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arXiv:2011.00197 · IEEE Transactions on Information Theory

Mitigating Coherent Noise by Balancing Weight-2 Z-Stabilizers

Jingzhen Hu, Qingzhong Liang, Narayanan Rengaswamy, Robert Calderbank

Published Nov 23, 2021• 16 citations
arXivHTMLPDF

Abstract

Physical platforms such as trapped ions suffer from coherent noise that does not follow a simple stochastic model. Stochastic errors in quantum systems occur randomly but coherent errors are more damaging since they can accumulate in a particular direction. We consider coherent noise acting transversally, giving rise to an effective error which is a$Z$-rotation on each qubit by some angle$\theta $. Rather than address coherent noise through active error correction, we investigate passive mitigation through decoherence free subspaces. In the language of stabilizer codes, we require the noise to preserve the code space, and to act trivially (as the logical identity operator) on the protected information. Thus, we develop necessary and sufficient conditions for all transversal$Z$-rotations to preserve the code space of a stabilizer code. These conditions require the weight-$2~Z$-stabilizers to cover all the qubits that are in the support of the$X$-component of some stabilizer. Furthermore, the weight-$2~Z$-stabilizers generate a direct product of single-parity-check codes with even block length. By adjusting the sizes of these components, we are able to construct a large family of QECC codes oblivious to coherent noise, one that includes the$[[4L^{2}, 1, 2L]]$Shor codes. The Shor codes are examples of constant excitation codes, where logical qubits are encoded as a code state that is a sum of physical states indexed by binary vectors with the same weight. Constant excitation codes are oblivious to coherent noise since a transversal$Z$-rotation acts as a global phase. We prove that a CSS code is oblivious to coherent noise if and only if it is a constant excitation code, and that if the code is error-detecting, then the (constant) weights in different cosets of the$X$-stabilizers are identical.

