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arXiv:2607.08200

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

Gui-Long Jiang

quant-phSubmitted Jul 9, 2026• Announced Jul 10, 2026• 0 citations
v1 · Jul 9, 2026v2 · Aug 18, 2026
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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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Quantum circuits of a general quantum gate acting on multiple ddd-level quantum systems play a prominent role in multi-valued quantum computation. We first propose a new recursive Cartan decomposition of semi-simple unitary Lie group U(3n)U(3^n)U(3n) (arbitrary nnn-qutrit gate). Note that the decomposition completely decomposes an n-qutrit gate into local and non-local operations. We design an explicit quantum circuit for implementing arbitrary two-qutrit gates, and the cost of our construction is 21 generalized controlled X (GCX) and controlled increment (CINC) gates less than the earlier best result of 26 GGXs. Moreover, we extend the program to the nnn-qutrit system, and the quantum circuit of generic nnn-qutrit gates contained 4196⋅32n−4⋅3n−1−(n22+n4−2932)\frac{41}{96}\cdot3^{2n}-4\cdot3^{n-1}-(\frac{n^2}{2}+\frac{n}{4}-\frac{29}{32})9641​⋅32n−4⋅3n−1−(2n2​+4n​−3229​) GGXs and CINCs is presented. Such asymptotically optimal structure is the best known result so far.
Implementation of high-dimensional (HD) quantum gates shows very promising perspectives for HD quantum computation. A bipartite quantum system with arbitrary dimensions nnn and mmm is termed a quNit-quMit. Here we propose a synthesis scheme to construct the quantum circuit for general quNit-quMit gates with controlled increment (CINC) gates and local gates. This shows that CINC gates combined with local gates form a universal gate set for HD quantum computation. An upper bound of O(n2)O(n^2)O(n2) CINC gates is achieved for arbitrary quNit-quMit gate implementation in the proposed scheme, which is the best known result. Especially for the controlled quNit-quMit gates, our scheme requires only 2 CINC gates, whereas the previous scheme required 2n2n2n.
We propose the generalized controlled XXX (gcx) gate as the two-qudit elementary gate, and based on Cartan decomposition, we also give the one-qudit elementary gates. Then we discuss the physical implementation of these elementary gates and show that it is feasible with current technology. With these elementary gates many important qudit quantum gates can be synthesized conveniently. We provide efficient methods for the synthesis of various kinds of controlled qudit gates and greatly simplify the synthesis of existing generic multi-valued quantum circuits. Moreover, we generalize the quantum Shannon decomposition (QSD), the most powerful technique for the synthesis of generic qubit circuits, to the qudit case. A comparison of ququart (d=4d=4d=4) circuits and qubit circuits reveals that using ququart circuits may have an advantage over the qubit circuits in the synthesis of quantum circuits.
Abstract Implementation of high-dimensional (HD) quantum gates shows very promising perspectives for HD quantum computation. A bipartite quantum system with arbitrary dimensions n and m is termed a quNit–quMit. Here we propose a synthesis scheme to construct the quantum circuit for general quNit–quMit gates with controlled increment (CINC) gates and local gates. This shows that CINC gates combined with local gates form a universal gate set for HD quantum computation. An upper bound of O ( n 2 ) CINC gates is achieved for arbitrary quNit–quMit gate implementation in the proposed scheme, which is the best known result. Especially for the controlled quNit–quMit gates, our scheme requires only 2 CINC gates, whereas the previous scheme required 2 n .
Although much work has been devoted to multivalued quantum logic synthesis, the question of whether multivalued quantum circuits are more efficient than conventional binary quantum circuits is still open. This article is devoted to the optimization of generic multivalued quantum circuits. Multivalued quantum Shannon decompositions are improved so that the circuits obtained are asymptotically optimal for all dimensionality ddd. The syntheses of uniformly multifold controlled Ry{R}_{y}Ry​ rotations are also optimized to make the circuits further simplified. Moreover, the theoretical lower bound of complexity for multivalued quantum circuits is investigated, and a quantity known as the efficiency index is proposed to evaluate the efficiency of the synthesis of various quantum circuits. The algorithm for qudit circuits given here is an efficient synthesis routine which produces improved results for all dimensionality ddd, whether the number of qudits nnn is small or large. The two facts, the leading factor of the lower bound of complexity for qudit circuits is small by a factor of $d\ensuremath{-}1$ in comparison to that for qubit circuits and the asymptotic efficiency index is increased with an increase of dimensionality ddd, reveal the potential advantage of qudit circuits over generic qubit circuits. Generic nnn-qudit circuits with $d\ensuremath{\ge}5$ and generic two-ququart circuits synthesized by the algorithm given here are practical circuits which are more efficient than the most efficient qubit circuits.
One of the challenges in quantum computing is the synthesis of unitary operators into quantum circuits with polylogarithmic gate complexity. Exact synthesis of generic unitaries requires an exponential number of gates in general. We propose a novel approximate quantum circuit synthesis technique by relaxing the unitary constraints and interchanging them for ancilla qubits via block encodings. This approach combines smaller block encodings, which are easier to synthesize, into quantum circuits for larger operators. Due to the use of block encodings, our technique is not limited to unitary operators and can be applied for the synthesis of arbitrary operators. We show that operators which can be approximated by a canonical polyadic expression with a polylogarithmic number of terms can be synthesized with polylogarithmic gate complexity with respect to the matrix dimension.