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

• URL: http://arxiv.org/abs/2210.16195v1
• Date: Fri, 28 Oct 2022 15:04:55 GMT
• Title: CNOT circuits need little help to implement arbitrary Hadamard-free Clifford transformations they generate
• Authors: Dmitri Maslov and Willers Yang
• Abstract summary: A Hadamard-free Clifford transformation is a circuit composed of quantum Phase (P), CZ, and CNOT gates. 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.
• Score: 5.672898304129217
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 implies the ability to implement arbitrary Clifford transformation over LNN in depth no more than $7n{+}2$, 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 constant 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.

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