Quantum Computing
Lecture on Linear Algebra
Sources: Angela Antoniu,
Bulitko, Rezania, Chuang,
Nielsen
Introduction to Quantum Mechanics
• This can be found in Marinescu and in Chuang and
Nielsen
• Objective
– To introduce all of the fundamental principles of Quantum
mechanics
• Quantum mechanics
– The most realistic known description of the world
– The basis for quantum computing and quantum information
• Why Linear Algebra?
– LA is the prerequisite for understanding Quantum Mechanics
• What is Linear Algebra?
– … is the study of vector spaces… and of
– linear operations on those vector spaces
Linear algebra -Lecture objectives
• Review basic concepts from Linear Algebra:
–
–
–
–
–
–
–
Complex numbers
Vector Spaces and Vector Subspaces
Linear Independence and Bases Vectors
Linear Operators
Pauli matrices
Inner (dot) product, outer product, tensor product
Eigenvalues, eigenvectors, Singular Value Decomposition (SVD)
• Describe the standard notations (the Dirac notations)
adopted for these concepts in the study of Quantum
mechanics
• … which, in the next lecture, will allow us to study the
main topic of the Chapter: the postulates of quantum
mechanics
Review: Complex numbers
• A complex number z  C is of the form a ,b  R
where z  a  ib
and i2=-1
• Polar representation
z  u e  , where u ,θ  R
n
n
n
i
n
n
n
n
n
n
n
n
• With un  a 2  b 2 the modulus or magnitude
n
• And the phase
n
b
 n  arctan
 na 


n 
• Complex conjugate
z  u cos  i sin
n
n
n
 zn  an  ibn   an  ibn

n

Review: The Complex Number System
• Another definitions: It is the extension of the real number
system via closure under exponentiation.
i  -1
c  a  bi
(c  C, a,b  R)
The “imaginary” Re [c ]  a
unit
Im [c]  b
• (Complex) conjugate:
+i
b

c* = (a + bi)*  (a  bi)
• Magnitude or absolute value:
|c|2 = c*c = a2+b2
c  c*c  (a  bi )(a  bi )  a 2  b2
c
a
+
“Real” axis
“Imaginary”
i
axis
Review: Complex
Exponentiation
ei
+i
• Powers of i are complex
units:
θi
e  cos   i sin 
• Note:
i
ei/2 = i
ei = 1
e3 i /2 =  i
e2 i = e0 = 1

1
+1
i
Z1=2 e
Z12 = (2 e i)2 = 2 2 (e i)2 = 4 (e i )2
= 4 e 2i
2
4
Recall: What is a qubit?
• A qubit has two possible states
0 or 1
• Unlike bits, a qubit can be in a state other than
0 or 1
• We can form linear combinations of states
  0   1
• A qubit state is a unit vector in a two-dimensional
complex vector space
Properties of Qubits
• Qubits are computational basis states
- orthonormal basis
0 for i  j
 ij  
1 for i  j
i j   ij
- we cannot examine a qubit to determine its quantum state
- A measurement yields
0 with probability 
1 with probability 
2
where     1
2
2
2
(Abstract) Vector Spaces
• A concept from linear algebra.
• A vector space, in the abstract, is any set of objects that can be
combined like vectors, i.e.:
– you can add them
• addition is associative & commutative
• identity law holds for addition to zero vector 0
– you can multiply them by scalars (incl. 1)
• associative, commutative, and distributive laws hold
• Note: There is no inherent basis (set of axes)
– the vectors themselves are the fundamental objects
– rather than being just lists of coordinates
Vectors
• Characteristics:
– Modulus (or magnitude)
– Orientation
• Matrix representation of a vector
 z1 
v     (a column), and its dual
 zn 

v  v  z1 ,, zn  (row vector)
This is adjoint, transpose and
next conjugate
Operations
on vectors
Vector Space, definition:
• A vector space (of dimension n) is a set of n vectors
satisfying the following axioms (rules):
– Addition: add any two vectors v and v ' pertaining to a
vector space, say Cn, obtain a vector,
 z1  z1' 
v  v'    
 z n  z n' 


•
•
•
•
the sum, with the properties :
v  v'  v'  v
Commutative:
Associative:  v  v'   v' '  v   v'  v' ' 
Any v has a zero vector (called the origin):
To every v in Cn corresponds a unique vector - v
such as v  0  v
v   v   0
– Scalar multiplication:  next slide
Operations
on vectors
Vector Space (cont)
Scalar multiplication: for any scalar
z  C and vector v  C n there is a vector
 zz1 
z v    , the scalar product,in
insuch
suchway
waythat
that
 zzn 
 Multiplication by scalars is Associative:
z z ' v    zz ' v

