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Quantum Computers
The basics
Introduction
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Introduction
• Quantum computers use quantum-mechanical phenomena
to represent and process data
• Quantum mechanics
can 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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Introduction
• Atomic orbitals - an example of quantum mechanics
Electrons, within an atom,
exist in quantized energy levels
(orbits)
Limiting the total energy…
...limits the electron
to k different levels
A hydrogen atom –
only one electron
This atom might be used
to store a number between 0 and k-1
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The superposition principle
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The superposition principle
• The superposition principle states
that 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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The superposition principle
• State of a system with k energy levels
“pure” states
“ket psi”
Bra-ket (Dirac) notation
amplitudes
Reminder:
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The superposition principle
• 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
• The measurement principle says
that measurement on the k state system
yields only one of at most k possible outcomes
and alters the state
to be exactly the outcome of the measurement
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The measurement principle
• It is said
that quantum state collapses to a classical state
as a result of the measurement
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The measurement principle
If we try to measure this state...
…the system will end up in this state…
…and we will also get it
as a result of the measurement
The probability of a system collapsing to this state
is given with
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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
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The measurement principle
• Probability of a system collapsing to a state j is given with
Does the equation
appear more natural now?
– 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 evolve
from one state to another
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“God does not play dice”
– Albert Einstein
“Don’t tell God what to do”
– Niels Bohr
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Qubit
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Qubit
• Isolating two individual levels in our hydrogen atom
and the qubit (quantum bit) is born
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Qubit
• Qubit state
• The measurement collapses the qubit state to a classical bit
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Vector reprezentation
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Vector representation
• Pure states of a qubit
can be interpreted as orthonormal unit vectors
in a 2 dimensional Hilbert space
– Hilbert space – N dimensional complex vector space
Reminder:
Another way to write a vector –
as a column matrix
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Vector representation
• Column vectors (matrices)
qubit state
pure states
a little bit of math
Reminder:
Scalar multiplication
Reminder:
Adding matrices
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Vector representation
• System with k energy levels
represented as a vector in k dimensional Hilbert space
system state
pure states
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Entanglement
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Entanglement
• Let’s consider a system of two qubits –
two hydrogen atoms,
each with one electron and two "pure" states
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Entanglement
• By the superposition principle,
the quantum state of these two atoms
can be any linear combination of the four classical states
Does this look familiar?
– Vector representation
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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
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Entanglement
– Cross product in Dirac notation
is often written in a bit different manner
• The joint state of A and B in Dirac notation
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Entanglement
Let’s try to decompose
to separate states of two qubits
all four have to be non-zero
at least one has to be zero
It’s impossible!
at least one has to be zero
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Entanglement
• States like the one from the previous example
are called entangled states
and 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 happen
if we try to measure only a single qubit
of an entangled quantum system?
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Entanglement
Let’s take a look at the same example once again
amplitudes
probabilities
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Entanglement
measuring the first qubit
value of the first qubit
measuring the second qubit
value of the second qubit
This remains true
no matter how large the distance between qubits is!
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“Spooky action at a distance”
– Albert Einstein
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Unitary evolution
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Unitary evolution
• Unitary evolution means
that transformation of the quantum system state
does not change the state vector length
– Geometrically,
unitary transformation is a rigid body rotation of the Hilbert space
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Unitary evolution
• It comes down
to mapping the old orthonormal basis states to new ones
– These new states
can be described as superpositions of the old ones
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Unitary evolution
• Unitary transformation of a single qubit
– Dirac notation
Replace the old basis states…
…with new ones
– Matrix representation
Multiply unitary matrix…
…with the old state vector
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Unitary evolution
• Example of calculus using Dirac notation
Qubit is in the state…
…applying following (Hadamard) transformation…
…results in the state
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Unitary evolution
• Example of calculus using matrix representation
Reminder:
matrix multiplication
Qubit is in the state…
…applying Hadamard transformation…
…results in the state
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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:
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Reversibility
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Reversibility
• Reversibility is an important property
of unitary transformation as a function –
knowing the output it is always possible to determine input
– What makes an operation reversible?
output
• AND circuit INPUT OUTPUT
A
0
0
1
1
• NOT circuit
B
0
1
0
1
A and B
0
0
0
1
INPUT OUTPUT
A
not A
0
1
1
0
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0
input
A=1
B=1
? irreversible
output
input
1
A=0
0
A=1
reversible
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Reversibility
– Reversible operation has to be one-to-one –
different inputs have to give different outputs and vice-versa
• Consequently, reversible operations
have the same number of inputs and outputs
• Are classical computers reversible?
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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
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Offtopic: Landauer’s principle
• Again, an irreversible operation
– NAND circuit
INPUT
A B
0 0
0 1
1 0
1 1
OUTPUT
A nand B
1
1
1
0
We say information is “erased”
every time output of NAND is 1
Whenever output of NAND is 1
– input cannot be determined
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Offtopic: Landauer’s principle
• Landauer’s principle says
that energy must be dissipated when information is erased,
in the amount
Boltzman's constant
Absolute temperature
– Even if all other energy loss mechanisms are eliminated
irreversible operations still dissipate energy
• Reversible operations
do not erase any information when they are applied
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References
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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.
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