Quantum Fast Hitting on Glued Trees Mapped on a Photonic chip

Zi-Yu Shi,1, 2 Hao Tang,1, 2 Zhen Feng,1, 2 Yao Wang,1, 3 Zhan-Ming Li,1, 2 Zhi-Qiang

Jiao,1, 2 Jun Gao,1, 3 Yi-Jun Chang,1, 2 Wen-Hao Zhou,1, 2 and Xian-Min Jin1, 2, 3, ∗

1State Key Laboratory of Advanced Optical Communication Systems and Networks,School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

2Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China

3Shenzhen Institute for Quantum Science and Engineering and Department of Physics,Southern University of Science and Technology, Shenzhen 518055, China

Hitting the exit node from the entrance node faster on a graph is one of the properties thatquantum walk algorithms can take advantage of to outperform classical random walk algorithms.Especially, continuous-time quantum walks on central-random glued binary trees have been investi-gated in theories extensively for their exponentially faster hitting speed over classical random walks.Here, using heralded single photons to represent quantum walkers and waveguide arrays writtenby femtosecond laser to simulate the theoretical graph, we are able to demonstrate the hitting ef-ficiency of quantum walks with tree depth as high as 16 layers for the first time. Furthermore, weexpand the graph’s branching rate from 2 to 5, revealing that quantum walks exhibit more supe-riority over classical random walks as branching rate increases. Our results may shed light on thephysical implementation of quantum walk algorithms as well as quantum computation and quantumsimulation.

Due to the unique features like superposition and en-tanglement, introducing quantum mechanical effects intoclassical algorithms has shown great speed advantage1–8.Classical random walks (CRWs) on graphs have be-come a powerful tool for designing classical algorithms.As its counterpart in quantum field, quantum walks9

(QWs) also find wide applications in not only quantumalgorithms10–16 but also quantum computation17,18 andquantum simulation19–22. Fast hitting23–25 is one of theproperties26,27 of QWs on graphs that quantum algo-rithms can take advange of to demonstrate remarkablequantum speedup, which focuses on the time that a par-ticle needs to reach the exit node from a entrance nodewith certainty as a function of the size of a graph.

An modified version25 of the regular glued binarytrees28 is one of the graphs through which using QWsto traverse is exponentially faster than not only CRWsbut also any classical algorithms people can come upwith25. This structure is constituted by two identicalbinary trees, and the depth n of one of the two trees isusually used to describe the size of the whole graph. Eachend of the left tree has two branches that are randomlyglued to two different ends of the right one, and viceversa (Fig.1a). The left root is the entrance node andthe right root is the exit node. We term this structure ascentral-random glued trees throughout this article.

The probability of a CRW hitting the right root28, i.e.the hitting efficiency, is less than 2−n. Due to the uni-tarity, the hitting efficiency of QWs always changes withevolving time28, and the first peak occurs in O(n) timeand scales as PQW ∼ n−2/3 for large n, meaning a QWcan almost certainly hit the exit in polynomial time25,29.What’s more, each node on the graph can be extendedto B branches rather than just two branches, exhibitinghigher complexity (Fig.1b)29. Unfortunately, mappingthe full central-random glued trees into a physical system

is impossible under existing studies, as the node numbergrows exponentially with n. There has yet been no exper-imental demonstration of QW’s hitting efficiency on thisgraph, not to mention the expansion to higher branch-ing rates. Even exploiting a deformation30 or simplifiedstructure31 is already of great interest.

On the other hand, on account of the coherent super-position of quantum particles as well as the symmetricalshape of this glued trees, the analysis of QW travers-ing the graph can be reduced to only concerned withthe graph’s column positions (see Supplementary Note1). The complex two-dimensional graph can be simpli-fied to a one-dimensional chain with nonuniform hoppingrates25 (Fig.1b). Nonetheless, the physical implementa-tion is still a big challenge compared with those reporteduniform structures20,32,33, for the relative difference ofthe chain’s hopping rates is the key to simulate differentbranching rates, which requires high-precision control ofthe coupling strength between waveguides.

Fortunately, after plenty of time and efforts, we haveconquered the difficulties. For the first time, we not onlyexperimentally investigate the variation of QW’s opti-mal hitting efficiency with n ranging from 2 to 16, whichdemonstrates that a quantum walker can find the exitwith polynomially high probability in linear time, butalso its variation with B going from 2 to 5, revealing thatthe branching rate can also contribute to the speed ad-vantage of QWs. Our work will inspire the experimentalrealization of fast hitting as well as the other propertiesof QWs10,11,26,27 in more scenarios, further facilitatingthe physical realization of quantum walk algorithms.

The row of Fig.1a shows the changing of tree depth,while the column exhibits the variation of branching rate.As shown in Fig.1c, in borosilicate chip substrate, we usefemtosecond laser direct writing technique34–38 to fabri-cate an array of waveguides, the cross section of which is

arX

iv:1

907.

