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arXiv:1302.0056 · Physical Review A

Synthesis of multivalued quantum logic circuits by elementary gates

Yao-Min Di, Hai‐Rui Wei

Published Jan 22, 2013• 73 citations
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Abstract

We propose the generalized controlled $X$ (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 ($d=4$) circuits and qubit circuits reveals that using ququart circuits may have an advantage over the qubit circuits in the synthesis of quantum circuits.

DOI
10.1103/physreva.87.012325

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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.
In this article the elementary gates for ternary quantum logic circuit are studied. We propose the ternary controlled X (TCX) gate or ternary controlled Z (TCZ) gate as two-qutrit elementary gate, which is universal when assisted by arbitrary one-qutrit gates. It is primitive, efficient and easy to implement. Based on Cartan decomposition, we also give the one-qutrit elementary gates. Then the synthesis of some important ternary gates is investigated and the scheme of physical implementation for these ternary gates is discussed. Finally we extend these elementary gates to more general qudit case, so it provides a unified description for the synthesis of the binary and multi-valued quantum circuits.
Circuit synthesis of multi-controlled gates is crucial for qudit (ddd-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)(n-1)(n−1)-controlled unitaries on nnn qudits, we reduce the controlled-increment (CINC) and generalized controlled-XXX (GCX) gate counts to O(n2)O(n^2)O(n2), improving upon existing O(n2+log⁡2d)O(n^{2+\log_2 d})O(n2+log2​d) CINC and O(n3)O(n^3)O(n3) GCX bounds. For (n−1)(n-1)(n−1)-controlled special unitaries, this complexity is further reduced to O(n)O(n)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 nnn to mmm qudits. Moreover, for the first time, we present a circuit synthesis scheme for single-controlled gates using SUM gates and single-qudit gates when ddd 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 nnn-qudit unitaries.
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.
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.
Recent research in generalizing quantum computation from 2-valued qudits to d-valued qudits has shown practical advantages for scaling up a quantum computer. A further generalization leads to quantum computing with hybrid qudits where two or more qudits have different finite dimensions. In both cases, a quantum computation is performed when a unitary evolution operator, acting as a quantum logic gate, transforms the state of qudits in a quantum system. Unitary operators can be represented by unitary matrices, and therefore for a quantum system consists of multiple qudits, a gate may be synthesized by matrix decomposition techniques such as QR factorization and the Cosine-sine Decomposition (CSD). In this article, we present a CSD based synthesis method for n qudit hybrid and d-valued quantum circuits.
As a model for quantum computation, quantum circuits have found their wide applications in communications, cryptography and information processing. In order to synthesize arbitrary quantum circuits, we present a new type of gate called quantum multiplex rotation gate, which is implemented by simply elementary gates. A method based on QR decomposition and two optimization rules are proposed to decompose general quantum circuit acting on n-qubits into quantum multiplex rotation gates. In comparison with other synthesis algorithms by QR decomposition, our methods achieve better performance in terms of elementary gate counts, 1.2 × 4n approximately.