Prsentation PowerPoint

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What Do You Mean Simulating a Quantum
Computation?
David Poulin IQC, University of
Waterloo Perimeter Institute
October 2002
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A second example of what Chris called a bad
question.
In condensed matter physics, it is often quite
useful to introduce the notion of
quasi-particles. These are excitations which
behave almost like free particles but have extra
weird features. For example, the mass of the
quasi-particle may depend on the direction of
its motion a mass tensor.
Yet, people dont organize meetings on the
interpretation of quasi-particles!
David Poulin, IQC University of Waterloo PI
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It is clear at this time that quantum mechanics
is not the final theory. In whatever turns out
to be the final theory (string theory, quantum
loop gravity, etc.), quantum mechanics will only
be a good approximation. It is also possible
that some of the weirdness disappears.
But we are here having this meeting!
David Poulin, IQC University of Waterloo PI
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What Do You Mean Simulating a Quantum
Computation?
How is this simulation business related to
foundation of QM?
A journey from ontic to epistemic
... with consequences Does this have
consequences on the way we think about simulation?
David Poulin, IQC University of Waterloo PI
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Outline
QS
QC
CC
David Poulin, IQC University of Waterloo PI
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Stuff about QS we usually compute with CC
simulations (at an exponential cost).

• Ground state energy
• Properties of the thermal/ground state
  (symmetries)
• Propagators
• Degeneracy of energy levels
• Transport properties
• Properties of spectral functions
• Properties of cross section
• Partition function
• etc.

David Poulin, IQC University of Waterloo PI
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The real thing should be at least as good as the
simulated one!
How much of the stuff on the previous slide can
we measure from the QS itself... ... or a
polynomial number of copies of it? Does there
exist physical quantities extractable from poly
copies of a QS which requires exponential CC?
The strongest argument indicating that the
simulation of QS is a hard problem is Gauss
failure at finding an efficient algorithm for
factoring.
---Gilles (maybe in a dream...)
David Poulin, IQC University of Waterloo PI
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So I know that quantum mechanics seems to
involve probability --- and I therefore want to
talk about simulating probability.
---Feynman

• There are two ways of addressing this problem
• 1. Simulate the wave packet dynamics ?(x,t)
  likeone would do with water waves.
• Use a probabilistic CC which reproduces
  somestatistical properties of the system.

David Poulin, IQC University of Waterloo PI
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One method for classically simulating a quantum
computation is to directly compute the state at
each step from the sequence of unitary
operations prescribed in the quantum
algorithm. --- Jozsa Linden
Entanglement is only related to simulatability
through the way we chose to represent the wave
function.
David Poulin, IQC University of Waterloo PI
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If we insist on computing an exponential amount
of extra unphysical information (?), the
exponential overhead is inevitable.
Slightly weaker notion of simulating
probabilities Reproduce the probabilities of a
fixed final measurement.
Unperformed experiments have no
results ---Peres
David Poulin, IQC University of Waterloo PI
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Simulate physics, not counterfactual experiments
p-blockness ? p-blockability!
F I, Q1 , Q2 , ..., QL , O
Qk
p-block states
L is the circuits depth
If F form a family of consistent histories, then
the measurements Qk can be carried out ---
collapsing the state to a p-block state ---
without changing the factual (physically
meaningful) probabilities pij .
David Poulin, IQC University of Waterloo PI
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Probabilistic simulation
If it is possible to simulate the wave packets
dynamics or the factual probabilities it is
possible to statistically reproduce the behavior
of the QS.
... but it seams otherwise impossible!
Physics Properties which require exponential
resources to be estimated are
practically not measurable.
David Poulin, IQC University of Waterloo PI
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But Avogadros number is so large!
It takes a while before the exponential kicks in.
Ex. Molecule N 50 hydrogen-like 2-levels
atoms. Sample m 1g. Number of
states 250 ltlt Number of molecules 1024/50 (7
orders of magnitude!)
If N 100, then m has to be gt 1Tonne!!!
Reproducing the statistics is not a fair
requirement... ... what about some coarse grained
version of it?
Coarse graining leads to consistency... which
leads to classical simulatability!
David Poulin, IQC University of Waterloo PI
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Beyond simulating!
When asking a CC to simulate a QS, we should only
ask about things we can actually measure on that
system.
Should we expect more from a QC?
... its not completely crazy. Ex. Is the ground
state of this QS degenerated?
David Poulin, IQC University of Waterloo PI