Multi-controlled gates are essential primitives in quantum algorithms, yet implementing them via standard gate-level decompositions remains resource-intensive. We develop efficient pulse-level implementations of multi-controlled gates in trapped-ion systems using the Cirac-Zoller scheme. We first show that the Cirac-Zoller construction admits a freedom in the sign choice of red-sideband (RSB) pulses, which leaves the logical operation invariant up to a local Pauli- $Z$ correction. By exploiting this freedom, we construct equivalent realizations of multi-controlled gates and develop pulse cancellation for more efficient implementations of successive gates. We perform numerical simulations and show that pulse cancellation reduces the gate time and improves the state fidelity. Furthermore, we propose ancilla-free circuits for general $N$ -controlled gates that use a single-controlled gate primitive and $\mathcal{O}(N)$ RSB pulses. As a key application, we apply our pulse cancellation to the linear combination of unitaries (LCU) method for block encoding. We show that the RSB-pulse cost of the select operator over $L$ unitaries can be reduced from $\mathcal{O}(L\log L)$ to $\mathcal{O}(L)$ , which improves the efficiency and scalability of LCU-based quantum circuits.Multi-controlled gates are essential primitives in quantum algorithms, yet implementing them via standard gate-level decompositions remains resource-intensive. We develop efficient pulse-level implementations of multi-controlled gates in trapped-ion systems using the Cirac-Zoller scheme. We first show that the Cirac-Zoller construction admits a freedom in the sign choice of red-sideband (RSB) pulses, which leaves the logical operation invariant up to a local Pauli-$Z$ correction. By exploiting this freedom, we construct equivalent realizations of multi-controlled gates and develop pulse cancellation for more efficient implementations of successive gates. We perform numerical simulations and show that pulse cancellation reduces the gate time and improves the state fidelity. Furthermore, we propose ancilla-free circuits for general $N$-controlled gates that use a single-controlled gate primitive and $\mathcal{O}(N)$ RSB pulses. As a key application, we apply our pulse cancellation to the linear combination of unitaries (LCU) method for block encoding. We show that the RSB-pulse cost of the select operator over $L$ unitaries can be reduced from $\mathcal{O}(L\log L)$ to $\mathcal{O}(L)$, which improves the efficiency and scalability of LCU-based quantum circuits.

quant-pharXiv:2605.04654Quantum Science and TechnologyMay 7, 20260 citations

Optimal Control and Circuit Synthesis of Quantum Gates

Hyeong-Jin Kim

Jan 1, 20210 citations

Efficient High-Dimensional Quantum Circuit Synthesis: From Multi-Controlled Gates to Isometries and Quantum Channels

Gui-Long Jiang

Circuit synthesis of multi-controlled gates is crucial for qudit ( $d$ -level) quantum computing. This paper presents efficient synthesis schemes that reduce the elementary gate count for multi-controlled single-qudit gates. For synthesizing general $(n-1)$ -controlled unitaries on $n$ qudits, we reduce the controlled-increment (CINC) and generalized controlled- $X$ (GCX) gate counts to $O(n^2)$ , improving upon existing $O(n^{2+\log_2 d})$ CINC and $O(n^3)$ GCX bounds. For $(n-1)$ -controlled special unitaries, this complexity is further reduced to $O(n)$ . Furthermore, we present a method for approximately synthesizing multi-controlled unitary gates with a linear complexity in the number of control qudits. By utilizing the proposed circuit, we present qudit-based circuit constructions for isometries and quantum channels from $n$ to $m$ qudits. Moreover, for the first time, we present a circuit synthesis scheme for single-controlled gates using SUM gates and single-qudit gates when $d$ is prime. This enables all CINC-based circuits for various quantum operations to be converted into SUM-gate circuits while preserving the same asymptotic complexity. Finally, we establish a theoretical lower bound on the number of SUM and CINC gates required to synthesize general $n$ -qudit unitaries.Circuit synthesis of multi-controlled gates is crucial for qudit ($d$-level) quantum computing. This paper presents efficient synthesis schemes that reduce the elementary gate count for multi-controlled single-qudit gates. For synthesizing general $(n-1)$-controlled unitaries on $n$ qudits, we reduce the controlled-increment (CINC) and generalized controlled-$X$ (GCX) gate counts to $O(n^2)$, improving upon existing $O(n^{2+\log_2 d})$ CINC and $O(n^3)$ GCX bounds. For $(n-1)$-controlled special unitaries, this complexity is further reduced to $O(n)$. Furthermore, we present a method for approximately synthesizing multi-controlled unitary gates with a linear complexity in the number of control qudits. By utilizing the proposed circuit, we present qudit-based circuit constructions for isometries and quantum channels from $n$ to $m$ qudits. Moreover, for the first time, we present a circuit synthesis scheme for single-controlled gates using SUM gates and single-qudit gates when $d$ is prime. This enables all CINC-based circuits for various quantum operations to be converted into SUM-gate circuits while preserving the same asymptotic complexity. Finally, we establish a theoretical lower bound on the number of SUM and CINC gates required to synthesize general $n$-qudit unitaries.

