Numerous quantum algorithms require the use of quantum error correction to overcome the intrinsic unreliability of physical qubits. However, error correction imposes a unique performance bottleneck, known as T-complexity, that can make an implementation of an algorithm as a quantum program run more slowly than on idealized hardware. In this work, we identify that programming abstractions for control flow, such as the quantum if-statement, can introduce polynomial increases in the T-complexity of a program. If not mitigated, this slowdown can diminish the computational advantage of a quantum algorithm. To enable reasoning about the costs of control flow, we present a cost model, using which a developer can analyze the T-complexity of a program under quantum error correction and pinpoint the sources of slowdown. We also present a set of program-level optimizations, using which a developer can rewrite a program to reduce its T-complexity, predict the T-complexity of the optimized program using the cost model, and then compile it to an efficient circuit via a straightforward strategy. We implement the program-level optimizations in Spire, an extension of the Tower quantum compiler. Using a set of 11 benchmark programs that use control flow, we show that the cost model is accurate, and that Spire's optimizations recover programs that are asymptotically efficient, meaning their runtime T-complexity under error correction is equal to their time complexity on idealized hardware. Our results show that optimizing a program before it is compiled to a circuit can yield better results than compiling the program to an inefficient circuit and then invoking a quantum circuit optimizer found in prior work. For our benchmarks, only 2 of 8 existing circuit optimizers recover circuits with asymptotically efficient T-complexity. Compared to these 2 optimizers, Spire uses 54x to 2400x less compile time.

翻译：众多量子算法需借助量子纠错来克服物理量子比特的固有不可靠性。然而，纠错会引入独特的性能瓶颈（即T复杂度），使得量子程序实现算法的运行速度远低于理想硬件的表现。本研究发现：诸如量子if语句等控制流编程抽象，可能使程序的T复杂度呈多项式级增长。若不加缓解，这种减速会削弱量子算法的计算优势。为量化控制流的代价，我们提出一种成本模型，开发者可通过该模型分析量子纠错下程序的T复杂度，并精准定位减速根源。我们还提出一组程序级优化策略：开发者可据此重写程序以降低T复杂度，利用成本模型预测优化后的T复杂度，最终通过简单策略编译为高效量子电路。我们基于Tower量子编译器的扩展系统Spire实现了这些优化。通过对11个含控制流的基准程序测试，证实成本模型准确，且Spire的优化能恢复渐近高效的程序——即其纠错下的运行时T复杂度等于理想硬件上的时间复杂度。实验表明：将程序编译为电路前进行优化，效果优于先编译为低效电路再调用经典电路优化器的方式。在已有8种电路优化器中，仅2种能恢复渐近高效T复杂度的电路；而相较于这两种优化器，Spire的编译时间减少54至2400倍。