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

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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.