# Decomposition of orthogonal matrix and synthesis of two-qubit and three-qubit orthogonal gates

## Abstract

The decomposition of matrices associated to two-qubit and
three-qubit orthogonal gates is studied, and based on the
decomposition the synthesis of these gates is investigated. The
optimal synthesis of general two-qubit orthogonal gate is obtained.
For two-qubit unimodular orthogonal gate, it requires at most 2 CNOT
gates and 6 one-qubit gates. For the general three-qubit
unimodular orthogonal gate, it can be synthesized by 16 CNOT gates
and 36 one-qubit and gates
in the worst case.

###### pacs:

03.67.Lx, 03.65.Fd## I Introduction

In quantum computing, the algorithms are commonly described by the quantum circuit model 1 (). The building blocks of quantum circuits are quantum gates, i.e., unitary transformations acting on a set of qubits. In 1995, Barenco et al showed that any qubit quantum circuit can be decomposed into a sequence of one-qubit gates and CNOT gates 2 (). The process of constructing quantum circuits by these elementary gates is called synthesis by some authors. The complexity of quantum circuit can be measured in terms of the number of CNOT and one-qubit elementary gates required. Achieving gate arrays of less complexity is crucial not only because it reduces resource, but it also reduces errors.

Decomposition of matrix plays very important role to synthesize and optimize quantum gates. Based on Cartan decomposition 3 (); 4 (), the synthesis, optimization and “small circuit” structure of two-qubit gate are well solved 5 (); 6 (); 7 (); 8 (); 9 (). To implement the general two-qubit gate, it requires at most 3 CNOT gates and 15 elementary one-qubit gates from the family 7 (); 8 ().

Unfortunately, the aforementioned optimal synthesis of most general two-qubit quantum gates have not yet led to similarly tight results for three-qubit gates. Based on one of Cartan decompositions for multi-qubit system, Khaneja-Glaser decomposition (KGD) 4 (), Vatan and Williams get the result that a general three-qubit quantum gate can be synthesized using at most 40 CNOT gates and 98 one-qubit and gates 10 (). Using the modified KGD, the results have been improved in 11 (), that is it requires at most 26 CNOT gates and 73 one-qubit and gates. Now the best known result is based on quantum Shannon decomposition (QSD) 12 () proposed by Shende, Bullock and Markov, it requires at most 20 CNOT gates. According to the result of multi-qubit case, the best known theoretical lower bound on CNOT gate cost for general three-qubit gates is 14 8 (). However, no circuit construction yielding these numbers of CNOT gates has been presented in the literature.

The orthogonal gate is an important class of gate, the matrix corresponding to the gate is orthogonal. For example, classical reversible logic circuits have a long history 13 () and are a necessary subclass whose realization is required for any quantum computer to be universal. The matrix elements of them are all real, so they are orthogonal. Utilizing the basic property of magic basis, in 2004, Vatan and Williams investigated the synthesis of two-qubit orthogonal gate in 7 (). The result is that the synthesis of the unimodular orthogonal gate requires at most 2 CNOT gates and 12 elementary one-qubit gates. As for the non-unimodular orthogonal gate, that is its matrix determinant is equal to minus one, it requires at most 3 CNOT gates and 12 elementary one-qubit gates 7 (). The number of the one-qubit gates required can still be reduced further. Moreover, no articles discuss the synthesis of general orthogonal three-qubit quantum gates yet.

In this work, we devote to investigating the synthesis of two-qubit and three-qubit orthogonal gates. For this purpose, we study the Cartan decomposition of matrix for these gates first. Based on the particular decompositions, the two kinds of synthesis are obtained. For two-qubit unimodular orthogonal gate, it requires at most 2 CNOT gates and 6 one-qubit gates, beating an earlier bound of 2 CNOT gates and 12 one-qubit elementary gates. The numbers required for one-qubit gate and CNOT gate are all reach the lower bound. For three-qubit unimodular orthogonal gate, it can be synthesized by 16 CNOT gates and 36 one-qubit and gates in the worst case.

This paper is organized as follows. The concept of Cartan decomposition and its application in quantum information science (QIS) are briefly introduced in Section II. Based on a kind of Cartan decomposition of special orthogonal group , we provide an optimal synthesis of general two-qubit orthogonal gate in Section III. The decomposition of the group associated to three-qubit unimodular orthogonal gate is investigated in Section IV. The synthesis of general three-qubit unimodular orthogonal gate is studied in Section V. It is first time to discuss the synthesis of this kind gate. A brief conclusion is made in Section VI.

## Ii Cartan Decomposition and Its Application in QIS

The Cartan decomposition of Lie group 3 () depends on the decomposition of its Lie algebra. A Cartan decomposition of a real semisimple Lie algebra is the decomposition

(1) |

where is the orthogonal complement of with respect to the Killing form, and satisfy the commutation relations:

(2) |

is a Lie subalgebra of . A maximal Abelian subalgebra contained in is called a Cartan subalgebra of the pair denoted as . Then using the relation between Lie group and Lie algebra, every element of the Lie group can be written as

(3) |

where , , and .

There are many kinds of Cartan decomposition for semisimple Lie groups. Now the main application in quantum information science is the decomposition of group for multi-qubit system, i.e. Khaneja-Glaser Decomposition (KGD) 4 (). Moreover there are some other decompositions, such as Concurrence Canonical Decomposition (CCD) 14 (); 15 () which is a decomposition of group too, the Odd-Even Decomposition (OED) 16 (), which is a generalization of CCD to more general multipartite quantum system case. Some kinds of Cartan decomposition for a bipartite high dimension quantum system were discussed in 17 (); 18 (); 19 (). These Cartan decompositions have been applied in the synthesis and implementation of quantum logic gates 10 (); 11 (); 20 (); 21 (), the entanglement of multipartite quantum systems 14 (); 15 (), etc. But we need to find new suitable algebraic structures of Cartan decomposition to meet the purpose here.

