Cosmology and Complexity Classes

Scott Aaronson (UC Berkeley)

ZPP

LGapP

W[P]

SZK

QAM

EEXP

Complexity Classes Not Needed For This Talk0-1-NPC - #L - #L/poly - #P - #W[t] - +EXP - +L - +L/poly - +P - +SAC1 - AC - AC0 - AC0[m] - ACC0 - AH - AL -

AM - AmpMP - AP - AP - APP - APX - AVBPP - AvE - AvP - AW[P] - AWPP - AW[SAT] - AW[*] - AW[t] - βP - BH - BPE - BPEE - BPHSPACE(f(n)) - BPL - BPPKT - BPP-OBDD - BPQP - BQNC - BQP-OBDD - k-BWBP - C=L

- C=P - CFL - CLOG - CH - CkP - CNP - coAM - coC=P - coMA - coModkP - coNE - coNEXP - coNL - coNP -

coNP/poly - coRE - coRNC - coRP - coUCC - CP - CSL - CZK - Δ2P - δ-BPP - δ-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)) - G[t] - HkP - HVSZK -

IC[log,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 - NPMV t - 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)) - Φ2P - PhP - Π2P - PK - PKC - PL - PL1 - PLinfinity - PLF - PLL - P/log - PLS - PNP - PNP[k] - PNP[log]

- 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 - QAC0[m] - 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 - Σ2P - SKC - SL - SLICEWISE PSPACE - SNP - SO-

E - SP - span-P - SPARSE - SPP - SUBEXP - symP - SZK - TALLY - TC0 - TFNP - Θ2P - TREE-REGULAR - UCC

- UL - UL/poly - UP - US - VNCk - VNPk - VPk - VQPk - W[1] - W[P] - WPP - W[SAT] - W[*] - W[t] - W*[t] - XP -

XPuniform - YACC - ZPE - ZPP - ZPTIME(f(n))

More at http://www.cs.berkeley.edu/~aaronson/zoo.html

Outline

• The Physics of Databases

• Quantum Search of Spatial Regions

• The Universe in 10 Minutes

• The Inflationary Turing Machine(work in progress)

Quantum Search of Spatial Regions

Joint work with Andris Ambainis (U. of Latvia)

quant-ph/0303041

Grover’s O(n) Quantum Search Algorithm:

Great for combinatorial search

But can it help search a physical region?

BWAHAHA! Look who

needs physics now!

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 we’re screwed!

So why not pack data in 3 dimensions?

Then the complexity would be n n1/3 = n5/6

Trouble: Suppose our “hard disk” has mass density

We saw Grover search of a 2D grid presented a problem…

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

Actually worse—even 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 (r3/2)

Holographic Principle: A region of space can’t store more than 1.41069 bits per meter2 of surface area

Quantum Mechanics and General Relativityboth yield a n lower bound on search

If space had d>3 dimensions, then relativity bound would be weaker: n1/(d-1)

Is that bound achievable? Apparently not, since even stronger limit (Bekenstein’s) applies for weakly-gravitating systems

What We Can Achieve

If n ~ rc bits are scattered in a 3D ball of radius r (where c3 and bits’ locations are known), search time is (n1/c+1/6) (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

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

Can we do anything better?

Benioff (2001): Guess we can’t…

Back to the Main Issue

Classical search takes (n) timeQuantum search takes (rn)

(r = maximum radius of region)

REVENGE OF COMPUTER SCIENCE

• We can.

Using amplitude amplification techniques of BHMT’2002, we get:

O(n log3n) for 2D grid

O(n) for 3 and higher dimensions

• Idea: Recursively divide into sub-squares

• Undirected connected graph G=(V,E)• Bit xi at each vertex vi

• Goal: Compute some Boolean f(x1…xn){0,1}

• State can have arbitrary ancilla z:

• Alternate query transforms with ‘local’ unitariesWhat does ‘local’ mean? Depends on your religion

, ,i z iv z , 1 ,ix

i iv z v z

What’s the Model?

Defining Locality: 3 Choices

(1) Unitary must be decomposable into commuting local operations, each acting on a single edge

(2) Just don’t “send amplitude” between non-adjacent vertices: if (i,j)E then

(3) Take U=eiH 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)

, , 0i z j zU

, , 0i z j zH

• Assume there’s a unique marked item• Divide into n1/5 subcubes, each of size n4/5 • Algorithm A:

If n=1, check whether you’re at a marked itemElse pick a random subcube and run A on itRepeat n1/11 times using amplitude amplification

• Running time:

1/11 4/5 1/

5/11

dT n n T n O n

O n

In More Detail: d3

• Success probability (unamplified):

• With amplification:

(since is negligible)

• Amplify whole algorithm n1/22 times to get

1/5 4/5P n n P n

d3 (continued)

2/11 1/5 4 /5

1/11

1P n n n P n

n

1/ 22 5/111 ,P n T n O n n O n

• For r marked items, we get

for d3, even if r is unknown

• For d=2, get T(n)=O(n log3n)

• For any graph that’s “d-dimensional” by expansion properties (d>2), if h “potential” marked items are scattered around (and their locations are known), get

Other Resultsto which I won’t subject you

1/ 2 1/ d

nT n O d

r

1/

logd

nT n O h poly h

h

• Razborov 2002:

• Problem: Alice has x1…xn{0,1}n, Bob has y1…yn

They want to know if xiyi=1 for some i

Application: Disjointness

• How many qubits must they communicate?

• Buhrman, Cleve, Wigderson 1998: logO n n

• Høyer, de Wolf 2002: log*nO nc

n

A B

, , , ,A Bi z z i A i Bv z v z

State at any time:

Communicating one of 6 directions takes only 3 qubits

Disjointness in O(n) Communication

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)

The Inflationary Turing Machine

Before we were asking how the nature of space affects query complexity

Now let’s ask how it affects computational complexity

And let’s ground ourselves in the firm soil of observation…

The New York Times Theory of Cosmology

Closed Flat Open

Source: Supernova Cosmology Project (Perlmutter et al.)

astro-ph/9812133

With a vacuum energy density >0, geometry is no longer destiny

Evidence for >0

Scale Factor a(t)(not to scale)

Matter-Dominated Eraa(t) ~ t2/3

-Dominated Eraa(t) ~ ct again

10 billion years ABB: Matter and contribute

equallyInflationa(t) ~ ct

14 billion years ABB:P=?NP problem posed

Tipler’s 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

Bousso’s boundhep-th/0010252

p

q 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

Lloyd’s 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 > 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…

The Inflationary Turing Machine

0 1 01 1 0 0 1

The Inflationary Turing Machine

0 1 01 1 0 0 1

The Inflationary Turing Machine

0 1 01 1 0 0 1

The Inflationary Turing Machine

0 1 01 1 0 0 1

The Inflationary Turing Machine

0 1 01 1 0 0 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 > 0 spacetime

Let INF(1/) be the class of languages decided by inflationary machine

Claim:

1 1 1DSPACE INF DSPACE

Same for quantum analogues, BQSPACE and BQINF

Open Problems In This Model

• In O(n) time, can we compute anything with an nn square worktape that we couldn’t with a nn 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”?

• 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 questionsQuantum computing is just one example

Not all strings have n bits