Search by quantum walk and extended hitting time Andris Ambainis, Martins Kokainis University of Latvia

Exhaustive search Finite search space. Some elements might be marked. Find a marked element!

Search with structure Finite search space. Some elements might be marked. Find a marked element! After checking A, it may be easier to check B than C.

Example: search on grids  N   N grid. In one step, we can: – check if vertex marked; – move 1 step;

Search by random walk Random walk, following the locality constraints. Stop after finding a marked vertex.

Szegedy’2004 Random walk: T steps Quantum walk: O(  T) steps

Szegedy’2004 (fine print) Random walk finds marked element: T steps Quantum walk detects if marked element exists: O(  T) steps

Quantum walk detects if marked element exists: O(  T) steps

Quantum walk detects if marked element exists: O(  T) steps |  start  - starting state; No marked element - |  start  unchanged; Marked elements - |  start  diverges to an almost orthogonal state | . |  not concentrated on marked element

Open question Random walk finds marked element: T steps Quantum walk finds marked element: O(  T) steps ?

Krovi, Ozols, Magniez, Roland (previous talk) Quantum algorithm that finds marked element in O(  HT + ) steps, HT + - extended hitting time. HT + = HT if there is 1 marked element; HT + can be larger than HT. How large can HT + be?

This talk Weak (upper) bound on HT+. Two big gaps between HT+ and HT.

DEFINITIONS

Markov chains /3 1/3 2 2/3

Classical hitting time

Matrix form Eigenvalues – real. 1 = 1, eigenvector – stationary distribution. 1 > 2 ...  n. Spectral gap: probability of transition i  j

UPPER BOUND ON HT +

Upper bound

1 - 2 

Corollary

How strong is this result?

Unstructured search (Grover, 1996)

Grover’s algorithm Query Q: check if an element marked; Diffusion D: – |  start   |  start  ; – |   -| , |   |  start . Repeat D, Q, D, Q,..., D, Q.

Diffusion Diffusion D: – |  start   |  start  ; – |   -| , |   |  start . Markov chain: – |v 1   |v 1  ; – |v i   i |v i , i  1- . Can implement diffusion with O(1/  ) steps of Markov chain.

Summary KMRO algorithm: – at least as good as Magniez-Nayak-Roland-Santha; – finds 1 marked element optimally. More general description when KMRO works well?

GAPS BETWEEN HT+ AND HT

Example 1 Stationary distribution: π x =1/3 for all x. M = {1, 2}. HT  10/9. HT+  1/(4  )   1-  0.1- 

Example 1 If  =0, two eigenvectors with i = 1. If  0, 2  1. Large contribution to HT +, causing HT +    1-  0.1- 

Gap between HT and HT+, for a natural search space?

2D grid  N  N grid. Spectral gap:  (1/N). Possible: HT +  N  HT.

2D grid: example 1  N  N grid. HT =  (1). HT + =  (N).  (1) fraction of vertices marked. Classical search easy – no need for quantum.

2D grid: example 2 Gap persists, unless the number of marked vertices small.

marked unmarked

2D grid: example 2 Outside: divide into k  k squares, mark corners. Inside: divide into (2k)  (2k) squares, mark corners. Regular pattern, with different densities outside and inside.

Classical hitting time Lower bound: hitting time with 1 marked vertex in each (2k)  (2k) square. 1 of 4k 2 vertices marked.

Extended hitting time Calculation, using eigenvectors of the grid. HT + =  (N), for any density of marked vertices.

KMRO algorithm Result: uniform superposition over marked vertices.

KMRO algorithm Pr=1/2 Pr=4/5

Marking more elements may increase HT + HT + =  (1)HT + =  (N)

What else can we try?

A, Bačkurs, et al., TQC’ D grid, 1 marked vertex. Standard quantum walk. After O(  N log N) steps, state orthogonal to |  start . Measurement: Pr[marked] = o(1); Pr[distance N  from marked]  const.

Idea 1 Does final state |  final  have large probability on vertices that are close to marked? If true - measure|  final , obtain v, search the neighbourhood of v classically.

Idea 2 If HT = T, classical walk P hits a marked vertex in O(T) steps, with probability  1- . G’ – neighbourhood of the starting vertex where P stays during O(T) steps. Quantum walk on G’ instead of the full space?

Conclusions Upper bound for HT +, via spectral gap. KMRO algorithm at least as good as Magniez- Nayak-Roland-Santha. Two examples of gaps between HT + and HT. Optimal quantum algorithm should not be producing the uniform superposition of marked vertices!