Cosmology and Complexity Classes

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Cosmology and Complexity Classes
SZK
ZPP
QAM

• Scott Aaronson (UC Berkeley)

GapP
?L
EEXP
WP
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Complexity Classes Not Needed For This Talk
0-1-NPC - L - L/poly - P - Wt - EXP - L -
L/poly - P - SAC1 - AC - AC0 - AC0m - ACC0 -
AH - AL - AM - AmpMP - AP - AP - APP - APX -
AVBPP - AvE - AvP - AWP - AWPP - AWSAT -
AW - AWt - ßP - BH - BPE - BPEE -
BPHSPACE(f(n)) - BPL - BPPKT - BPP-OBDD - BPQP -
BQNC - BQP-OBDD - k-BWBP - CL - CP - CFL - CLOG
- CH - CkP - CNP - coAM - coCP - coMA - coModkP
- coNE - coNEXP - coNL - coNP - coNP/poly - coRE
- coRNC - coRP - coUCC - CP - CSL - CZK - ?2P -
d-BPP - d-RP - DET - DisNP - DistNP - DP - E - EE
- EEE - EEXP - EH - ELEMENTARY - ELkP - EPTAS -
k-EQBP - EQP - EQTIME(f(n)) - ESPACE - EXP -
EXPSPACE - Few - FewP - FNL - FNL/poly - FNP -
FO(t(n)) - FOLL - FP - FPR - FPRAS - FPT - FPTnu
- FPTsu - FPTAS - F-TAPE(f(n)) - F-TIME(f(n)) -
GapL - GapP - GC(s(n),C) - GPCD(r(n),q(n)) - Gt
- HkP - HVSZK - IClog,poly - IP - L - LIN - LkP
- LOGCFL - LogFew - LogFewNL - LOGNP - LOGSNP -
L/poly - LWPP - MA - MAC0 - MA-E - MA-EXP - mAL -
MaxNP - MaxPB - MaxSNP - MaxSNP0 - mcoNL - MinPB
- MIP - MIPEXP - (Mk)P - mL - mNC1 - mNL - mNP -
ModkL - ModkP - ModP - ModZkL - mP - MP - MPC -
mP/poly - mTC0 - NC - NC0 - NC1 - NC2 - NE - NEE
- NEEE - NEEXP - NEXP - NIQSZK - NISZK - NL -
NLIN - NLOG - NL/poly - NPC - NPC - NPI - NP
intersect coNP - (NP intersect coNP)/poly - NPMV
- NPMV-sel - NPMVt - NPMVt-sel - NPO - NPOPB -
NP/poly - (NP,P-samplable) - NPR - NPSPACE - NPSV
- NPSV-sel - NPSVt - NPSVt-sel - NQP -
NSPACE(f(n)) - NTIME(f(n)) - OCQ - OptP - PBP -
k-PBP - PC - PCD(r(n),q(n)) - P-close -
PCP(r(n),q(n)) - PEXP - PF - PFCHK(t(n)) - F2P -
PhP - ?2P - PK - PKC - PL - PL1 - PLinfinity -
PLF - PLL - P/log - PLS - PNP - PNPk - PNPlog
- P-OBDD - PODN - polyL - PP - PPA - PPAD - PPADS
- P/poly - PPP - PPP - PR - PR - PrHSPACE(f(n)) -
PromiseBPP - PromiseRP - PrSPACE(f(n)) - P-Sel -
PSK - PSPACE - PT1 - PTAPE - PTAS -
PT/WK(f(n),g(n)) - PZK - QAC0 - QAC0m - QACC0 -
QAM - QCFL - QH - QIP - QIP(2) - QMA - QMA(2) -
QMAM - QMIP - QMIPle - QMIPne - QNC0 - QNCf0 -
QNC1 - QP - QSZK - R - RE - REG - RevSPACE(f(n))
- RHL - RL - RNC - RPP - RSPACE(f(n)) - S2P - SAC
- SAC0 - SAC1 - SC - SEH - SFk - S2P - SKC - SL -
SLICEWISE PSPACE - SNP - SO-E - SP - span-P -
SPARSE - SPP - SUBEXP - symP - SZK - TALLY - TC0
- TFNP - T2P - TREE-REGULAR - UCC - UL - UL/poly
- UP - US - VNCk - VNPk - VPk - VQPk - W1 -
WP - WPP - WSAT - W - Wt - Wt - XP -
XPuniform - YACC - ZPE - ZPP - ZPTIME(f(n)) More
at http//www.cs.berkeley.edu/aaronson/zoo.html
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Outline

• The Physics of Databases
• Quantum Search of Spatial Regions
• The Universe in ?10 Minutes
• The Inflationary Turing Machine (work in
  progress)

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Quantum Search of Spatial Regions

• Joint work with Andris Ambainis (U. of Latvia)
• quant-ph/0303041

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Grovers O(?n) Quantum Search Algorithm Great
for combinatorial search But can it help search
a physical region?
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Speed of light is finite
What even a dumb computer scientist knows
THE SPEED OF LIGHT IS FINITE
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We saw Grover search of a 2D grid presented a
problem
So why not pack data in 3 dimensions?
Then the complexity would be ?n ? n1/3 n5/6
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Holographic principle
Once radius exceeds Schwarzschild bound of
?(1/??), hard disk collapses to form a black
hole Makes things harder to retrieve
Actually worseeven a 2D hard disk would collapse
once radius exceeds ?(1/?) 1D hard disk would
not collapse
But we care about entropy, not mass
A ball of radiation of radius r has energy ?(r)
but entropy ?(r3/2)
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Holographic principle
Holographic Principle A region of space cant
store more than 1.4?1069 bits per meter2 of
surface area
Quantum Mechanics and General Relativity both
yield a ?n lower bound on search If space had
dgt3 dimensions, then relativity bound would be
weaker n1/(d-1)
Is that bound achievable? Apparently not, since
even stronger limit (Bekensteins) applies for
weakly-gravitating systems
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What We Can Achieve
If n rc bits are scattered in a 3D ball of
radius r (where c?3 and bits locations are
known), search time is ?(n1/c1/6) (up to
polylog factor) For radiation disk (n r3/2)
?(n5/6) ?(r5/4) For n r2 (saturating
holographic bound) ?(n2/3) ?(r4/3) To get
O(?n polylog n), bits would need to be
concentrated on a 2D surface
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Objections to the Model

