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Requirements On Mathematics Apparatus
• Physical states
• Mathematic entities
• Interference phenomena
• Nondeterministic predictions
• Model the effects of measurement
• Distinction between evolution and measurement
What’s Quantum Mechanics
• A mathematical framework
• Description of the world known
• Rather simple rules
but counterintuitive
applications
Introduction to Linear Algebra
• Quantum mechanics The basis for quantum computing and
quantum information
• Why Linear Algebra?Prerequisities
• What is Linear Algebra concerning?Vector spacesLinear operations
Basic linear algebra useful in QM
• Complex numbers
• Vector space
• Linear operators
• Inner products
• Unitary operators
• Tensor products
• …
Dirac-notation: Dirac-notation: Bra and Bra and KetKet
• For the sake of simplification
• “ket” stands for a vector in Hilbert
• “bra” stands for the adjoint of
• Named after the word “bracket”
Hilbert Space Hilbert Space FundamentalsFundamentals
• Inner product space: linear space equipped with inner product
• Hilbert Space (finite dimensional): can be considered as inner product space of a quantum system
• Orthogonality: • Norm: • Unit vector parallel to |v:
0wv
vvv
v
v
Hilbert Space (Cont’d)
• Orthonormal basis:
a basis set where
• Can be found from an arbitrary basis set by Gram-Schmidt Orthogonalization
nvv ,...,1 ijji vv
Inner Products
• Inner Product is a function combining two vectors
• It yields a complex number• It obeys the following rules
• •
C ),(
kkk
kkk wvawav ,,
*),(),( wvwv
0),( vv
Unitary Operator
• An operator U is unitary, if
• Preserves Inner product
IUUτ
Uofadjoint for the stands Uwhere
wvwUvU ,,
Tensor ProductTensor Product
• Larger vector space formed from two
smaller ones
• Combining elements from each in all
possible ways
• Preserves both linearity and scalar
multiplication
Mathematically, what is a qubit ? (1)
• We can form linear combinations of
states
• A qubit state is a unit vector in a two
dimensional complex vector space
Qubits Cont'd
• We may rewrite as…
• From a single measurement one obtains only a single bit of information about the state of the qubit
• There is "hidden" quantum information and this information grows exponentially
0 1
cos 0 sin 12 2
i ie e
cos 0 sin 12 2
ie
We can ignore ei as it has no
observable effect
Any pair of linearly independent Any pair of linearly independent vectors can be a basis!vectors can be a basis!
Postulates in QMPostulates in QM
• Why are postulates important?… they provide the connections between the
physical, real, world and the quantum mechanics mathematics used to model these systems
- Isaak L. Chuang
24242424
Physical Systems -Physical Systems - Quantum Mechanics Connections Quantum Mechanics Connections
Postulate 1Isolated physical
system Hilbert Space
Postulate 2Evolution of a
physical system
Unitary transformation
Postulate 3Measurements of a
physical system
Measurement operators
Postulate 4Composite physical
system
Tensor product of
components
entanglement
We assume the opposite Leads to contradiction, so we
cannot decompose as this
Entangled state as opposed to separable states
The Measurement Problem
Can we deduce postulate 3 from 1 and 2?
Joke. Do not try it. Slides are from MIT.
Quantum Computing Mathematics and Postulates
Advanced topic seminar SS02
“Innovative Computer architecture and concepts”Examiner: Prof. Wunderlich
Presented byPresented by
Chensheng QiuChensheng Qiu
Supervised bySupervised by
Dplm. Ing. Gherman Dplm. Ing. Gherman
Examiner: Prof. Wunderlich
Anuj Dawar , Michael Nielsen
Sources