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

Charles Yuan, Michael Carbin

TL;DR

This work examines how quantum error correction imposes a T-gate bottleneck that is exacerbated by control-flow abstractions like quantum if-statements. It introduces a syntax-level cost model to predict T-complexity, and two program-level optimizations—conditional flattening and conditional narrowing—to reduce this overhead. Implemented in Spire, an extension of the Tower compiler, the approach is validated on 11 benchmark programs, demonstrating asymptotically efficient T-complexity after optimization and substantial compile-time gains over circuit-optimizer-only workflows. The findings emphasize the importance of high-level language design and compiler strategies that account for error-correction costs to preserve quantum advantages in real hardware.

Abstract

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 that a developer can use to accurately analyze the T-complexity of a program and pinpoint the sources of slowdown. We also present a set of program-level optimizations, that a developer can use to 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 tested circuit optimizers recover circuits with asymptotically efficient T-complexity. Compared to these 2 optimizers, Spire uses 54x to 2400x less compile time.

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

TL;DR

This work examines how quantum error correction imposes a T-gate bottleneck that is exacerbated by control-flow abstractions like quantum if-statements. It introduces a syntax-level cost model to predict T-complexity, and two program-level optimizations—conditional flattening and conditional narrowing—to reduce this overhead. Implemented in Spire, an extension of the Tower compiler, the approach is validated on 11 benchmark programs, demonstrating asymptotically efficient T-complexity after optimization and substantial compile-time gains over circuit-optimizer-only workflows. The findings emphasize the importance of high-level language design and compiler strategies that account for error-correction costs to preserve quantum advantages in real hardware.

Abstract

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 that a developer can use to accurately analyze the T-complexity of a program and pinpoint the sources of slowdown. We also present a set of program-level optimizations, that a developer can use to 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 tested circuit optimizers recover circuits with asymptotically efficient T-complexity. Compared to these 2 optimizers, Spire uses 54x to 2400x less compile time.
Paper Structure (94 sections, 6 theorems, 9 equations, 28 figures, 2 tables)

This paper contains 94 sections, 6 theorems, 9 equations, 28 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Error Correction
  3. Resource Estimation
  4. T-Complexity Bottleneck
  5. T-Complexity Costs of Control Flow in Quantum Programs
  6. Control Flow
  7. Costs of Control Flow
  8. Cost Model for Accurately Predicting T-Complexity Costs
  9. Program-Level Optimizations for Mitigating T-Complexity Costs
  10. Contributions
  11. Implications
  12. Background on Quantum Computation
  13. Superposition
  14. Unitary Operator
  15. Measurement
  16. ...and 79 more sections

Key Result

theorem 1

If $s$ is well-formed, i.e. ${\Gamma} \vdash {s} \dashv {\Gamma'}$, then the number of arbitrarily controllable Clifford gates in $\mathcal{C}\!\left\llbracket{s}\right\rrbracket$ is equal to $C^{\textrm{MCX}}_{}(s)$, up to choices for $c^\textrm{MCX}_s$.

Figures (28)

  • Figure 1: Program computing the length of a list.
  • Figure 2: Number of gates in the circuit of \ref{['fig:length-tower']}.
  • Figure 3: Tower program that uses nested quantum if-statements.
  • Figure 4: Translation of \ref{['fig:simple-if']} to a circuit. On each multiply-controlled-NOT (MCX) gate, each orange control bit incurs T-complexity.
  • Figure 5: Decomposing MCX to Toffoli.
  • ...and 23 more figures

Theorems & Definitions (7)

  • theorem 1: MCX-Complexity Soundness
  • theorem 2: T-Complexity Soundness
  • theorem 3
  • definition 1: Circuit Equivalence
  • theorem 4: Conditional Flattening Soundness
  • theorem 5
  • theorem 6: Conditional Narrowing Soundness