higher fidelities than topological and concatenated architectures in these open-system settings. These findings establish a quantitative framework for evaluating QEC under realistic noise environments and provide guidance for developing noise-resilient quantum architectures in near-term quantum technologies.

Figures (14)

• Figure 1: Schematic of the system-ancilla and its respective thermal baths. $\mathcal{A}$ denotes the ancilla for the respective main qubit $\mathcal{M}$. The ancilla and the system qubit interact with local thermal baths with different temperatures. The different shades of red are used to designate the different temperatures of the thermal bath.
• Figure 2: (Top) Conceptual overview of the QEC process. Physical information is encoded into logical states before the noise acts and distorts them. After passing through a correction stage, the recovered logical information is finally decoded to reproduce the original physical information. (Bottom) A generic block diagram for $n$-qubit quantum error correcting code. A physical qubit with state $\ket\psi$ is encoded via $\mathcal{E(\cdot)}$ with $n-1$ auxiliary qubits initialized at state $\ket0$ to build one logical qubit, which undergoes some error channel, denoted in red. Then the original state is recovered via some recovery operation $\mathcal{R}(\cdot)$, which consists of syndrome measurement, i.e., correction operation $\mathcal{C}or(\cdot)$ and decoder $\mathcal{D}(\cdot)$. By discarding the auxiliary qubits, one can get back the original state $\ket{\psi}$.
• Figure 3: State fidelity $\mathcal{F}_{state}$ of the initial state $\ketbra{0}{0}$ as a function of time $t$ and coupling strength $\kappa$ with and without five-qubit QEC. Single and multiple cycle QEC are considered for the analysis. All qubits are coupled to baths with coupling strength (a) $\kappa/\omega=0.01$ and (b) $\kappa/\omega=0.1$, with the bath temperature $T=0.2$. The fidelity increases with the increase in the number of QEC cycles.
• Figure 4: State fidelity $\mathcal{F}_{state}$ of the initial state $p \ketbra{\psi^-}{\psi^-}+ (1-p) \frac{I}{4}$ for $p=0.5$ as a function of time $t$ and coupling strength $\kappa$ with and without five-qubit QEC. We have considered single as well as multiple cycles QEC. In the (a) weak coupling regime, i.e., $\kappa/\omega=0.01$ and (b) large system-bath coupling regime, i.e., $\kappa/\omega=0.1$, all the qubits are considered to be coupled to baths with a temperature $T=0.2$.
• Figure 5: The critical values $\kappa t_c$ where QEC fails to improve fidelity are plotted against mixing parameter $p$ of different Werner states $p\ketbra{\psi^-}{\psi^-}+(1-p)\frac{I}{4}$. Single and multiple cycle five-qubit QEC are considered for the process. All qubits are coupled to baths with coupling strength $\kappa/\omega=0.01$, with the bath temperature a) $T=0.2$ and (b) $T=10$. This indicates that this five-qubit QEC cannot protect entanglements. However, the critical value can be reduced by multiple cycles of QEC.