In this article we study n-qubit quantum gates which may be represented as direct sums of two-dimensional unitary operators. We call this class of gates uniformly controlled one-qubit gates. We present a quantum circuit which implements any gate of this type using 2^{n-1} - 1 controlled-NOT gates, 2^{n-1} one-qubit gates and a single n-qubit diagonal gate. The construction is based on a modified version of the so-called quantum multiplexor circuit. We illustrate the versatility of the uniformly controlled gates by applying them to the decomposition of a general n-qubit gate and an n-qubit state-preparation procedure. Moreover, we study the implementation of these circuits using only nearest-neighbor gates and give the worst-case one-qubit and CNOT gate counts for all of the aforementioned applications. In all four cases, the proposed method either improves or achieves the previously reported upper bounds for the gate counts.In this article we study n-qubit quantum gates which may be represented as direct sums of two-dimensional unitary operators. We call this class of gates uniformly controlled one-qubit gates. We present a quantum circuit which implements any gate of this type using 2^{n-1} - 1 controlled-NOT gates, 2^{n-1} one-qubit gates and a single n-qubit diagonal gate. The construction is based on a modified version of the so-called quantum multiplexor circuit. We illustrate the versatility of the uniformly controlled gates by applying them to the decomposition of a general n-qubit gate and an n-qubit state-preparation procedure. Moreover, we study the implementation of these circuits using only nearest-neighbor gates and give the worst-case one-qubit and CNOT gate counts for all of the aforementioned applications. In all four cases, the proposed method either improves or achieves the previously reported upper bounds for the gate counts.

Oct 9, 20045 citations

Quantum circuits with uniformly controlled one-qubit gates

Ville Bergholm, Juha J. Vartiainen, Mikko Möttönen +1 more

Uniformly controlled one-qubit gates are quantum gates which can be represented as direct sums of two-dimensional unitary operators acting on a single qubit. We present a quantum gate array which implements any $n$ -qubit gate of this type using at most ${2}^{n\ensuremath{-}1}\ensuremath{-}1$ controlled-NOT gates, ${2}^{n\ensuremath{-}1}$ one-qubit gates, and a single diagonal $n$ -qubit gate. To illustrate the versatility of these gates we then apply them to the decomposition of a general $n$ -qubit gate and a state preparation procedure. Moreover, we study their implementation using only nearest-neighbor gates. We give upper bounds for the one-qubit and controlled-NOT gate counts for all the aforementioned applications. In all four cases, the proposed circuit topologies either improve on or achieve the previously reported upper bounds for the gate counts. Thus, they provide the most efficient method for general gate decompositions currently known.Uniformly controlled one-qubit gates are quantum gates which can be represented as direct sums of two-dimensional unitary operators acting on a single qubit. We present a quantum gate array which implements any $n$-qubit gate of this type using at most ${2}^{n\ensuremath{-}1}\ensuremath{-}1$ controlled-NOT gates, ${2}^{n\ensuremath{-}1}$ one-qubit gates, and a single diagonal $n$-qubit gate. To illustrate the versatility of these gates we then apply them to the decomposition of a general $n$-qubit gate and a state preparation procedure. Moreover, we study their implementation using only nearest-neighbor gates. We give upper bounds for the one-qubit and controlled-NOT gate counts for all the aforementioned applications. In all four cases, the proposed circuit topologies either improve on or achieve the previously reported upper bounds for the gate counts. Thus, they provide the most efficient method for general gate decompositions currently known.

arXiv:quant-ph/0410066Physical Review AMay 27, 2005159 citations

A Unified Framework for Optimizing Uniformly Controlled Structures in Quantum Circuits

Chengzhuo Xu, Xiao Chen, Xi Li +2 more

Quantum unitaries of the form $Σ_{c}\ket{c}\bra{c}\otimes U_{c}$ are ubiquitous in quantum algorithms. This class encompasses not only standard uniformly controlled gates (UCGs) but also a wide range of circuits with uniformly controlled structures. However, their circuit-depth and gate-count complexities have not been systematically analyzed within a unified framework. In this work, we study the general decomposition problem for UCG and UCG-like structure. We then introduce the restricted Uniformly Controlled Gates (rUCGs) as a unified algebraic model, defined by a 2-divisible Abelian group that models the controlled gate set. This model captures uniformly controlled rotations, multi-qubit uniformly controlled gates, and diagonal unitaries. Furthermore, this model also naturally incorporates k-sparse version (k-rUCGs), where only a subset of control qubits participate in each multi-qubit gate. Building on this algebraic model, we develop a general framework. For an n-control rUCG, the framework reduce the gate complexity from ${O(n2^n)}$ to ${O(2^n})$ and the circuit depth from ${O(2^n\log n)}$ to ${O(2^n\log n/n)}$ . The framework further provides systematic size and depth bounds for k-rUCGs by exploiting sparsity in the control space, with same optimization coefficient as rUCG, respectively. Empirical evaluations on representative QAOA circuits confirm reductions in depth and size, which highlight that the rUCG model and its associated decomposition framework unify circuits previously considered structurally distinct under a single, asymptotically optimal synthesis paradigm.Quantum unitaries of the form $Σ_{c}\ket{c}\bra{c}\otimes U_{c}$ are ubiquitous in quantum algorithms. This class encompasses not only standard uniformly controlled gates (UCGs) but also a wide range of circuits with uniformly controlled structures. However, their circuit-depth and gate-count complexities have not been systematically analyzed within a unified framework. In this work, we study the general decomposition problem for UCG and UCG-like structure. We then introduce the restricted Uniformly Controlled Gates (rUCGs) as a unified algebraic model, defined by a 2-divisible Abelian group that models the controlled gate set. This model captures uniformly controlled rotations, multi-qubit uniformly controlled gates, and diagonal unitaries. Furthermore, this model also naturally incorporates k-sparse version (k-rUCGs), where only a subset of control qubits participate in each multi-qubit gate. Building on this algebraic model, we develop a general framework. For an n-control rUCG, the framework reduce the gate complexity from ${O(n2^n)}$ to ${O(2^n})$ and the circuit depth from ${O(2^n\log n)}$ to ${O(2^n\log n/n)}$. The framework further provides systematic size and depth bounds for k-rUCGs by exploiting sparsity in the control space, with same optimization coefficient as rUCG, respectively. Empirical evaluations on representative QAOA circuits confirm reductions in depth and size, which highlight that the rUCG model and its associated decomposition framework unify circuits previously considered structurally distinct under a single, asymptotically optimal synthesis paradigm.

