Rise of conditionally clean ancillae for efficient quantum circuit constructions

TL;DR

This paper introduces conditionally clean ancilla qubits as a versatile quantum resource that combines advantages of clean and dirty ancillae, enabling more efficient quantum circuit constructions with lower gate counts and depths.

Contribution

It presents new circuit constructions utilizing conditionally clean ancillae and introduces laddered toggle detection to replace clean ancillae with dirty ones, reducing resource overhead.

Findings

Lower gate counts and depths for key quantum operations.

Efficient constructions with limited ancilla availability.

Demonstrates the utility of conditionally clean ancillae in resource-constrained quantum computing.

Abstract

We introduce conditionally clean ancilla qubits, a new quantum resource, recently explored by [NZS24], that bridges the gap between traditional clean and dirty ancillae. Like dirty ancillae, they begin and end in an unknown state and can be borrowed from existing system qubits, avoiding the space overhead of explicit qubit allocation. Like clean ancillae, they can be treated as initialized in a known state within specific computations, thus avoiding the overhead of toggle detection required for dirty ancillae. We present new circuit constructions leveraging conditionally clean ancillae to achieve lower gate counts and depths, particularly with limited ancilla availability. Specifically, we provide constructions for: (a) $n$-controlled NOT using $2n$ Toffolis and $O(\log{n})$ depth given 2 clean ancillae. (b) $n$-qubit incrementer using $3n$ Toffolis given $\log_2^*{n}$ clean ancillae. (c) $n$-qubit quantum-classical comparator using $3n$ Toffolis given $\log_2^*{n}$ clean ancillae. (d) unary iteration over $[0,N)$ using $2.5N$ Toffolis given $\log_2^*{n}$ clean ancillae. (e) unary iteration via skew tree over $[0, N)$ using $1.25N$ Toffolis given $n$ dirty ancillae. We also introduce laddered toggle detection, a technique to replace clean ancillae with dirty ancillae in all our constructions, incurring a 2x Toffoli gate overhead. Our results demonstrate that conditionally clean ancillae are a valuable tool for quantum circuit design, especially in the resource-constrained early fault-tolerant era.