Improved Clifford operations in constant commutative depth
Richard Cleve, Zhiqian Ding, Luke Schaeffer
TL;DR
The paper advances the study of quantum circuits under a commutative-depth model by showing that every Clifford operation on $n$ qubits can be implemented in constant commutative depth $16$ with no ancillas and size $Θ(n^2)$, improving the prior bound of depth $23$. It achieves a constant-depthPrefix Sum construction with depth $16$ (or $17$ for odd $n$) and size $Θ(n \, log \, n)$ via a novel Ladner–Fischer–style decomposition into sparse left and right blocks, plus efficient combining tricks. The authors further refine the Bravyi–Maslov–Nam approach by providing a six-layer Clifford-decomposition that, together with a linear-depth $11$ subcircuit, yields overall depth $16$; they also establish lower bounds showing depth at least $4$ is necessary for some Cliffords and that constant-depth circuits can require $Ω(n^2)$ size. These results collectively tighten the landscape for constant-depth Clifford circuitry, highlighting both practical circuit-construction gains and fundamental limits in the commutative-depth paradigm.
Abstract
The commutative depth model allows gates that commute with each other to be performed in parallel. We show how to compute Clifford operations in constant commutative depth more efficiently than was previously known. Bravyi, Maslov, and Nam [Phys. Rev. Lett. 129:230501, 2022] showed that every element of the Clifford group (on $n$ qubits) can be computed in commutative depth 23 and size $O(n^2)$. We show that the Prefix Sum problem can be computed in commutative depth 16 and size $O(n \log n)$, improving on the previous depth 18 and size $O(n^2)$ bounds. We also show that, for arbitrary Cliffords, the commutative depth bound can be reduced to 16. Finally, we show some lower bounds: that there exist Cliffords whose commutative depth is at least 4; and that there exist Cliffords for which any constant commutative depth circuit has size $Ω(n^2)$.