Rise of conditionally clean ancillae for efficient quantum circuit constructions

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.