quant-pharXiv:2512.08675Dec 10, 20250 citations

Beyond Quantum Shannon: Circuit Construction for General n-Qubit Gates Based on Block ZXZ-Decomposition

Anna M. Krol, Zaid Al-Ars

This paper proposes a new optimized quantum block-ZXZ decomposition method [7,8,10] that results in more optimal quantum circuits than the quantum Shannon decomposition (QSD)[27], which was introduced in 2006 by Shende et al. The decomposition is applied recursively to generic quantum gates, and can take advantage of existing and future small-circuit optimizations. Because our method uses only one-qubit gates and uniformly controlled rotation-Z gates, it can easily be adapted to use other types of multi-qubit gates. With the proposed decomposition, a general 3-qubit gate can be decomposed using 19 CNOT gates (rather than 20). For general $n$ -qubit gates, the proposed decomposition generates circuits that have $\frac{22}{48}4^n - \frac{3}{2}2^n +\frac{5}{3}$ CNOT gates, which is less that the best known exact decomposition algorithm by $(4^{n-2} -1)/3$ CNOT gates.This paper proposes a new optimized quantum block-ZXZ decomposition method [7,8,10] that results in more optimal quantum circuits than the quantum Shannon decomposition (QSD)[27], which was introduced in 2006 by Shende et al. The decomposition is applied recursively to generic quantum gates, and can take advantage of existing and future small-circuit optimizations. Because our method uses only one-qubit gates and uniformly controlled rotation-Z gates, it can easily be adapted to use other types of multi-qubit gates. With the proposed decomposition, a general 3-qubit gate can be decomposed using 19 CNOT gates (rather than 20). For general $n$-qubit gates, the proposed decomposition generates circuits that have $\frac{22}{48}4^n - \frac{3}{2}2^n +\frac{5}{3}$ CNOT gates, which is less that the best known exact decomposition algorithm by $(4^{n-2} -1)/3$ CNOT gates.

arXiv:2403.13692Mar 20, 20241 citation

n

-qubit gate of this type using at most ${2}^{n\ensuremath{-}1}\ensuremath{-}1$ controlled-NOT gates, ${2}^{n\ensuremath{-}1}$ one-qubit gates, and a single diagonal

n

-qubit gate. To illustrate the versatility of these gates we then apply them to the decomposition of a general

n

-qubit gate and a state preparation procedure. Moreover, we study their implementation using only nearest-neighbor gates. We give upper bounds for the one-qubit and controlled-NOT gate counts for all the aforementioned applications. In all four cases, the proposed circuit topologies either improve on or achieve the previously reported upper bounds for the gate counts. Thus, they provide the most efficient method for general gate decompositions currently known.

Quantum unitaries of the form

Σ_{c}\ket{c}\bra{c}\otimes U_{c}

are ubiquitous in quantum algorithms. This class encompasses not only standard uniformly controlled gates (UCGs) but also a wide range of circuits with uniformly controlled structures. However, their circuit-depth and gate-count complexities have not been systematically analyzed within a unified framework. In this work, we study the general decomposition problem for UCG and UCG-like structure. We then introduce the restricted Uniformly Controlled Gates (rUCGs) as a unified algebraic model, defined by a 2-divisible Abelian group that models the controlled gate set. This model captures uniformly controlled rotations, multi-qubit uniformly controlled gates, and diagonal unitaries. Furthermore, this model also naturally incorporates k-sparse version (k-rUCGs), where only a subset of control qubits participate in each multi-qubit gate. Building on this algebraic model, we develop a general framework. For an n-control rUCG, the framework reduce the gate complexity from

{O(n2^n)}

{O(2^n})

and the circuit depth from

{O(2^n\log n)}

{O(2^n\log n/n)}

. The framework further provides systematic size and depth bounds for k-rUCGs by exploiting sparsity in the control space, with same optimization coefficient as rUCG, respectively. Empirical evaluations on representative QAOA circuits confirm reductions in depth and size, which highlight that the rUCG model and its associated decomposition framework unify circuits previously considered structurally distinct under a single, asymptotically optimal synthesis paradigm.

n

-qubit gates, the proposed decomposition generates circuits that have

\frac{22}{48}4^n - \frac{3}{2}2^n +\frac{5}{3}

CNOT gates, which is less that the best known exact decomposition algorithm by

(4^{n-2} -1)/3

CNOT gates.