Optimal synthesis of multivalued quantum circuits

Abstract

Although many of works have been done in multivalued quantum logic synthesis, the question whether multivalued quantum circuits are more efficient than the conventional binary quantum circuits is still open. In this article we devote to the optimization of generic multivalued quantum circuits. The multivalued quantum Shannon decompositions (QSD) are improved so that the circuits obtained are asymptotically optimal for all dimensionality d. The syntheses of uniformly multifold controlled R y 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 efficiency index is proposed to evaluate the efficiency of synthesis of various quantum circuits. The algorithm for qudit circuits given here is an efficient synthesis routine which produces best known results for all dimensionality d, and for both cases the number of qudit n is small and that is asymptotic. The multivalued quantum circuits are indeed more efficient than the binary quantum circuits. The facts, the leading factor of the lower bound of complexity for qudit circuits is small by a factor of d − 1 in comparison to that for qubit circuits and the asymptotic efficiency index is increased with the increase of dimensionality d, reveal the potential advantage of qudit circuits over generic qubit circuits. The generic nqudit circuits with d ≥ 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.

Quantifying the magic of quantum channels

“…[83][84][85][86][87][88][89][90]) and qudits (e.g. [91][92][93][94][95]). Elementary twoqudit gates include the controlled-increment gate [91] and the generalized controlled-X gate [93,94].…”

“…[91][92][93][94][95]). Elementary twoqudit gates include the controlled-increment gate [91] and the generalized controlled-X gate [93,94]. More recently, the synthesis of single-qutrit gates was studied in [96,97].…”

Normal form for single-qutrit Clifford+Toperators and synthesis of single-qutrit gates

“…Much work has been done in the area of gate synthesis for qudits, e.g., [12][13][14][15][16][17][18][19]. It has been shown in [13,14,20] that any multi-qudit gate may be implemented exactly by composing single-qudit gates with any two-qudit gate (known as the elementary two-qudit gate) that creates entanglement without ancillas.…”

“…It has been shown in [13,14,20] that any multi-qudit gate may be implemented exactly by composing single-qudit gates with any two-qudit gate (known as the elementary two-qudit gate) that creates entanglement without ancillas. Some examples of elementary two-qudit gates include the controlled-increment gate proposed by Brennan et al in [17], and the generalized controlled-X gate proposed by Di and Wei in [18,19]. These authors investigated the synthesis of multi-qudit gates using the proposed elementary two-qudit gate, assuming the ability to implement an arbitrary single-qudit gate.…”

Synthesis and Upper Bound of Schmidt Rank of Bipartite Controlled‐Unitary Gates

“…One of the central problems in quantum computing is to minimize the number of one-and two-partite gates required to implement a desired quantum gate. Utilizing Cartan decomposition, [44] Vatan et al [45] designed a controlled-NOT (CNOT)-optimized general two-qubit quantum circuit in 2004, Shende et al [46] presented the highest known lower bound on asymptotic CNOT cost required to implement an unstructured n-qubit quantum computing, Di and Wei [47,48] synthesized universal multiple-valued quantum circuits. Using higher-dimensional Hilbert spaces, Lanyon et al [1] reduced the cost of a Toffoli gate from six CNOTs to three CNOTs in 2009, Li et al [49] further optimize the n-qubit universal quantum circuit, Liu and Wei [14] decreased the complexity of a Fredkin gate to three entangling gates in 2020.…”