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

Intro Grovers 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?

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

Grovers 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

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

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

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

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 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…

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 )

Holographic Principle: A region of space cant store more than 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

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

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 cant… Back to the Main Issue Classical search takes (n) time Quantum search takes (r n) (r = maximum radius of region)

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

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?

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

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, Høyer, Mosca, Tapp 2002

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

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

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

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

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

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

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:

A B 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 log 3 n?) Promising numerical evidence (courtesy N. Shenvi) Random walk

Open Problem #2 Heres a graph of diameter n that takes (n 3/4 ) time to search (by BBBV96 hybrid argument): Does it also take (n 3/4 ) time to decide if every row of a 2D grid has a marked item? n Starfish

Open Problem #3 Cosmological constant > 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 ~ 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 (cont) 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 cant do with a n n worktape? What about a quantum TM? 2D Turing machine Related to Feiges 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 L 1 distance?

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