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Dec 20, 2015

- Slide 1
- Quantum Search of Spatial Regions Scott Aaronson (UC Berkeley) Joint work with Andris Ambainis (IAS / U. Latvia)
- Slide 2
- 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] - P - 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 - 2 P - -BPP - -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)) - 2 P - PhP - 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 - 2 P - SKC - SL - SLICEWISE PSPACE - SNP - SO- E - SP - span-P - SPARSE - SPP - SUBEXP - symP - SZK - TALLY - TC 0 - TFNP - 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
- Slide 3
- Quantum Computing Model of computation based on our best-confirmed physical theory State of computer is superposition over strings: Most famous algorithm: Shors algorithm for factoring This talk: Grovers algorithm for search
- Slide 4
- Grovers 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: Grovers algorithm is optimal
- Slide 5
- |000 Initial Superposition Grover Illustration |001 |101 |100 |011 |010
- Slide 6
- |000 Amplitude of Solution State Inverted Grover Illustration |001 |101 |100 |011 |010
- Slide 7
- |000 All Amplitudes Inverted About Mean Grover Illustration |001 |101 |100 |011 |010
- Slide 8
- Grovers Algorithm: Great for combinatorial search But can it help search a physical region? BWAHAHA! Look who needs physics now!
- Slide 9
- What even a dumb computer scientist knows: THE SPEED OF LIGHT IS FINITE Marked item Robot nn nn 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
- Slide 10
- Talk Outline The Physics of Databases Algorithm for Space Search Application: Disjointness Protocol Open Problems
- Slide 11
- 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
- Slide 12
- 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 )
- Slide 13
- Holographic Principle: A region of space cant store more than 1.4 10 69 bits per meter 2 of surface area So 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
- Slide 14
- What We Will 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
- Slide 15
- 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
- Slide 16
- 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)
- Slide 17
- 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
- Slide 18
- 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?
- Slide 19
- 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). Whether theyre equivalent is open Locality religions
- Slide 20
- 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 Amplification Brassard, Hyer, Mosca, Tapp 2002
- Slide 21
- 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
- Slide 22
- Success probability (unamplified): With amplification: (since is negligible) Amplify whole algorithm n 1/22 times to get d 3 (continued)
- Slide 23
- 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
- Slide 24
- If exactly r marked items: for d 3. Basically optimal: If at least r marked items, can use doubling trick of BBHT98 to get same bound for d 3. For d=2 we get Multiple Marked Items
- Slide 25
- 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{r d,n}) Can search in time Main idea: Build tree of subgraphs bottom-up Search on Irregular Graphs
- Slide 26
- 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
- Slide 27
- 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: Hyer, de Wolf 2002:
- Slide 28
- A B State at any time: Communicating one of 6 directions takes only 3 qubits Disjointness in O( n) Communication
- Slide 29
- Open Problem #1 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
- Slide

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