Error Correction 6

Abstract: A key challenge in fault-tolerant quantum computing is synthesising and optimising circuits in a noisy environment, as traditional techniques often fail to account for the effect of noise on circuits. In this work, we propose a framework for designing fault-tolerant quantum circuits that are correct by construction. The framework starts with idealised specifications of fault-tolerant gadgets and refines them using provably sound basic transformations. To reason about manipulating circuits while preserving their error correction properties, we define fault equivalence; two circuits are considered fault-equivalent if all undetectable faults on one circuit have a corresponding fault on the other. This guarantees that the effect of undetectable faults on both circuits is the same. We argue that fault equivalence is a concept that is already implicitly present in the literature. Many problems, such as state preparation and syndrome extraction, can be naturally expressed as finding an implementable circuit that is fault-equivalent to an idealised specification. To utilise fault equivalence in a computationally tractable manner, we adapt the ZX calculus, a diagrammatic language for quantum computing. We restrict its rewrite system to not only preserve the underlying linear map but also fault equivalence, i.e. the circuit's behaviour under noise. We show that this rewriting system is complete, meaning that two circuits are fault-equivalent if and only if one can be transformed into the other using fault-equivalent ZX rewrites. Enabled by our framework, we verify, optimise and synthesise new and efficient circuits for syndrome extraction and cat state preparation. We anticipate that fault equivalence can capture and unify different approaches in fault-tolerant quantum computing, paving the way for an end-to-end circuit compilation framework.

Abstract: We show how to perform scalable fault-tolerant non-Clifford gates in two dimensions by introducing domain walls between the surface code and a non-Abelian topological code whose codespace is stabilized by Clifford operators. We formulate a path integral framework which provides both a macroscopic picture for different logical gates as well as a way to derive the associated microscopic circuits. We present explicit protocols and planar non-Clifford circuits that implement non-Clifford logic gates on both surface codes as well as color codes on different geometries. The logical action of the protocol is determined by the spacetime geometry, using the same bulk circuit, composed of simple 2D local circuits of similar complexity to commonly used stabilizer-readout circuits. We present fault-tolerant schemes for logical Clifford measurements as well as diagonal unitary gates in the third level of the Clifford hierarchy such as T, CS and CCZ gate. We also show an equivalence between our approach and prior proposals where a 2D array of qubits reproduces the action of a transversal gate in a 3D stabilizer code over time, thus, establishing a new connection between 3D codes and 2D non-Abelian topological phases. We prove a threshold theorem for our protocols under local stochastic circuit noise using a just-in-time decoder to correct the non-Abelian code.

Abstract: We introduce a framework called spacetime concatenation for fault-tolerant compilation of syn- drome extraction circuits of stabilizer codes. This framework enables efficient compilation of syn-drome extraction circuits into dynamical codes through structured gadget layouts and encoding matrices, facilitating low-weight measurements while preserving logical information. This frame-work uses conditions that are sufficient for fault-tolerance of the dynamical code, including not measuring logical operators and preserving the spacetime distance. This framework goes beyond dynamical weight-reduction methods, which do not explicitly produce Floquet codes with nontrivial logical automorphisms. Using this fault-tolerant framework, we reproduce known Floquet codes such as the hon-eycomb Floquet code and the ruby lattice Floquet color code with nontrivial automorphisms and a fault-tolerant planar variant of the Floquet toric code. We also construct new explicit examples of dynamical codes, including a dynamical bivariate bicycle code and a dynamical Haah code. Be-yond constructing examples, we write a restricted equivalence relation which can be used to discuss classification and resource trade-offs of dynamical codes. Lastly, we demonstrate the adaptability of our framework to arbitrary qubit layouts and fabrication defects, making it well-suited for a hardware-first design approach.