MITIGATING COHERENT NOISE

Loading...
Thumbnail Image

Date

2023

Journal Title

Journal ISSN

Volume Title

Repository Usage Stats

20
views
30
downloads

Abstract

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.

Department

Description

Provenance

Citation

Citation

Liang, Qingzhong (2023). MITIGATING COHERENT NOISE. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/27638.

Collections


Dukes student scholarship is made available to the public using a Creative Commons Attribution / Non-commercial / No derivative (CC-BY-NC-ND) license.