Quantum Circuits Achieve GF() Multiplication and Division with Gate Count Reduction To, Improving on Previous Bounds of

Quantum computing promises to revolutionise fields from medicine to materials science, but realising this potential demands efficient algorithms and optimised circuits, particularly for fundamental mathematical operations. Noureldin Yosri and Dmitri Maslov, both from Google Quantum AI, have now achieved significant advances in optimising circuits for performing multiplication and division within Galois Fields, essential building blocks for many quantum algorithms. Their work delivers a substantial reduction in the number of quantum gates required for these operations, improving upon existing methods and achieving a factor of 100 or more improvement for practical parameter choices. This breakthrough not only simplifies the implementation of these critical operations, but also demonstrates that even seemingly simple operations, such as calculating square roots of linear transformations, can introduce unexpected complexities in quantum circuit design.

In this work, researchers focus on optimising fundamental quantum computing primitives within various quantum algorithms. The team achieves an improvement in the gate count complexity of its ancilla-free Galois Field multiplication circuit to O(m log₂ 3), surpassing previously established bounds. This advancement stems from the development of an efficient O(m) circuit designed for multiplication by the constant polynomial 1 + x⌈m/2⌉, a crucial element of Van Hoof’s construction, and translates to over a 100-fold improvement in CNOT gate counts for practical parameter sets.

Toom-3 Polynomial Multiplication over GF(2)

Scientists investigated the Toom-3 algorithm for multiplying polynomials over the Galois Field GF(2), a binary field, and explored its potential advantages and limitations. The team implemented and analysed a framework for constructing Toom-k circuits, focusing on the specific case of k=3, involving careful selection of evaluation points and optimisation of calculations to minimise CNOT gate counts. The research demonstrates that while Toom-3 theoretically offers potential benefits, it does not outperform Karatsuba until the degree of the polynomials becomes very large, exceeding 6,000, due to overhead associated with splitting polynomials into three parts and a high leading constant in the gate count.

Efficient Galois Field Multiplication and Division Circuits

Scientists achieved a significant reduction in gate count complexity for Galois Field (GF) multiplication, reaching a complexity of 3k for GF(2ᵐ) multiplication with m = 2k, an improvement over previously known bounds. This breakthrough stems from a novel circuit for multiplication by a constant polynomial, a key component of established construction methods. For GF division, researchers reduced gate count complexity by selecting irreducible polynomials that enable efficient implementation of both constant polynomial multiplication and field squaring operations, demonstrating practical advantages for cryptographically relevant field sizes. Numeric data confirms that this optimisation results in a CNOT gate count lower than the Toffoli count, a substantial improvement over existing methods.

Ancilla-Free Galois Field Arithmetic Optimizations

This research presents significant advances in the efficiency of performing calculations within Galois Fields, mathematical structures crucial for modern cryptography and coding theory. Scientists have developed optimised circuits for both multiplication and division operations, fundamental building blocks in many algorithms, achieving a reduced gate count complexity for multiplication. This enhancement stems from a particularly efficient circuit designed for multiplication by a constant polynomial, a key component of established construction methods. The practical impact of this optimisation translates to substantial reductions in gate counts for parameters commonly used in real-world applications.

Furthermore, the team reduced the complexity of Galois Field division by carefully selecting irreducible polynomials that streamline both constant polynomial multiplication and field squaring operations, offering a pathway to more efficient cryptographic systems. The researchers also investigated the complexity of implementing square roots of linear reversible unitaries, revealing that these operations can require deeper circuits than the original transformations themselves. While acknowledging a focus on Karatsuba-based methods, they suggest that many of the techniques developed could also optimise alternative circuit designs, providing a foundation for building more efficient and practical cryptographic systems and advancing the understanding of computational complexity within Galois Fields.