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Simulation analysis of quantum walks

Simulation analysis of quantum walks. Tomohiro YAMASAKI QCI, ERATO, JST / University of Tokyo. Background. Quantum computation is more powerful Shor’s factoring algorithm Grover’s searching algorithm Random walks are very efficient in various fields

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Simulation analysis of quantum walks

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

  2. Background • Quantum computation is more powerful • Shor’s factoring algorithm • Grover’s searching algorithm • Random walks are very efficient in various fields • Investigation of quantum walks has been prompted

  3. 1 1 1 2 5 2 5 2 5 3 4 3 4 3 4 Definition • Coined quantum walk (Aharonov et al., 2001) Coin-tossing Shift Direction Vertex One step of the quantum walk is given by

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

  5. Absorbing probability and absorbing time Initial state Final state Absorbed How long does it take? 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

  6. Random and quantum walkon 8-dimensional hypercube Hamming distance Absorbing time (random walk) Absorbing time (quantum walk) Absorbing prob. (quantum walk) 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

  7. Dimension and absorbing time

  8. Conjectures concerning random andquantum walks on n-dimensional hypercube • When the Hamming distance is k, absorbing probability is Exponential speed up

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

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

  11. Conjectures concerninggeneralized Hadamard walk on the line • When we use as a coin-tossing operator,

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

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