Quantum Computers
Post on 03-Jan-2016
52 Views
Preview:
DESCRIPTION
Transcript
Quantum Computers
The basics
Introduction
2/47Dušan Gajević
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
3/47Dušan Gajević
• 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
4/47Dušan Gajević
The superposition principle
5/47Dušan Gajević
• 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
The superposition principle
6/47Dušan Gajević
The superposition principle
• State of a system with k energy levels“pure” states
amplitudes“ket psi”Bra-ket (Dirac) notation
Reminder:
7/47Dušan Gajević
The superposition principle
• A system with 3 energy levels – examples of valid states
8/47Dušan Gajević
“Very interesting theory – it makes no sense at all”– Groucho Marx
9/47Dušan Gajević
The measurement principle
10/47Dušan 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
11/47Dušan Gajević
The measurement principle
• It is saidthat quantum state collapses to a classical stateas a result of the measurement
12/47Dušan Gajević
The measurement principle
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
13/47Dušan Gajević
The measurement principle
• 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
14/47Dušan 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?
15/47Dušan Gajević
“God does not play dice”– Albert Einstein“Don’t tell God what to do”– Niels Bohr
16/47Dušan Gajević
Qubit
17/47Dušan Gajević
• Isolating two individual levels in our hydrogen atomand the qubit (quantum bit) is born
Qubit
18/47Dušan Gajević
Qubit
• Qubit state
• The measurement collapses the qubit state to a classical bit
19/47Dušan Gajević
Vector reprezentation
20/47Dušan Gajević
• 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
21/47Dušan Gajević
Vector representation
• Column vectors (matrices)
qubit state pure states
a little bit of math
Reminder:Adding matrices
Reminder: Scalar multiplication
22/47Dušan Gajević
Vector representation
• System with k energy levelsrepresented as a vector in k dimensional Hilbert space
system state pure states
23/47Dušan Gajević
Entanglement
24/47Dušan Gajević
Entanglement
• Let’s consider a system of two qubits –two hydrogen atoms,each with one electron and two "pure" states
25/47Dušan 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?
26/47Dušan 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
27/47Dušan Gajević
Entanglement– Cross product in Dirac notation
is often written in a bit different manner
• The joint state of A and B in Dirac notation
28/47Dušan Gajević
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
29/47Dušan 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?
30/47Dušan Gajević
Entanglement
Let’s take a look at the same example once again
amplitudes probabilities
31/47Dušan Gajević
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!
32/47Dušan Gajević
“Spooky action at a distance”– Albert Einstein
33/47Dušan Gajević
Unitary evolution
34/47Dušan Gajević
• 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
35/47Dušan Gajević
• 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
36/47Dušan 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
37/47Dušan Gajević
Unitary evolution
• Example of calculus using Dirac notationQubit is in the state…
…applying following (Hadamard) transformation…
…results in the state
38/47Dušan Gajević
Unitary evolution
• Example of calculus using matrix representationQubit is in the state…
…applying Hadamard transformation…
Reminder:matrix multiplication
…results in the state
39/47Dušan Gajević
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:
40/47Dušan Gajević
Reversibility
41/47Dušan 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
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
42/47Dušan 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?
43/47Dušan Gajević
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
44/47Dušan Gajević
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
45/47Dušan 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
Absolute temperatureBoltzman's constant
46/47Dušan 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 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.
47/47Dušan Gajević
top related