EN2912C: Future Directions in Computing
Lecture 17: Postulates of Quantum Computing
Prof. Sherief Reda
Division of Engineering
Brown University
Fall 2008
S. Reda. Brown U. EN2912C Fall ‘08
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1. State space postulate
•  The state of a system is described by a unit vector in a
Hilbert space.


α0
 α1 

 where |α0 |2 + |α1 |2 + |α2 |2 + |α3 |2 = 1
 α2 
α3
α0 |00! + α1 |01! + α2 |10! + α3 |11!
complex amplitude
S. Reda. Brown U. EN2912C Fall ‘08
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1. State space postulate
Representations of a single qubit
|ψ! =
!
α
β
"
Bloch sphere
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2. Evolution postulate
A physical system changes in time, and so the state vector
system will actually by a function of time.
|ψ! of a
•  The time-evolution of the state of a closed quantum system is
described by a unitary operator. That is, for any evolution of the
closed system there exists a unitary operator U such that if the initial
state of the system is |ψ!, then after the evolution of the state of the
system will be
|ψ2 ! = U |ψ1 !
The evolution of the state vector of a closed system is linear
U
!
i
S. Reda. Brown U. EN2912C Fall ‘08
αi |ψi ! =
!
i
αi U |ψi !
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2. Evolution postulate
•  The only linear operators that preserve such norms of vectors are
the unitary operators
•  A linear operator is specified completely by its action on a basis
•  Example: Pauli operators
!
"
σ0 = I =
σy = Y =
!
!
1
0
0
i
0
1
−i
0
"
"
σx = X =
σz = Z =
!
0
1
1
0
1
0
0
−1
"
•  Unitary is implied by the continuous time evolution of a closed
quantum mechanical system as described by he Schrodinger
equation
|ψ(t)!
ih
dt
= H(t)|ψ(t)!
|ψ(t2 )!dt = e−ihH(t2 −t1 ) |ψ(t1 )!
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Unitary operator
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3. Composition of system postulate
•  When two physical systems are treated as one combined system,
the state space of the combined physical system is the tensor
product space H1 ⊗ H2 of the state space H1, H2 of the component
subsystems. In the first system is the state |ψ1 ! and the second
system in the state |ψ2 ! , then the state of the combined system is
|ψ1 ! ⊗ |ψ2 !
•  If two qubits are prepared independently, and kept isolated, then
each qubit forms a closed system, and the state can be written in
the product form.
•  If the qubits are allowed to interact, then the closed system includes
both qubits together, and it may not be possible to write the state in
the product form
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4. Measurement postulate
The evolution of the state of a system during a measurement is not
unitary, and an additional postulate is needed to describe measurement
•  For a given orthonormal basis B = {|ψi !} of a state space HA for a
given system A, it is possible to perform a Von Neumann
measurements on system HA with respect to the basis B that, given
a state
|ψ! =
!
αi |φi !
i 2
outputs a label i with probability|αi | and leaves the system in state
|φi !
α0 |0! + α1 |1!
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|b!
b
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4. Measurement postulate
|ψ! =
!
i
αi |φi !
2
∗
p
=
|α
|
=
α
αi = !φi |ψ" i
i
i αi = !ψ|φi "!φi |ψ"
Question: What are the result of measuring the first qubit?
!
!
!
!
1
5
2
3
|ψ! =
|0!|0! +
|0!|1! +
|1!|0! +
|1!|1!
11
11
11
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Two states |ψ! and eiθ |ψ! (differing only by a global phase) are
equivalent
eiθ |ψ! =
!
i
αi eiθ |ψi !
pi = αi∗ e−iθ αi eiθ = |αi |2
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