[Paper Review] High Performance Quantum Modular Multipliers

Rich Rines, Isaac L. Chuang|arXiv (Cornell University)|Jan 3, 2018

Quantum Computing Algorithms and Architecture23 references25 citations

TL;DR

This paper presents three novel reversible quantum modular multipliers based on classical techniques—standard division, Montgomery reduction, and Barrett reduction—achieving asymptotic resource complexity matching that of non-modular integer multiplication. The proposed designs, especially the Fourier-basis variants, deliver a circuit depth of 14n two-qubit gates and require only 2n + O(log n) qubits, outperforming prior exact quantum modular multipliers in depth and ancilla qubit usage while maintaining exact results without approximation.

ABSTRACT

We present a novel set of reversible modular multipliers applicable to quantum computing, derived from three classical techniques: 1) traditional integer division, 2) Montgomery residue arithmetic, and 3) Barrett reduction. Each multiplier computes an exact result for all binary input values, while maintaining the asymptotic resource complexity of a single (non-modular) integer multiplier. We additionally conduct an empirical resource analysis of our designs in order to determine the total gate count and circuit depth of each fully constructed circuit, with inputs as large as 2048 bits. Our comparative analysis considers both circuit implementations which allow for arbitrary (controlled) rotation gates, as well as those restricted to a typical fault-tolerant gate set.

Motivation & Objective

• To design reversible quantum modular multipliers with asymptotic resource complexity matching non-modular integer multiplication.
• To eliminate the need for repeated comparison operations and costly QFT/QFT† circuits found in prior Fourier-basis approaches.
• To achieve exact modular multiplication without approximation, preserving fidelity in quantum algorithms.
• To enable efficient implementation in the quantum Fourier transform basis, reducing ancilla qubit overhead.
• To provide a framework compatible with fault-tolerant gate sets and scalable quantum architectures.

Proposed method

• Adapts classical modular arithmetic techniques—standard division, Montgomery reduction, and Barrett reduction—into reversible quantum circuits.
• Employs reversible adders and subtracters as core primitives, enabling integration across different quantum arithmetic models.
• Designs Fourier-basis compatible versions of Montgomery and Barrett multipliers to avoid depth-intensive QFT/QFT† operations.
• Uses pre-computed reduction factors in Barrett reduction and N-residue representation in Montgomery arithmetic to eliminate division during runtime.
• Constructs out-of-place quantum-quantum modular multipliers by adapting input accumulators to second quantum inputs.
• Performs empirical resource analysis for 2048-bit inputs under both arbitrary rotation and fault-tolerant gate sets.

Experimental results

Research questions

• RQ1 Can exact quantum modular multiplication be achieved with resource complexity matching that of non-modular integer multiplication?
• RQ2 Can Fourier-basis arithmetic be leveraged to reduce circuit depth and ancilla qubit count in modular multipliers?
• RQ3 How do the proposed multipliers compare in performance and resource usage to prior exact and inexact quantum modular multipliers?
• RQ4 Can classical modular arithmetic techniques be effectively adapted into reversible quantum circuits without sacrificing accuracy?
• RQ5 What is the trade-off between gate count, circuit depth, and qubit overhead in Fourier-basis versus binary-representation quantum modular multipliers?

Key findings

• The proposed modular multipliers achieve an asymptotic circuit depth of 14n two-qubit gates when implemented in the quantum Fourier transform basis, significantly reducing latency compared to prior exact Fourier-basis multipliers with 1000n gate depth.
• The designs require only 2n + O(log n) qubits, matching the low ancilla count of Beauregard’s Fourier-basis modular adder and improving upon prior exact multipliers that require 9n and 3n ancilla qubits.
• The Barrett and Montgomery-based multipliers eliminate the need for repeated comparison operations and associated QFT/QFT† circuits, which dominate depth in modular addition-based approaches.
• The Fourier-basis versions of the multipliers achieve performance comparable to the fastest inexact quantum modular multiplier (12n gate depth), while maintaining exactness and avoiding fidelity loss from approximation.
• The resource analysis confirms that the proposed designs scale efficiently, with total gate count and circuit depth asymptotically equivalent to a single non-modular integer multiplier.
• The framework is extensible to other domains, such as Galois field arithmetic over GF(2^m), and can be combined with fast multiplication or inexact computation techniques.