Abstract
Abstract Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived as a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank 1 idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) U is a symmetric unitary matrix if and only if it has the form I − 2E for a symmetric idempotent E, (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available.
Highlights
Basic matrices are de ned and it is shown that any normal matrix is the product of basic commuting matrices and that the product is unique apart from the order
Unique builders for quantum logic gates are derived as a quantum logic gates is represented by, or is said to be, a unitary matrix
The class of normal matrices include the classes of unitary matrices (UU∗ = I), and Hermitian, called self-adjoint, matrices (H∗ = H). These occur in many applications: for example quantum logic gates are represented by unitary matrices and their properties and applications depend on the structure of unitary matrices
Summary
Basic matrices are de ned and it is shown that any normal matrix is the product of basic commuting matrices and that the product is unique apart from the order. The class of normal matrices include the classes of unitary matrices (UU∗ = I), and Hermitian, called self-adjoint, matrices (H∗ = H) These occur in many applications: for example quantum logic gates are represented by unitary matrices and their properties and applications depend on the structure of unitary matrices. An e cient algorithm is given for expressing a rank r idempotent matrix as a sum of r rank (pure) orthogonal idempotents with increasing initial zeros and such an expression is unique. A quantum logic gate is a unique, apart from order, product of basic logic gates and these basic logic gates are building blocks for quantum logic gates in general.
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