Quantum Computing

Shortcut to this page: ntrllog.netlify.app/quantum

Notes provided by Professor Negin Forouzesh (CSULA) and Quantum Computing for Computer Scientists

Apparently complex numbers are used a lot in quantum mechanics. So this will start off with some light review of complex numbers and linear algebra.

Complex Numbers (Algebraically)

A complex number `c` is an expression `c=a+bi` where `a,b in RR`. `a` is referred to as the real part of `c` and `b` is the imaginary part of `c`.

With this in mind, we can also define a complex number as an ordered pair of real numbers: `c=(a,b)`.

Addition and Subtraction

Let `c_1=(a_1,b_1)` and `c_2=(a_2,b_2)` be two complex numbers.

`c_1+c_2=(a_1,b_1)+(a_2,b_2)=(a_1+a_2,b_1+b_2)`

`c_1-c_2=(a_1,b_1)-(a_2,b_2)=(a_1-a_2,b_1-b_2)`

Multiplication

`c_1xxc_2=(a_1,b_1)xx(a_2,b_2)=(a_1a_2-b_1b_2,a_1b_2+a_2b_1)`

This comes from the same idea as FOIL.

`c_1xxc_2=(a_1+b_1i)xx(a_2+b_2i)`

`=a_1a_2+a_1b_2i+b_1ia_2+b_1b_2i^2`

`=a_1a_2+a_1b_2i+b_1ia_2-b_1b_2`

`=a_1a_2-b_1b_2+a_1b_2i+a_2b_1i`

`=(a_1a_2-b_1b_2)+(a_1b_2+a_2b_1)i`

Division

`c_1/c_2=\frac{(a_1,b_1)}{(a_2,b_2)}=(a_1a_2+b_1b_2)/(a_2^2+b_2^2)+(a_2b_1-a_1b_2)/(a_2^2+b_2^2)i`

Let `(x,y)=\frac{(a_1,b_1)}{(a_2,b_2)}` be the quotient of two complex numbers.

From this, we get

`(a_1,b_1)=(x,y)xx(a_2,b_2)=(a_2x-b_2y,a_2y+b_2x)`

So we have

`(1): a_1=a_2x-b_2y`

`(2): b_1=a_2y+b_2x`

We can solve for `x` by multiplying `(1)` by `a_2` and multiplying `(2)` by `b_2`:

`a_1a_2=a_2^2x-b_2a_2y`

`b_1b_2=a_2b_2y+b_2^2x`

`a_1a_2+b_1b_2=a_2^2x-b_2a_2y+a_2b_2y+b_2^2x`

`a_1a_2+b_1b_2=a_2^2x+b_2^2x`

`a_1a_2+b_1b_2=x(a_2^2+b_2^2)`

`x=\frac{a_1a_2+b_1b_2}{a_2^2+b_2^2}`

We can solve for `y` by multiplying `(1)` by `b_2` and multiplying `(2)` by `-a_2`:

`a_1b_2=a_2b_2x-b_2^2y`

`-b_1a_2=-a_2^2y-a_2b_2x`

`a_1b_2-b_1a_2=a_2b_2x-b_2^2y-a_2^2y-a_2b_2x`

`a_1b_2-b_1a_2=-b_2^2y-a_2^2y`

`a_1b_2-b_1a_2=-y(b_2^2y+a_2^2)`

`\frac{a_1b_2-b_1a_2}{a_2^2+b_2^2}=-y`

`y=\frac{a_2b_1-a_1b_2}{a_2^2+b_2^2}`

Conjugation

Changing the sign of only the imaginary part of a complex number is called conjugation. Formally, if `c=a+bi`, then the conjugate of `c` is

`bar c=a-bi`.

Modulus

The modulus of a complex number `c=a+bi` is

`+sqrt(a^2+b^2)`

and is denoted as `abs(c)`.

The modulus is a generalization of the absolute value operation for real numbers.