• Would need n parallel computing elements to
  maintain a quantum database
• Response Might have n passive elements, but
  many fewer active elements (i.e. robots), which
  we wish to place in superposition over locations
• (2) Must consider effects of time dilation
• Response For upper bounds, will have in mind
  weakly-gravitating systems, for which time
  dilation is by at most a constant factor

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Back to the Main Issue
Classical search takes ?(n) time Quantum search
takes ?(r?n) (r maximum radius of region)
Can we do anything better?
Benioff (2001) Guess we cant
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Revenge of computer science
REVENGE OF COMPUTER SCIENCE

• We can.

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Whats the Model?

• Undirected connected graph G(V,E)
• Bit xi at each vertex vi
• Goal Compute some Boolean f(x1xn)?0,1
• State can have arbitrary ancilla z
• Alternate query transforms
• with local unitaries
• What does local mean? Depends on your religion

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Locality religions
Defining Locality 3 Choices
(1) Unitary must be decomposable into commuting
local operations, each acting on a single
edge (2) Just dont send amplitude between
non-adjacent vertices if (i,j)?E then (3) Take
UeiH where H has eigenvalues of absolute value
at most ?, and if (i,j)?E then (1) ? (2),(3).
Upper bounds will work for (1) lower bounds for
(2),(3)
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In More Detail d?3

• Assume theres a unique marked item
• Divide into n1/5 subcubes, each of size n4/5
• Algorithm A
• If n1, check whether youre at a marked item
• Else pick a random subcube and run A on it
• Repeat n1/11 times using amplitude amplification
• Running time

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d?3 (continued)

• Success probability (unamplified)
• With amplification
• (since ? is negligible)
• Amplify whole algorithm n1/22 times to get

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Other Resultsto which I wont subject you

• For r marked items, we get
• for d?3, even if r is unknown
• For d2, get T(n)O(?n log3n)
• For any graph thats d-dimensional by
  expansion properties (dgt2), if h potential
  marked items are scattered around (and their
  locations are known), get

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Application Disjointness

• Problem Alice has x1xn?0,1n, Bob has y1yn
  They want to know if xiyi1 for some i

• How many qubits must they communicate?

• Buhrman, Cleve, Wigderson 1998

• Høyer, de Wolf 2002

• Razborov 2002

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Disjointness in O(?n) Communication
State at any time
Communicating one of 6 directions takes only 3
qubits
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Random walk
Searching by Walking
Can a quantum walk search a 2D grid efficiently?
(Maybe even ?n time instead of ?n log3n?)
Promising numerical evidence (courtesy N. Shenvi)
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The Inflationary Turing Machine
Before we were asking how the nature of space
affects query complexity Now lets ask how it
affects computational complexity And lets ground
ourselves in the firm soil of observation
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The New York Times Theory of Cosmology
Closed
Flat
Open
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The Chart
With a vacuum energy density ?gt0, geometry is no
longer destiny
Source Supernova Cosmology Project (Perlmutter
et al.) astro-ph/9812133
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Evidence for ?gt0
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Scale Factor a(t)(not to scale)
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Tiplers Theory
As the big crunch draws near, violent
oscillations cause O(1) computation steps to be
performed in shorter and shorter intervals, so
that time appears subjectively infinite
Advantage of theory Falsifiable
Disadvantage Falsified
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Boussos boundhep-th/0010252
Largest number of bits accessible to any one
observer 3?/? ? 10122 Idea Any experiment has a
beginning (p) and an end (q) Consider causal
diamond D intersection of future light-cone of p
with past light-cone of q Use holographic
principle to upper-bound entropy in D
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Lloyds boundquant-ph/0110141

• Largest number of bits accessible so far( of
  Planck times elapsed since the big bang)2 ?
  (1061)2 10122
• Also uses holographic principle, but does not
  depend on ? gt 0
• Why do the two bounds coincide? We live in a
  transitional era, when both ? and dust
  contribute significantly to net energy ?? ? 0.7,
  ?dust ? 0.3
• Why should that be so? Dunno

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The Inflationary Turing Machine
1
1
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The Inflationary Turing Machine
1
1
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The Inflationary Turing Machine
1
1
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The Inflationary Turing Machine
1
1
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The Inflationary Turing Machine
1
1
At each time step t, a new tape square
(initialized to 0) is created after square k/? -
t for each integer k Toy model for ? gt 0 spacetime
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Let INF(1/?) be the class of languages decided by
inflationary machine Claim
Same for quantum analogues, BQSPACE and BQINF
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Open Problems In This Model

• In O(n) time, can we compute anything with an
  n?n square worktape that we couldnt with a ?n??n
  square tape? I.e. how much of the observable
  universe could we take advantage of before it
  recedes?
• What about quantum-mechanically?
• What is the effect of including gravity?

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Conclusions

• In a ?gt0 spacetime, a quantum robot could search
  a larger region than a classical one (not
  assuming any time bound)

• Physics is a good source of pure CS
  questions Quantum computing is just one example