distributive with respect to vector addition:
z  v  v'   z v  z v'
1v  v
Operations
on vectors
 Multiplication by vectors is
distributive with respect to scalar addition:
 z  z ' v  z v  z ' v
A Vector subspace in an n-dimensional vector
space is a non-empty subset of vectors satisfying the same
axioms
Hilbert spaces
• A Hilbert space is a vector space in which the
scalars are complex numbers, with an inner
product (dot product) operation  : H×H  C
– Definition of inner product:
Black dot is
an inner
xy
=
(yx)*
(*
=
complex
conjugate)
product
xx  0
“Component” xx = 0 if and only if x = 0
picture:
xy is linear, under scalar multiplication
y
and vector addition within both x and y
x
xy/|x|
Another notation often used:
x y  x y
“bracket”
Vector Representation of States
• Let S={s0, s1, …} be a maximal set of
distinguishable states, indexed by i.
• The basis vector vi identified with the ith such state
can be represented as a list of numbers:
s0 s1 s2 si-1 si si+1
vi = (0, 0, 0, …, 0, 1, 0, … )
• Arbitrary vectors v in the Hilbert space can then be
defined by linear combinations of the vi:
v   ci vi  (c0 , c1 ,)
i
• And the inner product is given by:
x y   x*i yi
i
Dirac’s Ket Notation
You havr to be familiar
with these three notations
• Note: The inner product
x y   x* yi
definition is the same as the
“Bracket” i
 y1 
matrix product of x, as a
*
*




 x1 x2   y2 
conjugated row vector, times
  
y, as a normal column vector.
• This leads to the definition, for state s, of:
i
– The “bra” s| means the row matrix [c0* c1* …]
 c1 
– The “ket” |s means the column matrix

 
†
• The adjoint operator takes any matrix M
to its conjugate transpose M†  MT*, so
s| can be defined as |s†, and xy = x†y.
c
 2
  
Linear Algebra
Vector Spaces
Complex number
field
n
C
Spanning Set and Basis vectors
Or SPANNING SET for Cn: any set of n vectors such that
any vector in the vector space Cn can be written using the n
base vectors
Spanning set
Example for C2 (n=2):
is a set of all
such vectors
for any alpha
and beta
which is a linear combination of the 2-dimensional
basis vectors 0 and 1
Bases and Linear
Independence
Linearly
independent
vectors
in the space
Red and blue
vectors add to 0,
are not linearly
independent
Always exists!
Basis
Bases for
n
C
So far we talked only about vectors and operations on
them. Now we introduce matrices
Linear Operators
Linear Operators
New space
Pauli Matrices
X is like inverter
Properties: Unitary
and Hermitian
σ  σ  I, k

k
k
σ   σ

k
k
This is
adjoint
Matrices
Pay attention
to this
notation
Examples of operators
Inner Products of vectors
Complex
numbers
We already talked
about this when we
defined Hilbert space
Be able to prove these properties from definitions
Slightly other formalism for Inner
Products
Be familiar with
various formalisms
Example: Inner
n
Product on C
Norms
Outer Products of vectors
This is
Kronecker
operation
Outer Products of vectors
|u> <v| is an outer
product of |u> and
|v>
|u> is from U,
|v> is from V.
|u><v| is a map
V U
We will illustrate how this
can be used formally to
create unitary and other
matrices
Eigenvectors of linear operators and
their Eigenvalues
Eigenvalues of matrices are
used in analysis and
synthesis
Eigenvalues and Eigenvectors
versus diagonalizable matrices
Eigenvector of
Operator A
Diagonal Representations of matrices
Diagonal matrix
Adjoint Operators
This is very
important, we
have used it
many times
already
Normal and
Hermitian Operators
But not necessarily
equal identity
Unitary Operators
Exam Problems
• Diagonalization of unitary matrices
Unitary and Positive
Operators: some properties
Other notation for adjoint
(Dagger is also used
Positive operator
Positive definite
operator
Hermitian Operators: some
properties in different notation
These are important and
useful properties of our
matrices of circuits
Tensor Products of Vector Spaces
Notation for vectors
in space V
Note various
notations
Tensor Products of vectors and
Tensor Products of Operators
Properties of tensor
products for vectors
Tensor product for
operators
Tensor Product of two
Matrices
Properties of Tensor Products
of vectors and operators
These can be vectors of any size
We repeat them in
different notation
here
Functions of
Operators
I is the identity
matrix
Matrix of Pauli
rotation X
Spectral
decomposition
eigenvalues
Trace and
Commutator
Polar
Decomposition
Left polar decomposition
Right polar decomposition
Eigenvalues and Eigenvectors
More on Inner Products
Hilbert Space:
Orthogonality:
Norm:
Orthonormal basis:
Review to remember
Quantum Notation
(Sometimes denoted by bold fonts)
(Sometimes called Kronecker
multiplication)
Bibliography & acknowledgements
• Michael Nielsen and Isaac Chuang, Quantum
Computation and Quantum Information, Cambridge
University Press, Cambridge, UK, 2002
• R. Mann,M.Mosca, Introduction to Quantum
Computation, Lecture series, Univ. Waterloo, 2000
http://cacr.math.uwaterloo.ca/~mmosca/quantumcou
rsef00.htm
• Paul Halmos, Finite-Dimensional Vector Spaces,
Springer Verlag, New York, 1974