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arXiv:2212.07718 · Physical Review Research

Quantum error correction under numerically exact open-quantum-system dynamics

Aravind Plathanam Babu, Tuure Orell, Vasilii Vadimov, Wallace S. Teixeira, Mikko Möttönen, Matti Silveri

Published Nov 20, 2023• 9 citations
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Abstract

The known quantum error-correcting codes are typically built on approximative open-quantum-system models such as Born-Markov master equations. However, it is an open question how such codes perform in actual physical systems that, to some extent, necessarily exhibit phenomena beyond the limits of these models. To this end, we employ numerically exact open-quantum-system dynamics to analyze the performance of a five-qubit error-correction code where each qubit is coupled to its own bath. We first focus on the performance of a single error-correction cycle covering timescales and coupling strengths beyond those of Born-Markov models. We observe distinct power-law behavior of the error-corrected channel infidelity $\ensuremath{\propto}{t}^{2a}$: $a\ensuremath{\lesssim}2$ in the ultrashort times $t<3/{\ensuremath{\omega}}_{\mathrm{c}}$ and $a\ensuremath{\approx}1/2$ in the short-time range $3/{\ensuremath{\omega}}_{\mathrm{c}}<t<30/{\ensuremath{\omega}}_{\mathrm{c}}$, where ${\ensuremath{\omega}}_{\mathrm{c}}$ is the cutoff angular frequency of the bath. Furthermore, the observed scaling of the performance of error correction is found to be robust against various imperfections in physical qubit systems, such as perturbations in qubit-qubit coupling strengths and parametric disorder. For repeated error correction, we demonstrate the breaking of the five-qubit error-correction code and of the Born-Markov models if the repetition rate of the error-correction cycles exceeds $2\ensuremath{\pi}/\ensuremath{\omega}$ or the coupling strength $\ensuremath{\kappa}\ensuremath{\gtrsim}\ensuremath{\omega}/10$, where $\ensuremath{\omega}$ is the angular frequency of the qubit. Our results provide bounds of validity for the standard quantum error-correction codes and pave the way for applying numerically exact open-quantum-system methods for further studies of error correction beyond simple error models and for other strongly coupled many-body models.

