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

• URL: http://arxiv.org/abs/2607.08200v1
• Date: Thu, 09 Jul 2026 07:57:57 GMT
• Title: Efficient High-Dimensional Quantum Circuit Synthesis: From Multi-Controlled Gates to Isometries and Quantum Channels
• Abstract summary: Circuit of multi-controlled gates is crucial for qudit ($d$-level) quantum computing.<n>This paper presents efficient synthesis schemes that reduce the elementary gate count for multi-controlled single-qudit gates.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $O(n)$. By utilizing the proposed circuit, we present qudit-based circuit constructions for isometries and quantum channels from $n$ to $m$ qudits. When specialized to general $n$-qudit unitaries, our construction requires fewer CINC gates than previous results. 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.

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