EN2912C: Future Directions in Computing Lecture 17...

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S. Reda. Brown U. EN2912C Fall ‘08 EN2912C: Future Directions in Computing Lecture 17: Postulates of Quantum Computing Prof. Sherief Reda Division of Engineering Brown University Fall 2008 1

Transcript of EN2912C: Future Directions in Computing Lecture 17...

S. Reda. Brown U. EN2912C Fall ‘08

EN2912C: Future Directions in Computing Lecture 17: Postulates of Quantum Computing

Prof. Sherief Reda Division of Engineering

Brown University Fall 2008

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S. Reda. Brown U. EN2912C Fall ‘08

1. State space postulate

•  The state of a system is described by a unit vector in a Hilbert space.

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!

""#

!0

!1

!2

!3

$

%%& |!0|2 + |!1|2 + |!2|2 + |!3|2 = 1where

!0|00! + !1|01! + !2|10! + !3|11!complex amplitude

S. Reda. Brown U. EN2912C Fall ‘08

1. State space postulate

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Bloch sphere

Representations of a single qubit

|!! =!

"#

"

S. Reda. Brown U. EN2912C Fall ‘08

2. Evolution postulate

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•  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 , then after the evolution of the state of the system will be

|!!

|!2! = U |!1!

U!

i

!i|"i! =!

i

!iU |"i!The evolution of the state vector of a closed system is linear

A physical system changes in time, and so the state vector of a system will actually by a function of time.

|!!

S. Reda. Brown U. EN2912C Fall ‘08

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

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!0 = I =!

1 00 1

"!x = X =

!0 11 0

"

!y = Y =!

0 !ii 0

"!z = Z =

!1 00 !1

"

ih|!(t)!

dt= H(t)|!(t)!

•  Unitary is implied by the continuous time evolution of a closed quantum mechanical system as described by he Schrodinger equation

|!(t2)!dt = e!ihH(t2!t1)|!(t1)!Unitary operator

S. Reda. Brown U. EN2912C Fall ‘08

3. Composition of system postulate

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•  When two physical systems are treated as one combined system, the state space of the combined physical system is the tensor product space of the state space H1, H2 of the component subsystems. In the first system is the state and the second system in the state , then the state of the combined system is

H1 !H2

|!1!|!2!

|!1! " |!2!

•  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

S. Reda. Brown U. EN2912C Fall ‘08

4. Measurement postulate

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•  For a given orthonormal basis 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

outputs a label with probability and leaves the system in state

B = {|!i!}

|!! =!

i

"i|#i!

i |!i|2

|!i!

The evolution of the state of a system during a measurement is not unitary, and an additional postulate is needed to describe measurement

!0|0! + !1|1! |b!

b

S. Reda. Brown U. EN2912C Fall ‘08

4. Measurement postulate

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|!! =!

i

"i|#i!

!i = !"i|#"

ei!|!! =!

i

"iei!|!i!

pi = |!i|2 = !!i !i = !"|#i"!#i|""

Two states and (differing only by a global phase) are equivalent

|!! ei!|!!

pi = !!i e"i!!ie

i! = |!i|2

|!! =!

111

|0!|0! +!

511

|0!|1! +!

211

|1!|0! +!

311

|1!|1!

Question: What are the result of measuring the first qubit?