The Group Zoo of Classical Reversible Computing and Quantum Computing

Abstract

By systematically inflating the group of $n \times n$ permutation matrices to the group of $n \times n$ unitary matrices, we can see how classical computing is embedded in quantum computing. In this process, an important role is played by two subgroups of the unitary group U(n), i.e. XU(n) and ZU(n). Here, XU(n) consists of all $n \times n$ unitary matrices with all line sums (i.e. the n row sums and the n column sums) equal to 1, whereas ZU(n) consists of all $n \times n$ diagonal unitary matrices with upper-left entry equal to 1. As a consequence, quantum computers can be built from NEGATOR gates and PHASOR gates. The NEGATOR is a 1-qubit circuit that is a natural generalization of the 1-bit NOT gate of classical computing. In contrast, the PHASOR is a 1-qubit circuit not related to classical computing.

Appendix

For any matrix P from the group P(n), the following property holds:

$$ P = C \left( \begin{array}{cc} 1 &{} \\ &{} P\, ' \end{array} \right) , $$

where

• $P\, '$ is a member of P($n-1$) and

• C is a member of CP(n).

Here, CP(n) denotes the group of $n \times n$ circulant permutation matrices. It is a group isomorphic with the cyclic group Z $_n$, a finite group of order n. Remarkable is the fact that here ${\tiny \left( \begin{array}{cc} 1 &{} \\ &{} P\ ' \end{array} \right) }$ is only multiplied to the left with a circulant matrix, whereas in decomposition (18.5), the matrix ${\tiny \left( \begin{array}{cc} 1 &{} \\ &{} X\, ' \end{array} \right) }$ is multiplied both to the left and to the right with a circulant matrix.

The decomposition algorithm is very straightforward: suffice it to choose C such that it has the same leftmost column as the given matrix P. Subsequently, the matrix ${\tiny \left( \begin{array}{cc} 1 &{} \\ &{} P\, ' \end{array} \right) }$ follows automatically by computing $C^{-1}P$. Here follows an example from P(4):

$$ \left( \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \end{array} \right) = \left( \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \end{array} \right) \left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{array} \right) . $$

By applying the theorem again and again, we find the following decomposition:

$$\begin{aligned} P = C_n\ \left( \begin{array}{cc} 1 &{} \\ &{} C_{n-1} \end{array} \right) \ \ldots \ \ \left( \begin{array}{cc} {\mathbf 1}_{n-3} &{} \\ &{} C_3 \end{array} \right) \ \left( \begin{array}{cc} {\mathbf 1}_{n-2} &{} \\ &{} C_2 \end{array} \right) . \end{aligned}$$

(18.7)

where all $C_k$ are CP(k) matrices. We conclude: any $n \times n$ permutation matrix can be decomposed as a product of $n-1$ matrices of the form ${\tiny \left( \begin{array}{cc} {\mathbf 1}_{n-k} &{} \\ &{} C_k \end{array} \right) }$. We give an example:

$$ \left( \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \end{array} \right) = \left( \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \end{array} \right) \left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \end{array} \right) \left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 \end{array} \right) . $$

Decomposition (18.6) is not a straightforward generalization of (18.7). This constitutes an illustration of the fact that, in spite of an overall similarity between the group P(n) and the group XU(n), literal translations from P(n) properties to XU(n) properties sometimes fail [24].