1253

9v1

[qu

ant-

ph]

29

Jul 2

019

2

a

b

n=2n=3

n=4

B=3

B=2

B=4 entrance exit

branching rate B, tree depth n

entrance exit

γ

γ

γ

γ γ

γ

√Bγ √Bγ √Bγ √Bγ √Bγ √BγBγ

c

inject single photons

hitting efficiency

FIG. 1: Schematic diagram of central-random glued trees. a. Central-random glued trees with an increasing n (in row)and B (in column). b. Generalized central-random glued trees and its one-dimensional equivalence for QWs. The hoppingrate between any adjacent nodes of the two-dimensional graph is γ, while on the one-dimensional chain, the hopping rate is√Bγ except for the two adjacent nodes at the center, which becomes Bγ. c. Experimental implementation of the mapped

one-dimensional chain on a photonic chip using femtosecond laser direct writing technique. The cross section of the waveguidearray is consistent with the theoretical one-dimensional chain and the longitudinal direction of the waveguide corresponds tothe evolving time. Since the hopping rate of the two adjacent waveguides at the center is greater than others, the spacingbetween the two waveguides is reduced accordingly.

a strict mapping of the theoretical one-dimensional chain(Fig.1b). Each waveguide represents a node in the theo-retical graph and the longitudinal direction of the waveg-uides is for the evolving process in time. As evolvinglength z of photons is proportional to evolving time t forz = vt, in which v is the propagation speed of photonsin waveguides, we use length z to replace time t in thefollowing description.

We prepare and inject single photons into the entrancewaveguide, and observe the spatial photon number distri-bution when the waveguide lengths change. When prop-agating along the waveguides, a single photon will becoupled evanescently to the other waveguides33,39. It is

worth noting that, since the coupling coefficient decreasesexponentially with waveguide spacing, we only considerthe coupling effects between the nearest waveguides33,40.The evolving process of a continuous-time QW can bedescribed by the wavefunction of photons according to

|Ψ(z)〉 = e−iHz |Ψ(0)〉 , (1)

where |Ψ(0)〉 is the initial state and H is the Hamilto-nian. The equation can be calculated by matrix exponen-tial methods41. The Hamiltonian of photons propagating

3

n=4

n=6

n=8

n=10

n=12

n=14

n=16

a b

Hitt

ing

Effic

ienc

y

Evolving Length(mm)

FIG. 2: Spatial photon number distributions and hitting efficiencies for trees at B = 2. a. Variation of hittingefficiency with evolving length with n = 2. Several pictures of spatial photon number distributions for samples of differentevolving lengths are also shown as insets. The injecting waveguide is marked by a white circle. And corresponding hittingefficiencies of CRWs on central-random glued trees are also plotted. b. Spatial photon number distributions that show optimalhitting efficiencies at different n.

through a waveguide system can be described by20,32:

H =

N∑i

βia†iai +

N∑i6=j

Ci,ja†iaj , (2)

where a†i , ai are the creation and annihilation operatorfor waveguide i. βi = β is the propagation constant.Ci,j is the coupling coefficient between waveguide i andj, which is mainly determined by waveguide spacing.

For a central-random glued trees with depth n, its one-dimensional equivalence has 2n+ 2 nodes and the proba-bilities of a walker moving right or left (hopping rate) onthe chain are same on all the nodes except the two near-est nodes at the center. As Fig.1c shows, on the chain,the hopping rate, which corresponds to the coupling coef-ficient in experiments, of a quantum walker between thetwo nearest central nodes is

√B times of that between

the other nearest nodes, where γ is the hopping rate onthe trees. What value γ is doesn’t affect the optimal hit-ting efficiency, but the ratio of the hopping rate at thecenter to that at the non-center position, i.e.

√B, on

the chain does, therefore precisely mapping this ratio towaveguide arrays is essential to simulate glued trees withdifferent branching rates. If j represents column posi-tions, then the non-zero elements of the Hamiltonian forthe one-dimensional chain are29

〈j|H |j + 1〉 =

{ √Bγ 0 ≤ j < n, n < j ≤ 2n

Bγ j = n. (3)

The rest of the non-zero elements can be deduced by theHermiticity. The probability distribution of a CRW isuniform and stationary as long as the evolving time islong enough11, whereas the probability distribution of acontinuous time QW is always changing.29,30,42. Several

peaks of hitting efficiency which we term as optimal hit-ting efficiency will appear as the evolving length of pho-tons increases. However, compared with the time cost,the increase in hitting efficiency is negligible and the lossof photons is considerable, so it is desirable to only studythe first peak of hitting efficiency and calculate corre-sponding evolving length where such first peak occurs,i.e. optimal evolving length.

