Polylogarithmic-depth controlled-NOT gates without ancilla qubits

• Published in: Nature communications Vol. 15; no. 1; pp. 5886 - 8
• Main Authors: Claudon, Baptiste, Zylberman, Julien, Feniou, César, Debbasch, Fabrice, Peruzzo, Alberto, Piquemal, Jean-Philip
• Format: Journal Article
• Language: English
• Published: London Nature Publishing Group UK 13.07.2024; Nature Publishing Group; Nature Portfolio
• Subjects: 639/766/483; 639/766/483/481; Algorithms; Asymptotic methods; Asymptotic properties; Decomposition; Fault tolerance; Gates; Gates (circuits); Humanities and Social Sciences; Machine learning; multidisciplinary; Physics; Quantum chemistry; Quantum computing; Quantum Physics; Qubits (quantum computing); Science; Science (multidisciplinary); controlled-NOT gate; Quantum computing; Controlled operation; Toffoli; Fault tolerant quantum computation; Quantum physics; quantum information
• ISSN: 2041-1723; 2041-1723
• DOI: 10.1038/s41467-024-50065-x

Controlled operations are fundamental building blocks of quantum algorithms. Decomposing n -control-NOT gates ( C n ( X )) into arbitrary single-qubit and CNOT gates, is a crucial but non-trivial task. This study introduces C n ( X ) circuits outperforming previous methods in the asymptotic and non-asymptotic regimes. Three distinct decompositions are presented: an exact one using one borrowed ancilla with a circuit depth Θ ( log ( n ) 3 ) , an approximating one without ancilla qubits with a circuit depth O ( log ( n ) 3 log ( 1 / ϵ ) ) and an exact one with an adjustable-depth circuit which decreases with the number m ≤ n of ancilla qubits available as O ( log ( n / ⌊ m / 2 ⌋ ) 3 + log ( ⌊ m / 2 ⌋ ) ) . The resulting exponential speedup is likely to have a substantial impact on fault-tolerant quantum computing by improving the complexities of countless quantum algorithms with applications ranging from quantum chemistry to physics, finance and quantum machine learning. Multi-controlled operations are very common in quantum algorithms applications. Here, the authors propose three schemes for decomposing n-control-NOT gates into single-qubit and CNOT gates, which have polylog overhead as opposed to current linear schemes.