Fast Cliffords When Your Quantum Memory Is Full

Additional qubits can reduce the depth of a quantum circuit by providing workspace for parallel computation, but standard constructions assume that this workspace is initialized in a known state. In this work we study catalytic implementations, i.e. asking whether dirty qubits can instead be used provided that their joint state including any entanglement with other registers is restored exactly at the end of the computation. We show that every $n$-qubit Clifford circuit has a catalytic implementation of depth $O(\log n)$ using $O(n^2\mathrm{/}{\log }^{2}n)$ catalytic qubits and no clean qubits, matching the asymptotic depth achievable when clean workspace is available. We extend this approach to diagonal elements of any fixed level $C_k$ of the Clifford hierarchy, which admit catalytic implementations of depth $O(\log (n+1))$ with $O(n^k\mathrm{/}\log (n))$ gates and $O(n^k\mathrm{/}{\log }^2(n))$ catalytic qubits, as well as to semi-Clifford Gates.