In experiments, on the basis of coupled mode theory,we fabricate a series of waveguide pairs with differentwaveguide spacings to obtain the function of couplingcoefficient with respect to waveguide spacing40. We thenchoose suitable coupling coefficients as hopping rates tocalculate the variation of hitting efficiency with evolvinglength. Finally, for graphs of each size, we choose dif-ferent waveguide lengths as samples to fabricate waveg-uide arrays with waveguide spacings consistent with thecoupling coefficient used in theoretical calculation. Eachwaveguide is uniform and the distance between any twonearest waveguides is identical except the two waveguidesat the center (See Supplementary Note 2 for the fabri-cation details). With these waveguide arrays in hand,we study the transporting property of QWs by injectingvertically-polarized single photons with a wavelength of810nm into the entrance waveguide and monitoring theintensity pattern exiting the photonic chip with an inten-sified charge coupled device (ICCD) camera (See Supple-mentary Note 3 for details on single-photon imaging).The experimental hitting efficiency is obtained by calcu-lating the ratio of the intensity in the exit waveguide tothe total intensity (See Supplementary Note 4 for detailson data processing).

For trees with n ranging from 2 to 16, and B rangingfrom 2 to 5, we first work out the variation of hittingefficiency against evolving length from the experimental

4

Opt

imal

Hitt

ing

Effic

ienc

y

Tree Depth Tree Depth

Opt

imal

Hitt

ing

Effic

ienc

y

a b

B=2

B=3

B=5

B=5

B=4

B=2

B=3B=5

B=4

FIG. 3: Experimental optimal hitting efficiencies for different branching rates. a. Variation of optimal hittingefficiencies with n for B = 2, 3, 4, 5 respectively. As B increases, the optimal hitting efficiencies decrease slightly. b. Comparisonbetween CRWs and QWs in a half logarithmic coordinate. The ultimate steady hitting efficiencies of CRWs on central-randomblued trees decrease exponentially with n, in contrast with the polynomially decreasing trend of QWs. Though data of B = 5is the smallest among the four groups, it still outperforms CRWs.

patterns. Fig.2a exhibits the results of a 2-layer tree ata branching rate of 2. The experimental optimal hit-ting efficiency is 0.76 and corresponding evolving lengthis 15.5mm, agreeing very well with theoretical calcula-tion which suggests an optimal value of 0.82 at 15mm.However, the theoretical hitting efficiency of CRW25 at15mm is only 0.044 which is lower than QW for morethan one order. Some QW spatial photon number distri-butions are also added into Fig.2a to visually show thethe change of hitting efficiency. The leftmost light spot ineach sub-picture corresponds to the entrance waveguidewhich is marked by a white circle and the rightmost lightspot corresponds to the exit waveguide. It can be seenthat variation of the ratio of photons reaching the exitwaveguide in the spatial photon number distributions isconsistent with our calculated experimental curve.

Then, applying the method of finding experimental op-timal hitting efficiency as shown in Fig.2a, we investigatethe change of optimal hitting efficiency with n. Fig.2bshows spatial photon number distribution of optimal hit-ting efficiency with n going from 2 to 16 at B = 2. It isobvious that despite the number of waveguides increaseswith n, most of the photons gather at the exit waveguiderather than the other waveguides which represents thepositions of the other columns of the tree. All spatialphoton number distributions that show optimal hittingefficiency for graphs of different sizes as well as the lin-early increasing optimal evolving lengths are presentedin supplementary materials (See Supplementary Note 5and Supplementary Fig.S2-S5).

The optimal hitting efficiencies calculated from the-oretical models and experimental evolution results areplotted in Fig.3a, and the ultimate steady hitting effi-ciency of CRW when evolving length is long enough isadded in Fig.3b as a comparison. We emphasize that thisway of reducing to 1D chain is derived for QWs (as shownin Supplementary Note 1), but is not suitable for CRWs,therefore, all theoretical results about CRWs are basedon central-random glued trees. Fig.3a exhibits that, foreach B, QW’s optimal hitting efficiency decays polynomi-ally with the increase of n. In contrast, for CRW, whenevolving length is infinitely long, the hitting efficiencywill increase monotonously to an asymptotic value whichis the inverse of the number of nodes on the trees25, andthe scaling can be expressed as PCRW ∼ B−n, meaningits optimal hitting efficiency decays exponentially withn29 (Fig.3b). It can be seen from Fig.3b that though thehitting efficiency of QW at B = 5 is the smallest amongthe four kinds of branches, it is still exponentially higherthan CRW.