DOI
10.1109/tit.2021.3130155

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Physical platforms such as trapped ions suffer from coherent noise that does not follow a simple stochastic model. Stochastic errors in quantum systems occur randomly but coherent errors are more damaging since they can accumulate in a particular direction. We consider coherent noise acting transversally, giving rise to an effective error which is aZZZ-rotation on each qubit by some angleθ\theta θ. Rather than address coherent noise through active error correction, we investigate passive mitigation through decoherence free subspaces. In the language of stabilizer codes, we require the noise to preserve the code space, and to act trivially (as the logical identity operator) on the protected information. Thus, we develop necessary and sufficient conditions for all transversalZZZ-rotations to preserve the code space of a stabilizer code. These conditions require the weight-2 Z2~Z2 Z-stabilizers to cover all the qubits that are in the support of theXXX-component of some stabilizer. Furthermore, the weight-2 Z2~Z2 Z-stabilizers generate a direct product of single-parity-check codes with even block length. By adjusting the sizes of these components, we are able to construct a large family of QECC codes oblivious to coherent noise, one that includes the[[4L2,1,2L]][[4L^{2}, 1, 2L]][[4L2,1,2L]]Shor codes. The Shor codes are examples of constant excitation codes, where logical qubits are encoded as a code state that is a sum of physical states indexed by binary vectors with the same weight. Constant excitation codes are oblivious to coherent noise since a transversalZZZ-rotation acts as a global phase. We prove that a CSS code is oblivious to coherent noise if and only if it is a constant excitation code, and that if the code is error-detecting, then the (constant) weights in different cosets of theXXX-stabilizers are identical.
Physical platforms such as trapped ions suffer from coherent noise that does not follow a simple stochastic model. We view coherent errors as rotations about a particular axis, and observe that since they can accumulate coherently over time, they can be more damaging. It is natural to consider coherent noise acting transversally giving rise to an effective error, which is aZZZ-rotation on each qubit by some angleθ\thetaθ. Rather than addressing coherent noise through active error correction, we instead investigate passive mitigation through decoherence free subspaces. In the language of stabilizer codes, we require the noise to preserve the code space, and to act trivially (as the logical identity operator) on the protected information. Thus, we develop necessary and sufficient conditions for all transversalZZZ-rotations to preserve the code space of a stabilizer code. These conditions require the existence of a large number of weight 2ZZZ-stabilizers, and together, these weight 2ZZZ-stabilizers generate a direct product of single-parity-check codes. By adjusting the size of these components, we are able to construct a large family of CSS codes, oblivious to coherent noise, that includes the[[4L2,1,2L]][[4L^{2}, 1,2L]][[4L2,1,2L]]Shor codes. Givenmmmeven and given any[[n,k,d]][[n, k, d]][[n,k,d]]CSS code, we can construct an[[mn,k,d′≥d]][[mn, k, d^{\prime}\geq d]][[mn,k,d′≥d]]CSS code that is oblivious to coherent noise. This result is generalized to stabilizer codes in [Hu, Liang, Rengaswamy, and Calderbank 2020]. The MacWilliams Identities play a central role in the technical analysis, and classical coding theorists may be interested in connections to classical codes with all weights divisible by some integerddd.
Collective coherent (CC) errors are inevitable, as every physical qubit undergoes free evolution under its kinetic Hamiltonian. These errors can be more damaging than stochastic Pauli errors because they affect all qubits coherently, resulting in high-weight errors that standard quantum error-correcting (QEC) codes struggle to correct. In quantum memories and communication systems, especially when storage durations are long, CC errors often dominate over stochastic noise. Trapped-ion platforms, for example, exhibit strong CC errors with minimal stochastic Pauli components. In this work, we address the regime where immunity to CC errors, high code rate (due to limited qubit availability), and moderate distance (sufficient for correcting low-weight errors) are all essential. We construct a family of constant-excitation stabilizer codes with parameters $[[{2}^{r+1},{2}^{r}\ensuremath{-}(r+1),4]]$. The smallest instance, the [[8,1,4]] code, improves the code rate and error threshold of the best previously known CE code by factors of approximately two and four, respectively.
Coherent errors are a dominant noise process in many quantum computing architectures. Unlike stochastic errors, these errors can combine constructively and grow into highly detrimental overrotations. To combat this, we introduce a simple technique for suppressing systematic coherent errors in low-density parity-check stabilizer codes, which we call stabilizer slicing. The essential idea is to slice low-weight stabilizers into two equally weighted Pauli operators and then apply them by rotating in opposite directions, causing their overrotations to interfere destructively on the logical subspace. With access to native gates generated by three-body Hamiltonians, we can completely eliminate purely coherent overrotation errors, and for overrotation noise of 0.99 unitarity we achieve a 135-fold improvement in the logical error rate of surface-17. For more conventional two-body ion trap gates, we observe an 89-fold improvement for Bacon-Shor-13 with purely coherent errors which should be testable in near-term fault-tolerance experiments. This second scheme takes advantage of the prepared gauge degrees of freedom, and to our knowledge is the first example in which the state of the gauge directly affects the robustness of a code's memory. This Letter demonstrates that coherent noise is preferable to stochastic noise within certain code and gate implementations when the coherence is utilized effectively.
Abstract We study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In particular, we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment.
The performance of a given quantum error correction (QEC) code depends upon the noise model that is assumed. Independent Pauli noise, applied after each quantum operation, is a simplistic noise model that is easy to simulate and understand in the context of stabilizer codes. Although such a noise model is artificial, it is equivalent to independent, random, unbiased qubit rotations. What about spatially or temporally correlated qubit rotations? Such a noise model is applicable to global operations (e.g., NMR or ESR), common control sources (e.g., lasers), or slow drift (e.g., charge or magnetic noise) in various qubit technologies. In the worst case, such errors can combine constructively and result in a post-correction failure rate that increases with the number of error correction cycles. However, we show that this worst case does not generally arise unless taking active corrective actions while performing QEC. That is, by employing virtual Pauli frame updates ("passive" error correction) rather than physical corrections ("active" error correction), coherent errors do not compound appreciably. Starting in a random Pauli frame is also advantageous. In fact, through perturbation theory arguments and supporting numerical simulations, we show that the logical qubit performance beyond distance 3 for correlated single-qubit Hamiltonian noise models (i.e., global errant qubit rotations), when employing these "lazy" strategies, essentially matches the performance of Pauli noise model with the same process fidelity (fidelity after one application). In a more general circuit model of noise, correlations may add constructively within syndrome extraction rounds but Pauli frame randomization from passive error correction mitigates this effect across multiple rounds.
Abstract Coherent errors, which arise from collective couplings, are a dominant form of noise in many realistic quantum systems, and are more damaging than oft considered stochastic errors. Here, we propose integrating stabilizer codes with constant-excitation codes by code concatenation. Namely, by concatenating an [[n, k, d]] stabilizer outer code with dual-rail inner codes, we obtain a [[2n, k, d]] constant-excitation code immune from coherent phase errors and also equivalent to a Pauli-rotated stabilizer code. When the stabilizer outer code is fault-tolerant, the constant-excitation code has a positive fault-tolerant threshold against stochastic errors. Setting the outer code as a four-qubit amplitude damping code yields an eight-qubit constant-excitation code that corrects a single amplitude damping error, and we analyze this code’s potential as a quantum memory.