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### Quantum Computers

The basics

Introduction

Dušan Gajević

Introduction

- Quantum computers use quantum-mechanical phenomenato represent and process data
- Quantum mechanicscan be described with three basic postulates
- The superposition principle- tells uswhat states are possible in a quantum system
- The measurement principle - tells ushow much information about the state we can access
- Unitary evolution - tells ushow quantum system is allowed to evolve from one state to another

Dušan Gajević

Introduction

- Atomic orbitals- an example of quantum mechanics

Electrons, within an atom,exist in quantized energy levels (orbits)

Limiting the total energy…

...limits the electronto k different levels

A hydrogen atom – only one electron

This atom might be usedto store a number between 0 and k-1

Dušan Gajević

The superposition principle

Dušan Gajević

The superposition principle

- The superposition principle statesthat if a quantum system can be in one of k states,it can also be placed in a linear superposition of these states with complex coefficients
- Ways to think about superposition
- Electron cannot decide in which state it is
- Electron is in more than one state simultaneously

Dušan Gajević

The superposition principle

- State of a system with k energy levels

“pure” states

“ket psi”

Bra-ket (Dirac) notation

amplitudes

Reminder:

Dušan Gajević

The measurement principle

Dušan Gajević

The measurement principle

- The measurement principle saysthat measurement on the k state systemyields only one of at most k possible outcomesand alters the stateto be exactly the outcome of the measurement

Dušan Gajević

The measurement principle

- It is saidthat quantum state collapses to a classical stateas a result of the measurement

Dušan Gajević

The measurement principle

If we try to measurethis state...

…the system will end up inthis state…

…and we will also get itas a result of the measurement

The probability of a system collapsing to this state is given with

Dušan Gajević

The measurement principle

- This means:
- We can tell the state we will readonly with a certain probability
- Repeating the measurementwill always yield the same result we got this first time
- Amplitudes are lost as soon as the measurement is made, so amplitudes cannot be measured

Dušan Gajević

The measurement principle

- Probability of a system collapsing to a state j is given with
- One might ask,if amplitudes come down to probabilities when the state is measured,why use complex amplitudes in the first place?
- Answer to this will be given later,when we see how system is allowed to evolvefrom one state to another

Does the equation

appear more natural now?

Dušan Gajević

Qubit

Dušan Gajević

Qubit

- Isolating two individual levels in our hydrogen atomand the qubit(quantum bit) is born

Dušan Gajević

Vector reprezentation

Dušan Gajević

Vector representation

- Pure states of a qubitcan be interpreted as orthonormal unit vectorsin a 2 dimensional Hilbert space
- Hilbert space– N dimensional complex vector space

Reminder:

Another way to write a vector –

as a column matrix

Dušan Gajević

Vector representation

- Column vectors (matrices)

qubit state

pure states

a little bit of math

Reminder: Scalar multiplication

Reminder:Adding matrices

Dušan Gajević

Vector representation

- System with k energy levelsrepresented as a vector in k dimensional Hilbert space

system state

pure states

Dušan Gajević

Entanglement

Dušan Gajević

Entanglement

- Let’s consider a system of two qubits –two hydrogen atoms,each with one electron and two "pure" states

Dušan Gajević

Entanglement

- By the superposition principle,the quantum state of these two atomscan be any linear combination of the four classical states
- Vector representation

- Does this look familiar?

Dušan Gajević

Entanglement

- Let’s consider the separate states of two qubits, A and B
- Interpreting qubits as vectors,their joint state can be calculated as their cross (tensor) product

Reminder:Tensor product

Dušan Gajević

Entanglement

- Cross product in Dirac notationis often written in a bit different manner
- The joint state of A and B in Dirac notation

Dušan Gajević

Entanglement

Let’s try to decompose

to separate states of two qubits

all four have to be non-zero

It’s impossible!

at least one has to be zero

at least one has to be zero

Dušan Gajević

Entanglement

- States like the one from the previous exampleare called entangled statesand the displayed phenomenon is called the entanglement
- When qubits are entangled,state of each qubit cannot be determined separately,they act as a single quantum system
- What will happenif we try to measure only a single qubitof an entangled quantum system?

