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理论物理通讯：英文版

Realization of Two-Qutrit Quantum Gates with Control Pulses

Di Yao

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

We investigate the realization of 2-qutrit logic gate in a bipartite 3-level system with qusi-Ising interaction. On the basis of Cartan decomposition of matrices,the unitary matrices of 2-qutrit are factorized into products of a series of realizable matrices.It is equivalent to exerting a certain control field on the system,and the control goal is usually gained by a sequence of control pulses.The general discussion on the realization of 2-qutrit logic gate is made first,and then the realization of the ternary SWAP gate and the ternry SWAP~(1/2)gate are discussed specifically,and the sequences of control pulses and drift processes implementing these gates are given.

Related papers

Realization of Two-Qutrit Quantum Gates with Control Pulses

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We investigate the realization of 2-qutrit logic gate in a bipartite 3-level system with qusi-Ising interaction. On the basis of Cartan decomposition of matrices, the unitary matrices of 2-qutrit are factorized into products of a series of realizable matrices. It is equivalent to exerting a certain control field on the system, and the control goal is usually gained by a sequence of control pulses. The general discussion on the realization of 2-qutrit logic gate is made first, and then the realization of the ternary SWAP gate and the ternary gate are discussed specifically, and the sequences of control pulses and drift processes implementing these gates are given.

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In this paper, the synthesis and implementation of three-quhit SWAP gate is discussed. The three-quhit SWAP gate can be decomposed into product of 2 two-qubit SWAP gates, and it can be realized by 6 CNOT gates. Research illustrated that although the result is very simple, the current methods of matrix decomposition for multi-qubit gate can not get that. Then the implementation of three-qubit SWAP gate in the three spin system with Ising interaction is investigated and the sequence of control pulse and drift process to implement the gate is given. It needs 23 control pulses and 12 drift processes. Since the interaction can not be switched on and off at will, the realization of three-qubit SWAP gate in specific quantum system also can not simply come down to 2 two-qubit SWAP gates.

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Cartan decomposition of a two-qutrit gate

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我们与 bequsi 相互作用在一个由两部组成的 3 水平系统调查 2-qutrit 逻辑门的实现。根据矩阵的 Cartan 分解， 2-qutrit 的单一的矩阵是进一系列可实现的矩阵的产品的 factorized。它等价于在系统，和控制目标上施加某个控制域被控制脉搏的一个序列通常获得。逻辑门被成为第一的 2-qutrit 的实现的一般讨论，然后实现第三交换门，第三的门明确地被讨论，并且控制脉搏的序列和实现这些门的飘移过程被给。

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Although only two quantum states of a physical system are often used to encode quantum information in the form of qubits, many levels can in principle be used to obtain qudits and increase the information capacity of the system. To take advantage of the additional levels, a parameterization of unitary transformations in terms of experimentally realizable operations is needed. Many parameterizations of unitary 3 * 3 matrices (U(3)) exist. One decomposition of a general unitary matrix can be expressed as the product of an exponential of a diagonal matrix and an exponential of an off-diagonal matrix. This decomposition is relevant for controlling superconducting qutrits using fixed-frequency resonant control pulses. This decomposition is numerically confirmed to allow the parameterization of any element in U(3). It is shown that a simple setting of parameter ranges of parameters can easily lead to an over-parameterization, in the sense that several different sets of values for the parameters produce the same element in U(3). This fact is demonstrated using the Walsh-Hadamard (WH) matrix as an example, which is also a special qutrit gate of practical interest. The different decompositions are shown to be related, and the relationships between them are presented using general methods. The shortest path needed for the implementation of a qutrit gate is found. Other parameterizations obtained by other analytic means, which can be advantageous for various reasons, are also discussed.

arXiv:2506.17797•Physical Review A•Jun 15, 2026•0 citations
We investigate the realization of 2-qutrit logic gate in a bipartite 3-level system with qusi-Ising interaction. On the basis of Cartan decomposition of matrices,the unitary matrices of 2-qutrit are factorized into products of a series of realizable matrices.It is equivalent to exerting a certain control field on the system,and the control goal is usually gained by a sequence of control pulses.The general discussion on the realization of 2-qutrit logic gate is made first,and then the realization of the ternary SWAP gate and the ternry SWAP~(1/2)gate are discussed specifically,and the sequences of control pulses and drift processes implementing these gates are given.
We investigate the realization of 2-qutrit logic gate in a bipartite 3-level system with qusi-Ising interaction. On the basis of Cartan decomposition of matrices, the unitary matrices of 2-qutrit are factorized into products of a series of realizable matrices. It is equivalent to exerting a certain control field on the system, and the control goal is usually gained by a sequence of control pulses. The general discussion on the realization of 2-qutrit logic gate is made first, and then the realization of the ternary SWAP gate and the ternary gate are discussed specifically, and the sequences of control pulses and drift processes implementing these gates are given.
In this paper, the synthesis and implementation of three-quhit SWAP gate is discussed. The three-quhit SWAP gate can be decomposed into product of 2 two-qubit SWAP gates, and it can be realized by 6 CNOT gates. Research illustrated that although the result is very simple, the current methods of matrix decomposition for multi-qubit gate can not get that. Then the implementation of three-qubit SWAP gate in the three spin system with Ising interaction is investigated and the sequence of control pulse and drift process to implement the gate is given. It needs 23 control pulses and 12 drift processes. Since the interaction can not be switched on and off at will, the realization of three-qubit SWAP gate in specific quantum system also can not simply come down to 2 two-qubit SWAP gates.
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.
我们与 bequsi 相互作用在一个由两部组成的 3 水平系统调查 2-qutrit 逻辑门的实现。根据矩阵的 Cartan 分解， 2-qutrit 的单一的矩阵是进一系列可实现的矩阵的产品的 factorized。它等价于在系统，和控制目标上施加某个控制域被控制脉搏的一个序列通常获得。逻辑门被成为第一的 2-qutrit 的实现的一般讨论，然后实现第三交换门，第三的门明确地被讨论，并且控制脉搏的序列和实现这些门的飘移过程被给。
Although only two quantum states of a physical system are often used to encode quantum information in the form of qubits, many levels can in principle be used to obtain qudits and increase the information capacity of the system. To take advantage of the additional levels, a parameterization of unitary transformations in terms of experimentally realizable operations is needed. Many parameterizations of unitary 3 * 3 matrices (U(3)) exist. One decomposition of a general unitary matrix can be expressed as the product of an exponential of a diagonal matrix and an exponential of an off-diagonal matrix. This decomposition is relevant for controlling superconducting qutrits using fixed-frequency resonant control pulses. This decomposition is numerically confirmed to allow the parameterization of any element in U(3). It is shown that a simple setting of parameter ranges of parameters can easily lead to an over-parameterization, in the sense that several different sets of values for the parameters produce the same element in U(3). This fact is demonstrated using the Walsh-Hadamard (WH) matrix as an example, which is also a special qutrit gate of practical interest. The different decompositions are shown to be related, and the relationships between them are presented using general methods. The shortest path needed for the implementation of a qutrit gate is found. Other parameterizations obtained by other analytic means, which can be advantageous for various reasons, are also discussed.