Short Plenary 7: Code Switching Protocols

Abstract: It is a major challenge to perform addressable logical operations on constant-rate quantum LDPC (qLDPC) codes. Indeed, the overhead of targeting specific logical qubits represents a crucial bottleneck in many quantum fault-tolerance schemes. We introduce a protocol for performing fully addressable logical $CNOT$ or Hadamard gates with constant quantum space-time overhead, on a family of constant-rate and polynomial-distance qLDPC codes. Specifically, our gadgets perform Hadamard on any chosen logical qubit within a code block, and $CNOT$ between any pair of logical qubits, either within a block or across blocks. We also construct constant-overhead gadgets for highly parallel logical operations, including a large class of permutations of logical qubits. Prior protocols for such operations required polynomial space-time overhead with respect to the distance, or else relied on codes with certain symmetries that lack known asymptotic constructions. Our codes are given by tensor (i.e. hypergraph) products of classical codes constructed from lossless expander graphs. To address individual logical qubits, we develop a constant-overhead code-switching procedure between 2- and 3-dimensional product codes, which generalizes Bombin’s dimensional jump (arXiv:1412.5079). We provide rigorous fault-tolerance proofs for our gadgets, and specifically prove a constant threshold under locally stochastic noise. Along the way, we develop a small-set flip decoder for high-dimensional product codes from lossless expanders. Our techniques yield additional interesting consequences, such as single-shot state preparation of 2-dimensional product codes with constant space-time overhead.

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 which 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 that is characterized by (i) single-shot error correction, (ii) single-shot state preparation, as well as (iii) logical gates and logical measurements with constant depth circuits. We achieve this by showing how to perform single-shot code switching between high-rate homological product codes by developing a generalization of Bombin's dimensional jump for color codes and Hillmann et al.'s single-shot lattice surgery for higher-dimensional topological codes. We introduce a vastly simpler recipe to construct 3D homological product 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.