Presented by : Erik Cox, Shannon Hintzman, Mike Miller, Jacquie Otto,

Presented by: Erik Cox, Shannon Hintzman, Mike Miller, Jacquie Otto, Adam Serdar, Lacie Zimmerman

What’s to come… -Brief history and background of quantum mechanics and quantum computation -Linear Algebra required to understand quantum mechanics -Dirac Bra-ket Notation -Modeling quantum mechanics and applying it to quantum computation

History of Quantum Mechanics Classical (Newtonian) Physics Sufficiently describes everyday things and events. Breaks down for very small sizes (quantum mechanics) and very high speeds (theory of relativity).

Why do we need Quantum Mechanics? In short, quantum mechanics describes behaviors that classical (Newtonian) physics cannot. Some behaviors include: - Discreteness of energy - The wave-particle duality of light and matter - Quantum tunneling - The Heisenberg uncertainty principle - Spin of a particle

Spin of a Particle - Discovered in 1922 by Otto Stern and Walther Gerlach - Experiment indicated that atomic particles possess intrinsic angular momentum, called spin, that can only have certain discrete values.

The Quantum Computer Idea developed by Richard Feynman in 1982. Concept:Create a computer that uses the effects of quantum mechanics to its advantage.

Classical Computer Information Quantum Computer Information vs. - Qubit, exists in two states, 0 or 1, and superposition of both - Bit, exists in two states, 0 or 1

Why are quantum computers important? Recently, Peter Shor developed an algorithm to factor large numbers on a quantum computer. Since factoring is key to current encryption, quantum computers would be able to quickly break current cryptography techniques.

In the beginning, there was Linear Algebra… - Complex inner product spaces - Linear Operators - Unitary Operators - Projections - Tensor Products

An inner product space is a complex vector space , together with a map f : V x V → F where F is the ground field C. We write <x, y> instead of f(x, y) and require that the following axioms be satisfied: Complex inner product spaces

and iff denotes complex conjugate (Positive Definiteness) (Conjugate Bilinearity) (Conjugate Symmetry)

Let where Let Complex Conjugate: Example of Complex Inner Product Space:

Let and be vector spaces over , then is a linear operator if . The following properties exist: (Additivity) (Homogeneity) Linear Operators Example:

Unitary Operators Properties: Norm Preserving… Inner Product Preserving… t denotes adjoint

AdjointsMatrix Representation

Definition of Adjoint: Suppose (Inner Product Preserving) (Norm Preserving)

Projections • In quantum mechanics we use orthogonal projections. • Definition: Let V be an inner product space over F. Let M be a subspace of V. Given an element then the orthogonal projection of y onto M is the vector which satisfies where v is orthogonal to every element .

Projections A projection operator P on V satisfies We say P is the projection onto its range, i.e., onto the subspace

Tensor Products In quantum mechanics tensor products are used with : • Vectors • Vector Spaces • Operators • N-Fold tensor products.

Tensor Products If and , there is a natural mapping defined by We use notation w  v to symbolize T(w, v) and call w  v the tensor productof w and v.

Tensor Products • W  V means the vector space consisting of all finite formal sums: where and

Tensor Product with Operators If A, B are operators on W and V we define AB on WV by

Tensor Product Properties 4 Properties of Tensor Products • a(w  v) = (aw)  v = w  (av) for all a in C; • (x+ y)  v = x v + y  v; • w  (x + y) = w  x + w  y; •  w  x | y  z  =  w | y   x | v . Note: | is the notation used for inner products in quantum mechanics.

Property #1: Proof Property #1: a(w  v) = (aw)  v = w  (av) for all a in C Example in :

Property #1: Proof Example in :

Property #2: Proof Property #2: (x + y)  v = x v + y v Example in :

Property #3: Proof Property #3: w  (x + y) = w x + w y Example in :

Example in : Property #3: Proof

Property #4: Proof Property #4:  w  x | y  z  =  w | y   x | z  Example in :

Property #4: Proof

Dirac Bra-Ket Notation Notation Inner Products Outer Products Completeness Equation Outer Product Representation of Operators

Bra-Ket Notation Involves Vector Xn can be represented two ways Ket |n> Bra <n| = |n>t is the complex conjugate of m

Inner Products An Inner Product is a Bra multiplied by a Ket <x| |y> can be simplified to <x|y> = <x|y> =

Outer Products An Outer Product is a Ket multiplied by a Bra |y><x| = = By Definition

Completeness Equation is used to create a identity operator represented by vector products. Let |i>, i = 1, 2, ..., n, be a basis for V and v is a vector in V So Effectively

Proof for the Completeness Equation Using Linear Algebra, the basis of a vectors space can be represented series of vectors with a one in each successive position and zeros in every other (aka {1, 0, 0, ... }, {0, 1, 0, ...}, {0, 0, 1, ...}, ...) So |i><i| will create a matrix with a one in each successive position along the diagonal. etc.

Completeness Cont. Thus = + + + ... = = I

One application of the Dirac notation is to represent Operators in terms of inner and outer products. and

If A is an operator, we can represent A by applying the completeness equation twice this gives the following equation: • This shows that any operator has an outer product representation and that the entries of the associated matrix for the basis |i are:

Projections • Projection is a type of operator • Application of inner and outer products

We can represent graphically: Using the rule of dot products we know Given that we can say Linear Algebra View

Using these facts we can solve for and Again using the rule of dot products We get Linear Algebra View (Cont.)

So Plugging this back into the original equation Gives us: Linear Algebra View (Cont.)

If is a unit vector Linear Algebra View (Cont.)

Given that and This graph is a representation of Given and Projections in Quantum Mechanics

being the full basis of We can regard the full basis of as being For some On Basis Projections in QM (cont.)

Taking the inner products gives Therefore and more generally So Projections in QM (cont.)

Presented by : Erik Cox, Shannon Hintzman, Mike Miller, Jacquie Otto,