Google Quantum AI Slashes Gate Counts in Galois Field Operations by 100x
Researchers from Google Quantum AI have made significant progress in optimizing quantum circuits for Galois Field (GF) operations. Noureldin Yosri and Dmitri Maslov developed new methods that drastically cut the number of gates needed for multiplication and division in these fields.
Their work could lead to more efficient cryptographic systems by reducing the computational overhead in quantum algorithms.
The team focused on improving the efficiency of GF multiplication and division. For multiplication over GF(2^150), they lowered the gate count complexity to 3k when m = 2k, surpassing previous benchmarks. They also refined an ancilla-free multiplication circuit, reducing its complexity to O(m log₂ 3).
A key breakthrough came in GF division, where selecting specific irreducible polynomials enabled more efficient constant polynomial multiplication and field squaring. This choice led to a CNOT gate count lower than the Toffoli count, a notable improvement over existing techniques. Additionally, they designed an O(m) circuit for multiplying by the constant polynomial 1 + x⌈m/2⌉, achieving a 100-fold reduction in CNOT gates for practical parameters.
The researchers also analysed the Toom-3 algorithm for polynomial multiplication over GF(2). They found it only outperforms Karatsuba when polynomial degrees exceed 6,000 due to overhead and a high leading constant in gate count. Meanwhile, their optimisations delivered a 100-fold improvement in gate efficiency for realistic parameter sets.
However, implementing square roots of linear reversible unitaries still poses challenges, as these circuits often require greater depth than the original transformations.
The optimisations introduced by Yosri and Maslov mark a major step forward in quantum circuit efficiency. Their methods reduce gate counts by a factor of 100 or more for practical applications, particularly in cryptography. These improvements could accelerate the development of faster and more reliable quantum algorithms for field operations.
