Optimal Space-Depth Trade-Off of CNOT Circuits in Quantum Logic Synthesis

TL;DR

This paper establishes an asymptotically optimal trade-off between space (ancillae) and depth for CNOT quantum circuits, improving previous bounds and extending results to stabilizer circuits, with implications for near-term quantum computing.

Contribution

The authors derive a tight asymptotic bound for the space-depth trade-off in CNOT circuit synthesis, improving prior results and extending to stabilizer circuits.

Findings

Achieved an asymptotically optimal space-depth trade-off for CNOT circuits.

Reduced ancillae requirements for shallow CNOT circuit synthesis.

Extended results to stabilizer circuits via known reductions.

Abstract

Decoherence -- in the current physical implementations of quantum computers -- makes depth reduction a vital task in quantum-circuit design. Moore and Nilsson (SIAM Journal of Computing, 2001) demonstrated that additional qubits -- known as ancillae -- can be used to provide an extended space to parallelize quantum circuits. Specifically, they proved that, with $O(n^2)$ ancillae, any $n$-qubit CNOT circuit can be transformed into an equivalent one of $O(\log n)$ depth. However, the near-term quantum technologies can only support a limited amount of qubits, making space-depth trade-off a fundamental research subject for quantum-circuit synthesis. In this work, we establish an asymptotically optimal space-depth trade-off for CNOT circuits. We prove that any $n$-qubit CNOT circuit can be parallelized to $$O\left(\max\left\{\log n, \frac{n^2}{(n+m)\log (n+m)}\right\}\right)$$ depth with $m$ ancillae. This bound is tight even if the task is expanded from exact synthesis to the approximation of CNOT circuits with arbitrary two-qubit quantum gates. Our result can be extended to stabilizer circuits via the reduction by Aaronson and Gottesman (Physical Review A, 2004). Furthermore, we provide hardness evidence for optimizing CNOT circuits in terms of size or depth. Our result has improved upon two previous papers that motivated our work. Moore-Nilsson's construction (aforementioned) for $O(\log n)$-depth CNOT circuit synthesis: We have reduced their need for ancillae by a factor of $\log^2 n$ by showing that $m= O(n^2/\log^2 n)$ additional qubits -- which is asymptotically optimal -- suffice to build equivalent $O(\log n)$-depth $O(n^2/\log n)$-size CNOT circuits. Patel-Markov-Hayes's construction (Quantum Information & Computation 2008) for $m = 0$: We have reduced their depth by a factor of $n$ and achieved the asymptotically optimal bound of $O(n/\log n)$.