Quantum walks as a quantum simulators Ivn Mrquez Martn 23/10/2019 - PowerPoint PPT Presentation

Quantum walks as a quantum simulators Iván Márquez Martín 23/10/2019

Index I. Introduction to QWs II. From QWs to Dirac equation III. QWs in hexagonal and triangular lattices.

I. Introduction to QW In the CRW Same prob. to move left or right

I. Introduction to QW H = H p ⊗ H c Hilbert space | i i canonical basis i ∈ Z H p position sites Ψ ∈ H H c c ∈ ↑ , ↓ H c | c i canonical basis of coin state

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Introduction to QW Unitary operator U = S ( I ⊗ C ) C: coin operator ✓ cos θ C : | "i ! cos θ | "i + i sin θ | #i ◆ i sin θ C = Superposition i sin θ cos θ C : | #i ! i sin θ | "i + cos θ | #i S: shift operator X | i + 1 i h i | ⌦ | "i h" | + | i � 1 i h i | ⌦ | #i h# | S = i ∈ Z Time

I. Introduction to QW P ( i ; t ) = | Ψ t,i | 2 Probability density Spreading faster than CRW

I. Introduction to QW Quantum algorithm 𝒫 ( N ) e.g. Searching in graphs first search algorithm in a hypercube Quantum random-walk search algorithm, Phys. Rev. A 67, 052307 (2003) 𝒫 ( N log N ) In 2D time complexity 1 a2 b a1 G. Abal, R. Donangelo, F. L. Marquezino, et al. “Spatial search on a honeycomb network”. In: Mathematical Structures in Computer Science 20.6 (2010), pp. 999–1009 G. Abal, R. Donangelo, M. Forets, et al. “Spatial quantum search in a triangular network”. In: Mathematical Structures in Computer Science 22.3 (2012), pp. 521–531. issn: 09601295.

I. Introduction to QW Quantum simulation I. M. Georgescu, S. Ashhab, and Franco Nori Rev. Mod. Phys. 86 , 153

Index I. Introduction to QWs II. From QWs to Dirac equation III. QWs in hexagonal and triangular lattices.