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IEEE Conference on Decision and Control (CDC)

Robust Quantum Coding for Thermal Noise via Dissipative Dynamics

Kazuki Nishino, Kentaro Ohki, Koji Tsumura

Published Dec 13, 2023• 1 citation
Publisher

Abstract

This paper proposes an improved quantum coding method based on Markovian dissipative dynamics that is robust against thermal noise. A method for correcting information errors is indispensable for the practical use of quantum information devices, and stabilizer codes for quantum systems have been devised. One of the quantum coding methods, namely, quantum coding based on Markovian dissipative dynamics, has a favorable feature that it does not require strict time control, however it is known that these methods are susceptible to thermal noise. In this paper, we propose a quantum coding method that incorporates a new mechanism to correct the disturbance of the quantum state caused by thermal noise into the quantum coding method using dissipative dynamics. We also analyze the stability of the proposed quantum dynamics in the quantum state corresponding to the target code word. Numerical experiments confirm the effectiveness of the proposed method.

DOI
10.1109/cdc49753.2023.10383415

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This paper proposes an improved quantum coding method based on Markovian dissipative dynamics that is robust against thermal noise. A method for correcting information errors is indispensable for the practical use of quantum information devices, and stabilizer codes for quantum systems have been devised. One of the quantum coding methods, namely, quantum coding based on Markovian dissipative dynamics, has a favorable feature that it does not require strict time control, however it is known that these methods are susceptible to thermal noise. In this paper, we propose a quantum coding method that incorporates a new mechanism to correct the disturbance of the quantum state caused by thermal noise into the quantum coding method using dissipative dynamics. We also analyze the stability of the proposed quantum dynamics in the quantum state corresponding to the target code word. Numerical experiments confirm the effectiveness of the proposed method.
In this paper, we deal with the stability of the improved quantum coding method based on Markovian dissipative dynamics that is robust against thermal noise. This coding method was proposed by Nishino et al. and the suppression of the effect of thermal noise was demonstrated by numerical simulations, however the mathematical proof was not given. In this paper, we analyze the impact of the thermal noise in this method and mathematically prove that the quantum states converge to a neighborhood around the target code words. The bound of the neighborhood set is also given strictly. We also demonstrate numerical simulations, which confirm the theoretical results.
In this paper, we deal with quantum information decoding. Quantum codes are a known technique for correcting errors of quantum information caused by noise or operational errors and one of the most popular types of codes is the so-called stabilizer code. Various implementation methods have been proposed to encode information into a stabilizer code, but there have been few studies on decoding to extract the original information from the code words. In this paper, we propose a decoding method for stabilizer codes using dissipation dynamics and show that the original information can be extracted from the code words.
In order to ensure reliable quantum information processing in realistic noisy devices, it is necessary to resort to suitable error correction and avoidance strategies. A key aspect of these techniques is how the relevant information is logically mapped (encoded) in the noise-protected codewords. Here, we study how to engineer Markov dynamics that transfers the desired information from an "upload" subsystem to the target error-correcting or noiseless quantum code, effectively acting as a dissipative quantum encoder. Dissipative encoders offer advantages with respect to standard unitary encoding protocols based on quantum circuits, as they can associate the target states to non-trivial basins of attraction, and thus tolerate more general initializations in the upload qubits. In particular, we show that devising continuous-time dissipative encoders requires the target code to be invariant, making this task more delicate as compared to its discrete-time counterpart. Nonetheless, we show how this is always possible for stabilizer quantum error-correcting codes.
We formalize the problem of dissipative quantum encoding, and explore the advantages of using Markovian evolution to prepare a quantum code in the desired logical space, with emphasis on discrete-time dynamics and the possibility of exact finite-time convergence. In particular, we investigate robustness of the encoding dynamics and their ability to tolerate initialization errors, thanks to the existence of non-trivial basins of attraction. As a key application, we show that for stabilizer quantum codes on qubits, a finite-time dissipative encoder may always be constructed, by using at most a number of quantum maps determined by the number of stabilizer generators. We find that even in situations where the target code lacks gauge degrees of freedom in its subsystem form, dissipative encoders afford nontrivial robustness against initialization errors, thus overcoming a limitation of purely unitary encoding procedures. Our general results are illustrated in a number of relevant examples, including Kitaev's toric code.
Quantum error correction plays a critical role in enabling fault-tolerant quantum computing by protecting fragile quantum information from noise. While general-purpose quantum error correction codes are designed to address a wide range of noise types, they often require substantial resources, making them impractical for near-term quantum devices. Approximate quantum error correction provides an alternative by tailoring codes to specific noise environments, reducing resource demands while still maintaining noise-robustness. Dynamical codes, including Floquet codes, introduce a dynamic approach to quantum error correction, employing time-dependent operations to stabilize logical qubits. In this work, we combine the flexibility of dynamical codes with the versatility of approximate quantum error correction to offer a promising avenue for addressing dominant noise in quantum systems. We construct several approximate dynamical codes using the recently developed strategic code framework. As a special case, we recover the approximate static codes widely studied in the existing literature. By analyzing these approximate dynamical codes through semidefinite programming, we establish the uniqueness and robustness of the optimal encoding, decoding, and check measurements. We also develop a temporal Petz recovery map suited to approximate dynamical codes.
It is by now well established that noise itself can be useful for performing quantum information processing tasks. We present results which show how one can effectively reduce the error rate associated with a noisy quantum channel by counteracting its detrimental effects with another form of noise. In particular, we consider the effect of adding on top of a purely Markovian (Lindblad) dynamics, a more general form of dissipation, which we refer to as generalized-Markovian noise. This noise has an associated memory kernel and the resulting dynamics are described by an integrodifferential equation. The overall dynamics are characterized by decay rates which depend not only on the original dissipative time scales but also on the new integral kernel. We find that one can engineer this kernel such that the overall rate of decay is lowered by the addition of this noise term. We illustrate this technique for the case where the bare noise is described by a dephasing Pauli channel. We analytically solve this model and show that one can effectively double (or even triple) the length of the channel, while achieving the same fidelity, entanglement, and error threshold. We numerically verify this scheme can also be used to protect against thermal Markovian noise (at nonzero temperature), which models spontaneous emission and excitation processes. A physical interpretation of this scheme is discussed, whereby the added generalized-Markovian noise causes the system to become periodically decoupled from the background Markovian noise.