`abs(a)=+sqrt(a^2)`

Complex Numbers (Geometrically)

As we've seen, we can represent complex numbers as ordered pairs. This naturally lends itself to being graphable on a plane.

Cartesian Representation

For example, the complex number `2+3i` would look like

The complex plane (where the real numbers sit on the horizontal axis and the imaginary numbers sit on the vertical axis) is also called the Argand plane.

So we can think of complex numbers as vectors on the complex plane.

Notice that if we calculate the length of the vector, we get the modulus.

Polar Representation

To convert a complex number from Cartesian coordinates to polar coordinates, we need `rho` and `theta`:

`rho=sqrt(a^2+b^2)`

`theta=tan^(-1)(b/a)`

To go back from polar to Cartesian:

`a=rho cos(theta)`

`b=rho sin(theta)`

Multiplication

Let `c_1=(rho_1,theta_1)` and `c_2=(rho_2,theta_2)`.

`c_1xxc_2=(rho_1,theta_1)xx(rho_2,theta_2)=(rho_1rho_2,theta_1+theta_2)`

Complex Vector Spaces

A complex vector space is a set of vectors where each element of a vector is a complex number.

For example, `CC^4` is the set of `4`-element vectors and an element of `CC^4` would look like this:

`[[6-4i],[7+3i],[4.2-8.1i],[-3i]]`

Just like with regular vectors, adding complex vectors is the same.

`CC^n` with vector addition is an Abelian group, which means:

`CC^n` is closed under vector addition
vector addition is associative
`CC^n` has an identity element: the zero vector
every vector in `C^n` has an inverse
vector addition is commutative

(The first four properties are what makes a group and the last property is what makes a group Abelian.)

And so is scalar multiplication.

`CC^n` with scalar multiplication (and vector addition) is a complex vector space, which means:

`1*V=V`
`c_1*(c_2*V)=(c_1xxc_2)*V`
`c*(V+W)=c*V+c*W`
`(c_1+c_2)*V=c_1*V+c_2*V`

`CC^(mxxn)` is the set of `mxxn` matrices, and it is also a complex vector space.

Matrix addition, scalar multiplication, and the transpose is the same as regular matrices.

For a complex matrix `A`,

the conjugate of `A`, denoted `bar A`, is obtained by conjugating each element of `A`
the adjoint of `A`, denoted `A^†`, is obtained by transposing and conjugating `A`

Basis

A set of complex vectors `B` is a basis of a complex vector space if

every vector can be written as a linear combination of vectors from `B`
`B` is linearly independent

Dimension

The dimension of a complex vector space is the number of elements in a basis of the space.

Inner Products

For two vectors `V_1=[[r_0],[r_1],[vdots],[r_n]]` and `V_2=[[r_0'],[r_1'],[vdots],[r_n']]`, the inner product is defined as

`(:V_1,V_2:)=V_1^T***V_2=sum_(j=0)^nr_jr_j'`

For two matrices `A,B in RR^(nxxn)`, the inner product is defined as

`(:A,B:)=Trace(A^T***B)`

where the trace of a matrix is the sum of the diagonal elements of the matrix.

For complex vectors, the inner product is defined as

`(:V_1,V_2:)=V_1^†***V_2`

For two matrices `A,B in CC^(nxxn)`, the inner product is defined as

`(:A,B:)=Trace(A^†***B)`

Norm

The norm (length) of a vector `V` is defined as

`abs(V)=sqrt((:V,V:))`

Orthogonal/Orthonormal

Two vectors are orthogonal if their inner product is equal to `0`.

A basis is orthogonal if all the vectors in the basis are pairwise orthogonal to each other.

`(:V_j,V_k:)=0` for `j ne k`

An orthogonal basis is orthonormal if all the vectors in the basis is of norm `1`.

`(:V_j,V_k:)={(1,if j=k),(0,if j ne k):}`

Trig

In `RR^3`, the inner product `(:V,V':)` is the same as

`abs(V)abs(V')costheta`

where `theta` is the angle between `V` and `V'`.