Finally, we investigate the variation of optimal hittingefficiency with B. As B increases, the complexity ofgraphs rises dramatically and the increasing branches ateach node provide ever more challenges for CRW particlesto choose from. On the other hand, a quantum walker isable to explore the landscape it traverses with superpo-sition, and hence has even more evident superiority overCRW at a higher branching rate. From Fig.3a, the QW’soptimal hitting efficiency for n = 16 at B = 5 is still morethan 50% of that at B = 2, while the ratio for the same

5

Opt

imal

Hitt

ing

Effic

ienc

y

Branching rate

Enhancement R

atio

FIG. 4: Variation of experimental optimal hitting ef-ficiencies with branching rate for n = 4. Scatter plotscorrespond to the left y axis, representing optimal hitting ef-ficiencies of QWs in theory and experiment respectively. Col-umn plots correspond to the right y axis, representing theenhancement rate, i.e. optimal hitting efficiency ratio of QWagainst CRW. The tree depth remains to be 4 when this mea-surement is taken.

pair drops to approximately (2/5)16 = 4.29 × 10−7 forCRW (see Fig.3b), as PCRW ∼ B−n.

In Fig.4, we plot the QW’s optimal hitting efficiency(left y axis) and the enhancement ratio over CRW’s opti-mal hitting efficiency (right y axis) for different branchingrates from 2 up to 10. The results all come from 4-layerglued trees. Even for the shallow glued tree without pur-suing a large tree depth, the enhancement of optimalhitting efficiency by QW over CRW is impressive at ahigh branching rate. It has been suggested in theory29

that when B is considerably large, QW’s optimal hittingefficiency scales with B by PQW ∼ 1/B, and then the en-hancement ratio would scale as r = PQW /PCRW ∼ Bn−1

for a certain n. We have observed the increasing enhance-ment ratio for a few branching rates, and provided thefirst experiment demonstration of central-random gluedtrees of high branching rates. The experimental resultsprove that branching rate can be another useful resourcebesides the more commonly discussed tree depth, to in-crease the complexity of central-random glued trees forstrengthening the quantum superiority.

We also exploit the waveguide arrays used in Fig.4as samples to measure the second-order anti-correlationparameter43 α of the photons exiting the exit waveguides,which tends to be 0 for ideal single photons and 1 forclassical coherent light. For graphs with B=2, 3, 4, 5at n = 4, the results are shown in TABLE I, demon-strating excellent quantum property, making our setupdistinguishable from those using classical coherent light.(See Supplementary note 6 for details)

In summary, we demonstrate the fast-hitting propertyof QWs on central-random glued trees in a single-photonregime. Harnessing femtosecond laser direct writing tech-nique, we precisely control the coupling coefficient be-

TABLE I: The measured α of the single photons exist-ing from the exit waveguide for graphs with B=2, 3,4, 5 at n = 4.

B 2 3 4 5

α 0.086 0.114 0.091 0.093

errorbar 0.001 0.001 0.001 0.003

tween waveguides and fabricate waveguide arrays withhigh tree depths and branching rates. We employ ultra-low-noise single-photon imaging techniques to measurethe probability of heralded single photons reaching theexit waveguides. Using single photons instead of clas-sical coherent light, we manage to observe the accumu-lated process from the individual single photons to thethe eventual spatial photon number distribution, realiz-ing a true sense of quantum-particle experiment insteadof the simulation results by laser beams. Besides, us-ing single photons also gives a possible direction for theexperimental expansion to a bigger Hilbert space, i.e. in-creasing the number of particles walking on the graph,which classical coherent light cant achieve.

We experimentally demonstrate that the optimal hit-ting efficiency of a quantum walker hitting the exit onlydecays polynomially as tree depth increases, which isin contrast with CRW’s exponentially decreasing trend.We further demonstrate that the enhancement ratio ofthe QW’s optimal hitting efficiency over CRW becomeshigher when branching rate increases, suggesting thebranching rate as another useful approach besides thetree depth to introduce more evident quantum advan-tages.

In fabricating high-branched glued trees, we manage toprecisely set coupling coefficients as

√Bγ or Bγ using the

advanced waveguide writing techniques, without whichit is impossible to implement such glued trees on pho-tonic chips. This precise writing techniques can be fur-ther utilized for richer explorations of on-chip quantumalgorithms and quantum simulation. Another inspiringpoint from this work is the idea of reducing a complexgraph to a simpler equivalence that can be mapped inthe physical systems. For many complex graphs such ashypercubes and other high-dimensional structures, thismay shed light upon the experimental implementation ina highly feasible way. Further, it would also be interest-ing to map the structure of high branching rates and treedepths to optimization problems in different fields24, andbring up quantum advantages by these hitting protocolsinto real-life applications.

Acknowledgements

The authors thank Andrew Childs and Jian-Wei Panfor helpful discussions. This work was supported by Na-

6

tional Key R&D Program of China (2017YFA0303700);National Natural Science Foundation of China (NSFC)(61734005, 11761141014, 11690033); Science and Tech-nology Commission of Shanghai Municipality (STCSM)(15QA1402200, 16JC1400405, 17JC1400403); Shang-

hai Municipal Education Commission (SMEC)(16SG09,2017-01-07-00-02-E00049); X.-M.J. acknowledges sup-port from the National Young 1000 Talents Plan.

