It is a major challenge to perform addressable and parallel 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 fault-tolerant protocols for performing various addressable as well as parallel logical operations with constant space-time overhead, on a family of constant-rate and polynomial-distance qLDPC codes. Specifically, we construct gadgets for a large class of permutations of logical qubits. We apply these logical permutations to construct gadgets for applying a targeted Hadamard (or $CNOT$) gate on any chosen logical qubit (pair). We also construct gadgets for preparing logical code states, and for applying Hadamard gates on all logical qubits in a codeblock. All of our gadgets use constant quantum space-time overhead along with polynomially bounded classical computation. Prior protocols for such operations required larger overhead, or else relied on codes with certain symmetries that lack known asymptotic constructions. Our codes are given by tensor products of classical codes constructed from lossless expander graphs. Our core technical contribution is a constant-overhead code-switching procedure between 2- and 3-dimensional product codes, which generalizes Bombin's dimensional jump (arXiv:1412.5079). We prove that all of our gadgets exhibit 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. We also propose a method for performing parallel non-Clifford gates by extending our techniques to codes supporting transversal application of such gates.

翻译：在恒定速率的量子低密度奇偶校验（qLDPC）码上执行可寻址且并行的逻辑操作是一个重大挑战。事实上，针对特定逻辑量子位的开销是许多量子容错方案中的关键瓶颈。本文针对一族恒定速率且具有多项式距离的qLDPC码，提出了以恒定时空开销执行各类可寻址及并行逻辑操作的容错协议。具体而言，我们为逻辑量子位的大类置换结构构建了功能组件。通过应用这些逻辑置换，我们构建了可在任意选定逻辑量子位（对）上实施目标哈达玛门（或$CNOT$门）的功能组件。此外，我们还构建了用于制备逻辑码态以及在码块中所有逻辑量子位上实施哈达玛门的功能组件。所有组件均采用恒定量子时空开销及多项式有界经典计算。先前实现此类操作的协议需要更大开销，或依赖具有特定对称性但缺乏已知渐近构造的编码。本文所采用的编码基于由无损扩展图构造的经典码的张量积。核心技术创新是二维与三维乘积码间的恒定开销码切换协议，该协议推广了Bombin的维度跃迁方法（arXiv:1412.5079）。我们证明所有功能组件在局部随机噪声下均具有恒定阈值。在此过程中，我们针对基于无损扩展图的高维乘积码开发了小集合翻转解码器。本技术还产生若干重要推论，例如以恒定时空开销实现二维乘积码的单次态制备。通过将本技术扩展至支持此类门横向应用的编码，我们还提出了执行并行非克利福德门的方法。