Quantum
Computing
Corby Ziesman
Future of Computing?
• Transistor-based Computing
– Move towards parallel architectures
• Biological Computing
– DNA computing / Peptide computing
• Optical Computing
• Quantum Computing
Quantum Mechanics is Weird
• Prism will reflect
laser at internal
angles of 45°
• Light travels freely
through prism
material
• We see no dot on
the vertical paper,
only the bottom
• We conclude that
no light passes that
angled prism edge
Quantum Mechanics is Weird
• Place a 2nd prism close, but
not touching
• The gap is a wall in the
probability of where that
laser will travel
• There is a small probability
that the laser can tunnel
through that wall
• So we now see a dot on the
vertical paper, even though
we just concluded that no
light passes the angled
prism edge
• Quantum mechanics is all
about probabilities, and
quantum computing takes
this into account
What’s a Quantum Computer?
• A quantum computer uses quantum mechanics in some way to perform
calculations
• There are a number of quantum computing candidates, among those:
– "Superconductor-based quantum computers (including SQUID-based quantum
computers)
– "Trapped ion quantum computer
– "Electrons on helium quantum computers
– "Nuclear magnetic resonance on molecules in solution"-based
– "Quantum dot on surface"-based (e.g. the Loss-DiVincenzo quantum computer)
– "Cavity quantum electrodynamics" (CQED)-based
– "Molecular magnet"-based
– Fullerene-based ESR quantum computer
– Solid state NMR Kane quantum computers
– Optic-based quantum computers (Quantum optics)
– Topological quantum computer
– Spin-based quantum computer
– Adiabatic Quantum Computing
• So what quantum properties could be used?
Probability
• Probability governs quantum
mechanics and the world around us
• Classical models suggest that it may
be possible to completely know the
state of a system, and that – the
state being known – it should be
possible to predict with 100%
accuracy the behavior of the system.
Probability
• We know that it is impossible to
completely know all aspects of a system
– e.g. Heisenberg’s uncertainty principle, states
that as we know the position of a particle with
higher accuracy, we know the momentum with
less accuracy, and vice-versa
• We can only say what the probability is
(e.g. we now view electron orbits as “fuzzy
clouds” of where the electron is likely to
be
Wavefunctions
• Wavefunctions
describe probabilities
• There are areas with
very low or zero
probability
– When an electron
changes energy states,
it does not physically
move from one orbit to
another, it
instantaneously jumps
to the other orbit
Superposition and Wavefunction Collapse
• Wavefunctions can be combined in superposition
• Each wavefunction corresponds to a state, with a
complex number coefficient describing its
probability among other states
• Coefficient is complex because each element can
interact with or destroy other states
• When observed, the wavefunction collapses
randomly into an observable state with probability
based on the square of the amplitude of that
wavefunction in the combination
Qubits
• A quantum bit is a two-state
unit of quantum information
• Can have superposition
however (both states at once)
• The information in a qubit is
equal to one bit, but may be
handled more efficiently when
processing information
• This efficiency may make
many previously difficult
computing tasks easy
• Shor’s algorithm is a quantum algorithm to factor a number in
polynomial time (major consequences for cryptography).
– Factoring is reduced to the problem of order-finding, which can be done
on a normal computer.
– Order-finding problem is done on a quantum computer.
• Quantum algorithms tend to only probably give the right answer, but
confidence can be increased by repeating the computation
Complexity
• As mentioned, Shor’s quantum algorithm can factor numbers
in polynomial time
• Normally we are familiar with NP, NP-Complete, P, etc.
which deal with deterministic and non-deterministic Turing
machines
• There is also BPP (Bounded-error, Probabilistic, Polynomial-
time) which relates to probabilistic Turing machines
– BPP defines algorithms that can flip coins and make random
decisions, as long as the algorithm has at least a 50% chance of
getting it right
– The algorithm can be repeated, and then the probability of
having a wrong answer drops off exponentially
• Since quantum computing (and quantum mechanics) is all
about probabilities, the analog to BPP is BQP (Bounded-
error, Quantum, Polynomial-time)
• Suspected relationship to BQP:
Quantum Algorithms
• A normal 3-bit register can store up to 8 possible sequences of
number, such as 000, 001, 010, etc.
• A quantum computer can keep all possible states at once, as
described by a wavefunction:
• The data can then by transformed by multiplying it by a unitary
matrix described by the physics of the computer
– May mean the computer is specifically designed for solving only one
problem, not a general quantum computer
• Quantum computing is reversible
• When measured, the result is one of the states, according to the
probability coefficients
• By measuring, the stored data has been altered and becomes
useless
• However, by repeating the algorithm, the correct result will occur
most often, and so the most frequent result among runs of the
algorithm can be selected as the correct answer
– It’s also possible (as in factoring) to simply verify the result using a
classical computer, and then no more trials need to be done once the
correct result is found
How to Encode Data
• Atomic Spin (“Spintronics”)
– Up, Down, or Up/Down in superposition
• Quantum Entanglement Properties
– Two particles are in a quantum state and are described
in relation to each other
– Can prepare particles so that when one is observed to be
spin-up, the other will always be spin-down and vice-
versa
– This is despite the fact the particles may be spatially
separated by incredible distances
– Does not violate causality, which states – in the most
general sense – that information can not travel faster
than the speed of light, because these observations are
a result of wavefunction collapse
– Particles’ behavior is related and intertwined, but do not
influence each other
Quantum Gates
• Quantum gates derive from reversible
computing
• They are described by unitary matrices
such as the Hadamard gate or Controlled
NOT gate:
• From these, reversible quantum circuits
may be created
– Physically connecting the gates may lead to
problems relating to quantum decoherence
Challenges
• Decoherence
– As mentioned, the states in a system
can interfere with each other
– If the external environment interacts
with the system, the quantum
superpositions in the new wavefunction
(that includes external influences) may
not be able to interfere with each other
– Need to isolate the system from the
environment and remove all noise
Challenges
• As mentioned, the physics of the
device may be specific to solving one
problem
• May be some time before a general
quantum computer comes along with
the flexibility of modern computers
Recent News
• D-Wave Systems demoed last
week a quantum computer, which
it plans to make into a
commercial product
• There are questions of whether
or not the computer actually
makes use of quantum phenomena
or if it is merely an analog
computer
• D-Wave states that progress is
continuing, and that isolating the
system from the external
environment is a major concern in
the design
Timeline
Wikipedia Article as on Feb 19,
2007 on the history of Quantum
Computing
The End
Questions
References
• PHY360 (Modern Physics) at ASU
• Wikipedia articles:
– Quantum computer, Quantum superposition,
Quantum entanglement, Quantum information,
Quantum state, Quantum gate, Quantum
circuit, Uncertainty principle, Quantum leap,
Wavefunction, Shor’s algorithm, Quantum
mechanics, Qubit, BQP
• Google news search results for “dwave
quantum”
• Slashdot and Scientific American articles
I’ve read over the years