Quantum Search of Spatial Regions Scott Aaronson (UC Berkeley) Joint work with Andris Ambainis (IAS / U. Latvia)

Quantum Search of Spatial Regions

Scott Aaronson (UC Berkeley)

Joint work with Andris Ambainis (IAS / U. Latvia)

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

Quantum Computing

Model of computation based on our best-confirmed physical theory

State of computer is superposition over strings:

Most famous algorithm: Shor’s algorithm for factoring

This talk: Grover’s algorithm for search

2

0,1 0,1

, , 1m m

x x x

x x

x

Grover’s Search Algorithm

Unsorted database of n items

Goal: Find one “marked” item

• Classically, order n queries to database needed

• Grover 1996: Quantum algorithm using order n queries

• BBBV 1996: Grover’s algorithm is optimal

|000

Initial Superposition

|001 |101|100|011|010

|000

Amplitude of Solution State Inverted

|001 |101|100

|011|010

|000

All Amplitudes Inverted About Mean

|001 |101|100|011|010

Grover’s 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!

Talk Outline

• The Physics of Databases

• Algorithm for Space Search

• Application: Disjointness Protocol

• Open Problems

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

So 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 Will 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). Whether they’re equivalent is open

, , 0i z j zU

, , 0i z j zH

• Generalization of Grover search

If a quantum algorithm has success probability , then by invoking it 2m+1 times (m=O(1/)), we can make the success probability

Amplitude AmplificationBrassard, Høyer, Mosca, Tapp 2002

2

22 11 2 1

3

mm

• 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

• Here diameter of grid (n) exactly matches time for Grover search

• So we have to recurse more, breaking into squares of size n/log n

• Running time suffers correspondingly:

(best we could get)

d=2

2log

log log

nT n O n

n

• If exactly r marked items:

for d3. Basically optimal:

• If at least r marked items, can use “doubling trick” of BBHT’98 to get same bound for d3. For d=2 we get

Multiple Marked Items

1/ 2 1/ d

nT n O d

r

/ 2 1/ 2 1/2d d

nT n

r

3log

log log

nT n O n

n

• Our algorithm can be adapted to any graph with good expansion properties (not just hypercubes)

• Say G is d-dimensional if for any v, number of vertices at distance r from v is (min{rd,n})

• Can search in time

• Main idea: Build tree of subgraphs bottom-up

Search on Irregular Graphs

log

log , 2

2 , 2O n

T n O npoly n d

T n n d

• If G is >2-dimensional, and has h possible marked items (whose locations are known), then

• Intuitively: Worst case is when bits are scattered uniformly in G

Bits Scattered on a Graph

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

Open Problem #1

Can a quantum walk search a 2D grid efficiently? (Maybe even n time instead of n log3n?)

Promising numerical evidence (courtesy N. Shenvi)

Childs, Farhi, Goldstone et al.: Rigorous proof that random walk searches 5-D hypercube in O(n) time, 4-D hypercube in O(n polylog n) time

(or so they tell me)

Update (1 month ago)

Open Problem #2Here’s a graph of diameter n that takes (n3/4) time to search (by BBBV’96 hybrid argument):

Does it also take (n3/4) time to decide if every row of a 2D grid has a marked item?

n

Starfish

Open Problem #3

Cosmological constant 10-122 > 0(type-Ia supernova observations)

Number of bits accessible to any one observer is at most 3/ (Bousso 2000, Lloyd 2002)

How many of those ~10123 bits could a computer “use” before they recede past its horizon?

Our result shows a quantum computer could search more of the bits than a classical one

But what about using them as memory?

2D Turing machine

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

Open Problem #3 (con’t)

Consider a 2D Turing machine with O(n) time, a square worktape, and a separate input tape

Is there anything it can do with an nn worktape that it can’t do with a nn worktape?

What about a quantum TM?

2D Turing machine

Related to Feige’s embedding problem: Given n checkers on an nn checkerboard, can we move them to an O(n)O(n) board so that no 2 checkers become farther apart in L1 distance?

• 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