[Paper Review] Mitigating Coherent Noise by Balancing Weight-2 $Z$-Stabilizers

Jingzhen Hu, Qingzhong Liang+2 more|UA Campus Repository (The University of Arizona)|Oct 31, 2020

Quantum Computing Algorithms and Architecture41 references7 citations

TL;DR

This paper proposes a passive mitigation strategy for coherent noise in quantum error-correcting codes by ensuring transversal Z-rotations act trivially on the code space. It establishes that a stabilizer code is oblivious to coherent noise—specifically, transversal Z-rotations—if and only if its weight-2 Z-stabilizers form a direct product of even-length single-parity-check codes, enabling construction of constant excitation codes like the [[4L²,1,2L]] Shor codes that are inherently resilient to coherent noise accumulation.

ABSTRACT

Physical platforms such as trapped ions suffer from coherent noise where errors manifest as rotations about a particular axis and can accumulate over time. We investigate passive mitigation through decoherence free subspaces, requiring the noise to preserve the code space of a stabilizer code, and to act 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, which require the weight-$2$ $Z$-stabilizers to cover all the qubits that are in the support of some $X$-component. Further, the weight-$2$ $Z$-stabilizers generate a direct product of single-parity-check codes with even block length. By adjusting the size of these components, we are able to construct a large family of QECC codes, oblivious to coherent noise, that includes the $[[4L^2, 1, 2L]]$ Shor codes. Moreover, given $M$ even and any $[[n,k,d]]$ stabilizer code, we can construct an $[[Mn, k, \ge d]]$ stabilizer code that is oblivious to coherent noise. If we require that transversal $Z$-rotations preserve the code space only up to some finite level $l$ in the Clifford hierarchy, then we can construct higher level gates necessary for universal quantum computation. The $Z$-stabilizers supported on each non-zero $X$-component form a classical binary code C, which is required to contain a self-dual code, and the classical Gleason's theorem constrains its weight enumerator. The conditions for a stabilizer code being preserved by transversal $2π/2^l$ $Z$-rotations at $4 \le l \le l_{\max} <\infty$ level in the Clifford hierarchy lead to generalizations of Gleason's theorem that may be of independent interest to classical coding theorists.

Motivation & Objective

• To identify conditions under which transversal Z-rotations preserve the code space of a stabilizer code.
• To develop a passive error-mitigation strategy for coherent noise without active error correction.
• To characterize stabilizer codes that are invariant under all transversal Z-rotations, ensuring logical identity action.
• To establish a structural characterization of codes oblivious to coherent noise using classical coding theory.
• To construct a large family of quantum error-correcting codes (QECCs) that are inherently resilient to coherent Z-rotations.

Proposed method

• Derives necessary and sufficient conditions for transversal Z-rotations to preserve the code space using trigonometric constraints and MacWilliams identities.
• Requires that weight-2 Z-stabilizers cover all qubits in the support of any X-stabilizer component.
• Constructs product codes from even-length single-parity-check codes to achieve coherence resilience.
• Uses the generator coefficient framework to describe codes preserved under Z-rotations of angle π/2^l for finite l.
• Applies classical coding theory tools, particularly MacWilliams identities, to analyze weight enumerators and trigonometric sums.
• Demonstrates that a CSS code is oblivious to coherent noise if and only if it is a constant excitation code.

Experimental results

Research questions

• RQ1 What conditions ensure that transversal Z-rotations act trivially on the logical state of a stabilizer code?
• RQ2 How can we construct stabilizer codes that are inherently robust to coherent noise from transversal Z-rotations?
• RQ3 What structural properties must a stabilizer code possess to be invariant under all transversal Z-rotations?
• RQ4 Is there a characterization of stabilizer codes that are oblivious to coherent noise in terms of classical coding structures?
• RQ5 Can we relax the requirement of invariance under all Z-rotations while still achieving effective noise mitigation?

Key findings

• A stabilizer code is oblivious to coherent noise if and only if it is a constant excitation code, where logical states are superpositions of computational basis states with equal weight.
• The necessary and sufficient condition for invariance under transversal Z-rotations is that the weight-2 Z-stabilizers generate a direct product of even-length single-parity-check codes.
• The construction yields a family of codes, including the [[4L²,1,2L]] Shor codes, that are robust to coherent Z-rotations.
• For error-detecting codes, the weights in different cosets of the X-stabilizers are identical, a key structural constraint.
• The minimal resource cost to achieve coherence resilience is doubling the number of physical qubits (scaling by 2), as shown by the product code construction.
• The paper proves that if a code is invariant under all transversal Z-rotations of angle π/2^l for l ≥ 2, then the logical action is trivial, and this property extends to smaller l via composition.