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The Effects of Quantum Error Correction on Noisy Systems

Stefanie J. Beale

Published Aug 23, 2018• 0 citations
OpenAlex

Abstract

Full accuracy simulations of quantum systems are very costly, and as a result most studies \nof quantum error correction assume a probabilistic Pauli error model, largely because such \nerrors can be e ciently simulated. Therefore, the behaviour of more general noise in a \nquantum error correcting code is poorly characterized. In this thesis, we present results which \ndemonstrate the scaling of the logical noise with respect to the physical delity, and argue \nthat the e ective logical noise approaches a Pauli channel as the code distance increases, \neven when no recovery operations are applied. As a result, we argue that the average logical \n delity can be used to accurately quantify the e ective logical noise, and to select recovery \noperations appropriate to the system. We further demonstrate that when physical noise \nacts on fewer than d qubits in an [[n; k; d]] code, the resultant noise is Pauli, and develop \na method for approximating the dominant contributions to the e ective logical noise up to \na speci ed precision in terms of the physical in delity. We derive conditions under which \nsets of recovery operations will produce equivalent logical noise channels, with examples of \nequivalencies in the 3 qubit repetition code, the 5 qubit code, the Steane code, and the Shor \ncode. We also provide a general expression for the e ective logical noise when the physical \nqubits undergo depolarizing or Pauli noise in a quantum error correcting code, examine the \nbehaviour of depolarizing noise under concatenation of the 5 qubit and Steane codes, and \npresent an algorithm for soft decoding which is not subject to statistical sampling, with an \nemphasis on the e ective behaviour of a concatenated 5 qubit code undergoing depolarizing \nnoise after applying a specialized version our soft decoding algorithm.

