Quantum Codes with Addressable and Transversal Non-Clifford Gates

2025-02-03

AI Summary

Implementing addressable, transversal non-Clifford gates—such as CCZ and T—on quantum error-correcting codes remains a key challenge for low-overhead fault-tolerant quantum computation; existing schemes only support synchronous, all-qubit operations without logical qubit selectivity. Method: We introduce the framework of “addressable orthogonality”, unifying and generalizing classical constructions (e.g., triorthogonality) to jointly achieve transversality and logical addressability. Leveraging high-dimensional binary orthogonality, algebraic coding theory, and Clifford-based correction techniques, we construct the first asymptotically good qubit code. Results: Our code enables depth-one transversal CCZ gates on arbitrary subsets of logical qubits—within single or multiple code blocks—and extends to pre-specified disjoint triples and T gates. This significantly reduces the physical resource overhead for fault-tolerant computation and opens a new pathway toward scalable, fault-tolerant quantum computing.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Data transparency and provenance
• Responsible Web: Algorithmic accountability and transparency on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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 $ extit{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 $ extit{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 $ell eq 2$. Going beyond this, we develop a formalism for constructing quantum codes with $ extit{addressable and transversal}$ gates. Our framework, called $ extit{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 $ extit{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.

Problem

• Develop quantum codes with addressable transversal non-Clifford gates
• Construct asymptotically good codes for targeted logical qubit operations
• Extend orthogonality frameworks to enable addressable gate design

Innovation

• Addressable logical gates for chosen qubits
• Depth-one physical CCZ gates circuit
• Addressable orthogonality framework extension