CNOT circuits need little help to implement arbitrary Hadamard-free Clifford transformations they generate

TL;DR

This paper demonstrates that implementing Hadamard-free Clifford transformations with CNOT circuits can be significantly optimized in depth, especially over linear nearest neighbor architectures, reducing overall circuit depth.

Contribution

The paper provides new depth bounds for implementing Hadamard-free Clifford transformations, improving previous results and offering heuristic evidence for near-optimal implementations over all-to-all architectures.

Findings

LNN implementation depth is $5n$, matching the CNOT stage alone.

Arbitrary Clifford transformations over LNN can be implemented in depth no more than $7n-4$.

Depth reduction from $2n + O( ext{log}^2 n)$ to approximately $1.5n + O( ext{log}^2 n)$ over unrestricted architectures.

Abstract

A Hadamard-free Clifford transformation is a circuit composed of quantum Phase (P), CZ, and CNOT gates. It is known that such a circuit can be written as a three-stage computation, -P-CZ-CNOT-, where each stage consists only of gates of the specified type. In this paper, we focus on the minimization of circuit depth by entangling gates, corresponding to the important time-to-solution metric and the reduction of noise due to decoherence. We consider two popular connectivity maps: Linear Nearest Neighbor (LNN) and all-to-all. First, we show that a Hadamard-free Clifford operation can be implemented over LNN in depth $5n$, i.e., in the same depth as the -CNOT- stage alone. This allows us to implement arbitrary Clifford transformation over LNN in depth no more than $7n{-}4$, improving the best previous upper bound of $9n$. Second, we report heuristic evidence that on average a random uniformly distributed Hadamard-free Clifford transformation over $n{>}6$ qubits can be implemented with only a tiny additive overhead over all-to-all connected architecture compared to the best-known depth-optimized implementation of the -CNOT- stage alone. This suggests the reduction of the depth of Clifford circuits from $2n\,{+}\,O(\log^2(n))$ to $1.5n\,{+}\,O(\log^2(n))$ over unrestricted architectures.