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Experimental Quantum Fast Hitting on Hexagonal Graphs
Hao Tang, Carlo Di Franco, Zi-Yu Shi, Tian-Shen He, Zhen Feng, Jun Gao, Ke Sun, Zhan-Ming Li, Zhi-Qiang Jiao, Tian-Yu Wang, M. S. Kim, Xian-Min Jin
TL;DR
Experimental implementation of quantum fast hitting on complex glued trees is constrained by their exponentially growing node counts. This paper maps modified hexagonal graphs onto photonic chips and experimentally demonstrates that optimal quantum hitting time grows linearly with depth, unlike the quadratic classical relation.
Problem
Glued-tree quantum-walk implementations are challenging because their node counts grow exponentially, motivating experiments on simpler graphs with polynomial growth.
Source: p. 1Method
The authors map modified hexagonal graphs onto photonic chips and study quantum hitting as graph depth increases from 2 to 8 layers.
Source: pp. 1, 3Results
Optimal quantum hitting time increases linearly with layer depth, while the classical relation is quadratic.
Source: p. 4Takeaways & Limitations
Femtosecond-laser-written photonic chips provide a route toward broader use of quantum walks on complex graphs.
Source: p. 5
Abstract
from arXiv · showhide
Quantum walks are powerful kernels in quantum computing protocols that possess strong capabilities in speeding up various simulation and optimisation tasks. One striking example is given by quantum walkers evolving on glued trees for their faster hitting performances than in the case of classical random walks. However, its experimental implementation is challenging as it involves highly complex arrangements of exponentially increasing number of nodes. Here we propose an alternative structure with a polynomially increasing number of nodes. We successfully map such graphs on quantum photonic chips using femtosecond laser direct writing techniques in a geometrically scalable fashion. We experimentally demonstrate quantum fast hitting by implementing two-dimensional quantum walks on these graphs with up to 160 nodes and a depth of 8 layers, achieving a linear relationship between the optimal hitting time and the network depth. Our results open up a scalable way towards quantum speed-up in complex problems classically intractable.
Methods
The Methods describe fabricating hexagonal waveguide chips, preparing heralded single photons, and measuring hitting efficiency from the light at the Exit site.
Methods
- Each hexagonal graph is mapped into a waveguide array, with the designed propagation length setting the evolution length.The arrays are fabricated in borosilicate using femtosecond laser direct writing; power and SLM compensation improve uniformity.Source: Methods · p. 5
- Vertically polarized photons enter the Entry waveguide, while their partners trigger heralding on the ICCD camera.The photon pairs are generated by spontaneous parametric down conversion and separated by a polarized beam splitter.Source: Methods · p. 5
- Hitting efficiency is the proportion of normalized light intensity measured at the Exit site.A pixel mask sums intensity within each waveguide, and the resulting proportions estimate waveguide probabilities.Source: Methods · p. 5
Supplementary Note 1 - Calculation of the hitting
The supplementary methods model classical walks as the classical endpoint of a quantum stochastic walk and specify how classical convergence and experimental comparisons are assessed.
efficiency for classical random walks
- Setting ω to 1 in the quantum stochastic walk yields classical hitting dynamics; ω interpolates between quantum and classical evolution.The Hamiltonian term represents quantum evolution, while Lindblad operators describe classical evolution.Source: efficiency for classical random walks · p. 7
- Classical hitting time is measured when every node's population is within 10^-4 Pa of the uniform stationary population.The reported error-bar bounds use thresholds of 10^-5 Pa and 10^-3 Pa; Pa is the inverse number of graph nodes.Source: efficiency for classical random walks · p. 7
- In a one-dimensional walk, laser and heralded-photon evolution patterns were highly similar, with longer-length discrepancies attributed to their different bandwidths [3-4].Variance slopes were 1.98 for laser and 1.95 for heralded single photons.Source: efficiency for classical random walks · p. 7
- Scanning nine samples per graph depth identifies the evolution length with optimal hitting efficiency, and measured curves overall agree with theory.The passage reports laser-beam measurements for hexagonal graphs with 2–8 layers.Source: efficiency for classical random walks · p. 7