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A quantum compiler design method by using linear combinations of permutations

Ammar Daskin

TL;DR

The paper addresses how to translate a general $A$ into a quantum circuit by first embedding $A$ into a block-encoded unitary and then transforming it into a doubly stochastic matrix $S$, which can be written as a linear combination of permutation matrices via the Birkhoff–von Neumann theorem. It then details how to map permutation matrices to quantum circuits, decompose permutations into transpositions, and construct a circuit for $S$ using an ancilla and controlled permutation blocks. The authors discuss practical compiler-oriented optimizations, including simplifications, term truncation, and integration into quantum compiler design, as well as extensions to Hamiltonian simulation. This framework provides a generic route to exact or approximate circuit synthesis for arbitrary matrices and lays groundwork for permutation-based abstractions in future quantum programming languages and compilers.

Abstract

A matrix can be converted into a doubly stochastic matrix by using two diagonal matrices. And a doubly stochastic matrix can be written as a sum of permutation matrices. In this paper, we describe a method to write a given generic matrix in terms of quantum gates based on the block encoding. In particular, we first show how to convert a matrix into doubly stochastic matrices and by using Birkhoff's algorithm, we express that matrix in terms of a linear combination of permutations which can be mapped to quantum circuits. We then discuss a few optimization techniques that can be applied in a possibly future quantum compiler software based on the method described here.

A quantum compiler design method by using linear combinations of permutations

TL;DR

The paper addresses how to translate a general into a quantum circuit by first embedding into a block-encoded unitary and then transforming it into a doubly stochastic matrix , which can be written as a linear combination of permutation matrices via the Birkhoff–von Neumann theorem. It then details how to map permutation matrices to quantum circuits, decompose permutations into transpositions, and construct a circuit for using an ancilla and controlled permutation blocks. The authors discuss practical compiler-oriented optimizations, including simplifications, term truncation, and integration into quantum compiler design, as well as extensions to Hamiltonian simulation. This framework provides a generic route to exact or approximate circuit synthesis for arbitrary matrices and lays groundwork for permutation-based abstractions in future quantum programming languages and compilers.

Abstract

A matrix can be converted into a doubly stochastic matrix by using two diagonal matrices. And a doubly stochastic matrix can be written as a sum of permutation matrices. In this paper, we describe a method to write a given generic matrix in terms of quantum gates based on the block encoding. In particular, we first show how to convert a matrix into doubly stochastic matrices and by using Birkhoff's algorithm, we express that matrix in terms of a linear combination of permutations which can be mapped to quantum circuits. We then discuss a few optimization techniques that can be applied in a possibly future quantum compiler software based on the method described here.
Paper Structure (16 sections, 3 theorems, 17 equations, 1 algorithm)

This paper contains 16 sections, 3 theorems, 17 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Doubly Stochastic(bistochastic) Matrices
  3. Difficulty of Finding This Decomposition
  4. Birkhoff's Algorithm
  5. Quantum Circuit Implementations
  6. Quantum Circuit of a Permutation Matrix
  7. Composition and Commutation
  8. Decomposition into Transpositions
  9. Circuit Implementation of a Transposition
  10. Circuit Implementation of a Doubly Stochastic Matrix
  11. Discussion and Future Directions
  12. Simplifications of Permutations and their Combinations
  13. Approximation by Cutting off the Terms in the Summation
  14. Quantum Compiler Design
  15. Hamiltonian Simulation
  16. ...and 1 more sections

Key Result

Theorem 1

If the Hamming distance between $a_i$ and $a_j$ is one, then a transposition $(a_ia_j)$ can be implemented by using a multi-controlled $X$ gate whose target is the qubit where $a_i$ and $a_j$ are different and control qubits are all the other qubits.

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Lemma 1
  • Theorem 2
  • proof