Continuous Time Quantum Walk on finite dimensions Shanshan Li Joint - - PowerPoint PPT Presentation

Shanshan Li Joint work with Stefan Boettcher Emory University QMath13, 10/11/2016

Continuous Time Quantum Walk on finite dimensions

Grover Algorithm: Unstructured Search

Oracle f(x) ! #$%&' $()*%$

f (x) = ( 1 x = w

• therwise

Initialize the system to the state

|si = 1 p N

N−1

X

x=0

|xi

Apply Grover Iteration

(UsUw)

√ N |si

Us = 2 |si hs| I Uw = I 2 |wi hw|

θ = 2 arcsin 1 √ N

Quantum Walk Basics for Spatial Search

Continuous time quantum walk

H = γL |ωi hω|

initial state Random Walk

d dtpx =

N−1

X

y=0

Lxypy

Unstructured search: f(x) is a computable function Spatial Search: N items stored in a d-dimensional physical space Laplacian = Degree Matrix - Adjacency Matrix

L = D − A

marked state |wi

dΨx (t) dt = X

y

HxyΨy (t)

|si = 1 N X

x

|xi

Critical Point in the Hamiltonian

• A. Childs et al. Spatial Search by Quantum Walk (2004)

γ = 0 H = |wi hw| 0th, 1st = |wi , |si γ = 1 H = γL 0th, 1st = |si , |wi

γ = γc 0th, 1st = (|wi ± |si) / p 2 T = π 2 p N

CTQW: optimal performance

Quadratic Speedup

• complete graph, hypercube, strongly regular graph

(E. Farhi and S. Gutmann 1998, A. M. Childs et al 2002, J. Janmark et al, 2014)

• Erdös Renyi graph

( SS. Chakraborty et al, 2016)

• lattices

(A. M. Childs et al 2004)

p ≥ log

3 2 N/N

❖ We extend CTQW to fractal graphs with real fractal dimension ❖ Spectral dimension of graph Laplacian determines the computational complexity

Grover efficiency

O ⇣√ N ⌘

d > 4

Finite Dimensional Fractals

we generalize to arbitrary real (fractal) dimension

❖ Hierarchical networks ❖ Sierpinski Gasket, Migdal-Kadanoff network with regular degree 3 ❖ Diamond fractals based on the Migdal-Kadanoff renormalization group scheme

Dimensions in fractal networks

Fractal dimensions Spectral dimension

0.0001 0.001 0.01 0.1 1

i/N

0.0001 0.001 0.01 0.1 1

i

N = 16 N = 32 N = 64 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 N = 16384

5000 10000 15000 i 1 2 3 4 5 6

i

N = 16384

Hierarchical Network with regular degree 3

N ∼ ldf

λi ∼ N −2/ds

Migdal-Kadanoff renormalization group(MKRG)

Model regular lattices closely arbitrary real dimension

Bond-moving scheme on square lattices with rescaling length l=2, branching factor b=2

b

Procedure to build the Diamond Fractals

k=0 k=1 k=2

Measure Critical Point

When the CTQW is optimal for search, the critical point takes place (numerically true for almost all sites in fractals we consider) The spectral Zeta function The transition probability

• A. Childs et al. Spatial Search by Quantum Walk (2004)

Assumption on fractal Laplacian eigenvector

0.1 1 10 0.1 1 10

N |<w|i>|

2

1 10 100 1000

i

th eigenvector

0.1 1 10

Highest Level 2

nd-Highest Level

3

rd-Highest Level

MK renormalization group with b=2

Renormalization Group Argument

The spectral Zeta function

Ij ∼ ✓ @ @✏ ◆j ln 1 ✏ det (L + ✏)

• |✏→0

∼ ( N

2j ds −1

ds < 2j const ds > 2j

det h L(k) ⇣ q(k)

i

, p(k)

i

, · · · ⌘i

det h L(k+1) ⇣ q(k+1)

i

, p(k+1)

i

, · · · ⌘i

Computational Complexity of CTQW

❖ spectral dimension of network Laplacian determines the computational complexity ❖ complement the discussions on regular lattices and mean-field networks

References:

❖ Shanshan Li and Stefan Boettcher, arXiv preprint arXiv: 1607.05317 , 2016 ❖ Stefan Boettcher and Shanshan Li, arXiv preprint arXiv: 1607.05168 , 2016

Thank you !

RG calculation for spectral determinant

b

• Stefan Boettcher and Shanshan Li, arXiv preprint arXiv: 1607.05168 , 2016