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IEEE International Symposium on Information Theory (ISIT)

CSS Codes that are Oblivious to Coherent Noise

Jingzhen Hu, Qingzhong Liang, Narayanan Rengaswamy, Robert Calderbank

Published Jul 12, 2021• 3 citations
Publisher

Abstract

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 a$Z$-rotation on each qubit by some angle$\theta$. 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 transversal$Z$-rotations to preserve the code space of a stabilizer code. These conditions require the existence of a large number of weight 2$Z$-stabilizers, and together, these weight 2$Z$-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$[[4L^{2}, 1,2L]]$Shor codes. Given$m$even and given any$[[n, k, d]]$CSS code, we can construct an$[[mn, k, d^{\prime}\geq d]]$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 integer$d$.

DOI
10.1109/isit45174.2021.9518206

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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.
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 angle$\theta .Ratherthanaddresscoherentnoisethroughactiveerrorcorrection,weinvestigatepassivemitigationthroughdecoherencefreesubspaces.Inthelanguageofstabilizercodes,werequirethenoisetopreservethecodespace,andtoacttrivially(asthelogicalidentityoperator)ontheprotectedinformation.Thus,wedevelopnecessaryandsufficientconditionsforalltransversal. Rather than address coherent noise through active error correction, we investigate passive mitigation through decoherence free subspaces. In the language of stabilizer codes, we require the noise to preserve the code space, and to act trivially (as the logical identity operator) on the protected information. Thus, we develop necessary and sufficient conditions for all transversal.Ratherthanaddresscoherentnoisethroughactiveerrorcorrection,weinvestigatepassivemitigationthroughdecoherencefreesubspaces.Inthelanguageofstabilizercodes,werequirethenoisetopreservethecodespace,andtoacttrivially(asthelogicalidentityoperator)ontheprotectedinformation.Thus,wedevelopnecessaryandsufficientconditionsforalltransversalZ$-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.
Quantum error correction codes are needed to protect quantum systems from interference. An important family of quantum error correction codes are stabilizer codes. Many stabilizer codes in the literature belong to a subclass of stabilizer codes, referred to as CSS codes since they are based on the so-called CSS formalism. CSS codes have received significant attention because the dual-containing condition for the CSS formalism is relatively straightforward to satisfy and also because it is easy to produce CSS codes from classical error correction codes. However, some stabilizer codes in the literature show superior error correction performance than existing CSS codes, and are based on binary low-density parity-check (LDPC) codes. Since it has been shown that CSS codes based on nonbinary LDPC codes have good performance over quantum qubit channels, in this paper we investigate stabilizer codes based on nonbinary LDPC codes. In particular, we propose constructions of stabilizer codes based on nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes for qubit channels. The simulation results show that our QC-LDPC stabilizer codes decoded by a nonbinary sum-product algorithm have better performance than their binary counterparts.
Realizing Logical Diagonal Gates via Transversal Physical ZZZ-Rotations in CSS Codes
Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes C2⊆C1C_2 \subseteq C_1C2​⊆C1​, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs (C1,C2)(C_1, C_2)(C1​,C2​) whose resulting CSS codes realize a target logical diagonal gate via transversal physical ZZZ-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit ZZZ-rotations and multi-qubit controlled ZZZ-rotations via transversal physical ZZZ-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an [[n′,k′]][[n',k']][[n′,k′]] CSS code Q′Q'Q′ and a target logical ZZZ-rotation (single-qubit or multi-controlled) ULU_LUL​, and extends Q′Q'Q′ by systematically appending n′′n''n′′ physical qubits to obtain an [[n,k]][[n,k]][[n,k]] CSS code QQQ with n=n′+n′′n = n'+n''n=n′+n′′ and k=k′k=k'k=k′. The target logical gate ULU_LUL​ is realized in QQQ by applying a well-chosen physical transversal ZZZ-rotation to the n′′n''n′′ appended physical qubits. Moreover, any logical gate realized via transversal physical ZZZ-rotations in the input code Q′Q'Q′ remains transversally realizable in the extended code QQQ. The CSS code QQQ may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code Q′Q'Q′ to obtain a CSS code QQQ that supports fault-tolerant implementations of multiple desired logical ZZZ-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled ZZZ-rotations.
Stabilizer codes lie at the heart of modern quantum-error-correcting codes (QECCs). Of particular importance is a class called Calderbank-Shor-Steane (CSS) codes, which includes many important examples such as toric codes, color codes, and fractons. Recent studies have revealed that the decoding transition for these QECCs could be intrinsically captured by calculating information-theoretic quantities from the mixed state. Here we perform a simple analytic calculation of the coherent information for general CSS codes under local incoherent Pauli errors via diagonalization of the density matrices and mapping to classical statistical mechanical (SM) models. Our result establishes a rigorous connection between the decoding transition of the quantum code and the phase transition in the random classical SM model. It is also directly confirmed for CSS codes that exact error correction is possible if and only if the maximum-likelihood decoder always succeeds in the thermodynamic limit. Thus, the fundamental threshold is saturated by the optimal decoder.
In order to perform universal fault-tolerant quantum computation, one needs to implement a logical non-Clifford gate. Consequently, it is important to understand codes that implement such gates transversally. In this paper, we adopt an algebraic approach to characterize all stabilizer codes for which transversal T and T†gates preserve the codespace. Our Heisenberg perspective reduces this question to a finite geometry problem that translates to the design of certain classical codes. We prove three corollaries of this result: (a) For any non-degenerate [n, k, d] stabilizer code supporting a physical transversal T (which might not be logical T), there exists an [n, k, d]] CSS code with the same property; (b) Triorthogonal codes form the most general family of CSS codes that realize logical transversal T via physical transversal T; (c) Triorthogonality is necessary for physical transversal T on a CSS code to realize the logical identity. The main tool we use is a recent characterization of a particular family of diagonal gates in the Clifford hierarchy that are efficiently described by symmetric matrices over rings of integers [N. Rengaswamy et al., Phys. Rev. A 100, 022304]. We refer to these operations as Quadratic Form Diagonal (QFD) gates. Our framework generalizes all existing stabilizer code constructions that realize logical gates via transversal T. We provide several examples of codes and briefly discuss connections to decreasing monomial codes, pin codes, generalized triorthogonality and quasitransversality. We partially extend these results towards characterizing all stabilizer codes that support transversal π/2ℓZ-rotations. In particular, using Ax's theorem on residue weights of polynomials, we provide an alternate characterization of logical gates induced by transversal π/2ℓZ-rotations on a family of quantum Reed-Muller codes. We also briefly discuss a general approach to analyze QFD gates that might lead to a characterization of all stabilizer codes that support any given physical transversal 1- or 2-local diagonal gate.
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.