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arXiv:1105.5485

Elementary gates for ternary quantum logic circuit

Yao-Min Di, Hai‐Rui Wei

Published May 27, 2011• 29 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1105.5485

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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.
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.
Because ternary computer has more superiority than other d-ary number systems, we focus on the investigation of ternary elementary quantum gates and the synthesis algorithm of ternary quantum logic circuits. Above all, Pauli operators and their matrices on qutrit are introduced. Then eight qutrit operators are selected as elementary operators and eight qutrit quantum logic gates are defined. Permutation groups are introduced to characterize the quantum gates and quantum logic circuits. Some important qutrit quantum logic gates are defined also, such as QNOT, QKCXi, EQKCXi, QSwap, QCNOT and EQCNOT. Based on these elementary gates, we prove two very important theorems: (1) all qutrit quantum reversible logic circuit can be generated by Xi gate and QKCXi gate; (2) all qutrit quantum reversible logic circuits can be generated by Xi gate and QCNOT gate. The two theorems indicate that any complicated qutrit quantum reversible circuit can be constructed by the simplest ternary quantum gate. This will greatly simplify the implementation difficulty of quantum circuit. Subsequently, we propose a synthesis algorithm for qutrit quantum reversible logic circuit, which is verified through simulation experiment by the computer program we have designed.
Basic logic gates and their operations in ternary quantum domain are involved in the synthesis of ternary quantum circuits. Only a few works define ternary algebra for ternary quantum logic realization. In this paper, a ternary logic function is expressed in terms of projection operations including a new one. A method to realize new multi-qutrit ternary gates in terms of generalized ternary gates and projection operations is also presented. We also introduced ten simplification rules for reducing ancilla qutrits and gate levels. Our method yields lower gate cost and fewer gate levels and ancilla qutrits than that obtained by earlier methods for the ternary benchmark circuits. The nnn qutrit ternary sum function is synthesized without any ancilla qutrit by our proposed methodology.
Synthesis of ternary quantum circuits involves basic ternary gates and logic operations in the ternary quantum domain. Works that define ternary algebra and their applications for ternary quantum logic realization, are very few. In this paper, we express a ternary logic function in terms of projection operations including a new one. We demonstrate how to realize a few new multi-qutrit ternary gates in terms of generalized ternary gates and projection operations. We then employ our synthesis method to design ternary adder circuits which have better cost than that obtained by earlier method.
Abstract — This paper investigates a universal approach of synthesizing arbitrary ternary logic circuits in quantum computation based on the truth table technology. It takes into account of the relationship of classical logic and quantum logic circuits. By adding inputs with constant value and garbage outputs, the classical non-reversible logic can be transformed into reversible logic. Combined with group theory, it provides an algorithm using the ternary Swap gate, ternary NOT gate and ternary Toffoli gate library. Simultaneously, the main result shows that the numbers of qutrits we use are minimal compared to other methods. We also illustrate with two examples to test our approach. Index Terms — ternary quantum computation, reversible logic, circuit synthesis, group theory I.
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.