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Numerous quantum algorithms require the use of quantum error correction toovercome the intrinsic unreliability of physical qubits. However, errorcorrection imposes a unique performance bottleneck, known as T-complexity, thatcan make an implementation of an algorithm as a quantum program run more slowlythan on idealized hardware. In this work, we identify that programmingabstractions for control flow, such as the quantum if-statement, can introducepolynomial increases in the T-complexity of a program. If not mitigated, thisslowdown 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 quantumerror correction and pinpoint the sources of slowdown. We also present a set ofprogram-level optimizations, using which a developer can rewrite a program toreduce its T-complexity, predict the T-complexity of the optimized programusing the cost model, and then compile it to an efficient circuit via astraightforward strategy.  We implement the program-level optimizations in Spire, an extension of theTower quantum compiler. Using a set of 11 benchmark programs that use controlflow, we show that the cost model is accurate, and that Spire's optimizationsrecover programs that are asymptotically efficient, meaning their runtimeT-complexity under error correction is equal to their time complexity onidealized hardware.  Our results show that optimizing a program before it is compiled to a circuitcan yield better results than compiling the program to an inefficient circuitand then invoking a quantum circuit optimizer found in prior work. For ourbenchmarks, only 2 of 8 existing circuit optimizers recover circuits withasymptotically efficient T-complexity. Compared to these 2 optimizers, Spireuses 54x to 2400x less compile time.

The T-Complexity Costs of Error Correction for Control Flow in Quantum Computation