∗ Electronic address: xianmin.jin@sjtu.edu.cn1. Montanaro, A. Quantum algorithms: an overview. NPJ

Quantum Inf. 2, 15023 (2016).2. Shor, P. W. Algorithms for quantum computation: Dis-

crete logarithms and factoring. In Foundations of Com-puter Science, 1994 Proceedings., 35th Annual Sympo-sium on, 124–134 (Ieee, 1994).

3. Lanyon, B. P. et al. Experimental demonstration of acompiled version of shors algorithm with quantum entan-glement. Phys. Rev. Lett. 99, 250505 (2007).

4. Grover, L. K. A fast quantum mechanical algorithm fordatabase search. In Proceedings of the twenty-eighth an-nual ACM symposium on Theory of computing, 212–219(ACM, 1996).

5. Grover, L. K. Quantum mechanics helps in searching fora needle in a haystack. Phys. Rev. Lett. 79, 325 (1997).

6. Jones, J. A., Mosca, M. & Hansen, R. H. Implementationof a quantum search algorithm on a quantum computer.Nature 393, 344 (1998).

7. Farhi, E. et al. A quantum adiabatic evolution algorithmapplied to random instances of an np-complete problem.Science 292, 472–475 (2001).

8. Aaronson, S. & Shi, Y. Quantum lower bounds for thecollision and the element distinctness problems. J. ACM51, 595–605 (2004).

9. Aharonov, Y., Davidovich, L. & Zagury, N. Quantumrandom walks. Phys. Rev. A 48, 1687–1690 (1993).

10. Ambainis, A. Quantum walks and their algorithmic appli-cations. International Journal of Quantum Information1, 507–518 (2003).

11. Kempe, J. Quantum random walks: an introductoryoverview. Contemporary Physics 44, 307–327 (2003).

12. Ambainis, A. Quantum walk algorithm for element dis-tinctness. SIAM J. Comput. 37, 210–239 (2007).

13. Douglas, B. L. & Wang, J. B. A classical approach tothe graph isomorphism problem using quantum walks.Journal of Physics A: Mathematical and Theoretical 41,075303 (2008).

14. Childs, A. M. & Goldstone, J. Spatial search by quantumwalk. Phys. Rev. A 70, 022314 (2004).

15. Shenvi, N., Kempe, J. & Whaley, K. B. Quantumrandom-walk search algorithm. Phys. Rev. A 67, 052307(2003).

16. Tulsi, A. Faster quantum-walk algorithm for the two-dimensional spatial search. Phys. Rev. A 78, 012310(2008).

17. Childs, A. M. Universal computation by quantum walk.Phys. Rev. Lett. 102, 180501 (2009).

18. Childs, A. M., Gosset, D. & Webb, Z. Universal com-putation by multiparticle quantum walk. Science 339,791–794 (2013).

19. Schreiber, A. et al. A 2d quantum walk simulation oftwo-particle dynamics. Science 1218448 (2012).

20. Peruzzo, A. et al. Quantum walks of correlated photons.

Science 329, 1500–1503 (2010).21. Sansoni, L. et al. Two-particle bosonic-fermionic quan-

tum walk via integrated photonics. Phys. Rev. Lett. 108,010502 (2012).

22. Crespi, A. et al. Anderson localization of entangled pho-tons in an integrated quantum walk. Nat. Photon. 7, 322(2013).

23. Kempe, J. Discrete quantum walks hit exponentiallyfaster. Probability theory and related fields 133, 215–235(2005).

24. Farhi, E. & Gutmann, S. Quantum computation and de-cision trees. Phys. Rev. A 58, 915–928 (1998).

25. Childs, A. M. et al. Exponential algorithmic speedup by aquantum walk. In Proceedings of the Thirty-fifth AnnualACM Symposium on Theory of Computing, STOC ’03,59–68 (ACM, New York, NY, USA, 2003).

26. Aharonov, D., Ambainis, A., Kempe, J. & Vazirani, U.Quantum walks on graphs. In Proceedings of the thirty-third annual ACM symposium on Theory of computing,50–59 (ACM, 2001).

27. Moore, C. & Russell, A. Quantum walks on the hyper-cube. In International Workshop on Randomization andApproximation Techniques in Computer Science, 164–178(Springer, 2002).

28. Childs, A. M., Farhi, E. & Gutmann, S. An exampleof the difference between quantum and classical randomwalks. Quantum Information Processing 1, 35–43 (2002).

29. Carneiro, I. et al. Entanglement in coined quantum walkson regular graphs. New J. Phys. 7, 156 (2005).

30. Tang, H. et al. Experimental quantum fast hitting onhexagonal graphs. Nat. Photon. 1 (2018).

31. Qi, F., Ma, Q., Wang, Y. & Zheng, W. Silicon photonicchips for search on improved-glued-binary-tree based oncontinuous-time quantum walk. In Quantum and Nonlin-ear Optics IV, vol. 10029, 100291C (International Societyfor Optics and Photonics, 2016).

