[Paper Review] Constant-optimized quantum circuits for modular multiplication and exponentiation

Igor L. Markov, Mehdi Saeedi|arXiv (Cornell University)|May 1, 2012

Quantum Computing Algorithms and Architecture25 references80 citations

TL;DR

This paper presents constant-optimized quantum circuits for modular multiplication and exponentiation, focusing on practical efficiency rather than asymptotic complexity. By modeling the search for compact circuits as a shortest-path problem when ancilla qubits are available, it achieves significant constant-factor and additive improvements—especially beneficial for small-scale quantum experiments—without altering the Θ(n²) worst-case scaling.

ABSTRACT

Reversible circuits for modular multiplication Cx%M with x < M arise as components of modular exponentiation in Shor's quantum number-factoring algorithm. However, existing generic constructions focus on asymptotic gate count and circuit depth rather than actual values, producing fairly large circuits not optimized for specific C and M values. In this work, we develop such optimizations in a bottom-up fashion, starting with most convenient C values. When zero-initialized ancilla registers are available, we reduce the search for compact circuits to a shortest-path problem. Some of our modular-multiplication circuits are asymptotically smaller than previous constructions, but worst-case bounds and average sizes remain Θ(n2). In the context of modular exponentiation, we offer several constant-factor improvements, as well as an improvement by a constant additive term that is significant for few-qubit circuits arising in ongoing laboratory experiments with Shor's algorithm.

Motivation & Objective

• To develop practical, constant-optimized quantum circuits for modular multiplication Cx%M, tailored to specific values of C and M.
• To address the gap in existing quantum circuit constructions that prioritize asymptotic performance over actual gate count and depth for real-world quantum hardware.
• To improve the efficiency of modular exponentiation circuits used in Shor's algorithm by reducing constant factors and additive terms.
• To enable more feasible implementations of Shor's algorithm in current and near-term quantum computers with limited qubit counts and gate fidelity.

Proposed method

• Transform the problem of finding compact quantum circuits into a shortest-path problem by leveraging zero-initialized ancilla qubits.
• Use dynamic programming or graph search techniques to explore circuit configurations with minimal gate count for fixed C and M.
• Construct modular multiplication circuits bottom-up, starting from the most convenient values of C to build efficient building blocks.
• Apply these optimized modular multiplication circuits as subroutines in modular exponentiation circuits to reduce overall resource overhead.
• Focus on minimizing constant factors and additive terms in gate count and circuit depth, especially relevant for few-qubit systems.
• Preserve reversibility and unitarity throughout circuit design to maintain compatibility with quantum algorithms like Shor's factorization.

Experimental results

Research questions

• RQ1 How can quantum circuits for modular multiplication be optimized for specific values of C and M, rather than asymptotic behavior?
• RQ2 What is the achievable reduction in constant factors and additive terms in gate count and circuit depth for modular multiplication circuits?
• RQ3 To what extent can these optimizations improve the efficiency of modular exponentiation circuits in Shor's algorithm for small-scale quantum systems?
• RQ4 Can the search for compact circuits be effectively modeled as a shortest-path problem when ancilla qubits are available?
• RQ5 How do the optimized circuits compare in size and depth to generic constructions in practical, near-term quantum computing scenarios?

Key findings

• The proposed method reduces the constant factor in gate count and circuit depth for modular multiplication circuits, particularly beneficial for small-scale implementations.
• The use of zero-initialized ancilla qubits enables a reformulation of the circuit search as a shortest-path problem, leading to more compact solutions.
• Some of the resulting modular-multiplication circuits are asymptotically smaller than prior constructions, though worst-case and average-case scaling remains Θ(n²).
• The modular exponentiation circuits benefit from both constant-factor reductions and additive improvements, which are significant in few-qubit systems used in current laboratory experiments.
• The approach enables more efficient and practical implementations of Shor's algorithm on near-term quantum hardware by minimizing resource overhead.