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arXiv:2203.11882 · Physical Review A

Linear-depth quantum circuits for multiqubit controlled gates

Adenilton José da Silva, Daniel K. Park

Published Oct 4, 2022• 49 citations
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Abstract

Quantum circuit depth minimization is critical for practical applications of circuit-based quantum computation. In this work, we present a systematic procedure to decompose multiqubit controlled unitary gates, which is essential in many quantum algorithms, to controlled-not and single-qubit gates with which the quantum circuit depth only increases linearly with the number of control qubits. Our algorithm does not require any ancillary qubits and achieves a quadratic reduction of the circuit depth against known methods. We show the advantage of our algorithm with proof-of-principle demonstrations on the IBM quantum cloud platform.

DOI
10.1103/physreva.106.042602

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Quantum circuit depth minimization is critical for practical applications of circuit-based quantum computation. In this work, we present a systematic procedure to decompose multiqubit controlled unitary gates, which is essential in many quantum algorithms, to controlled-not and single-qubit gates with which the quantum circuit depth only increases linearly with the number of control qubits. Our algorithm does not require any ancillary qubits and achieves a quadratic reduction of the circuit depth against known methods. We show the advantage of our algorithm with proof-of-principle demonstrations on the IBM quantum cloud platform.
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.
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 an improved decomposition of multi-controlled NOT gates with logarithmic depth using a single ancilla qubit while reducing the ancillary resource requirements compared to previous work. We further introduce a relative-phase multi-controlled NOT gate that eliminates the need for ancillas. Building on these results, 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, which do not require 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.
Multicontrolled unitary gates have been a subject of interest in quantum computing since their conception and are widely used in quantum algorithms. The current state-of-the-art approach to implementingnnn-qubit multicontrolled gates with a single target without relying on auxiliary qubits or approximate results 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 special unitary, SU(2). The decomposition of annnn-qubit multicontrolled SU(2) gate requires a circuit with a number of CNOT gates proportional to28n28n28n. In this work, we present a new decomposition ofnnn-qubit multicontrolled SU(2) gates that require a circuit with a number of CNOT gates proportional to20n20n20nand proportional to16n16n16nif the SU(2) gate has at least one real-valued diagonal. The proposed algorithms produce the most efficient known circuits and improve 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.
We present a scalable set of universal gates and multiply controlled gates in a qudit basis through a bijective mapping from N qubits to qudits with D = 2^N levels via rotations in U(2). For each of the universal gates (H, CNOT, and T), as well as the NOT gate and multiply-controlled-Z gates, we describe a systematic approach to identifying the set of U(2) rotations required to implement each gate for any qudit of size N and with minimal use of an ancilla level. The qudit gates are analyzed in terms of the total rotation count and gate depth as the system scales with D. We apply the qudit-basis to Grover's Algorithm and compare the circuit depth vs. system size to a qubit-based circuit. The results show that there is a dramatic reduction in circuit depth as the size of the system increases for the qudit circuit compared to qubit circuit. In particular, multiply controlled gates are the driving factor in the reduction of circuit complexity for qudit-based system scales with D.
We provide a method for compiling approximate multicontrolled single qubit gates into quantum circuits. Without ancilla qubits, the total number of elementary gates to decompose an n-qubit multicontrolled gate is proportional to 32n elementary operations. The proposed decomposition depends on an optimization technique that minimizes the CNOT gate count for multitarget and multicontrolled CNOT and SU(2) gates. We also provide an approximate decomposition with ancilla qubits with lower-circuit complexity. Computational experiments show the reduction of CNOT gates when multicontrolled U(2) gates are applied. As multicontrolled single-qubit gates serve as fundamental components of quantum algorithms, the proposed decomposition offers a comprehensive solution that can significantly decrease the count of elementary operations employed in quantum computing applications.
Efficient decomposition of multiqubit operators is critical in the execution of quantum algorithms. In this paper, we introduce decompositions of the multiple controlled-ZZZ (ccz and cccz) gates tailored to various qubit-coupling graphs. Specifically, we demonstrate that the ccz gate is realized with cz-depth 4 on a square-shaped qubit-coupling graph utilizing one auxiliary qubit. As for the cccz gate, previous research indicated that the decomposition requires 14 cz gates in a fully connected topology. However, our findings reveal that only four specific qubit couplings are needed to achieve a decomposition using the same number of cz gates, 14. Our research employs an optimization method to improve the alignment of parametrized quantum circuits with their intended quantum gates, which facilitates these efficient decompositions. This methodology is versatile and can be applied to decompose any quantum gates, not just the ccz and cccz gates. These advancements in decomposing multiqubit gates, coupled with our ccz and cccz decompositions, are poised to reduce quantum circuit execution times and enhance the efficiency of complex quantum algorithms in imminent quantum computing applications.