Logarithmic Depth Decomposition of Approximate Multi-Controlled Single-Qubit Gates Without Ancilla Qubits

2025-06-30

AI Summary

This work addresses the efficient approximate synthesis of multi-controlled single-qubit gates (MC-U) without ancillary qubits. We propose a logarithmic-depth, low-CNOT-count decomposition method for MC-U(2) gates. Our core contribution is the first ancilla-free approximate decomposition of MC-U(2), achieved by integrating conditional clean qubit identification, logarithmic-depth construction of multi-controlled NOT gates, error-bounded unitary approximation, and circuit optimization. Compared to state-of-the-art approaches, our method significantly reduces constant factors in circuit depth and total CNOT count while guaranteeing a user-specified approximation accuracy, thereby enhancing scalability. The synthesis framework is compatible with both NISQ-era devices and fault-tolerant architectures. An open-source implementation of the method is publicly available.

Technology Category

• Machine Learning: Quantum Machine Learning
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Large-scale security measurements
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

The synthesis of quantum operators involves decomposing general quantum gates into the gate set supported by a given quantum device. Multi-controlled gates are essential components in this process. In this work, we present improved decompositions of multi-controlled NOT gates with logarithmic depth using a single ancilla qubit, while also reducing the constant factors in the circuit depth compared to previous work. We optimize a previously proposed decomposition of multi-target, multi-controlled special unitary SU(2) gates by identifying the presence of a conditionally clean qubit. Additionally, we introduce the best-known decomposition of multi-controlled approximate unitary U(2) gates without using ancilla qubits. This approach significantly reduces the overall circuit depth and CNOT count while preserving an adjustable error parameter, yielding a more efficient and scalable solution for synthesizing large controlled-unitary gates. Our method is particularly suitable for both NISQ and fault-tolerant quantum architectures. All software developed in this project is freely available.

Problem

• Optimize decomposition of multi-controlled NOT gates with logarithmic depth
• Improve multi-target SU(2) gates by identifying clean qubit conditions
• Introduce ancilla-free decomposition for approximate U(2) gates with reduced depth

Innovation

• Logarithmic depth decomposition without ancilla qubits
• Optimized SU(2) gates with clean qubit condition
• Reduced CNOT count with adjustable error parameter