[Paper Review] Decomposition of Multi-controlled Special Unitary Single-Qubit Gates

Rafaella Vale, Thiago Melo D. Azevedo+3 more|arXiv (Cornell University)|Feb 13, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper presents a new decomposition method for n-qubit multi-controlled special unitary (SU(2)) gates that reduces the number of CNOT gates from the prior state-of-the-art 28n to 20n, and further to 16n when the SU(2) gate has at least one real diagonal. The approach leverages a unitary decomposition framework without auxiliary qubits, enabling significant circuit depth and gate count reductions for quantum algorithms.

ABSTRACT

Multi-controlled unitary gates have been a subject of interest in quantum computing since its inception, and are widely used in quantum algorithms. The current state-of-the-art approach to implementing n-qubit multi-controlled gates involves the use of a quadratic number of single-qubit and CNOT gates. However, linear solutions are possible for the case where the controlled gate is a special unitary SU(2). The most widely-used decomposition of an n-qubit multi-controlled SU(2) gate requires a circuit with a number of CNOT gates proportional to 28n. In this work, we present a new decomposition of n-qubit multi-controlled SU(2) gates that requires a circuit with a number of CNOT gates proportional to 20n, and proportional to 16n if the SU(2) gate has at least one real-valued diagonal. This new approach significantly improves the existing algorithm by reducing the number of CNOT gates and the overall circuit depth. As an application, we show the use of this decomposition for sparse quantum state preparation. Our results are further validated by demonstrating a proof of principle on a quantum device accessed through quantum cloud services.

Motivation & Objective

• To reduce the number of CNOT gates and circuit depth in multi-controlled SU(2) quantum circuits for near-term quantum devices.
• To overcome numerical instability in existing decomposition methods that rely on computing high-order roots of unitary operators.
• To develop a decomposition scheme that avoids auxiliary qubits while maintaining linear gate scaling.
• To improve efficiency in quantum algorithms requiring multi-controlled single-qubit operations, such as sparse state preparation.
• To validate the proposed method on real quantum hardware via cloud-based quantum services.

Proposed method

• Proposes a novel decomposition of n-qubit multi-controlled SU(2) gates using a unitary matrix factorization approach based on [8], without requiring auxiliary qubits.
• Employs a mathematical framework to decompose the controlled unitary into a sequence of single-qubit and controlled-phase operations, minimizing CNOT usage.
• Introduces a parameterization of SU(2) matrices that exploits real-valued diagonal entries to further reduce gate count by leveraging symmetry.
• Derives closed-form expressions for the decomposition parameters using real and imaginary parts of the target unitary matrix, avoiding numerical root-finding.
• Applies a recursive decomposition strategy that reduces the control qubit count iteratively, maintaining linear scaling in the number of controls.
• Validates the circuit design through simulation and implements it on IBM Quantum cloud devices to confirm practical feasibility and error resilience.

Experimental results

Research questions

• RQ1 Can the number of CNOT gates in n-qubit multi-controlled SU(2) gates be reduced below the current 28n threshold without using auxiliary qubits?
• RQ2 Does exploiting real-valued diagonal entries in SU(2) matrices enable a further reduction in CNOT count, and if so, to what extent?
• RQ3 Can the proposed decomposition avoid the numerical errors inherent in methods requiring computation of $2^n$-th roots of unitary operators?
• RQ4 How does the proposed method compare in circuit depth and gate count to existing approaches like those in [10], [11], and [12]?
• RQ5 To what extent can this decomposition improve the efficiency of quantum algorithms such as sparse quantum state preparation?

Key findings

• The proposed decomposition reduces the number of CNOT gates to at most $20n - 38$ for general multi-controlled SU(2) gates, improving upon the prior $28n - 88$ (even $n$) or $28n - 92$ (odd $n$) CNOTs.
• For SU(2) gates with at least one real-valued diagonal, the method achieves a further reduction to at most $16n - 40$ CNOTs.
• The method avoids numerical errors by eliminating the need to compute high-order roots of unitary matrices, such as $\sqrt[2^n]{U}$, which plague prior approaches.
• The circuit depth scales linearly with the number of control qubits, maintaining efficiency suitable for Noisy Intermediate-Scale Quantum (NISQ) devices.
• The decomposition was successfully implemented and validated on real quantum hardware via IBM Quantum cloud services, demonstrating practical feasibility.
• The method enables more efficient sparse quantum state preparation, reducing the required CNOT count in such circuits by a significant margin.