DOI
10.1103/physrevresearch.5.043161

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The known quantum error-correcting codes are typically built on approximative open-quantum-system models such as Born-Markov master equations. However, it is an open question how such codes perform in actual physical systems that, to some extent, necessarily exhibit phenomena beyond the limits of these models. To this end, we employ numerically exact open-quantum-system dynamics to analyze the performance of a five-qubit error-correction code where each qubit is coupled to its own bath. We first focus on the performance of a single error-correction cycle covering timescales and coupling strengths beyond those of Born-Markov models. We observe distinct power-law behavior of the error-corrected channel infidelity \ensuremath∝t2a\ensuremath{\propto}{t}^{2a}\ensuremath∝t2a: a\ensuremath≲2a\ensuremath{\lesssim}2a\ensuremath≲2 in the ultrashort times t<3/\ensuremathωct<3/{\ensuremath{\omega}}_{\mathrm{c}}t<3/\ensuremathωc​ and a\ensuremath≈1/2a\ensuremath{\approx}1/2a\ensuremath≈1/2 in the short-time range 3/\ensuremathωc<t<30/\ensuremathωc3/{\ensuremath{\omega}}_{\mathrm{c}}<t<30/{\ensuremath{\omega}}_{\mathrm{c}}3/\ensuremathωc​<t<30/\ensuremathωc​, where \ensuremathωc{\ensuremath{\omega}}_{\mathrm{c}}\ensuremathωc​ is the cutoff angular frequency of the bath. Furthermore, the observed scaling of the performance of error correction is found to be robust against various imperfections in physical qubit systems, such as perturbations in qubit-qubit coupling strengths and parametric disorder. For repeated error correction, we demonstrate the breaking of the five-qubit error-correction code and of the Born-Markov models if the repetition rate of the error-correction cycles exceeds 2\ensuremathπ/\ensuremathω2\ensuremath{\pi}/\ensuremath{\omega}2\ensuremathπ/\ensuremathω or the coupling strength \ensuremathκ\ensuremath≳\ensuremathω/10\ensuremath{\kappa}\ensuremath{\gtrsim}\ensuremath{\omega}/10\ensuremathκ\ensuremath≳\ensuremathω/10, where \ensuremathω\ensuremath{\omega}\ensuremathω is the angular frequency of the qubit. Our results provide bounds of validity for the standard quantum error-correction codes and pave the way for applying numerically exact open-quantum-system methods for further studies of error correction beyond simple error models and for other strongly coupled many-body models.
We analyze the performance of a quantum error-correction code subject to physically motivated noise modeled by a Lindblad master equation. We consider dissipative and coherent single-qubit terms and two-qubit crosstalk, studying how different approximations of the noise capture the success rate of a code. Focusing on the five-qubit code and adapting it to partially correct two-qubit errors in relevant parameter regimes according to the noise model, we find that a composite-channel approximation where every noise term is considered separately captures the behavior in many physical cases up to long timescales, eventually failing due to the effect of noncommuting terms. In contrast, we find that single-qubit approximations do not properly capture the error-correction dynamics with two-qubit noise, even for short times. A Pauli approximation going beyond a single-qubit channel is sensitive to the details of the noise, state, and decoder and succeeds at short timescales relative to the noise strength, beyond which it fails. Furthermore, we point out a mechanism for a Pauli model failure where it underestimates the success rate of a code even with frequent syndrome projection and correction cycles. These results shed light on the performance of error correction in the presence of realistic noise and can advance the ongoing efforts towards useful quantum error correction.
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We investigate continuous quantum error correction, comparing performance under a Markovian error model to two distinct non-Markovian models. The first non-Markovian model involves an interaction Hamiltonian between the system and an environmental qubit via an X-X coupling, with a "cooling" bath acting on the environment qubit. This model is known to exhibit abrupt transitions between Markovian and non-Markovian behavior. The second non-Markovian model uses the post-Markovian master equation (PMME), which represents the bath correlation through a memory kernel; we consider an exponentially decaying kernel and both underdamped and overdamped dynamics. We systematically compare these non-Markovian error models against the Markovian case and against each other, for a variety of different codes. We start with a single qubit, which can be solved analytically. We then consider the three-qubit repetition code and the five-qubit "perfect" code. In all cases, we find that the fidelity decays more rapidly in the Markovian case than in either non-Markovian model, suggesting that continuous quantum error correction has enhanced performance against non-Markovian noise. We attribute this difference to the presence of a quantum Zeno regime in both non-Markovian models.
We compare the performance of quantum error correcting codes when memory errors are unitary with the more familiar case of dephasing noise. For a wide range of codes, we analytically compute the effective logical channel that results when the error correction steps are performed noiselessly. Our examples include the entire family of repetition codes, the five-qubit, Steane, Shor, and surface codes. When errors are measured in terms of the diamond norm, we find that the error correction is typically much more effective for unitary errors than for dephasing. We observe this behavior for a wide range of codes after a single level of encoding, and in the thresholds of concatenated codes using hard decoders. We show that this holds with great generality by proving a bound on the performance of any stabilizer code when the noise at the physical level is unitary. By comparing the diamond norm error ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}$ of the logical qubit with the same quantity at the physical level ${D}_{\ensuremath{\diamond}}$, we show that ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\le}c{D}_{\ensuremath{\diamond}}^{d}$ where ddd is the distance of the code and ccc is a constant that depends on the code but not on the error. This bound compares very favorably to the performance of error correction for dephasing noise and other Pauli channels, where an error correcting code of odd distance ddd will exhibit a scaling ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\sim}{D}_{\ensuremath{\diamond}}^{(d+1)/2}$.
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