1. Simulation analysis of quantum walks Tomohiro YAMASAKI QCI, ERATO, JST / University of Tokyo

2. Background zQuantum computation is more powerful yShor’s factoring algorithm yGrover’s searching algorithm zRandom walks are very efficient in various fields zInvestigation of quantum walks has been prompted

3. Definition zCoined quantum walk (Aharonov et al., 2001) 1 2 34 5 1 2 34 5 1 2 34 5 Coin-tossingShift One step of the quantum walk is given by Direction Vertex

4. Previous works (Mixing time) Initial state Uniform distribution For general graphs, at most polynomial speed up (Aharonov et al., 2001) n -dimensional hypercube How long does it take?

5. Absorbing probability and absorbing time Initial state Final state Absorbed Absorbing point Quantum walks on hypercube Solving k -SAT by using quantum walks Vertex Truth assignment Absorbing point Truth assignment which satisfies the instance Absorbing time Expected run-time How long does it take?

6. Random and quantum walk on 8-dimensional hypercube 0 0.0000 0.0000 = 0.0000 / 1.0000 1 255.0000 29.0000 = 29.0000 / 1.0000 2 290.2857 59.0000 = 16.8571 / 0.2857 3 300.7143 97.2444 = 13.8921 / 0.1429 4 305.3714 115.5175 = 13.2020 / 0.1143 5 308.0286 95.7844 = 13.6835 / 0.1429 6 309.7905 56.3111 = 16.0889 / 0.2857 7 311.0762 26.5603 = 26.5603 / 1.0000 8 312.0762 22.3137 = 22.3137 / 1.0000 Hamming distance Absorbing time (random walk) Absorbing time (quantum walk) Absorbing prob. (quantum walk)

7. Dimension and absorbing time

8. Conjectures concerning random and quantum walks on n -dimensional hypercube zWhen the Hamming distance is k, absorbing probability is z Exponential speed up

9. Previous works (Hadamard walk on the line) xx+1x-1xx+1x-1 xx+1x-1xx+1x-1 Let P ( x, t ) be the probability of being at location x at time t (Ambainis et al., 2001)

10. Previous works (Hadamard walk on the line) Let r n be the probability that the particle is eventually absorbed by boundary at location n 0 n n -1 Initial state Absorbing boundary (Random walks) (Quantum walks)

11. Conjectures concerning generalized Hadamard walk on the line zWhen we use as a coin-tossing operator,

12. Discussion zQuantum walks can be asymmetric and nonrecurrent, while classical counterparts symmetric and recurrent. zFor particular absorbing points, quantum walks seem to be exponentially faster than classical counterparts.