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These points may be joined forming a shape like a spider. Joined points represent an "or" condition, also known as a logical disjunction. A spider diagram is a boolean expression involving unitary spider diagrams and the logical symbols ,,. For example, it may consist of the conjunction of two spider diagrams, the disjunction of two spider ...
An example ZX-diagram. This one has two inputs (wires coming from the left), and three outputs (wires exiting to the right), and hence it represents a linear map from to . ZX-diagrams consist of green and red nodes called spiders, which are connected by wires. Wires may curve and cross, arbitrarily many wires may connect to the same spider, and ...
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U −1 equals its conjugate transpose U *, that is, if = =, where I is the identity matrix.. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger ( † ), so the equation above is written
In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication.The unitary group is a subgroup of the general linear group GL(n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant 1.
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G.The general theory is well-developed in the case that G is a locally compact topological group and the representations are strongly continuous.
The unitary property of the S-matrix is directly related to the conservation of the probability current in quantum mechanics. The probability current density J of the wave function ψ ( x ) is defined as J = ℏ 2 m i ( ψ ∗ ∂ ψ ∂ x − ψ ∂ ψ ∗ ∂ x ) . {\displaystyle J={\frac {\hbar }{2mi}}\left(\psi ^{*}{\frac {\partial \psi ...
On the other hand, if is compact, then every finite-dimensional representation of admits an inner product with respect to which is unitary, showing that decomposes as a sum of irreducibles. [9] Similarly, if g {\displaystyle {\mathfrak {g}}} is a complex semisimple Lie algebra, every finite-dimensional representation of g {\displaystyle ...
As unitary matrices are useful in quantum computation, and Householder transformations are unitary, they are very useful in quantum computing. One of the central algorithms where they're useful is Grover's algorithm, where we are trying to solve for a representation of an oracle function represented by what turns out to be a Householder ...