MITIGATING COHERENT NOISE

Stochastic errors in quantum systems occur randomly but coherent errors may be more damaging since they can accumulate in a particular direction. We develop a framework for designing decoherence free subspaces (DFS), that are unperturbed by coherent noise. We consider a particular form of coherent $Z$-errors and construct stabilizer codes that form DFS for such noise (``Z-DFS''). More precisely, we develop conditions for transversal $\exp(\imath \theta \sigma_Z)$ to preserve a stabilizer code subspace for all $\theta$. 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 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 constant rate family of CSS Z-DFS codes. Invariance under transversal $\exp(\frac{\imath \pi}{2^l} \sigma_Z)$ translates to a trigonometric equation satisfied by $\tan\frac{2\pi}{2^l}$, and for every non-zero $X$-component of a stabilizer, there is a trigonometric equation that must be satisfied. The $Z$-stabilizers supported on this non-zero $X$-component form a classical binary code C, and the trigonometric constraint connects signs of $Z$-stabilizers to divisibility of weights in $C^{\perp}$. This construction may be of independent interest to classical coding theorists who have long been interested in codes $C$ with the property that all weights are divisible by some integer $d$. If we require that transversal $\exp(\frac{\imath \pi}{2^l} \sigma_Z)$ preserves 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 aforesaid code $C$ contains a self-dual code and the classical Gleason's theorem constrains its weight enumerator.

The trigonometric conditions corresponding to higher values of $l$ lead to generalizations of Gleason's theorem that may be of independent interest to classical coding theorists. The $[[16, 4, 2]]$ Reed-Muller code and the family of $[[4L^2, 1, 2L]]$ Shor codes are included in our general framework.