Single-Shot Universality in Quantum LDPC Codes via Code-Switching

TL;DR

The paper presents a framework for universal fault-tolerant quantum computation with low space-time overhead by using single-shot code-switching between high-rate 2D and 3D hypergraph product (HGP) codes built from Sipser–Spielman expander codes. It introduces a suite of constant-depth gadgets, including a 1-limited homomorphic CNOT between 2D and 3D HGP codes, and a dimensional-expansion/contraction protocol that teleports logical information acrossCode-switching between 2D and 3D codes enables transversal CCZ gates in the 3D block and Hadamard in the 2D block, together with single-shot state preparation and error correction. The authors prove fault tolerance under both adversarial and local-stochastic noise models and provide a construction that yields constant spatial overhead while maintaining universal computational power, thereby offering a path toward scalable fault-tolerant quantum computation without magic-state distillation. The work broadens the design space for high-rate LDPC codes with transversal non-Clifford gates and highlights practical avenues for reducing overhead in quantum architectures using code-switching and single-shot techniques.

Abstract

Code-switching is a powerful technique in quantum error correction that allows one to leverage the complementary strengths of different codes to achieve fault-tolerant universal quantum computation. However, existing code-switching protocols that encapsulate recent generalized lattice surgery approaches often either require many rounds of measurements to ensure fault-tolerance or suffer from low code rates. We present a single-shot, universal protocol that uses code-switching between high-rate quantum codes to perform fault-tolerant quantum computation. To our best knowledge, our work contains the first universal fault-tolerant quantum computation protocol that achieves what we term single-shot universality on high-rate codes that is characterized by (i) single-shot error correction, (ii) single-shot state preparation, as well as (iii) universal logical gates and logical measurements with constant depth circuits. We achieve this feat with single-shot code-switching between constant-rate 2D hypergraph product (HGP) codes and high-rate 3D HGP codes that can be viewed as a generalization of Bombin's dimensional jump for color codes and Hillmann et al.'s single-shot lattice surgery for higher-dimensional topological codes. In addition, we prove the fault-tolerance of our code-switching protocol under both the adversarial and local-stochastic noise models. We introduce a vastly simpler recipe to construct high-rate 3D HGP codes with transversal CCZ gates that grants immense flexibility in the choice of expander graphs and local codes, allowing us to expand the search space for codes with good parameters and interesting logical gates. Our work opens an alternative path towards universal fault-tolerant quantum computation with low space-time overhead by circumventing the need for magic state distillation.

Figures (7)

• Figure 1: Overview of the single-shot universal protocol for fault-tolerant quantum computation using 2D and 3D HGP codes. The scheme leverages single-shot error correction, state preparation, and code-switching to achieve addressable Clifford and non-Clifford operations with low spatial overhead and time complexity.
• Figure 2: Comparison of three repetition code structures: (a) standard repetition code $\mathcal{C}_S$, (b) circular repetition local code $\mathcal{C}_0$, and (c) generalized repetition code $\mathcal{C}_G$.
• Figure 3: Logical quantum circuit for performing logical teleportation between two code blocks $\mathcal{Q}$ and $\mathcal{Q}_G$ using Pauli-based measurements and gates.
• Figure 4: Logical quantum circuit for performing logical teleportation from a code block $\mathcal{Q}$ to another code block $\mathcal{Q}_G$ using an adapter ancilla code block $\mathcal{Q}$. The sequence of gates including the two CNOTs as well as the $Z$ measurement is effectively the $ZZ$ measurement. The first CNOT (colored in green) is the homomorphic CNOT described in Section \ref{['sec:homomorphic_cnot']}. The last CNOT is the standard transversal logical CNOT between two 2D HGP codes.
• Figure 5: Logical quantum circuit for performing logical teleportation from a code block $\mathcal{Q}_G$ to another code block $\mathcal{Q}$ using an adapter ancilla code block $\mathcal{Q}$. The sequence of gates including the two CNOTs as well as the $Z$ measurement is effectively the $ZZ$ measurement. The first CNOT (colored in green) is the homomorphic CNOT described in Section \ref{['sec:homomorphic_cnot']}. The last CNOT is the standard transversal logical CNOT between two 2D HGP codes.

Theorems & Definitions (75)

• theorem 1.1: Informal Statement of Theorem \ref{['thm:universal_FTQC']}
• theorem 1.2: Informal Statement of Theorem \ref{['thm:SSCS']}
• theorem 1.3: Informal Statement of Theorem \ref{['thm:satisfaction_of_local_multiplication_property']}
• remark 1.4
• definition 2.1: Spectral expander
• definition 2.1: Spectral expander
• definition 2.2: Classical Linear Code
• definition 2.3
• definition 2.4: Quantum CSS Codes
• definition 2.5: Chain Complexes