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The Weird World of Quantum

Information

Marianna Safronova

Department of Physics and Astronomy

What do we need to build a computer?

Memory

Initialization: ability to prepare one certain state

repeatedly on demand, for example put all to zero at the

start.

Ability to perform (universal) logical operations.

No or very small error rate (that can be fixed).

Ability to efficiently read out the result.

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Why quantum information?

Information is physical!

Any processing of information

is always performed by physical means

Bits of information obey laws of classical physics.

Why quantum information?

Information is physical!

Any processing of information

is always performed by physical means

Bits of information obey laws of classical physics.

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Why Quantum Computers?

Computer technology is

making devices smaller

and smaller… …reaching a point where classical

physics is no longer a suitable model for

the laws of physics.

Fundamental building blocks

of classical computers:

BITS

STATE:

Definitely

0 or 1

Bits & Qubits

Fundamental building blocks

of quantum computers:

Quantum bits

or

QUBITS

Basis states: and

0 1

Superposition:

0 1ψ α β= +

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A very brief introduction into quantum mechanics

Problem: indeterminacy of the quantum mechanics. Even if you know

everything that theory (i.e. quantum mechanics ) has to tell you about the

particle (i.e. wave function), you can not predict with certainty where this

particle is going to be found by the experiment.

Quantum mechanics provides statistical information about possible results.

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1 2 3, , andψ ψ ψ

One of the biggest difference between

classical and quantum physics: superposition

If your quantum system (particle) has three possible

states,

it may be in superposition of these three states

1 1 2 2 3 3a a aψ ψ ψ ψ= + +

If you make a measure the wave function will collapse to “eigenstate”

1 2 3, , andψ ψ ψ

The probability to “catch” particle in state 1 is . 2

1a

The probability to “catch” particle in state 2 is . 2

2a

The probability to “catch” particle in state 3 is . 2

3a

Superposition

Bits & Qubits: primary differences

0 1ψ α β= +

Example: two spin states of spin ½ particle

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Example: spin and measurements

In 1922, O. Stern and W. Gerlach conducted experiment to measure the

magnetic dipole moments of atoms. The results of these experiments

could not be explained by classical mechanics. First, let's discuss why

would atom poses a magnetic moment.

Even in Bohr's model of the hydrogen atom, an electron, which is a charged

particle, occupies a circular orbit, rotating with orbital angular momentum L.

A moving charge is equivalent to electric current, so an electron moving in a

closed orbit forms a current loop and this, therefore, creates a magnetic

dipole. The corresponding magnetic dipole moment is given by:

If the atom with a magnetic moment is placed in a magnetic field B, it will experiencea net force F,

Stern suggested to measure the magnetic moments of atoms by deflecting atomic beam byinhomogeneous magnetic field. In the experimental setup, the only force on the atomsis in z direction and

The direction of magnetic moment in the beam is random, so every value of in therange is expected. As a result, the deposit on the collectingplate is expected to be spread continuously over a symmetrical region about the point of nodisplacement.

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Electronic configurations of atoms in Stern-Gerlach experiments:

Conclusion: elementary particles carry intrinsic angular momentum S in addition to L.Spin of elementary particles has nothing to do with rotation, does not depend oncoordinates and , and is purely a quantum mechanical phenomena.

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Spin

, therefore and there are two eigenstates

We will call them spin up and spin down .

Taking these eigenstates to be basis vectors, we can express any spin state of aparticle with spin as:

0 1ψ α β= +

( ), ,x y zS S S S=�

m=1/2

m=-1/2

Two states deflected

differently in magnetic field.

In atoms, such states have

different energy levels in

magnetic field.

Modern version of Stern-Gerlach experiment

Measuring expectation values of

( ), ,x y zS S S S=�

, ,x y zS S S

Note on spin quantum numbers: S and M (or M

z

)

To fully described spin quantum number, one of the direction (z) is picked

Simulation:

https://phet.colorado.edu/en/simulation/legacy/stern-gerlach

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https://phet.colorado.edu/en/simulation/build-an-atom

Electron spin and periodic table

Single electron states are labeled by quantum numbers: n, l, ml, s, m

s

Rule:

In an atoms, all electrons have to differ in at least one quantum number

Electrons are fermions and have to be in different states – remember that this

leads to electron degeneracy pressure in white dwarfs.

n is principal quantum numbers, 1, 2, 3, 4, …

l is orbital angular momentum quantum numbers 0< l < n-1

ml is corresponding magnetic quantum number - l ≤ ml ≤ l

s is spin s=1/2

ms corresponding magnetic quantum number - s ≤ ms≤ s, so ms=-1/2, +1/2

H: electron is in n=1, l=0 state (1s)

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Measurement

Classical bit: we can find out if it is in state 0 or 1 and the

measurement will not change the state of the bit.

Qubit:

Bits & Qubits: primary differences

QOFR

Quantum calculation:

number of parallel processes

due to superposition

Superposition

Measurement

Classical bit: we can find out if it is in state 0 or 1 and the

measurement will not change the state of the bit.

Qubit: we cannot just measure α and β and thus determine its

state! We get either or with corresponding

probabilities |α|2 and |β|2.

The measurement changes the state of the qubit!

Bits & Qubits: primary differences

0 1

0 1ψ α β= +

2 2

1α β+ =

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Multiple qubits

Hilbert space is a big place!Hilbert space is a big place!Hilbert space is a big place!Hilbert space is a big place!- Carlton Caves

Multiple qubits

Two bits with states 0 and 1 form four definite states 00,

01, 10, and 11.

Two qubits: can be in superposition of four

computational basis set states.

Hilbert space is a big place!Hilbert space is a big place!Hilbert space is a big place!Hilbert space is a big place!- Carlton Caves

00 01 10 11α β γ δψ = + + +

2 qubits 4 amplitudes

3 qubits 8 amplitudes

10 qubits 1024 amplitudes

20 qubits 1 048 576 amplitudes

30 qubits 1 073 741 824 amplitudes

500 qubits

500 qubits500 qubits

500 qubits More amplitudes than our estimate of

number of atoms in the Universe!!!

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Entanglement

Entanglement

0 1

2

0 1ψ

+=

Results of the measurement

First qubit 0 1

Second qubit 0 1

ψ α β≠ ⊗ Entangledstates

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Quantum computer animation

https://www.youtube.com/watch?v=T2DXrs0OpHU