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Full accuracy simulations of quantum systems are very costly, and as a result most studies \nof quantum error correction assume a probabilistic Pauli error model, largely because such \nerrors can be e ciently simulated. Therefore, the behaviour of more general noise in a \nquantum error correcting code is poorly characterized. In this thesis, we present results which \ndemonstrate the scaling of the logical noise with respect to the physical delity, and argue \nthat the e ective logical noise approaches a Pauli channel as the code distance increases, \neven when no recovery operations are applied. As a result, we argue that the average logical \n delity can be used to accurately quantify the e ective logical noise, and to select recovery \noperations appropriate to the system. We further demonstrate that when physical noise \nacts on fewer than d qubits in an [[n; k; d]] code, the resultant noise is Pauli, and develop \na method for approximating the dominant contributions to the e ective logical noise up to \na speci ed precision in terms of the physical in delity. We derive conditions under which \nsets of recovery operations will produce equivalent logical noise channels, with examples of \nequivalencies in the 3 qubit repetition code, the 5 qubit code, the Steane code, and the Shor \ncode. We also provide a general expression for the e ective logical noise when the physical \nqubits undergo depolarizing or Pauli noise in a quantum error correcting code, examine the \nbehaviour of depolarizing noise under concatenation of the 5 qubit and Steane codes, and \npresent an algorithm for soft decoding which is not subject to statistical sampling, with an \nemphasis on the e ective behaviour of a concatenated 5 qubit code undergoing depolarizing \nnoise after applying a specialized version our soft decoding algorithm.
Quantum error correction protocols have been developed to offset the high sensitivity to noise inherent in quantum systems. However, much is still unknown about the behavior of a quantum error-correcting code under general noise, including noisy measurements. This lack of knowledge is largely due to the computational cost of simulating quantum systems large enough to perform nontrivial encodings. In this paper, we develop general methods for incorporating noisy measurement operations into simulations of quantum error-correcting codes and show that measurement errors on readout qubits manifest as a renormalization on the effective logical noise. We also derive general methods for reducing the computational complexity of calculating the exact effective logical noise by many orders of magnitude. This reduction is achieved by determining when different recovery operations produce equivalent logical noise. These methods could also be used to better approximate soft decoding schemes for concatenated codes or to reduce the size of a lookup table to speed up the error correction step in implementations of quantum error-correcting codes. We give examples of such reductions for the three-qubit, five-qubit, Steane, concatenated, and toric codes.
Analysis of quantum error correcting codes is typically done using a stochastic, Pauli channel error model for describing the noise on physical qubits. However, it was recently found that coherent errors (systematic rotations) on physical data qubits result in both physical and logical error rates that differ significantly from those predicted by a Pauli model. Here we examine the accuracy of the Pauli approximation for noise containing coherent errors (characterized by a rotation angle ) under the repetition code. We derive an analytic expression for the logical error channel as a function of arbitrary code distance d and concatenation level n , in the small error limit. We find that coherent physical errors result in logical errors that are partially coherent and therefore non-Pauli. However, the coherent part of the logical error is negligible at fewer than error correction cycles when the decoder is optimized for independent Pauli errors, thus providing a regime of validity for the Pauli approximation. Above this number of correction cycles, the persistent coherent logical error will cause logical failure more quickly than the Pauli model would predict, and this may need to be combated with coherent suppression methods at the physical level or larger codes.
In quantum error correction, the description of the noise channel cannot be completely accurate and fluctuation always appears in the noise channel. It is found that when fluctuation of the physical noise channel is considered, the average effective channel is dependent only on the average of the physical noise channel, and the average of the physical noise channel here plays the role of the independent error model in previous works. Now one may conclude that in the independent error model, the results in previous works are also valid for the average channel where fluctuation exists. In some typical cases, our numerical simulations in the concatenated quantum error-correction protocol with a five-qubit code, seven-qubit Steane code, and nine-qubit Shor confirm this conjecture. For a five-qubit code, the effective channels approximate to the depolarizing channel as the concatenated level increases. For the Steane code, the effective channels approximate to one Pauli channel as the concatenated level increases. For the Shor code, the effective channels approximate to either a Pauli-XXX or Pauli-ZZZ channel in each level, and in the next concatenated level, the effective channels approximate to the other. The numerical results for these codes indicate that the degree of approximation increases as the concatenated level increases, and the fluctuation of the noise channel decays exponentially as concatenated quantum error correction is performed. On the error-correction threshold, the attenuation ratio of the standard deviation of channel fidelity has a roughly stable value. In contrast, standard deviations of off-diagonal elements of the quantum process matrix (Pauli form) decay more quickly than standard deviations of diagonal elements.
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 ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\sim}{D}_{\ensuremath{\diamond}}^{(d+1)/2}$.
What must a quantum error-correction decoder know about temporally correlated noise, and what does it cost to learn? We study this question for fixed stabilizer memory experiments with known system-environment interactions. We formulate calibration in terms of the fidelity differences that select a Pauli recovery and relate missing information to the resulting fidelity loss. Our main example uses weak coherent XXX rotations in a class of CSS codes including Shor, Steane, and odd-distance rotated surface codes. All physical qubits share a sign either fixed throughout a run or independently redrawn each interval. The two models have identical syndrome statistics and the same spatiotemporal Pauli process (SPP), obtained by Pauli-twirling every noise interval. Yet at suitable storage times, they require different logical corrections. In the weak-noise limit, each model is almost perfectly correctable when known. Without model information, even the best final recovery has a worst-case fidelity loss approaching 1/21/21/2 relative to the noise-aware optimum, despite access to the full syndrome record. For this pair of models, at fixed code and fixed positive target loss below 1/21/21/2, we prove matching calibration bounds. With logical preparation and final readout, complete QEC cycles require a number of noise intervals scaling as the inverse rotation angle raised to the code's XXX distance. Allowing two uninterrupted intervals before one extraction reduces the optimal cost to inverse-square scaling. Physical Pauli twirling of every noise interval changes both recovery performance and its information requirements. Discarding the record of applied random Pauli operations makes the SPP sufficient for optimal recovery. Keeping this record can improve the attainable fidelity, but reaching that optimum may require additional noise information.
Typical studies of quantum error correction assume probabilistic Pauli noise, largely because it is relatively easy to analyze and simulate. Consequently, the effective logical noise due to physically realistic coherent errors is relatively unknown. Here, we prove that encoding a system in a stabilizer code and measuring error syndromes decoheres errors, that is, causes coherent errors to converge toward probabilistic Pauli errors, even when no recovery operations are applied. Two practical consequences are that the error rate in a logical circuit is well quantified by the average gate fidelity at the logical level and that essentially optimal recovery operators can be determined by independently optimizing the logical fidelity of the effective noise per syndrome.