Divisible Codes for Quantum Computation

TL;DR

This paper explores how divisible codes can be used to design quantum error-correcting codes that support transversal logical gates, providing new constructions and methods to circumvent the Eastin-Knill Theorem.

Contribution

It introduces conditions for transversal gates in CSS codes, connects divisible codes to quantum code design, and proposes layered stabilizer codes for universal fault-tolerant quantum computation.

Findings

Derived necessary and sufficient conditions for transversal gates in CSS codes.

Constructed new CSS codes using cosets of Reed Muller codes.

Developed a layered code architecture to bypass the Eastin-Knill Theorem.

Abstract

Divisible codes are defined by the property that codeword weights share a common divisor greater than one. They are used to design signals for communications and sensing, and this paper explores how they can be used to protect quantum information as it is transformed by logical gates. Given a CSS code $\mathcal{C}$, we derive conditions that are both necessary and sufficient for a transversal diagonal physical operator $U_Z$ to preserve $\mathcal{C}$ and induce $U_L$. The group of $Z$-stabilizers in a CSS code $\mathcal{C}$ is determined by the dual of a classical $[n, k_1]$ binary code $\mathcal{C}_1$, and the group of $X$-stabilizers is determined by a classical $[n, k_2]$ binary code $\mathcal{C}_2$ that is contained in $\mathcal{C}_1$. The requirement that a diagonal physical operator $U_Z$ fixes a CSS code $\mathcal{C}$ leads to constraints on the congruence of weights in cosets of $\mathcal{C}_2$. These constraints are a perfect fit to divisible codes, and represent an opportunity to take advantage of the extensive literature on classical codes with two or three weights. We construct new families of CSS codes using cosets of the first order Reed Muller code defined by quadratic forms. We provide a simple alternative to the standard method of deriving the coset weight distributions (based on Dickson normal form) that may be of independent interest. Finally, we develop an approach to circumventing the Eastin-Knill Theorem which states that no QECC can implement a universal set of logical gates through transversal gates alone. The essential idea is to design stabilizer codes in layers, with $N_1$ inner qubits and $N_2$ outer qubits, and to assemble a universal set of fault tolerant gates on the inner qubits.