Quantum Search of Spatial Regions

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

Intro Grover’s O(n) Quantum Search Algorithm: Great for combinatorial search But can it help search a physical region? Why is a computer scientist asking such a thing?

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

Grover’s 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

Grover Illustration Initial Superposition |000 |001 |010 |011 |100 |101

Grover Illustration Amplitude of Solution State Inverted |100 |000 |001 |010 |011 |101

Grover Illustration All Amplitudes Inverted About Mean |000 |001 |010 |011 |100 |101

Talk Outline • The Physics of Databases • Algorithm for Space Search • Application: Disjointness Protocol • Open Problems

Trouble: Suppose our “hard disk” has mass density  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

Holographic principle Once radius exceeds Schwarzschild bound of (1/), hard disk collapses to form a black hole Makes things harder to retrieve… Actually worse—even 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)

Holographic principle Holographic Principle:A region of space can’t store more than 1.41069 bits per meter2 of surface area So Quantum Mechanics and General Relativity bothyield 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 c3 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 • 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

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 can’t…

Revenge of computer science • Idea: Recursively divide into sub-squares Using amplitude amplification techniques of BHMT’2002, we get: O(n log3n) for 2D grid O(n) for 3 and higher dimensions REVENGE OF COMPUTER SCIENCE • We can.

What’s the Model? • 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’ unitaries • What does ‘local’ mean? Depends on your religion

Locality religions FARHIOLOGY H THERE IS ONLY THE HAMILTONIAN 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)

Amplitude AmplificationBrassard, Høyer, Mosca, Tapp 2002 • 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

In More Detail: d3 • 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 item • Else pick a random subcube and run A on it • Repeat n1/11 times using amplitude amplification • Running time:

d3 (continued) • Success probability (unamplified): • With amplification: • (since  is negligible) • Amplify whole algorithm n1/22 times to get

d=2 • 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)

Multiple Marked Items • If exactly r marked items: • for d3. Basically optimal: • If at least r marked items, can use “doubling trick” of BBHT’98 to get same bound for d3. For d=2 we get

Search on Irregular Graphs • 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

Bits Scattered on a Graph • 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

Application: Disjointness • Problem: Alice has x1…xn{0,1}n, Bob has y1…yn They want to know if xiyi=1 for some i • How many qubits must they communicate? • Buhrman, Cleve, Wigderson 1998: • Høyer, de Wolf 2002: • Razborov 2002:

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

Random walk 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)

Starfish n Open Problem #2 Here’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?

2D Turing machine 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 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 nn worktape that it can’t do with a nn worktape? What about a quantum TM? Related to Feige’s embedding problem: Given n checkers on an nn checkerboard, can we move them to an O(n)O(n) board so that no 2 checkers become farther apart in L1 distance?

Conclusions Not all strings have n bits • No fundamental obstacle to quantum speedup for search of physical regions • We should look for other “pure” CS theory questions inspired by laws of physics Quantum computing is just one example

Quantum Search of Spatial Regions