Toward simulation of topological phenomena with one-, two-, and three-dimensional quantum walks

Panahiyan, S.; Fritzsche, S.

doi:10.1103/physreva.103.012201

Cited by 9 publications

(3 citation statements)

References 48 publications

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“…This periodically driven system simulates an effective (Floquet) Hamiltonian that is topologically nontrivial [63]. This system replicates the effective Hamiltonians from all universal classes of 1-to 3-D topological insulators [64,17,65]. Interestingly, topological properties of Floquet topological insulators could be controlled via an external periodic drive rather than an external magnetic field.…”

Section: Introductionmentioning

confidence: 83%

Discrete-Time Quantum-Walk & Floquet Topological Insulators via Distance-Selective Rydberg-Interaction

Khazali

2022

Quantum

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This article proposes the first discrete-time implementation of Rydberg quantum walk in multi-dimensional spatial space that could ideally simulate different classes of topological insulators. Using distance-selective exchange-interaction between Rydberg excited atoms in an atomic-array with dual lattice-constants, the new setup operates both coined and coin-less models of discrete-time quantum walk (DTQW). Here, complicated coupling tessellations are performed by global laser that exclusively excite the site at the anti-blockade region. The long-range interaction provides a new feature of designing different topologically ordered periodic boundary conditions. Limiting the Rydberg population to two excitations, coherent QW over hundreds of lattice sites and steps are achievable with the current technology. These features would improve the performance of this quantum machine in running the quantum search algorithm over topologically ordered databases as well as diversifying the range of topological insulators that could be simulated.

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Section: Introductionmentioning

confidence: 83%

Discrete-Time Quantum-Walk & Floquet Topological Insulators via Distance-Selective Rydberg-Interaction

Khazali

2022

Quantum

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“…Such models can simulate dynamics of various physical systems, e.g. [15][16][17][18][19][20][21][22][23][24], and are known to be capable of universal quantum computation [25][26][27][28], and as a consequence, of universal quantum simulation [29]. Moreover, they were already implemented on many experimental platforms [30].…”

Section: Introductionmentioning

confidence: 99%

Complementarity in quantum walks

Grudka

Kurzyński

Polak

et al. 2023

J. Phys. A: Math. Theor.

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The eigenbases of two quantum observables, $\{|a_i\rangle\}_{i=1}^D$ and $\{|b_j\rangle\}_{j=1}^D$, form Mutually Unbiassed Bases (MUB) if $|\langle a_i |b_j \rangle| = 1/\sqrt{D}$ for all $i$ and $j$. In realistic situations MUB are hard to obtain and one looks for Approximate MUB (AMUB), in which case the corresponding eigenbases obey $|\langle a_i |b_j \rangle| \leq c/\sqrt{D}$, where $c$ is some positive constant independent of $D$. In majority of cases observables corresponding to MUB and AMUB do not have clear physical interpretation. Here we study discrete-time quantum walks on $d$-cycles with a position and coin-dependent phase-shift. Such a model simulates a dynamics of a quantum particle moving on a ring with an artificial gauge field. In our case the amplitude of the phase-shift is governed by a single discrete parameter $q$. We solve the model analytically and observe that for prime $d$ the eigenvectors of two quantum walk evolution operators form AMUB. Namely, if $d$ is prime the corresponding eigenvectors of the evolution operators, that act in the $D$-dimensional Hilbert space ($D=2d$), obey $|\langle v_q|v'_{q'} \rangle| \leq \sqrt{2}/\sqrt{D}$ for $q\neq q'$ and for all $|v_q\rangle$ and $|v'_{q'}\rangle$. Finally, we show that the analogous AMUB relation still holds in the continuous version of this model, which corresponds to a one-dimensional Dirac particle.

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“…Owing to quantum superposition effects, these walks display distinct statistical properties compared to their classical counterparts. Quantum walks have proved to be of immense utility in varied domains, like in quantum computation [6][7][8], quantum search [9], quantum algorithms [10,11], generating random numbers [12], modeling topological phenomena [13][14][15][16][17] etc. In the DTQWs, the walker is modeled as a qudit, belonging to a potentially infinite-dimensional space called the "walk space", while the coin is modeled as a qubit, belonging to two-dimensional space called the "coin-space".…”

Section: Introduction To Quantum Walksmentioning

confidence: 99%

Steered discrete-time quantum walks for engineering of quantum states

Kadiri¹

2022

Preprint

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We analyze the strengths and limitations of steered discrete time quantum walks in generating quantum states of bipartite quantum systems comprising of a qubit coupled to a qudit system. We demonstrate that not all quantum states in the composite space are accessible through quantum walks, even under the most generalized definition of a quantum step, leading to a bifurcation of the composite Hilbert space into "walk-accessible states" and the "walk-inaccessible states". We give an algorithm for generating any walk-accessible state from a simple-to-realize product state, in a minimal number of walk steps, all of unit step size. We further give a prescription towards constructing minimal quantum walks between any pair of such walk-accessible states. Linear optics has been a popular physical system for implementing coin-based quantum walks, where the composite space is built up of spin and orbital angular momenta of light beams. We establish that in such an implementation, all normalized quantum states are "walk-accessible". Furthermore, any generalized quantum step can be implemented upto a global phase using a single q-plate and a pair of homogeneous waveplates. We then give a quantum walk based scheme for realizing arbitrary vector beams, using only q-plates and waveplates.

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Toward simulation of topological phenomena with one-, two-, and three-dimensional quantum walks

Cited by 9 publications

References 48 publications

Discrete-Time Quantum-Walk & Floquet Topological Insulators via Distance-Selective Rydberg-Interaction

Discrete-Time Quantum-Walk & Floquet Topological Insulators via Distance-Selective Rydberg-Interaction

Complementarity in quantum walks

Steered discrete-time quantum walks for engineering of quantum states

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