Improved quantum circuits for division

Quantum division circuits just got a lot cheaper. Researchers from IBM and the University of Waterloo have published new fault-tolerant circuits for integer division that achieve up to 76% fewer T-gates and 68% fewer CNOT gates than prior constructions — with an asymptotic depth improvement from O(n squared) to O(n log n) in the bargain.

The paper, posted to arXiv last week by Priyanka Mukhopadhyay, Alexandru Gheorghiu, and Hari Krovi, introduces a circuit primitive called COMP-N-SUB that combines integer comparison with conditional subtraction in a single operation. The key finding: doing both steps together costs roughly as much as one addition. The previous approach performed them separately, costing roughly one controlled addition plus a regular addition.

The authors — two with IBM affiliations (Krovi runs IBM's fault-tolerant quantum algorithms group), one with Waterloo/IQC — tested their constructions in the Clifford+T gate set, the standard gate vocabulary for fault-tolerant quantum computation. T-gates are the expensive non-Clifford resource; cutting them by more than three-quarters is not a marginal improvement.

Division is a building block, not a destination. It shows up in Shor's algorithm, in quantum chemistry simulation via Trotterization, and in range-theoretic algorithms that underpin post-quantum cryptographic work. When you're running those algorithms at scales where fault-tolerant overhead dominates, better arithmetic primitives cascade upward.

The 76% T-count reduction is the headline number. The asymptotic depth improvement is the structural story. Whether either translates to real algorithm performance depends on implementation details this paper doesn't address — no benchmarking against actual hardware or compiler toolchains. Fair enough for a circuit-design paper, but the gap between circuit-level optimization and realized runtime is real.

What's notable beyond the numbers is the affiliation. IBM's fault-tolerant algorithms team has been relatively quiet in public about their roadmap. This paper suggests the team is actively working on the low-level circuit primitives that underpin larger algorithms — unglamorous infrastructure work that makes everything else possible. If the COMP-N-SUB primitive makes it into IBM's Qiskit or equivalent compiler stack, the gains documented here could propagate to any user running division-heavy quantum algorithms through their platform.

The paper is titled "Improved quantum circuits for division" (arXiv:2603.18110).