Download Quantum Mathematics

Copenhagen May 25, 2015
Michael Freedman
Microsoft Research—Station Q
1 Quantum Mathematics
and the
Relationship Between Math and Physics
2 Eugene Wigner
“The Unreasonable Effectiveness of
Mathematics in the Natural Sciences” 1960
I’d like to propose a “dual” aphorism:
​𝐖𝐢𝐠𝐧𝐞𝐫 “. . . the unreasonable effectiveness of physics in mathematics . . .”
3 Dualities in Mathematics
Poincaré Duality
(in topology)
Fourier Duality
(in analyis)
p 4 Physics
•  Particle ↔ wave
•  ADS/CFT
•  Donaldson ↔ Seiberg/Witten
Field Theory
Effective Infrared Limit
Effective Ultraviolet Limit
5 Both ​
and Wigner are supported by the work of
Wign and Vaughan Jones others in the past 30
Ed Witten
years.er Story of:
Jones polynomial (topology)
History Quantum Mechanics ↓ Operator algbebras ↓ von Neumann algebras ↓ braid representa?on Link invariants ‖ Jones Polynomial ↓ “Topological Quantum Field Theory” ↓ Topological quantum Categorifica?on 3-­‐manifold computa?on ↓ topology Khovanov homology Five branes, equa?ons in 4 & 5D 6 More​ 𝐖𝐢𝐠𝐧𝐞𝐫: Langlands Program
N=4 supersymmetric Yang-Mills → 4D families of TQFTs …
↓
Algebraic number theory / Galois groups
⇕
Automorphic forms / rep theory
Robert
Langlands
7 Energy of loops in non-­‐linear Σ-­‐model ↓ BoX periodicity ↓ Alexei Kitaev Raoul BoX Kitaev’s classifica?on of free fermions according to dimension and symmetry ↓ Controlled k-­‐theory 8 The central player in string theory and perhaps
mathematics as a whole is the complex curve.
9 Things sounding highly specialized to mathematicians:
super-symmetric string field theories, turn out to be
fundamental. They enumerate basic algebraic-geometric
objects, as shown by Candelas et. al. using a duality
between Calabi-Yau manifolds.
Shing-Tung Yau
Candelas, de la Ossa, Green, Parkes
10 •  “Wigner” says that the universe is regular.
​
•  Wign
suggests that the universe is not a
er
realization
of an arbitrary consistent system but
rather a system that is maximal or even unique.
•  Otherwise math would far outreach physics.
11 Leibniz spoke of “the best of all possible worlds.”
•  Maybe there is only one possible world.
Gottfried Leibniz
Could the universe
have been different?
12 •  Could there be a world where NP-complete
problems can be solved efficiently?
•  What about a world where Grover
search runs in cube root rather than square root time?
•  Many (Aaronson) think not – that just like perpetual motion,
such worlds cannot be consistent.
Free Will! Descarte Indeed. Aaronson 13 I’d like to expand the discussion to include
Information and Computation as promising,
younger colleagues of Physics and Mathematics.
•  The Godel, Turing and Shannon’s theories of proof,
computation and communication evolved in the
1960’s into the theory of computational complexity
14 Good computational models are rare.
Modern Church-Turing (MCT) Thesis:
•  There are only two maximal physically realistic
models of computation:
–  One based on Classical Physics
–  One based on Quantum Physics
Alonzo Church P
BQP
Alan Turing 15 We believe BQP is stronger than P
(evidence Shor’s factoring Algorithm
)
•  Why is the quantum world superior?
•  Our Classical world emerges through neglect, that
is failure to observe an entire system but merely a
piece of it.
16 •  Mathematically this neglect is called partial trace and
averages the unobserved degrees of Freedom. Physically the
process is called decoherence.
•  Planck’s contstant ћ ≤ ∆ p ∆ x is the quantum of phase space
volume and neglecting a portion of phase space large with
respect to ћ produces “classical outcomes.”
Quantum Classical Corollary of MCT: All we will ever know (or at least compute) will lie in BQP 17 Despite quandaries involving unitarity and
black holes I am happy in believing
Quantum Mechanics governs the universe.
Schrödinger (Amplitudes NOT probablities)
18 •  Amplitudes are square roots of probabilities
•  Square roots of probabilities are not intuitive.
•  Nothing in our large scale classical world, nothing in
our evolutionary experience, prepares our mind for
superposition of amplitudes within a Hilbert space.
•  Superposition was born amid mystery and paradox in
the period 1900-1927.