Eigenvalues and Eigenvectors

For a matrix `A in CC^(nxxn)`, if there is a number `c in CC` and a vector `V ne 0` such that

`AV=c cdotV`

then `c` is an eigenvalue of `A` and `V` is an eigenvector of `A` associated with `c`.

Hermitian Matrices

A complex matrix `A` is hermitian if `A^†=A`.

Unitary Matrices

A matrix is unitary if the matrix is invertible, and its inverse is its adjoint.

`U***U^†=U^†***U=I`

A matrix `A` is invertible if there exists `A^(-1)` such that

`A***A^(-1)=A^(-1)***A=I`

`I` is the identity matrix, which means only `1`s along the diagonal and `0`s everywhere else.

Unitary matrices preserve inner products: `(:UV,UV':)=(:V,V':)`.

`(:UV,UV':)`

`=(UV)^†***UV'`

`=V^†U^†***UV'`

`=V^†***I***V'`

`=V^†***V'`

`=(:V,V':)`

Unitary matrices also preserve norms: `abs(UV)=abs(V)`

`abs(UV)`

`sqrt((:UV,UV:))`

`=sqrt((:V,V:))`

`=abs(V)`

Tensor Product

For vectors,

`[[x],[y]]ox[[a],[b],[c]]=[[x cdot [[a],[b],[c]]],[y cdot [[a],[b],[c]]]]`

For matrices,

`[[a_(0,0),a_(0,1)],[a_(1,0),a_(1,1)]]ox[[b_(0,0),b_(0,1),b_(0,2)],[b_(1,0),b_(1,1),b_(1,2)],[b_(2,0),b_(2,1),b_(2,2)]]=[[a_(0,0) cdot [[b_(0,0),b_(0,1),b_(0,2)],[b_(1,0),b_(1,1),b_(1,2)],[b_(2,0),b_(2,1),b_(2,2)]], a_(0,1) cdot [[b_(0,0),b_(0,1),b_(0,2)],[b_(1,0),b_(1,1),b_(1,2)],[b_(2,0),b_(2,1),b_(2,2)]]], [a_(1,0) cdot [[b_(0,0),b_(0,1),b_(0,2)],[b_(1,0),b_(1,1),b_(1,2)],[b_(2,0),b_(2,1),b_(2,2)]], a_(1,1) cdot [[b_(0,0),b_(0,1),b_(0,2)],[b_(1,0),b_(1,1),b_(1,2)],[b_(2,0),b_(2,1),b_(2,2)]]]]`

`=[[a_(0,0)xxb_(0,0),a_(0,0)xxb_(0,1),a_(0,0)xxb_(0,2),a_(0,1)xxb_(0,0),a_(0,1)xxb_(0,1),a_(0,1)xxb_(0,2)],[a_(0,0)xxb_(1,0),a_(0,0)xxb_(1,1),a_(0,0)xxb_(1,2),a_(0,1)xxb_(1,0),a_(0,1)xxb_(1,1),a_(0,1)xxb_(1,2)],[a_(0,0)xxb_(2,0),a_(0,0)xxb_(2,1),a_(0,0)xxb_(2,2),a_(0,1)xxb_(2,0),a_(0,1)xxb_(2,1),a_(0,1)xxb_(2,2)],[a_(1,0)xxb_(0,0),a_(1,0)xxb_(0,1),a_(1,0)xxb_(0,2),a_(1,1)xxb_(0,0),a_(1,1)xxb_(0,1),a_(1,1)xxb_(0,2)],[a_(1,0)xxb_(1,0),a_(1,0)xxb_(1,1),a_(1,0)xxb_(1,2),a_(1,1)xxb_(1,0),a_(1,1)xxb_(1,1),a_(1,1)xxb_(1,2)],[a_(1,0)xxb_(2,0),a_(1,0)xxb_(2,1),a_(1,0)xxb_(2,2),a_(1,1)xxb_(2,0),a_(1,1)xxb_(2,1),a_(1,1)xxb_(2,2)]]`