32. Perets, H. B. et al. Realization of quantum walks withnegligible decoherence in waveguide lattices. Phys. Rev.Lett. 100, 170506 (2008).

33. Tang, H. et al. Experimental two-dimensional quantumwalk on a photonic chip. Sci. Adv. 4 (2018).

34. Davis, K. M., Miura, K., Sugimoto, N. & Hirao, K. Writ-ing waveguides in glass with a femtosecond laser. Opt.Lett. 21, 1729–1731 (1996).

35. Thomson, R. R., Birks, T. A., Leon-Saval, S., Kar, A. K.& Bland-Hawthorn, J. Ultrafast laser inscription of anintegrated photonic lantern. Opt. Express 19, 5698–5705(2011).

36. Crespi, A. et al. Integrated multimode interferometerswith arbitrary designs for photonic boson sampling. Nat.Photon. 7, 545 (2013).

37. Chaboyer, Z., Meany, T., Helt, L., Withford, M. J. &Steel, M. Tunable quantum interference in a 3d integratedcircuit. Sci. Rep. 5, 9601 (2015).

7

38. Feng, Z. et al. Invisibility cloak printed on a photonicchip. Sci. Rep. 6, 28527 (2016).

39. Gao, J. et al. Non-classical photon correlation in a two-dimensional photonic lattice. Opt. Express 24, 12607–12616 (2016).

40. Szameit, A., Dreisow, F., Pertsch, T., Nolte, S. &Tunnermann, A. Control of directional evanescent cou-pling in fs laser written waveguides. Opt. Express 15,1579–1587 (2007).

41. Izaac, J. A. & Wang, J. B. pyctqw: A continuous-timequantum walk simulator on distributed memory comput-ers. Comput. Phys. Commun. 186, 81 – 92 (2015).

42. Sanchez-Burillo, E., Duch, J., Gomez-Gardenes, J. &Zueco, D. Quantum navigation and ranking in complexnetworks. Sci. Rep. 2, 605 (2012).

43. Spring, J. B. et al. On-chip low loss heralded sourceof pure single photons. Opt. Express 21, 13522–13532(2013).

8

Supplemental Information: Quantum Fast Hitting on Glued Trees Mapped on aPhotonic chip

Supplementary Note 1 - Simplification process of the central-random glued trees to a 1D chain

We will first give the general derivation process [S1], then we will give a simple example to make the generalderivation easily understood.

Consider the Hilbert space of the 1D chain spanned by | col j〉, which can be represented as the uniform superpositionover the nodes in column j, i.e.,

| col j〉 =1√Nj

∑a∈column j

|a〉 (S1)

where |a〉 represents the state of a node on the glued trees, and Nj is the number of nodes in column j.

Nj =

{Bj 0 ≤ j ≤ nB2n+1−j n+ 1 ≤ j ≤ 2n+ 1

(S2)

If the adjacency matrix of the central-random glued trees is represented by A, in which, if two nodes i and j areconnected, Ai,j = 1, otherwise, Ai,j = 0, then we have the following derivation process. We assume the hopping rateγ of central-random glued trees is 1 in the whole derivation process. For any 0 < j < n, we have

A| col j〉 =1√Nj

∑a∈column j

A|a〉 (S3)

=1√Nj

B ∑a∈column j−1

|a〉+∑

a∈column j+1

|a〉

=

1√Nj

(B√Nj−1| col j − 1〉+

√Nj+1| col j + 1〉

)=√B(| col j − 1〉+ | col j + 1〉).

In a similar way, for any n+ 1 < j < 2n+ 1, we have

A| col j〉 =1√Nj

∑a∈column j−1

|a〉+B∑

a∈column j+1

|a〉

(S4)

=1√Nj

(√Nj−1| col j − 1〉+B

√Nj+1| col j + 1〉

)=√B(| col j − 1〉+ | col j + 1〉).

We can see that the hopping rates on the chain which correspond to the mapping of the left tree and the right treeare uniform, and are

√B times of that on the original graph.

However, when comes to the mapping of the random-glued part of the glued trees, the results are different. Therandom-glued part has to satisfy the condition that each node in column n should be connected to B different nodesin column n+ 1, and vice versa. Then, if j = n, we have

A| coln〉 =1√Nn

(B

∑a∈columnn−1

|a〉+B∑

a∈columnn+1

|a〉

)(S5)

=1√Nn

(B√Nn−1| coln− 1〉+B

√Nn+1| coln+ 1〉

)=√B| coln− 1〉+B| coln+ 1〉.

