Mitigating Coherent Noise by Balancing Weight-2 Z-Stabilizers
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Stochastic errors on quantum systems occur randomly but coherent errors are more damaging since they can accumulate in a particular direction. While active error correction can address coherent noise, a passive approach is to design decoherence free subspaces (DFS) that remain unperturbed by this noise. This paper considers a form of coherent Z-errors and constructs stabilizer codes that form DFS for such noise ("Z-DFS"). We develop conditions for transversal (θ Z) to preserve a stabilizer code subspace for all θ. If the code is error-detecting, then this implies a trivial action on the logical qubits. These conditions require the existence of a large number of weight-2 Z-stabilizers, and together, these generate a direct product of single-parity-check codes. By adjusting the size of these components, we construct a linear rate family of CSS Z-DFS codes. Invariance under transversal (π/2^l Z) translates to a trigonometric equation satisfied by tan2π/2^l, and there is an equation for each non-zero X-component of stabilizers. The Z-stabilizers on the support of a stabilizer's X-component form a code C, and the trigonometric constraint connects signs of the Z-stabilizers to divisibility of weights in C^⊥. This connection to divisibility might be of independent interest to classical coding theorists. Next, to induce a non-trivial logical operation, we impose that transversal (π/2^l Z) preserve the code space only up to a finite level l in the Clifford hierarchy. The aforesaid code C contains a self-dual code and the classical Gleason's theorem constrains its weight enumerator. Surprisingly, the finite l constraint makes the trigonometric equation generalize Gleason's theorem. Several examples, e.g. the [[4L^2,1,2L]] Shor codes, are described to illuminate our general results.
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