Quantum Computers

Quantum Computers

The basics

Introduction

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Introduction

• Quantum computers use quantum-mechanical phenomenato represent and process data

• Quantum mechanicscan be described with three basic postulates– The superposition principle - tells us

what states are possible in a quantum system– The measurement principle - tells us

how much information about the state we can access– Unitary evolution - tells us

how quantum system is allowed to evolve from one state to another

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• Atomic orbitals - an example of quantum mechanics

Introduction

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

A hydrogen atom – only one electron

Limiting the total energy…

...limits the electronto k different levels

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

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The superposition principle

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• 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


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• State of a system with k energy levels“pure” states

amplitudes“ket psi”Bra-ket (Dirac) notation

Reminder:

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• A system with 3 energy levels – examples of valid states

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“Very interesting theory – it makes no sense at all”– Groucho Marx

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The measurement principle

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• 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



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



If we try to measure this state...

…the system will end up in this state…

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

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



• This means:– We can tell the state we will read

only with a certain probability– Repeating the measurement

will 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



• 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?


“God does not play dice”– Albert Einstein“Don’t tell God what to do”– Niels Bohr


Qubit


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

Qubit


Qubit

• Qubit state

• The measurement collapses the qubit state to a classical bit


Vector reprezentation


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

Vector representation

Reminder:Another way to write a vector –

as a column matrix



• Column vectors (matrices)

qubit state pure states

a little bit of math

Reminder:Adding matrices

Reminder: Scalar multiplication



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

system state pure states


Entanglement


Entanglement

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


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?


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


Entanglement– Cross product in Dirac notation

is often written in a bit different manner

• The joint state of A and B in Dirac notation


Entanglement

It’s impossible!all four have to be non-zero

at least one has to be zero

at least one has to be zero

Let’s try to decompose

to separate states of two qubits


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?


Entanglement

Let’s take a look at the same example once again

amplitudes probabilities


Entanglement

measuring the first qubit

measuring the second qubit

value of the first qubit

value of the second qubit

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


“Spooky action at a distance”– Albert Einstein


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

Unitary evolution


• It comes downto mapping the old orthonormal basis states to new ones

– These new statescan be described as superpositions of the old ones

Unitary evolution


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


Unitary evolution

• Example of calculus using Dirac notationQubit is in the state…

…applying following (Hadamard) transformation…

…results in the state


Unitary evolution

• Example of calculus using matrix representationQubit is in the state…

…applying Hadamard transformation…

Reminder:matrix multiplication

…results in the state


Unitary matrices

• Unitary matrices satisfy the condition

Conjugate-transpose of U“U-dagger”

Reminder:Conjugate-transpose matrix

Reminder:Complex conjugate

Inverse of UReminder:

Inverse matrix

Identity matrixReminder:


Reversibility


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

INPUT OUTPUTA B A and B0 0 00 1 01 0 01 1 1

INPUT OUTPUTA not A0 11 0

output

1

0

A=1 B=1

input

?irreversible

output

1

0

input

A=0

A=1reversible


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?


Reversibility– Similar to AND circuit

applies to OR, NAND and NOR,the usual building blocks of classical computers

• Hence, in general,classical computers are not reversible


Offtopic: Landauer’s principle

• Again, an irreversible operation– NAND circuit

INPUT OUTPUTA B A nand B0 0 10 1 11 0 11 1 0

Whenever output of NAND is 1– input cannot be determined

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


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

Absolute temperatureBoltzman's constant


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 document

University 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.


http://www-inst.eecs.berkeley.edu/~cs191/sp12/

Quantum Computers

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