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Science in China. Series G, Physics, mechanics & astronomy/Science in China. Series G, Physics, Mechanics & Astronomy

Cartan decomposition of a two-qutrit gate

Di Yao

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

The decomposition of matrices corresponding to the 2-qutrit logic gate by succes-sive Cartan decomposition is investigated, and written in an exponential form based on the relationship between Lie group and Lie algebra, thus making them able to relate with the control field and the Hamiltonian of the system to perform the gate. Finally the decomposition of the ternary SWAP gate is presented in detail.

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The decomposition of matrices corresponding to the 2-qutrit logic gate by succes-sive Cartan decomposition is investigated, and written in an exponential form based on the relationship between Lie group and Lie algebra, thus making them able to relate with the control field and the Hamiltonian of the system to perform the gate. Finally the decomposition of the ternary SWAP gate is presented in detail.
The relationship between quantum state control and matrix decomposition is reviewed.The matrix decomposition newly applied in quantum information science,i.e.Cartan decomposition based on group theory and cosine-sine decomposition based on numerical linear algebra,is introduced firstly.Then the current research situation and the results in this area are reviewed.These results play important roles in quantum entanglement dynamics,the control of quantum states and optimization of quantum circuit Finally,the Cartan decomposition of a 2-qutrit logic gate is 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.
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.
This chapter discusses in detail the case of two-level systems and the associated Lie group. It gives an example of a general decomposition of SU(n), namely the decomposition into planar rotation, which is quite simple and can be obtained by straightforward computations. This decomposition gives a direct way to parametrize the special unitary group. The chapter presents the basics of Cartan decompositions of semisimple Lie groups. It gives some examples of applications of Lie group decompositions and introduces some notions of Lie group and Lie algebra theory. The chapter discusses Cartan decompositions of Lie groups. Cartan decompositions appear to be ubiquitous in quantum control and dynamics and in quantum information theory.
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.