9

Similarly,

A| coln+ 1〉 =1√Nn+1

(B

∑a∈columnn

|a〉+B∑

a∈columnn+2

|a〉

)(S6)

=1√Nn+1

(B√Nn| coln〉+B

√Nn+2| coln+ 2〉

)= B| coln〉+

√B| coln+ 2〉.

We can see that the hopping rate on the chain corresponding to the random-glued part is B times of the hoppingrate on the glued trees, different from the rest of the hopping rates. It is obvious that what exactly the random-gluedpart is doesn’t affect the geometry of the 1D chain. Next, we will use a simple example to illustrate the abovederivation process.

Col 0 Col 1 Col 2 Col 3 Col 4 Col 5

1

2

3

4

5

6

7

√2γ √2 √2 √22γ γ γ γ

γ

8

9

10

11

12

1314

Supplementary Figure 1: Schematic diagram of central-random glued trees with B=2, n=2.

If | col 0〉, | col 1〉, | col 2〉, | col 3〉 can be represented as

| col 0〉 = |1〉, (S7)

| col 1〉 =1√2

(|2〉+ |3〉), (S8)

| col 2〉 =1√4

(|4〉+ |5〉+ |6〉+ |7〉), (S9)

| col 3〉 =1√4

(|8〉+ |9〉+ |10〉+ |11〉), (S10)

then,

A| col 1〉 =1√2A(|2〉+ |3〉) (S11)

=1√2

(|1〉+ |4〉+ |5〉+ |1〉+ |6〉+ |7〉〉

=1√2

(2| col 0〉+√

4| col 2〉)

=√

2(| col 0〉+ | col 2〉).

10

A| col 2〉 =1√4A(|4〉+ |5〉+ |6〉+ |7〉) (S12)

=1√4

(|2〉+ |9〉+ |10〉+ |2〉+ |10〉+ |11〉+ |3〉+ |8〉+ |11〉+ |3〉+ |8〉+ |9〉〉

=1√4

(2√

2| col 1〉+ 2√

4| col 3〉)

=√

2| col 1〉+ 2| col 3〉.

The hopping rates between | col 0〉 and | col 1〉, | col 1〉 and | col 2〉, | col 2〉 and | col 3〉 on the chain are√

2,√

2, 2respectively. It is obvious that, as long as the random-glued part satisfies that each node in column 2 is connected to2 different nodes in column 3, and vice versa for column 3, the specific connection way doesn’t affect the structure ofthe 1D chain. The hopping rates between the remaining nodes on the chain can be derived in a similar way.

Supplementary Note 2 - Fabrication of the waveguide arrays

In the femtosecond laser direct writing process, the writing laser with a wavelength of 513nm is up-converted froma femtosecond laser with a wavelength of 1026nm, a pulse duration of 290 fs, and a repetition rate of 1 MHz, and isfirstly sent through a spatial light modulation (SLM) and then focused on a borosilicate chip substrate through a 50×objective lens. The waveguide is written at a depth of 380um with a velocity of 10mm/s. To insure the uniformity ofthe waveguide arrays, power and depth compensation are also used.

Supplementary Note 3 - Single-photon imaging of spatial photon number distributions

To carry out the experiment in a quantum regime, we use heralded single photons as the photon source (FIG.S6). The 810nm photon pairs are generated via a type-II spontaneous parametric down conversion (SPDC) process,pumped by an ultraviolet laser with a wavelength of 405nm in a PPKTP crystal. The generated photon pairs are sentthrough a long-pass filter and then separated into horizontal and vertical component by utilizing a polarized beamsplitter(PBS). The vertically-polarized photons are injected into the waveguide arrays, playing the role of quantumwalkers, while the horizontally-polarized photons are used to give the ICCD camera a photographing command viaa single-photon detector (APD). The photographing commands triggered by the horizontally-polarized photons areused to avoid the impact of thermal light statistics.

Supplementary Note 4 - Calculation of the hitting efficiency

By photographing the spatial photon number distributions of single photons via an ICCD, we obtain the pictureas well as a corresponding ASCII file which records the photon intensity of each pixel. We first find the center pixelcoordinate for each waveguide shown in the picture, and then measure the radius in terms of pixel counts for thelight spot in each waveguide. By summing up the intensity value within the radius and normalizing them, we get theprobabilities of photons distributed at each waveguide.

