Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2404.02279 · Physical Review Applied

Efficient implementation of multicontrolled quantum gates

Ben Zindorf, Sougato Bose

Published Sep 15, 2025• 17 citations
arXivHTMLPDF

Abstract

We present an implementation of multicontrolled () quantum gates that provides significant reductions of cost compared to state-of-the-art methods. The operator applied on the target qubit is a unitary, special unitary, or the Pauli X operator (multicontrolled Toffoli), and requires one clean ancilla, no ancilla, or one dirty ancilla, respectively. We generalize our methods for any number of target qubits, and provide further cost reductions if additional ancilla qubits are available. For each type of gate, we provide implementations for unrestricted (all-to-all) connectivity and for linear-nearest-neighbor (LNN) connectivity. All of the methods use a linear cost of gates from the Clifford+ T (fault-tolerant) set. In the context of LNN architecture, the cost and depth of our circuits scale linearly irrespective of the position of the qubits on which the gate is applied. Our methods directly improve the compilation process of many quantum algorithms, providing optimized circuits. Given the scale of our improvements, for example, the quadratic-to-linear controlled- count for LNN, they will naturally result in a large reduction of errors.

DOI
10.1103/8blx-nfcr

Related papers

Multi-Controlled Quantum Gates in Linear Nearest Neighbor

Ben Zindorf, Sougato Bose

Multi-controlled single-target (MC) gates are some of the most crucial building blocks for varied quantum algorithms. How to implement them optimally is thus a pivotal question. To answer this question in an architecture-independent manner, and to get a worst-case estimate, we should look at a linear nearest-neighbor (LNN) architecture, as this can be embedded in almost any qubit connectivity. Motivated by the above, here we describe a method which implements MC gates using no more than $\sim 4k+8n$ CNOT gates -- up-to $60\%$ reduction over state-of-the-art -- while allowing for complete flexibility to choose the locations of $n$ controls, the target, and a dirty ancilla out of $k$ qubits. More strikingly, in case $k \approx n$, our upper bound is $\sim 12n$ -- the best known for unrestricted connectivity -- and if $n = 1$, our upper bound is $\sim 4k$ -- the best known for a single long-range CNOT gate over $k$ qubits -- therefore, if our upper bound can be reduced, then the cost of one or both of these simpler versions of MC gates will be immediately reduced accordingly. In practice, our method provides circuits that tend to require fewer CNOT gates than our upper bound for almost any given instance of MC gates.

arXiv:2506.00695•May 31, 2025•1 citation

Linear Decomposition of Approximate Multicontrolled Single Qubit Gates

Jefferson D. S. Silva, Thiago Melo D. Azevedo, Israel F. Araujo +1 more

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.

arXiv:2310.14974•IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems•Oct 1, 2024•5 citations

Adaptive Clifford+T Decomposition of Large Toffoli Gates with One Clean Ancilla

Abhoy Kole, Majd Assaad, Till Schnittka +1 more

Multi-controlled Toffoli gates are fundamental building blocks in quantum computation, with applications in quantum arithmetic, simulation, and search algorithms. In fault-tolerant architectures, their realization is constrained by the high cost of non-Clifford resources, particularly in terms of T-count and T-depth. Recent advances have demonstrated that the use of ancillary qubits, relative-phase Toffoli gates, and dynamic circuit techniques can substantially reduce this overhead. In this work, we investigate the decomposition of large Toffoli gates using 3- and 4-input relative-phase Toffoli gates in the presence of a single clean ancilla and conditionally clean ancillas. We derive explicit resource bounds for Clifford+T implementations incorporating dynamic-circuit-based uncomputation and measurement-conditioned corrections. Our analysis emphasizes T-depth reduction under fixed CX and T-count overhead, ensuring relevance for near-term devices. We show that introducing 4-input relative-phase Toffoli gates enables significant T-depth reductions through enhanced parallelism while maintaining favorable ancilla requirements. We further validate our theoretical results through experimental evaluation and comparative analysis with existing approaches.

quant-pharXiv:2605.18169•May 19, 2026•0 citations

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

Jefferson D. S. Silva, Adenilton José da Silva

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.

arXiv:2507.00400•Jul 1, 2025•0 citations

Toffoli-porttien toteutuskustannuksien alentaminen approksimaatioilla

Simo Puttonen

The Toffoli gate (controlled-controlled-NOT) and its multi-controlled generalizations are fundamental logical building blocks of quantum algorithms and utilized extensively in pivotal algorithms such as Grover’s algorithm and Shor’s algorithm. Despite being a fundamental logical primitive of quantum algorithms, Toffoli gates must be decomposed to primitive single-qubit and two-qubit gates which are available in physical quantum hardware. These decompositions incur significant implementation costs in circuit depth and gate count, costs that propagate directly to the entire circuit. Consequently, reducing the implementation cots of Toffoli gates provides a direct method of reducing the implementation costs of many quantum algorithms. This thesis evaluates state-of-the-art methods for reducing the implementation costs of Toffoli gates by introducing approximations. While approximations introduce errors to the output of the Toffoli gate, they can reduce the depth and gate count of the implementation. This thesis finds that discrete errors in two-qubit and three-qubit controlled Toffoli gates have been utilized to reduce the gate counts in such a way that the implementation is exact cancelling the errors. Additionally, continuous errors are utilized to reduce the depth of a multi-controlled Toffoli gate decomposition in ancilla-free settings. However, this review also demonstrates other, non-approximate, construction of multi-controlled Toffoli gate, which achieves comparable implementation costs demonstrating marginal gains for the approximation techniques currently.

