Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multi-strategy Based Quantum Cost Reduction of Quantum Boolean Circuits

Taghreed Ahmed, Ahmed Younes, and Islam Elkabani

TL;DR

This paper tackles the challenge of reducing quantum cost for circuits implementing Boolean functions expressed in the $PPRM$ form. It introduces two strategies: (1) generating a simple algebraic form by factorization and a degree-based rearrangement to minimize $MCT$ terms before synthesis, and (2) mapping the resulting $MCT$ circuit to the $NCV$ gate set via decomposition and simplification rules. The authors demonstrate substantial cost reductions across multiple benchmarks, including RevLib, reorder-based, GA-based, and recent datasets, and discuss practical IBM quantum hardware compatibility by addressing negative controls and $ ext{Controlled-}V^{\dagger}$ substitutions. The work provides both methodological advances and practical guidance for implementing these circuits on near-term quantum hardware, with code available in a public repository.

Abstract

The construction of quantum computers is based on the synthesis of low-cost quantum circuits. The quantum circuit of any Boolean function expressed in a Positive Polarity Reed-Muller $PPRM$ expansion can be synthesized using Multiple-Control Toffoli ($MCT$) gates. This paper proposes two algorithms to construct a quantum circuit for any Boolean function expressed in a Positive Polarity Reed-Muller $PPRM$ expansion. The Boolean function can be expressed with various algebraic forms, so there are different quantum circuits can be synthesized for the Boolean function based on its algebraic form. The proposed algorithms aim to map the $MCT$ gates into the $NCV$ gates for any quantum circuit by generating a simple algebraic form for the Boolean function. The first algorithm generates a special algebraic form for any Boolean function by rearrangement of terms of the Boolean function according to a predefined degree of term $d_{term}$, then synthesizes the corresponding quantum circuit. The second algorithm applies the decomposition methods to decompose $MCT$ circuit into its elementary gates followed by applying a set of simplification rules to simplify and optimize the synthesized quantum circuit. The proposed algorithms achieve a reduction in the quantum cost of synthesized quantum circuits when compared with relevant work in literature. The proposed algorithms synthesize quantum circuits that can applied on IBM quantum computer.

Multi-strategy Based Quantum Cost Reduction of Quantum Boolean Circuits

TL;DR

This paper tackles the challenge of reducing quantum cost for circuits implementing Boolean functions expressed in the form. It introduces two strategies: (1) generating a simple algebraic form by factorization and a degree-based rearrangement to minimize terms before synthesis, and (2) mapping the resulting circuit to the gate set via decomposition and simplification rules. The authors demonstrate substantial cost reductions across multiple benchmarks, including RevLib, reorder-based, GA-based, and recent datasets, and discuss practical IBM quantum hardware compatibility by addressing negative controls and substitutions. The work provides both methodological advances and practical guidance for implementing these circuits on near-term quantum hardware, with code available in a public repository.

Abstract

The construction of quantum computers is based on the synthesis of low-cost quantum circuits. The quantum circuit of any Boolean function expressed in a Positive Polarity Reed-Muller expansion can be synthesized using Multiple-Control Toffoli () gates. This paper proposes two algorithms to construct a quantum circuit for any Boolean function expressed in a Positive Polarity Reed-Muller expansion. The Boolean function can be expressed with various algebraic forms, so there are different quantum circuits can be synthesized for the Boolean function based on its algebraic form. The proposed algorithms aim to map the gates into the gates for any quantum circuit by generating a simple algebraic form for the Boolean function. The first algorithm generates a special algebraic form for any Boolean function by rearrangement of terms of the Boolean function according to a predefined degree of term , then synthesizes the corresponding quantum circuit. The second algorithm applies the decomposition methods to decompose circuit into its elementary gates followed by applying a set of simplification rules to simplify and optimize the synthesized quantum circuit. The proposed algorithms achieve a reduction in the quantum cost of synthesized quantum circuits when compared with relevant work in literature. The proposed algorithms synthesize quantum circuits that can applied on IBM quantum computer.
Paper Structure (17 sections, 9 equations, 19 figures, 9 tables, 2 algorithms)

This paper contains 17 sections, 9 equations, 19 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Boolean Functions
  4. Boolean Quantum Circuits
  5. Decomposition of Quantum Circuits
  6. Background Methods
  7. The Direct Method
  8. The Factorization Algorithm
  9. The Reorder Method
  10. The Proposed Algorithm
  11. Formation of an Algebraic Form
  12. Mapping MCT Circuit into NCV Circuit
  13. Basic MCT Gates Decomposition
  14. Optimization by Simplification Rules
  15. Experimental Results
  16. ...and 2 more sections

Figures (19)

  • Figure 1: $MCT$ gates with positive condition (closed circle) and negative condition (open circle)
  • Figure 2: $NCV$ quantum gate library
  • Figure 3: $Clifford +T$ quantum gate library
  • Figure 4: Decomposition of $CNOT(c_{1},c_{2},t)$ with positive conditions on control qubits.
  • Figure 5: Decomposition of $CNOT(\bar{c}_{1},\bar{c}_{2},t)$ with negative conditions on control qubits
  • ...and 14 more figures

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1