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arXiv:0909.1466 · IEEE Transactions on Information Theory

Approximate Quantum Error Correction for Correlated Noise

Avraham Ben-Aroya, Amnon Ta‐Shma

Published May 25, 2011• 10 citations
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Abstract

Most of the research on quantum error-correcting codes studies an error model in which each noise operator acts on a bounded number of qubits. In this paper we study a different noise model where the noise operators act on all qubits together, but are otherwise restricted in their action. One example to such an operator is a controlled bit-flip operator, where the control depends on all qubits, i.e., we allow restricted, highly correlated noise. We show both positive and negative results. On the positive side, we show that even though controlled bit-flip errors cannot be perfectly corrected, they can be approximately corrected with a subconstant approximation error. On the negative side, we show that no nontrivial quantum error-correcting code can approximately correct controlled phase error with a subconstant approximation error.

DOI
10.1109/tit.2011.2134410

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Most of the research on quantum error-correcting codes studies an error model in which each noise operator acts on a bounded number of qubits. In this paper we study a different noise model where the noise operators act on all qubits together, but are otherwise restricted in their action. One example to such an operator is a controlled bit-flip operator, where the control depends on all qubits, i.e., we allow restricted, highly correlated noise. We show both positive and negative results. On the positive side, we show that even though controlled bit-flip errors cannot be perfectly corrected, they can be approximately corrected with a subconstant approximation error. On the negative side, we show that no nontrivial quantum error-correcting code can approximately correct controlled phase error with a subconstant approximation error.
We study the performance of simple error correcting and error avoiding quantum codes together with their concatenation for correlated noise models. Specifically, we consider two error models: (i) a bit-flip (phase-flip) noisy Markovian memory channel (model I); (ii) a memory channel defined as a memory degree dependent linear combination of memoryless channels with Kraus decompositions expressed solely in terms of tensor products of X-Pauli (Z-Pauli) operators (model II). The performance of both the three-qubit bit flip (phase flip) and the error avoiding codes suitable for the considered error models is quantified in terms of the entanglement fidelity. We explicitly show that while none of the two codes is effective in the extreme limit when the other is, the three-qubit bit flip (phase flip) code still works for high enough correlations in the errors, whereas the error avoiding code does not work for small correlations. Finally, we consider the concatenation of such codes for both error models and show that it is particularly advantageous for model II in the regime of partial correlations.
Quantum error correction comes from the marriage of quantum mechanics with the theory of error-correcting codes, which is still at the rudimentary stage tackling only 1 qubit to date. As, unlike the classical world, the extent of error in the quantum world is substantially high and unavoidable, scaling up the quantum error correction from 1 qubit to higher is of utmost significane. Therefore, to achive the scaling up, we present a general quantum error correction code that can be adapted for any number of quantum errors (bit-flip error to be specific in our case). This generalized code for bit-flip error correction realizes a delicate blending of using controlled-NOT gates with the notion of quantum entanglement. We augment the blending with majority voting in the final stage of the error correction. We demonstrate efficacy of our proposed error correction code through real experimentation in IBM Qiskit. Our experimental results up to 3-qubit bit-flip errors in the available Qiskit infrastructure confirm a total escape from errors using our proposed error correction code.
We study the performance of simple quantum error correcting codes with respect to correlated noise errors characterized by a finite correlation strength. Specifically, we consider bit flip (phase flip) noisy quantum memory channels and use repetition and noiseless quantum codes. We characterize the performance of the codes by means of the entanglement fidelity as function of the error probability and degree of memory. Finally, comparing the entanglement fidelities of repetition and noiseless quantum codes, we find a threshold for the correlation strength that allows to select the code with better performance.
Noise is a major challenge for current quantum computers. It can be broadly categorized into bit-flip and phase-flip errors. These two types do not necessarily affect the executed algorithm, thus also the application, in the same way. We illustrate this general effect for the example of the quantum approximate optimization algorithm (QAOA) applied to a small instance of the flight-gate assignment (FGA) problem. We compare bit-flip and phase-flip Pauli noise under both layer-level and gate-level noise models, using two circuit decompositions of the same ideal QAOA unitary: a CNOT-based decomposition and a native-RZZR_{ZZ}RZZ​ decomposition. In the simulations, bit-flip noise produces the larger degradation in the performance of the quantum optimization. The asymmetry is most visible in the layer-level and native-RZZR_{ZZ}RZZ​ simulations. We explain this by how the errors affect mixing, final measurements, and how they propagate inside the circuit. We then exploit these insights to tackle noise particularly efficiently using asymmetric error-correcting codes. As an illustration, we use the quantum parity code (QPC), a generalization of the 9-qubit Shor code, and show that a smaller asymmetric code can achieve nearly the same improvement as a larger symmetric choice. This demonstrates that error-correction resources should be assigned not only according to physical error rates, but also according to how strongly each error channel affects the application. As a result, asymmetric quantum error correction proves useful even in cases where the noise model is symmetric. Finally, we discuss how information about the noise obtained through calibration can be exploited in our approach.
We investigate the efficacy of topological quantum error-correction in correlated noise model which permits collective coupling of all the codeword qubits to the same non-Markovian environment. In this noise model, the probability distribution over set of phase-flipped qubits, decays sub-exponentially in the size of the set and carries non-trivial likelihood of the occurring large numbers of qubits errors. We find that in the presence of noise correlation, one cannot guarantee arbitrary high computational accuracy simply by incrementing the codeword size while retaining constant noise level per qubit operation. However, if instead, per-operation qubit error probability in an n-qubits long codeword is reduced O(\sqrt{n}) times below the accuracy threshold, arbitrarily accurate quantum computation becomes feasible with acceptable scaling of the codeword size. Our results suggest that progressively reducing noise level in qubits and gates is as important as continuously integrating more qubits to realize scalable and reliable quantum computer.