•Apr 24, 2026•0 citations

Circuit Decomposition of Multicontrolled Special Unitary Single-Qubit Gates

Rafaella F. Vale, Thiago Melo D. Azevedo, Ismael C. S. Araújo +2 more

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 implementing$n$-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 an$n$-qubit multicontrolled 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 multicontrolled SU(2) gates that require 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. 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.

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems•Oct 24, 2023•23 citations
We present an implementation of multicontrolled () quantum gates that provides significant reductions of cost compared to state-of-the-art methods. The operator applied on the target qubit is a unitary, special unitary, or the Pauli X operator (multicontrolled Toffoli), and requires one clean ancilla, no ancilla, or one dirty ancilla, respectively. We generalize our methods for any number of target qubits, and provide further cost reductions if additional ancilla qubits are available. For each type of gate, we provide implementations for unrestricted (all-to-all) connectivity and for linear-nearest-neighbor (LNN) connectivity. All of the methods use a linear cost of gates from the Clifford+ T (fault-tolerant) set. In the context of LNN architecture, the cost and depth of our circuits scale linearly irrespective of the position of the qubits on which the gate is applied. Our methods directly improve the compilation process of many quantum algorithms, providing optimized circuits. Given the scale of our improvements, for example, the quadratic-to-linear controlled- count for LNN, they will naturally result in a large reduction of errors.
Multi-controlled single-target (MC) gates are some of the most crucial building blocks for varied quantum algorithms. How to implement them optimally is thus a pivotal question. To answer this question in an architecture-independent manner, and to get a worst-case estimate, we should look at a linear nearest-neighbor (LNN) architecture, as this can be embedded in almost any qubit connectivity. Motivated by the above, here we describe a method which implements MC gates using no more than ∼4k+8n\sim 4k+8n∼4k+8n CNOT gates -- up-to 60%60\%60% reduction over state-of-the-art -- while allowing for complete flexibility to choose the locations of nnn controls, the target, and a dirty ancilla out of kkk qubits. More strikingly, in case k≈nk \approx nk≈n, our upper bound is ∼12n\sim 12n∼12n -- the best known for unrestricted connectivity -- and if n=1n = 1n=1, our upper bound is ∼4k\sim 4k∼4k -- the best known for a single long-range CNOT gate over kkk qubits -- therefore, if our upper bound can be reduced, then the cost of one or both of these simpler versions of MC gates will be immediately reduced accordingly. In practice, our method provides circuits that tend to require fewer CNOT gates than our upper bound for almost any given instance of MC gates.
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 Toffoli gates are fundamental building blocks in quantum computation, with applications in quantum arithmetic, simulation, and search algorithms. In fault-tolerant architectures, their realization is constrained by the high cost of non-Clifford resources, particularly in terms of T-count and T-depth. Recent advances have demonstrated that the use of ancillary qubits, relative-phase Toffoli gates, and dynamic circuit techniques can substantially reduce this overhead. In this work, we investigate the decomposition of large Toffoli gates using 3- and 4-input relative-phase Toffoli gates in the presence of a single clean ancilla and conditionally clean ancillas. We derive explicit resource bounds for Clifford+T implementations incorporating dynamic-circuit-based uncomputation and measurement-conditioned corrections. Our analysis emphasizes T-depth reduction under fixed CX and T-count overhead, ensuring relevance for near-term devices. We show that introducing 4-input relative-phase Toffoli gates enables significant T-depth reductions through enhanced parallelism while maintaining favorable ancilla requirements. We further validate our theoretical results through experimental evaluation and comparative analysis with existing approaches.
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.
The Toffoli gate (controlled-controlled-NOT) and its multi-controlled generalizations are fundamental logical building blocks of quantum algorithms and utilized extensively in pivotal algorithms such as Grover’s algorithm and Shor’s algorithm. Despite being a fundamental logical primitive of quantum algorithms, Toffoli gates must be decomposed to primitive single-qubit and two-qubit gates which are available in physical quantum hardware. These decompositions incur significant implementation costs in circuit depth and gate count, costs that propagate directly to the entire circuit. Consequently, reducing the implementation cots of Toffoli gates provides a direct method of reducing the implementation costs of many quantum algorithms. This thesis evaluates state-of-the-art methods for reducing the implementation costs of Toffoli gates by introducing approximations. While approximations introduce errors to the output of the Toffoli gate, they can reduce the depth and gate count of the implementation. This thesis finds that discrete errors in two-qubit and three-qubit controlled Toffoli gates have been utilized to reduce the gate counts in such a way that the implementation is exact cancelling the errors. Additionally, continuous errors are utilized to reduce the depth of a multi-controlled Toffoli gate decomposition in ancilla-free settings. However, this review also demonstrates other, non-approximate, construction of multi-controlled Toffoli gate, which achieves comparable implementation costs demonstrating marginal gains for the approximation techniques currently.
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.