## Iii Optimal Synthesis of General Two-Qubit Orthogonal Gates

We now consider the decomposition of 4 dimensional special orthogonal group associated to the two-qubit unimodular orthogonal gate. Difference from that in 7 (), the Lie algebra is constructed as

(4) |

in which each basis vector involves a matrix. A kind of Cartan decomposition of algebra is that

(5) |

with

(6) |

(7) |

where is a Lie subalgebra and . Its Cartan subalgebra is

(8) |

Utilizing the relation between Lie group and Lie algebra, the Cartan decomposition of Lie group can be obtained. For every element , we have

(9) |

where , and is a two-qubit operation of the form

(10) |

where .

The can be represented by the synthesis of elementary gates as

(11) |

Here and afterwards denotes the CNOT gate that control on the -th qubit and target on the -th qubit, and is an elementary one-qubit gate acting on the -th qubit. Combing Eqs.(9), (10) and (11), we can get the synthesis of general two-qubit orthogonal gate as in Fig.1, it requires at most 2 CNOT gates and 6 one-qubit gates. As for the non-unimodular orthogonal gate (the determinant is equal to minus one), it requires at most 3 CNOT gates and 6 one-qubit gates. The 2 CNOT gates is optimal for CNOT gate cost of two-qubit orthogonal gate, and it has been proved in 9 (). Since a matrix has 6 independent parameters, it needs at least 6 elementary one-qubit gates to load them. So the synthesis of general two-qubit orthogonal gate here is optimal both for CNOT gates and elementary one-qubit gates.

## Iv Decomposition of General Three-qubit Unimodular Orthogonal Gate

The matrices of any general three-qubit unimodular orthogonal gate are elements of special orthogonal group . We construct the Lie algebra first. Taking AI type of Cartan decomposition 3 () on and Lie algebra,

(12) |

(13) |

with

(14) |

(15) |

(16) |

A set of basis for a Lie algebra is given by 28 tensor products of the form

(17) |

Using the transformation matrix in 22 (); 23 ():

(18) |

the Lie algebra can be obtained by . So the Lie algebra is isomorphic to , and the basis in Eq.(18) can be called as magic basis of algebra.

Then we take Cartan decomposition of the Lie algebra as Eq.(1), with

(19) |

(20) |

The is isomorphic to . The Cartan subalgebra of the pair can be chosen as

(21) |

Using the formula

(22) |

it is easy to verify that the and in Eqs.(19, 20) satisfy the conditions of the Cartan decomposition in Eq.(2).

Lie subalgebra could be decomposed further

(23) |

with

(24) |

(25) |

The is isomorphic to . Its Cartan subalgebra can be chosen as

(26) |

From the correspondence between Lie group and Lie algebra and the conjugative transformation, we get the Cartan decomposition of Lie group : any element of the group can be decomposed as

(27) |

Here , and are the Abelian subgroup associated to Cartan subalgebra and respectively.

## V Synthesis of General Three-qubit Unimodular Orthogonal Gate

Based on the discussion in Section IV, the decomposition of general three-qubit orthogonal gate is shown in Fig.2, where and

(28) |

(29) |

The synthesis of transformation matrix is given in 7 () shown in Fig.3, that is

(30) |

The can be expressed as

(31) |

here

(32) | |||||

Since the Cartan subalgebra is commutative, we can break down the synthesis of into the following operations:

(33) |

(34) |

(35) |

And we have

(36) |

(37) |

(38) |

By putting Eqs.(36), (37) and (38) together, we get

(39) |

Here the identity is used. Combining Eqs.(30), (31) and (39), we have

(40) | |||||

and its circuit shown in Fig.4. Since gates commute with the control qubit of the CNOT gate, here the gates in and are canceled. The synthesis of is shown in Fig.5, that is

(41) |

Putting all these pieces together, we get that 16 CNOT gates and 36 one-qubit and gates at most are sufficient to synthesize general three-qubit orthogonal gate. For the same reason mentioned above, the two gates acting on same qubit neighbored have been combined to one. Here, each requires 1 CNOT gates and 3 one-qubit and gates, requires 6 CNOT gates and 6 one-qubit gates, each requires 4 CNOT gates and 2 one-qubit and gates, and 8 gates require 20 one-qubit and gates.

## Vi Conclusions

Based on the decomposition of matrices, the synthesis of two-qubit and three-qubit orthogonal gates is investigated. For two-qubit orthogonal gate, we get optimal result, which requires at most 2 CNOT gates and 6 one-qubit gates, beating an earlier bound of 2 CNOT gates and 12 one-qubit gates. For the three-qubit unimodular orthogonal gate, it requires 16 CNOT gates and 36 one-qubit gates from the family in the worst case. There are abundant algebraic structures for matrix decomposition of three-qubit orthogonal gate. We have many ways to investigate the synthesis of general three-qubit orthogonal gate. The result given here is the best one we have got, although we can not affirm that is optimal yet. The synthesis of general three-qubit gate has been studied in some literatures 10 (); 11 (); 12 (), the orthogonal gate is an important class of gate of them. So the work here is essentially on the “small circuit” issue of three-qubit gates, which is first investigated in this paper. Different from two-qubit gate, how to get optimal quantum circuit for general three-qubit gate has not been well solved and is worthy studying further.

## Acknowledgements

The work was supported by the Project of Natural Science Foundation of Jiangsu Education Bureau, China(Grant No.09KJB140010).

### Footnotes

- Email address: yaomindi@sina.com

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