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Branch Sequentialization in Quantum Polytime

Emmanuel Hainry, Romain Péchoux, Mário Alberto Machado da Silva

TL;DR

This work addresses branch sequentialization in quantum programming, where automated compilation of quantum conditionals can blow up circuit size. It introduces a restricted language, pbp (Polynomially Branching Programs), with quantum case and first-order recursion, and a compilation method (compile) that uses anchoring and merging to avoid exponential growth. The main results show that for pbp programs, the generated circuit size is bounded by the program's time complexity, with depth $O( ext{Time_P}(n))$ scaled by a factor of $O( abla(n))$ leading to $O( ext{Time_P}(n))$-type behavior and, in the standard setting, depth $O( ext{Time_P}(n) imes ext{log} n)$ and size $O(n imes ext{Time_P}(n))$. Moreover, pbp is sound and complete for the class fbqp, establishing polytime quantum computability within this framework. The approach improves upon prior compilation strategies by achieving polynomial-size circuits for quantum control and enabling practical encoding of polytime quantum algorithms like QFT within polynomial resource bounds.

Abstract

Quantum algorithms leverage the use of quantumly-controlled data in order to achieve computational advantage. This implies that the programs use constructs depending on quantum data and not just classical data such as measurement outcomes. Current compilation strategies for quantum control flow involve compiling the branches of a quantum conditional, either in-depth or in-width, which in general leads to circuits of exponential size. This problem is coined as the branch sequentialization problem. We introduce and study a compilation technique for avoiding branch sequentialization on a language that is sound and complete for quantum polynomial time, thus, improving on existing polynomial-size-preserving compilation techniques.

Branch Sequentialization in Quantum Polytime

TL;DR

This work addresses branch sequentialization in quantum programming, where automated compilation of quantum conditionals can blow up circuit size. It introduces a restricted language, pbp (Polynomially Branching Programs), with quantum case and first-order recursion, and a compilation method (compile) that uses anchoring and merging to avoid exponential growth. The main results show that for pbp programs, the generated circuit size is bounded by the program's time complexity, with depth scaled by a factor of leading to -type behavior and, in the standard setting, depth and size . Moreover, pbp is sound and complete for the class fbqp, establishing polytime quantum computability within this framework. The approach improves upon prior compilation strategies by achieving polynomial-size circuits for quantum control and enabling practical encoding of polytime quantum algorithms like QFT within polynomial resource bounds.

Abstract

Quantum algorithms leverage the use of quantumly-controlled data in order to achieve computational advantage. This implies that the programs use constructs depending on quantum data and not just classical data such as measurement outcomes. Current compilation strategies for quantum control flow involve compiling the branches of a quantum conditional, either in-depth or in-width, which in general leads to circuits of exponential size. This problem is coined as the branch sequentialization problem. We introduce and study a compilation technique for avoiding branch sequentialization on a language that is sound and complete for quantum polynomial time, thus, improving on existing polynomial-size-preserving compilation techniques.

Paper Structure

This paper contains 25 sections, 10 theorems, 16 equations, 13 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Contribution
  4. A bird's-eye view of our compilation strategy
  5. Related work
  6. First-Order Programming Language with Quantum Case
  7. Syntax of Programs
  8. Semantics of Programs
  9. Polynomial Branching Programs
  10. Compilation Strategy
  11. Anchoring and Merging
  12. The compile Algorithm
  13. Soundness of the algorithm
  14. Illustrating Example
  15. Main Results
  16. ...and 10 more sections

Key Result

Lemma 1

During the compilation of a pbp program $\mathrm{P}$, for each optimization of a (recursive) procedure $\text{\color{procedures}proc}$, all controlled statements in the union of list $l$ and the contextual list $l_\text{M}$ are pairwise orthogonal.

Figures (13)

  • Figure 1: Program $\texttt{PAIRS}$.
  • Figure 2: Compilation strategies.
  • Figure 3: Syntax of programs.
  • Figure 4: Semantics of statements.
  • Figure 5: Program $\texttt{QFT}$ for the quantum Fourier transform.
  • ...and 8 more figures

Theorems & Definitions (24)

  • Definition 1
  • Example 1
  • Definition 2
  • Definition 3: Polynomially Branching Programs
  • Example 2
  • Lemma 1
  • Theorem 1: Soundness of compilation
  • Theorem 2: No branch sequentialization
  • Example 3
  • Theorem 3: Soundness and Completeness
  • ...and 14 more