quant-pharXiv:2607.08200Jul 10, 20260 citations

Systematic construction of multi-controlled Pauli gate decompositions with optimal $T$ -countSystematic construction of multi-controlled Pauli gate decompositions with optimal $T$-count

Ken M. Nakanishi, Synge Todo

Multi-controlled Pauli gates are typical high-level qubit operations that appear in the quantum circuits of various quantum algorithms. We find multi-controlled Pauli gate decompositions with smaller CNOT-count or $T$ -depth while keeping the currently known minimum $T$ -count. For example, for the CCCZ gate, we find decompositions with CNOT-count 7 or $T$ -depth 2 while keeping the $T$ -count at the currently known minimum of 6. The discovery of these efficient decompositions improves the computational efficiency of many quantum algorithms. What led to this discovery is the systematic procedure for constructing multi-controlled Pauli gate decompositions. This procedure not only deepens our theoretical understanding of quantum gate decomposition but also leads to more efficient decompositions that have yet to be discovered.Multi-controlled Pauli gates are typical high-level qubit operations that appear in the quantum circuits of various quantum algorithms. We find multi-controlled Pauli gate decompositions with smaller CNOT-count or $T$-depth while keeping the currently known minimum $T$-count. For example, for the CCCZ gate, we find decompositions with CNOT-count 7 or $T$-depth 2 while keeping the $T$-count at the currently known minimum of 6. The discovery of these efficient decompositions improves the computational efficiency of many quantum algorithms. What led to this discovery is the systematic procedure for constructing multi-controlled Pauli gate decompositions. This procedure not only deepens our theoretical understanding of quantum gate decomposition but also leads to more efficient decompositions that have yet to be discovered.

arXiv:2410.00910Oct 1, 20241 citation

GEOMETRIC CONSTRUCTION OF MULTI-BIT QUANTUM GATES

Klaus Mølmer, Anders Søndberg Sørensen, Xiaoguang Wang

Solvay Conference on PhysicsAug 1, 20030 citations

Z

correction. By exploiting this freedom, we construct equivalent realizations of multi-controlled gates and develop pulse cancellation for more efficient implementations of successive gates. We perform numerical simulations and show that pulse cancellation reduces the gate time and improves the state fidelity. Furthermore, we propose ancilla-free circuits for general

N

-controlled gates that use a single-controlled gate primitive and

\mathcal{O}(N)

RSB pulses. As a key application, we apply our pulse cancellation to the linear combination of unitaries (LCU) method for block encoding. We show that the RSB-pulse cost of the select operator over

L

unitaries can be reduced from

\mathcal{O}(L\log L)

\mathcal{O}(L)

, which improves the efficiency and scalability of LCU-based quantum circuits.

Circuit synthesis of multi-controlled gates is crucial for qudit (

d

-level) quantum computing. This paper presents efficient synthesis schemes that reduce the elementary gate count for multi-controlled single-qudit gates. For synthesizing general

(n-1)

-controlled unitaries on

n

qudits, we reduce the controlled-increment (CINC) and generalized controlled-

X

(GCX) gate counts to

O(n^2)

, improving upon existing

O(n^{2+\log_2 d})

CINC and

O(n^3)

GCX bounds. For

(n-1)

-controlled special unitaries, this complexity is further reduced to

O(n)

. Furthermore, we present a method for approximately synthesizing multi-controlled unitary gates with a linear complexity in the number of control qudits. By utilizing the proposed circuit, we present qudit-based circuit constructions for isometries and quantum channels from

n

m

qudits. Moreover, for the first time, we present a circuit synthesis scheme for single-controlled gates using SUM gates and single-qudit gates when

d

is prime. This enables all CINC-based circuits for various quantum operations to be converted into SUM-gate circuits while preserving the same asymptotic complexity. Finally, we establish a theoretical lower bound on the number of SUM and CINC gates required to synthesize general

n

-qudit unitaries.

Systematic construction of multi-controlled Pauli gate decompositions with optimal

T

-count

T

-depth while keeping the currently known minimum

T

-count. For example, for the CCCZ gate, we find decompositions with CNOT-count 7 or

T

-depth 2 while keeping the

T

-count at the currently known minimum of 6. The discovery of these efficient decompositions improves the computational efficiency of many quantum algorithms. What led to this discovery is the systematic procedure for constructing multi-controlled Pauli gate decompositions. This procedure not only deepens our theoretical understanding of quantum gate decomposition but also leads to more efficient decompositions that have yet to be discovered.