Supplementary Note 5 - Spatial photon number distributions and scalings of optimal evolving length

The experimental results for trees with B =2, 3, 4, 5 are presented in FIG. S2-S5 respectively. Each figure showsspatial photon number distributions of optimal hitting efficiencies and scalings of corresponding optimal evolvinglengths with n ranging from 2 to 16. Optimal evolving length refers to the evolving length at which optimal hittingefficiency occurs. Note that hopping rate γ between two adjacent nodes on central-random glued trees will influencethe optimal evolving length, hence we set the same hopping rate for all the four scaling plots shown in FIG. S2-S5. Thelinear scaling is clearly suggested by theoretical calculation and is well agreed by experimental results. For samplesat the same B, though optimal hitting efficiency decreases slightly with the increasing number of waveguides as nincreases, most of the photons would gather at the exit waveguide when optimal hitting efficiency occurs. Comparingspatial photon number distributions of the same n in different Bs, there is an overall drop of the light intensity at the

11

exit waveguide from B = 2 to B = 5. This leads to the slight decrease of optimal hitting efficiency as B increases,and is consistent with theoretical studies [S2].

Supplementary Figure 2: Spatial photon number distributions and scalings of optimal evolving lengths for samplesat B = 2. a, Spatial photon number distributions of optimal hitting efficiency from 2-layer to 16-layer at B = 2. The injectingwaveguide is marked by a white circle. b, Variation of optimal evolving lengths with n ranging from 2 to 16 in theory andexperiment respectively. Error bars for the experimental optimal evolving length are the intervals of evolving length valuesbetween two adjacent samples at the same n and B used in experiments. Since evolving length values are discrete in experiment,the optimal evolving length may lie between two adjacent length values. The error bar descriptions also apply to FIG. S2, S3and S4.

Supplementary Figure 3: Spatial photon number distributions and scalings of optimal evolving lengths for samplesat B = 3. a, Spatial photon number distributions of optimal hitting efficiency from 2-layer to 16-layer at B = 3. The injectingwaveguide is marked by a white circle. b, Variation of optimal evolving lengths with n ranging from 2 to 16 in theory andexperiment respectively.

12

Supplementary Figure 4: Spatial photon number distributions and scalings of optimal evolving length for samplesat B = 4. a, Spatial photon number distributions of optimal hitting efficiency from 2-layer to 16-layer at at B = 4. Theinjecting waveguide is marked by a white circle. b, Variation of optimal evolving lengths with n ranging from 2 to 16 in theoryand experiment respectively.

Supplementary Figure 5: Spatial photon number distributions and scalings of optimal evolving length for samplesat B = 5. a, Spatial photon number distributions of optimal hitting efficiency from 2-layer to 16-layer at B = 5. The injectingwaveguide is marked by a white circle. b, Variation of optimal evolving lengths with n ranging from 2 to 16 in theory andexperiment respectively.

13

Supplementary Note 6 - Measurements of second-order anti-correlation parameter α

Supplementary Figure 6: Setup of measuring α. A 405nm laser pumping a PPKTP crystal can generate 810nm correlatedphoton pairs via the type-II SPDC process. A long pass filter (LPF) is inserted to block the pump laser. Then the photon pairspass through a PBS and are separated into vertically-polarized photons and horizontally-polarized photons. The horizontally-polarized photons play the rule of trigger signal and are detected by an avalanched photo diode (APD3); the vertically-polarizedphotons are injected into the photonic chip. Then, an iris is used to filter out the photons coming from the exit waveguide.Finally, the out-coming photons are coupled into a fiber beam splitter and detected by two separate APDs (APD1 and APD2).A photon coincidence counter module (not shown in the picture) is utilized to record the coincidence events. This setup canbe switched into the single-photon imaging of spatial photon number distribution by replacing APD1, APD2 and fiber beamsplitter with an ICCD camera. QWP, quarter-wave plate; HWP, half-wave plate; PBS, polarized beam splitter; PPKTP,periodically poled KTP crystal; LPF, long-pass filter; APD, avalanched photo diode.

The second-order anti-correlation parameter α, which tends to be 0 for ideal single photon and 1 for classicalcoherent light, can be described as [S3]

α =N3N123

N13N23, (S13)

where N3 represents the photon number of trigger signal; N23(N13), N123 represent the number of two- and three-foldcoincidence detection events.

As shown in FIG. S6, one photon of the single photon pairs that generated in the SPDC single photon source is usedas the trigger signal, and another is injected into the waveguide arrays. The number of trigger signal can be detectedby an avalanched photo diode (APD3), through which we can obtain N3. As for the photons being injected into thewaveguide arrays, when they have exited the photonic chip, those coming from the exit waveguide are filtered out byan iris inserted after the chip, then the out-coming photons are coupled into a fiber beam splitter and detected bytwo separate APDs (APD1 and APD2), hence we can obtain the rest of the coincidence detection events. A photoncoincidence counter module (not shown in the picture) is utilized to record the coincidence events.

Supplementary Reference

[S1] Retrieved from: https://www.math.uwaterloo.ca/~amchilds/teaching/w08/l13.pdf[S2] Carneiro, I., et al. Entanglement in coined quantum walks on regular graphs. New J. Phys. 7, 156 (2005)[S3] Spring, J. B., et al. On-chip low loss heralded source of pure single photons. Opt. Express 21, 13522-13532 (2013)