the limits of knowledge: philosophical and practical...
TRANSCRIPT
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The Limits of Knowledge:
Philosophical and practical
aspects of building a quantum
computer
Simons Center
November 19, 2010
Michael H. Freedman, Station Q, Microsoft Research
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• To explain quantum computing, I will offer some parallels between philosophical concepts, specifically from Catholicism on the one hand, and fundamental issues in math and physics on the other.
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Why in the world would I do that?
• I gave the first draft of this talk at the Physics Department of Notre Dame.
• There was a strange disconnect.
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The audience was completely secular.
• They couldn’t figure out why some guy named Freedman was talking to them about Catholicism.
• The comedic potential was palpable.
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I Want To Try Again
With a rethought talk and broader audience
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• Mathematics and Physics have been in their modern rebirth for 600 years, and in a sense both sprang from the Church (e.g. Roger Bacon)
• So let’s compare these two long term enterprises:
- Methods
- Scope of Ideas
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Common Points: Catholicism and Math/Physics
• Care about difficult ideas
• Agonize over systems and foundations
• Think on long time scales
• Safeguard, revisit, recycle fruitful ideas and methods
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Dali Lully Tolkien Descartes Dante Aquinas
Some of my favorites
• Lully may have been the first person to try to build a computer.
• He sought an automated way to distinguish truth from falsehood, doctrine from heresy
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• Philosophy and religion deal with large questions.
• In Math/Physics we also have great overarching questions which might be considered the social equals of: Omniscience, Free Will, Original Sin, and Redemption.
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• P/NP ↔ Omniscience
• Quantum Mechanics ↔ Redemption
• Universality ↔ Original Sin
• Unicity ↔ Free Will
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(1) P/NP (Omniscience)
• The limits of knowledge
• The limits of computation
• The scaling of effort required to:
▫ solve problems (factor numbers)
▫ discover proofs
Kurt Gödel
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Quantum Mechanics (Redemption)
• Does it provide a complete framework for all physics? Does it redeem our understanding and give us a theanthropic perspective of the world?
• What about gravity? • What about the classical world?
▫ the measurement problem ▫ where do unrealized probabilities go?
Schrodinger
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Universality (Original Sin)
• Physical systems (when viewed from a distance) can be grouped into a small number of classes with identical scaling laws.
• “Curse of Universality.” By looking at the emergent structure, one may never know what microscopics caused it.
Ken Wilson
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Unicity (Free Will)
• Did our universe have to be roughly as it is?
▫ 3+1 large dimensions
▫ stable matter
▫ weakly chaotic dynamics
• Did our universe have to be exactly as it is? Is it preordained?
Descartes “The best of all possible worlds.” – Gottfried Leibniz
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The topic today is:
QC
Math Physics
Computer Science
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And also:
QC
Math Physics
Computer Science
Everything Else
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• Along the way we’ll encounter our dancers:
Gödel
P/NP Quantum Mechanics Universality
Ken Wilson Scott Aaronson
Unicity
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• What we can hope to compute is limited by the scaling of effort inherent in each type of problem.
• Obviously it is a lot more work to factor a large number than a small one.
• But exactly how fast does the work load grow?
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P NP
Q
PSPACE
factoring
The class Q depends on a different way, a quantum mechanical way, of storing numbers.
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Quantum computation is a new paradigm in which computational work obeys different scaling laws than those that are known to hold in present day “classical” computers.
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Modern Church-Turing (MCT) Thesis:
• There are only two physically realistic models of computation:
▫ One based on Classical Physics
▫ One based on Quantum Physics
Alonzo Church Alan Turing
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Corollary of MCT
• All we will ever know (or at least compute) will lie in Q.
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Unicity (Free Will): Could the universe
have been different? • Is there a world where NP-complete
problems can be solved efficiently?
• Many (Aaronson) think not – that just like perpetual motion, such worlds cannot be consistent.
Free Will! Indeed.
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So perhaps there is only one possible
theory governing our universe:
Quantum Mechanics
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Q: What new power is conveyed by computing quantum mechanically?
A: Superposition
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• Superposition means that a general state ψ may be written as a linear combination of eigen-states ψi, which typically are classical configurations
• The coefficients αi, called amplitudes, are “square roots” of probabilities:
∑ |αi|2 = 1
ψ = ∑ αi ψi
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• Square roots of probabilities are not intuitive.
• Nothing in our large scale clumsy world, nothing in our evolutionary experience, prepares our mind for superposition.
• Superposition was born amid mystery and seeming paradox in the period 1900-1927.
Born Bohr Heisenberg Schrodinger Plank
Radiation, Diffraction, Scattering, Atomic Spectra
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• The double slit experiment shows amplitudes at work.
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Source
Blind Screen
Observed pattern “Classical expectation”
|α + β|2 = |α|2 + |β|2
+ All closed - 1 open + 2 open
0
-|α|2
-|β|2
+|α + β|2
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How do amplitudes, opposed to probabilities,
enhance computational power?
In a cleverly designed algorithm
factoring
useless computational paths can often be arranged to cancel out – like the dark spots (“nodes”) in the double slit experiment – and not consume computational resources.
This is possible because amplitudes, unlike probabilities, can be negative (or even imaginary).
With quantum effects, “factoring” goes from exponential to polynomial time.
Peter Shor
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What did Shor do?
• Classical part: Order finding: factoring
▫ Suppose f(x) = ax mod N has period r (even).
▫ Then (usually):
(ar/2 + 1)(ar/2 - 1) = ar – 1 = kN
will “separate” factors of N.
