Michael Beverland

We present the bicycle architecture, a modular quantum computing framework based on high-rate, low-overhead quantum LDPC codes identified in prior work. For two specific bivariate bicycle codes with distances 12 and 18, we construct explicit fault-tolerant logical instruction sets and estimate the logical error rate of the instructions under circuit noise. We develop a compilation strategy adapted to the constraints of the bicycle architecture, enabling large-scale universal quantum circuit execution. Integrating these components, we perform end-to-end resource estimates demonstrating that an order of magnitude larger logical circuits can be implemented with a given number of physical qubits on the bicycle architecture than on surface code architectures. We anticipate further improvements through advances in code constructions, circuit designs, and compilation techniques.

Stabilizer channels are stabilizer circuits that implement logical operations while mapping from an input stabilizer code to an output stabilizer code. They are widely used to implement fault tolerant error correction and logical operations in stabilizer codes such as surface codes and LDPC codes, and more broadly in subsystem, Floquet and space-time codes. We introduce a rigorous and general formalism to analyze the fault tolerance properties of any stabilizer channel under a broad class of noise models. This includes rigorous but easy-to-work-with definitions and algorithms for the fault distance and hook faults for stabilizer channels. The generalized notion of hook faults which we introduce, defined with respect to an arbitrary subset of a circuit’s faults rather than a fixed phenomenological noise model, can be leveraged for fault-tolerant circuit design. Additionally, we establish necessary conditions such that channel composition preserves the fault distance. We apply our framework to design and analyze fault tolerant stabilizer channels for surface codes, revealing novel aspects of fault tolerant circuits.

Abstract Estimating the reducing overhead of existing fault tolerance schemes is a crucial step toward realizing scalable quantum computers. Many of the most promising schemes are based upon two-dimensional (2D) topological codes such as the surface and color codes. In these schemes, universal computation is typically achieved using readily implementable Clifford operations along with a less convenient and more costly implementation of the $T$ gate. In our work, we compare the cost of fault-tolerantly implementing the $T$-gate in 2D color codes using two leading approaches: state distillation and code switching to a 3D color code. We report that state distillation is more resource-efficient than code switching, in terms of both qubit overhead and space-time overhead. In particular, we find a $T$ gate threshold via code switching of $0.07(1)\%$ under circuit noise, almost an order of magnitude below that for distillation with 2D color codes. To arrive at this result, we provide and implement a simplified end-to-end recipe for code switching, detailing each step and providing important optimization considerations. We not only find numerical overhead estimates of this code switching protocol, but also lower bound various conceivable improvements. We also optimize the 2D color code for circuit noise yielding it's largest threshold to date $0.37(1)\%$, and adapt and optimize the restriction decoder and find a threshold of $0.80(5)\%$ for the 3D color code with perfect measurements under $Z$ noise. We foresee that this analysis will influence the choice of which FT schemes and which salable hardware designs should be pursued in future.