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Cosmology and Complexity Classes Scott Aaronson (UC Berkeley) ZPP L GapP W[P] SZK QAM EEXP

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Complexity Classes Not Needed For This Talk 0-1-NP C - #L - #L/poly - #P - #W[t] - +EXP - +L - +L/poly - +P - +SAC 1 - AC - AC 0 - AC 0 [m] - ACC 0 - AH - AL - AM - AmpMP - AP - AP - APP - APX - AVBPP - AvE - AvP - AW[P] - AWPP - AW[SAT] - AW[*] - AW[t] - βP - BH - BPE - BPEE - BP H SPACE(f(n)) - BPL - BPP KT - BPP-OBDD - BPQP - BQNC - BQP-OBDD - k-BWBP - C=L - C=P - CFL - CLOG - CH - C k P - CNP - coAM - coC=P - coMA - coMod k P - coNE - coNEXP - coNL - coNP - coNP/poly - coRE - coRNC - coRP - coUCC - CP - CSL - CZK - Δ 2 P - δ-BPP - δ-RP - DET - DisNP - DistNP - DP - E - EE - EEE - EEXP - EH - ELEMENTARY - EL k P - EPTAS - k-EQBP - EQP - EQTIME(f(n)) - ESPACE - EXP - EXPSPACE - Few - FewP - FNL - FNL/poly - FNP - FO(t(n)) - FOLL - FP - FPR - FPRAS - FPT - FPT nu - FPT su - FPTAS - F-TAPE(f(n)) - F-TIME(f(n)) - GapL - GapP - GC(s(n),C) - GPCD(r(n),q(n)) - G[t] - H k P - HVSZK - IC[log,poly] - IP - L - LIN - L k P - LOGCFL - LogFew - LogFewNL - LOGNP - LOGSNP - L/poly - LWPP - MA - MAC 0 - MA-E - MA-EXP - mAL - MaxNP - MaxPB - MaxSNP - MaxSNP 0 - mcoNL - MinPB - MIP - MIP EXP - (M k )P - mL - mNC 1 - mNL - mNP - Mod k L - Mod k P - ModP - ModZ k L - mP - MP - MPC - mP/poly - mTC 0 - NC - NC 0 - NC 1 - NC 2 - NE - NEE - NEEE - NEEXP - NEXP - NIQSZK - NISZK - NL - NLIN - NLOG - NL/poly - NPC - NP C - NPI - NP intersect coNP - (NP intersect coNP)/poly - NPMV - NPMV-sel - NPMV t - NPMV t -sel - NPO - NPOPB - NP/poly - (NP,P-samplable) - NP R - NPSPACE - NPSV - NPSV-sel - NPSV t - NPSV t -sel - NQP - NSPACE(f(n)) - NTIME(f(n)) - OCQ - OptP - PBP - k-PBP - P C - PCD(r(n),q(n)) - P-close - PCP(r(n),q(n)) - PEXP - PF - PFCHK(t(n)) - Φ 2 P - PhP - Π 2 P - P K - PKC - PL - PL 1 - PL infinity - PLF - PLL - P/log - PLS - P NP - P NP[k] - P NP[log] - P-OBDD - PODN - polyL - PP - PPA - PPAD - PPADS - P/poly - PPP - P PP - PR - P R - Pr H SPACE(f(n)) - PromiseBPP - PromiseRP - PrSPACE(f(n)) - P-Sel - PSK - PSPACE - PT 1 - PTAPE - PTAS - PT/WK(f(n),g(n)) - PZK - QAC 0 - QAC 0 [m] - QACC 0 - QAM - QCFL - QH - QIP - QIP(2) - QMA - QMA(2) - QMAM - QMIP - QMIP le - QMIP ne - QNC 0 - QNC f 0 - QNC 1 - QP - QSZK - R - RE - REG - RevSPACE(f(n)) - R H L - RL - RNC - RPP - RSPACE(f(n)) - S 2 P - SAC - SAC 0 - SAC 1 - SC - SEH - SF k - Σ 2 P - SKC - SL - SLICEWISE PSPACE - SNP - SO- E - SP - span-P - SPARSE - SPP - SUBEXP - symP - SZK - TALLY - TC 0 - TFNP - Θ 2 P - TREE-REGULAR - UCC - UL - UL/poly - UP - US - VNC k - VNP k - VP k - VQP k - W[1] - W[P] - WPP - W[SAT] - W[*] - W[t] - W * [t] - XP - XP uniform - 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? BWAHAHA! Look who needs physics now!