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• Quantum Part: Study f(x) = ax mod N
▫ In Fourier space: f(k) = Mk,x f(x), Mk,x = e2𝜋kxi/M,
M is approximately N2
▫ Observing f returns information on the periodicity of f
▫ The nodes or “dark spots” are the places where f(k) is small.
^
^
^
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What might a quantum computer do in the
real world?
• (1) Wreak havoc: Break all classical codes
Panic on Wall Street
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• (2) Allow physicists to explore exotic states of matter
▫ Strongly correlated electron systems
High Tc superconductors
2-dimensional electron gasses (2-DEGS)
Exotic magnets
• ? Compute string theories ?
(good research problem!)
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• (3) Sample from the solution space of exponentially large linear systems.
Many engineering applications:
Electrical engineering and communication
Optimization
Fluid flow?
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• (4) Allow chemists / pharmacologists to design drugs?
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• (5) Artificial intelligence?
▫ In 1950, Alan Turing predicted :
Computing power would grow fast (it grew faster)
By 2000 we would have a hard time saying that machines were not thinking. (did not happen)
Quantum computing may give AI a second chance
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Topology
• There is a topological approach to quantum computation that avoids local degrees of freedom: nuclear spin, electron spin, photon polarization, etc.
• We need a two dimensional
electron gas (2DEG)– with a
special property
• Majorana fermions localized in “vortices”
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There are two prime candidates for Majoranas: • Fractional Quantum Hall Effect (FQHE) at ν = 5/2
• px + ipy superconductors
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• We can execute operations (“gates”) on system state by braiding Majoranas.
• Or “quasi particle interferometry.”
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Fractional Quantum Hall Effect
• 2DEG
• large B field (~ 10T)
• low temp (< 1K)
• gapped (incompressible)
• quantized filling fractions
2
1, , 0n hxy xxm e
R R
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A topological state of matter: the quantum Hall state
h
enxy
2
• Topological origin of the quantized Hall conductance: • Bulk gap (Landau level gap) • The first Chern number
(Laughlin PRB 1981, Thouless, Kohmoto,
Nightingale, den Nijs (TKNN), PRL 1982)
QH
• Chiral edge states on the boundary
E
x
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5/2 Hall Fluid
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• FQHE physics is topological, meaning that distance plays no role (or at least an inferior role).
• Topology is “rubber sheet geometry” and FQHE is “rubber sheet physics.”
• Controlled by the Chern-Simons lagranian which does not mention distance!
• It is topologically invariant
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Topological-invariance is clearly not a symmetry of the underlying Hamiltonian.
How can Chern-Simons theory possibly describe the low energy physics of the above Hamiltonian?
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The answer goes back to:
1970: Wilson, Renormalization: How does the Langrangian evolve when
re-expressed using longer length scales, lower frequencies, colder
temperatures?
The terms with the fewest derivatives dominate: This is because in
momentum space, differentiation becomes multiplication by k and:
k>>k2
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• Chern-Simons Action: A d A + (A A A) has one
derivative, • while kinetic energy p2/2m is written with two derivatives.
(pi = 1/2 m d/dxi ) • Thus, in condensed matter at low enough temperatures, we
may expect to see systems in which topological effects dominate and geometric detail becomes irrelevant.
• FQHE is such a system.
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A new proposal: Majoranas in a px + ipy superconductor within a “conventional” semi-conductor device
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• In the topological approach, interferometers will play a key role:
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FQH interferometer
Willett et al. `08 for =5/2
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A lot of theory [Bonderson, et al] has been devoted to using interferometers to:
• Measure topological charge • Manipulate quantum information
• Simulate quasi particle braiding
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FQH fluid (blue)
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ie
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• Interferometry creates probabilistic combinations of quasi-particle world lines. •Let me take you through some cartoons:
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• Braiding, and hence calculation, can be simulated by measurement.
a a
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Measurement Simulated Braiding!
a a
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• Bob Willett of Bell Labs has presented evidence of Majoranas in ν=5/2 FQHE systems
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Robert Willett
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24 hrs/run
Bob Willett
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Conclusions:
• In building a topological quantum computer, universality is our friend.
▫ It allows us to model and study exotic states of matter such as the fractional quantum Hall effect.
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▫ However, even if quantum computers are enormously successful and can compute LHC energy scale physics from string theory,
the same universality may cloak the true microscopics: Many competing theories may all work.
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• Building a quantum computer will test quantum mechanics (QM) in new regimes.
▫ We may find QM breaks down with complexity and cannot “redeem” the entire physical world.
▫ If QM is fundamentally correct, it is now only a matter of technology to build a quantum computer. The mathematical theory is completely convincing.
▫ Is the technology ready?
Charles Babbage 1792-1871
John von Neumann 1903-1957
Gordon E. Moore
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• Moving from P to Q
▫ The universe speaks to itself in the language of QM.
▫ We are on the verge of fluency in this internal language of QM.
▫ The quantum computer will give wings to our thoughts.
Athena
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• But a doubt arises!
▫ Why did biology not exploit this language first?
▫ After all, we think biology and also neurobiology only processes information classically.
=
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• Probably the universe is everywhere too warm for quantum mechanical thought.
▫ The microwave background is approximately 3.8K.
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• But now on Earth within dilution refrigerators it is colder than anywhere else (we know of) in space.
• The deep cold of a dilution refrigerator will be the home of our quantum computers. From this extreme stillness we will lift the ultimate tool out of our tool box.
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6.23 × 109
decimal
number
computer nuclear
biology
a |0> + b |1> quantum number/
quantum computer
fire
machines
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• May we use it well!
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