Planck Born Bohr Heisenberg Schrodinger Radiation, Diffraction, Scattering, Atomic Spectra
19 •  We need something like the double slit experiment to
see amplitudes at work
20 It is amplitudes not probabili?es which add All closed Blind 1 open 2 open -|α|2
Screen +|α + β|2
0
-|β|2
Source |α + β|2 = |α|2 + |β|2
Observed paXern 21 This has lead us to a new type of numeral : Let us “hash” Mankind’s history into a Brief History of Numbers •  -­‐13,000 years: Coun?ng in unary Possible futures contract for sheep in Anatolia •  -­‐3000 years: Place nota?on •  Hindu-­‐Arab, Chinese 7,123,973,713
•  1982: Configura?on numbers as basis of a Hilbert space of states 22 •  Quantum computers manipulate numbers in superposi?on – essen?ally crea?ng a new kind of numeral. •  We believe that quantum computers will do amazing things. 23 •  But we’re not sure exactly what. •  Prominent possibili?es: •  Quantum Chemistry •  Drug Design •  High Tc •  Machine Learning 24 •  Quantum mechanics does not permit copying of informa?on (no cloning theorem). Thus •  Long quantum mechanical computa?on requires either –  painful error correc?on: For big problems 99.9%-­‐99.99% of resources go to s?fle error, even given physical gates with 99.9% fidelity, or –  extreme accuracy (topology) 25 Topology •  Charlie Marcus of KU and NBI is making breathtaking advances in the topological direc?on: Sankar Das Sarma Stanescu, Lutchyn, Das Sarma, PRB’11 Charlie Marcus •  Within our life?mes a new tool will lie within our or collec?ve toolbox. 26 6.23 × 109
decimal
computer nuclear machines number biology α |0> + β |1>
quantum number/
quantum computer
fire 27 But a skep?c might ask: •  If topological quantum systems are so great at processing informa?on, why don’t natural biological systems exploit them? •  Quantum effects are most pronounced in cold environments, T« gap. •  Maybe biological systems will, but we’ll have to wait 1011 years for the cosmic background temperature to drop low enough. •  Our joint endeavor with Marcus and others is designed short-­‐cut this tedious hundred billion-­‐year wait. 28 Let me finish by revisi?ng two old ideas with “quantum thinking.” The first is modest and sober, the second is not. Max Flow = Min Cut (1956 Shannon, and Ford-­‐Fulkerson) Classical in Max Flow = Min Cut = 2 out Quantum
T’ 2
2
2
T
2
2
3
(2015 Cui, F., Stong, R.) T” 3
Max (rank (network)) = 7 ˂ 8 T, T’, T” 29 Now a less sober idea
•  Taking​Wigner seriously, let’s reverse a fifty-year effort to
construct a mathematical foundation for field theory and
instead seek a field theoretic foundation for mathematics.
•  A Feynman diagram (let’s take cubic interactions) has the same
structure as a proof in a formal system X: two things come
together and a third thing gets “spit out.”
A⇒ B
​Wigner B
A
Physics (field theory)
Logic (modus ponens)
30 •  We should attempt to “reverse engineer” the
“field theory” whose perturbative
expansions are the deductions of some fixed
formal system—X.
•  For such a theory, the partition function Zi, f
would (perturbatively) be a weighted sum of
all possible proofs from the initial conditions i,
the axioms, to the final condition f, the
statement in question.
i
f
31 But a field contains more than perturbative
information (think of kinks, instantons, phase
transition, etc.), so one would expect situations
in which
Zi, f > 0
even in the absence of a formal proof, in system
X, of statement f, from the axioms i.
David Thouless
Gerard t’Hooft
32 •  In a field based system, more things would be
“provable.” Corresponding to nonperturbative effects.
•  Perhaps, Gödel’s incompleteness theorems disappear:
there seems no enumeration scheme for our more
general “proofs.”
•  “Proofs” would no longer be something you can “write
down”, but merely accumulate evidence about.
33 •  The early 20th century paradoxes within set theory
might eventually be interpreted as “pushing a low
energy effective theory beyond its limits.”
•  With luck, we might undo all the good work of the
twentieth century on logic and set theory and return to
the world of Hilbert and Bohr.
David Hilbert Niels Bohr 34 •  There are two types of numbers (integers) in our experience that are effec?vely non-­‐overlapping: 0, 1, 2, …, 1070
Number of things
(22)
10(10 )
Number of configurations
or system states
35