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What even a dumb computer scientist knows: THE SPEED OF LIGHT IS FINITE Marked item Robot n n Consider a quantum robot searching a 2D grid: We need n Grover iterations, each of which takes n time, so were screwed! Speed of light is finite

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So why not pack data in 3 dimensions? Then the complexity would be n n 1/3 = n 5/6 Trouble: Suppose our hard disk has mass density We saw Grover search of a 2D grid presented a problem…

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Once radius exceeds Schwarzschild bound of (1/ ), hard disk collapses to form a black hole Makes things harder to retrieve… But we care about entropy, not mass Holographic principle Actually worseeven a 2D hard disk would collapse once radius exceeds (1/ ) 1D hard disk would not collapse… A ball of radiation of radius r has energy (r) but entropy (r 3/2 )

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Holographic Principle: A region of space cant store more than 1.4 10 69 bits per meter 2 of surface area Quantum Mechanics and General Relativity both yield a n lower bound on search If space had d>3 dimensions, then relativity bound would be weaker: n 1/(d-1) Holographic principle 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 ~ r c bits are scattered in a 3D ball of radius r (where c 3 and bits locations are known), search time is (n 1/c+1/6 ) (up to polylog factor) For radiation disk (n ~ r 3/2 ): (n 5/6 ) = (r 5/4 ) For n ~ r 2 (saturating holographic bound): (n 2/3 ) = (r 4/3 ) To get O( n polylog n), bits would need to be concentrated on a 2D surface

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Objections to the Model (1)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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Can we do anything better? Benioff (2001): Guess we cant… Back to the Main Issue Classical search takes (n) time Quantum search takes (r n) (r = maximum radius of region)

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REVENGE OF COMPUTER SCIENCE We can. Using amplitude amplification techniques of BHMT2002, we get: O( n log 3 n) for 2D grid O( n) for 3 and higher dimensions Idea: Recursively divide into sub-squares Revenge of computer science

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Undirected connected graph G=(V,E) Bit x i at each vertex v i Goal: Compute some Boolean f(x 1 …x n ) {0,1} State can have arbitrary ancilla z: Alternate query transforms with local unitaries What does local mean? Depends on your religion Whats the Model?

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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 U=e iH 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) Locality religions

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Assume theres a unique marked item Divide into n 1/5 subcubes, each of size n 4/5 Algorithm A: If n=1, check whether youre at a marked item Else pick a random subcube and run A on it Repeat n 1/11 times using amplitude amplification Running time: In More Detail: d 3

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Success probability (unamplified): With amplification: (since is negligible) Amplify whole algorithm n 1/22 times to get d 3 (continued)

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For r marked items, we get for d 3, even if r is unknown For d=2, get T(n)=O( n log 3 n) For any graph thats d-dimensional by expansion properties (d>2), if h potential marked items are scattered around (and their locations are known), get Other Results to which I wont subject you

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Razborov 2002: Problem: Alice has x 1 …x n {0,1} n, Bob has y 1 …y n They want to know if x i y i =1 for some i Application: Disjointness How many qubits must they communicate? Buhrman, Cleve, Wigderson 1998: Høyer, de Wolf 2002:

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A B State at any time: Communicating one of 6 directions takes only 3 qubits Disjointness in O( n) Communication

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Searching by Walking Can a quantum walk search a 2D grid efficiently? (Maybe even n time instead of n log 3 n?) Promising numerical evidence (courtesy N. Shenvi) Random walk

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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 ClosedFlatOpen

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The Chart Source: Supernova Cosmology Project (Perlmutter et al.) astro-ph/9812133 With a vacuum energy density >0, geometry is no longer destiny

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Evidence for >0

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Scale Factor a(t) (not to scale) Matter-Dominated Era a(t) ~ t 2/3 -Dominated Era a(t) ~ c t again 10 billion years ABB: Matter and contribute equally Inflation a(t) ~ c t 14 billion years ABB: P=?NP problem posed

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Tiplers Theory Advantage of theory: Falsifiable Disadvantage: Falsified 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

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Boussos bound hep-th/0010252 p q Largest number of bits accessible to any one observer: 3 / 10 122 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 bound quant-ph/0110141 Largest number of bits accessible so far: (# of Planck times elapsed since the big bang) 2 (10 61 ) 2 = 10 122 Also uses holographic principle, but does not depend on > 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 0 1 0 1 1001

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0 1 0 1 1001

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0 1 0 1 1001

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0 1 0 1 1001

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0 1 0 1 1001 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 > 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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In a >0 spacetime, a quantum robot could search a larger region than a classical one (not assuming any time bound) Conclusions Physics is a good source of pure CS questions Quantum computing is just one example Not all strings have n bits

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