Quantum Codes with Addressable and Transversal Non-Clifford Gates

TL;DR

This paper introduces quantum codes supporting addressable and transversal non-Clifford gates, enabling efficient logical gate implementation on chosen qubits, with explicit constructions and a new formal framework extending previous theories.

Contribution

It presents the first explicit construction of asymptotically good quantum codes supporting addressable triple-qubit CCZ gates and develops a formalism for designing such codes.

Findings

Constructed a code allowing any triple of logical qubits to be addressed with a depth-one CCZ circuit.

Developed the addressable orthogonality framework extending triorthogonality.

Demonstrated codes supporting addressable and transversal Z, CZ, CCZ, and T gates.

Abstract

The development of quantum codes with good error correction parameters and useful sets of transversal gates is a problem of major interest in quantum error-correction. Abundant prior works have studied transversal gates which are restricted to acting on all logical qubits simultaneously. In this work, we study codes that support transversal gates which induce $\textit{addressable}$ logical gates, i.e., the logical gates act on logical qubits of our choice. As we consider scaling to high-rate codes, the study and design of low-overhead, addressable logical operations presents an important problem for both theoretical and practical purposes. Our primary result is the construction of an explicit qubit code for which $\textit{any}$ triple of logical qubits across one, two, or three codeblocks can be addressed with a logical $\mathsf{CCZ}$ gate via a depth-one circuit of physical $\mathsf{CCZ}$ gates, and whose parameters are asymptotically good, up to polylogarithmic factors. The result naturally generalizes to other gates including the $\mathsf{C}^{\ell} Z$ gates for $\ell \neq 2$. Going beyond this, we develop a formalism for constructing quantum codes with $\textit{addressable and transversal}$ gates. Our framework, called $\textit{addressable orthogonality}$, encompasses the original triorthogonality framework of Bravyi and Haah (Phys. Rev. A 2012), and extends this and other frameworks to study addressable gates. We demonstrate the power of this framework with the construction of an asymptotically good qubit code for which $\textit{pre-designed}$, pairwise disjoint triples of logical qubits within a single codeblock may be addressed with a logical $\mathsf{CCZ}$ gate via a physical depth-one circuit of $\mathsf{Z}$, $\mathsf{CZ}$ and $\mathsf{CCZ}$ gates. In an appendix, we show that our framework extends to addressable and transversal $T$ gates, up to Clifford corrections.