Dušan Gajević

Entanglement

measuring the first qubit

value of the first qubit

measuring the second qubit

value of the second qubit

This remains trueno matter how large the distance between qubits is!

Dušan Gajević

Unitary evolution

Dušan Gajević

Unitary evolution

- Unitary evolution meansthat transformation of the quantum system statedoes not change the state vector length
- Geometrically,unitary transformation is a rigid body rotation of the Hilbert space

Dušan Gajević

Unitary evolution

- It comes downto mappingthe old orthonormal basis states to new ones
- These new statescan be described as superpositions of the old ones

Dušan Gajević

Unitary evolution

- Unitary transformation of a single qubit
- Dirac notation
- Matrix representation

Replace the old basis states…

…with new ones

Multiply unitary matrix…

…with the old state vector

Dušan Gajević

Unitary evolution

- Example of calculus using Dirac notation

Qubit is in the state…

…applying following (Hadamard) transformation…

…results in the state

Dušan Gajević

Unitary evolution

- Example of calculus using matrix representation

Reminder:matrix multiplication

Qubit is in the state…

…applying Hadamard transformation…

…results in the state

Dušan Gajević

Unitary matrices

- Unitary matrices satisfy the condition

Conjugate-transpose of U

“U-dagger”

Inverse of U

Reminder:Conjugate-transpose matrix

Reminder:Inverse matrix

Reminder:Complex conjugate

Identity matrix

Reminder:

Dušan Gajević

Reversibility

Dušan Gajević

Reversibility

- Reversibility is an important propertyof unitary transformation as a function –knowing the output it is always possible to determine input
- What makes an operation reversible?
- AND circuit
- NOT circuit

output

input

1

A=1 B=1

?

0

irreversible

output

input

1

A=0

0

A=1

reversible

Dušan Gajević

Reversibility

- Reversible operation has to be one-to-one –different inputs have to give different outputs and vice-versa
- Consequently, reversible operationshave the same number of inputs and outputs
- Are classical computers reversible?

Dušan Gajević

Reversibility

- Similar to AND circuitapplies to OR, NAND and NOR,the usual building blocks of classical computers
- Hence, in general,classical computers are not reversible

Dušan Gajević

Offtopic: Landauer’s principle

- Again, an irreversible operation
- NAND circuit

We say information is “erased” every time output of NAND is 1

Whenever output of NAND is 1

– input cannot be determined

Dušan Gajević

Offtopic: Landauer’s principle

- Landauer’s principle saysthat energy must be dissipated when information is erased,in the amount
- Even if all other energy loss mechanisms are eliminatedirreversible operations still dissipate energy
- Reversible operationsdo not erase any information when they are applied

Boltzman\'s constant

Absolute temperature

Dušan Gajević

References

- University of California, Berkeley,Qubits and Quantum Measurement and Entanglement, lecture notes,http://www-inst.eecs.berkeley.edu/~cs191/sp12/
- Michael A. Nielsen, Isaac L. Chuang,Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2010.
- Colin P. Williams, Explorations in Quantum Computing, Springer, London, 2011.
- Samuel L. Braunstein, Quantum Computation Tutorial, electronic documentUniversity of York, York, UK
- Bernhard Ömer, A Procedural Formalism for Quantum Computing, electronic document, Technical University of Vienna, Vienna, Austria, 1998.
- Artur Ekert, Patrick Hayden, Hitoshi Inamori,Basic Concepts in Quantum Computation, electronic document,Centre for Quantum Computation, University of Oxford, Oxford, UK, 2008.
- Wikipedia, the free encyclopedia, 2014.

Dušan Gajević

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