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arXiv:2407.05162

Low-depth Quantum Circuit Decomposition of Multi-controlled Gates

Thiago Melo D. Azevedo, Jefferson D. S. Silva, Adenilton José da Silva

Published Jul 6, 2024• 0 citations
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Abstract

Multi-controlled gates are fundamental components in the design of quantum algorithms, where efficient decompositions of these operators can enhance algorithm performance. The best asymptotic decomposition of an n-controlled X gate with one borrowed ancilla into single qubit and CNOT gates produces circuits with degree 3 polylogarithmic depth and employs a divide-and-conquer strategy. In this paper, we reduce the number of recursive calls in the divide-and-conquer algorithm and decrease the depth of n-controlled X gate decomposition to a degree of 2.799 polylogarithmic depth. With this optimized decomposition, we also reduce the depth of n-controlled SU(2) gates and approximate n-controlled U(2) gates. Decompositions described in this work achieve the lowest asymptotic depth reported in the literature. We also perform an optimization in the base of the recursive approach. Starting at 52 control qubits, the proposed n-controlled X gate with one borrowed ancilla has the shortest circuit depth in the literature. One can reproduce all the results with the freely available open-source code provided in a public repository.

DOI
10.48550/arxiv.2407.05162

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Multi-controlled gates are fundamental components in the design of quantum algorithms, where efficient decompositions of these operators can enhance algorithm performance. The best asymptotic decomposition of an n-controlled X gate with one borrowed ancilla into single qubit and CNOT gates produces circuits with degree 3 polylogarithmic depth and employs a divide-and-conquer strategy. In this paper, we reduce the number of recursive calls in the divide-and-conquer algorithm and decrease the depth of n-controlled X gate decomposition to a degree of 2.799 polylogarithmic depth. With this optimized decomposition, we also reduce the depth of n-controlled SU(2) gates and approximate n-controlled U(2) gates. Decompositions described in this work achieve the lowest asymptotic depth reported in the literature. We also perform an optimization in the base of the recursive approach. Starting at 52 control qubits, the proposed n-controlled X gate with one borrowed ancilla has the shortest circuit depth in the literature. One can reproduce all the results with the freely available open-source code provided in a public repository.
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.
Abstract Controlled operations are fundamental building blocks of quantum algorithms. Decomposing n-control-NOT gates (Cn(X)) into arbitrary single-qubit and CNOT gates, is a crucial but non-trivial task. This study introduces Cn(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)\Theta (\log {(n)}^{3})Θ(log(n)3) Θ ( log ( n ) 3 ) , an approximating one without ancilla qubits with a circuit depth O(log⁡(n)3log⁡(1/ϵ)){{{{{{{\mathcal{O}}}}}}}}(\log {(n)}^{3}\log (1/\epsilon ))O(log(n)3log(1/ϵ)) 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⌋)){{{{{{{\mathcal{O}}}}}}}}(\log {(n/\lfloor m/2\rfloor )}^{3}+\log (\lfloor m/2\rfloor ))O(log(n/⌊m/2⌋)3+log(⌊m/2⌋)) 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.
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.
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.
In quantum computation every unitary operation can be decomposed into quantum circuits---a series of single-qubit rotations and a single type entangling two-qubit gates, such as controlled-not(cnot) gates. Two measures are important when judging the complexity of the circuit: the total number of cnot gates needed to implement it and the depth of the circuit, measured by the minimal number of computation steps needed to perform it. Here we give an explicit and simple quantum circuit scheme for preparation of arbitrary quantum states, which can directly utilize any decomposition scheme for arbitrary full quantum gates, thus connecting the two problems. Our circuit reduces the depth of the best currently known circuit by a factor of 222. It also reduces the total number of cnot gates from 2n{2}^{n}2n to 23242n\frac{23}{24}{2}^{n}2423​2n in the leading order for even number of qubits. Specifically, the scheme allows us to decrease the upper bound from 111111 cnot gates to 999 and the depth from 111111 to 555 steps for four qubits. Our results are expected to help in designing and building small-scale quantum circuits using present technologies.
Efficient Synthesis of Multi-Controlled Toffoli Gates with Ternary Clifford+P9+P_9+P9​ Gates
Temporary occupation of qutrit levels can reduce the width and depth required for binary circuit logic in quantum computing. We introduce an efficient intermediate-qutrit decomposition of multi-controlled Toffoli gates with binary-subspace inputs and outputs. For balanced control widths n=2h−1n=2^h-1n=2h−1, the proposed decomposition is evaluated hierarchically through a tree, allowing independent subtree computations to proceed in parallel and reducing the depth from linear to logarithmic. The balanced low-depth construction requires 6n+36n+36n+3 logical P9P_9P9​ injections and n−34\frac{n-3}{4}4n−3​ clean ancillary qutrits. It matches the direct P9P_9P9​ count of a recursively extended Clifford+P9P_9P9​ baseline derived from the state-of-the-art work while using asymptotically one-quarter as many clean ancillas and replacing linear depth with logarithmic depth. For arbitrary control widths, a sequential extension preserves the 6n+36n+36n+3 P9P_9P9​ count but has depth O(log⁡m+n−m)O(\log m+n-m)O(logm+n−m) for a balanced core m=2h−1≤nm=2^h-1\leq nm=2h−1≤n, which is linear in the worst case over nnn. By reducing the resource overhead of a widely used reversible primitive, the proposed decomposition provides a practical building block for the design and compilation of more resource-efficient quantum algorithms in the fault-tolerant regime.