The Grover search as a naturally occurring phenomenon

Mathieu Roget,1 Stephane Guillet,1 Pablo Arrighi,2 and Giuseppe Di Molfetta3, ∗

1Aix-Marseille Univ, Universite de Toulon, CNRS, LIS, Marseille, France2Aix-Marseille Univ, Universite de Toulon, CNRS, LIS, Marseille, France and IXXI, Lyon, France

3Universite Publique, CNRS, LIS, Marseille, France and Quantum Computing Center, Keio University(Dated: 6 mai 2020)

We provide first evidence that under certain conditions, 1/2-spin fermions may naturally behave like a Groversearch, looking for topological defects in a material. The theoretical framework is that of discrete-time quantumwalks (QW), i.e. local unitary matrices that drive the evolution of a single particle on the lattice. Some QWare well-known to recover the (2 + 1)–dimensional Dirac equation in continuum limit, i.e. the free propagationof the 1/2-spin fermion. We study two such Dirac QW, one on the square grid and the other on a triangulargrid reminiscent of graphene-like materials. The numerical simulations show that the walker localises aroundthe defects in O(

√N) steps with probability O(1/ logN), in line with previous QW search on the grid. The

main advantage brought by those of this paper is that they could be implemented as ‘naturally occurring’ freelypropagating particles over a surface featuring topological—without the need for a specific oracle step. Froma quantum computing perspective, however, this hints at novel applications of QW search : instead of usingthem to look for ‘good’ solutions within the configuration space of a problem, we could use them to look fortopological properties of the entire configuration space.

Quantum Computing has three main fields of applications :quantum cryptography ; quantum simulation ; and quantum al-gorithmics (e.g. Grover, Shor...). Some quantum cryptogra-phic devices are already commercialized, and we may hopethat some quantum simulation devices will also reach thisstage within the next decade. Quantum algorithms, however,are generally considered to be a long-term application. Thisis because of the common understanding that we will needto build scalable implementations of universal quantum gatesets with fidelity 10−3 first, and implement quantum error cor-rections then, in order to finally be able to run our preferredquantum algorithm on the thereby obtained universal quantumcomputer. This seems feasible, yet long way to go.

In this letter we argue that this may be a pessimistic view.Scientists may get luckier than this and find out that nature ac-tually implements some of these quantum algorithms ‘sponta-neously’. Indeed, the hereby presented research suggests thatthe Grover search may in fact be a naturally occurring pheno-menon, e.g. when fermions propagate in crystalline materialsunder certain conditions.

Amongst all quantum algorithms, the reasons to focus onthe Grover search [17] are many. First of all because of itsremarkable generality, as it speeds up any brute force O(N)problem into a O(

√N) problem. Having just this quantum

algorithm would already be extremely useful. Second of all,because of its remarkable robustness : the algorithm comesin many variants and has been rephrased in many ways, in-cluding in terms of resonance effects [27] and quantum walks[12].

Remember that a quantum walks (QW) are essentially localunitary gates that drive the evolution of a particle on a lattice.They have been used as a mathematical framework to expressmany quantum algorithms e.g. [2, 32], but also many quantumsimulation schemes e.g. [3, 14, 15]. This is where things getinteresting. Indeed, it has been shown that many of these QWadmit, as their continuum limit, the Dirac equation [8, 13, 18,

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FIGURE 1: Defects in triangular (left) and square (right) lattices.

24], providing ‘quantum simulation schemes’, for the futurequantum computers, to simulate all free spin-1/2 fermions.

Recall that the Grover search is the alternation of a diffu-sion step, with an oracle step. Here we provide evidence that1/ these Dirac QW, in (2 + 1)–dimensions, work fine to im-plement the diffusion step of the Grover search and 2/ Topo-logical defects also work fine to implement the oracle step ofthe Grover search.

The second point is, on the practical side, probably moreimportant than the first. Indeed, whilst there are several expe-rimental realizations of QW, including 2D QW, [16, 29], thesehave not been considered as scalable substrates for implemen-ting the Grover search so far—probably due to the lack of aneasy way of implementing the oracle step.

From a theoretical perspective, that point is strongly sug-gestive also. Indeed, recall that many quantum algorithms areformulated as a QW search on a graph, whose nodes representelements the configuration space of a problem, and whoseedges represent the existence of a local transformation bet-ween two configurations—see [9] for a recent example of

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FIGURE 2: Quantum Walks scheme on triangular (left) and square(right) lattice. On the triangular grid, an anti-clockwise rotation R isimplemented by a partial shift T0,ε (blue) along a direction u0, T1,ε

(red) along a direction u1 and T2,ε (green) along a direction u2. Onthe square grid, an anti-clockwise rotation R is implemented by onlytwo partial shifts Tx,ε (blue) along a direction ux, Ty,ε (red) along adirection uy .

that. So far, the QW search has only been used to look for‘marked nodes’, i.e. good configurations within the configu-ration space, as specified by an oracle. Here, in contrast, theQW search is used to look for topological defects, which areproperties of the configuration space itself. This suggests ai-ming beyond recognizing simple hole defects in 2D crystals,to target more general topological classification problems—e.g. seeking to characterize homotopy equivalence over confi-guration spaces that represent manifolds as CW-complexes[7, 23].

DIRAC QUANTUM WALKS

We consider QW both over the square and the triangu-lar grid. More precisely we consider periodic tilings of theplane, where the tiles are either squares or equilateral tri-angles, of alternating grey and white colours, as in Fig. 2. Thewalker lives over the middle points of each side (aka facet)of each tile. For the square grid we can label these pointsby their positions in Z2, for the triangular grid this wouldbe a subset of Z2. To any such point x we assign a com-plex number representing the amplitude of the walker beingthere, which we denote by ψ+(x) (resp. ψ−(x)) if the tile iswhite (resp. grey). Of course wherever two facets are glued,so are their middle points, and so the two complex numbersform a spinor ψ(x) = (ψ+(x) ψ−(x))> in H2. Letting|v+〉 = (1 0)> and |v−〉 = (0 1)> we may then writeψ(x) = ψ+(x) |v+〉 + ψ−(x) |v−〉. This degree of freedomat a single point is referred to as the walker’s ‘coin’ or ‘spin’.For the full square grid, the overall state of the walker there-fore lies in the composite Hilbert space H2 ⊗ HZ2 and canbe written |ψ〉 =

∑x ψ−(x) |v−〉 ⊗ |x〉+ ψ+(x) |v+〉 ⊗ |x〉.

For the full triangular grid the amplitude of one in every two

position needs be zero. For a grid with a missing white (resp.grey) tile the corresponding ψ+(x) (resp. ψ−(x)) for x on aside of the tile, needs be zero.The class of evolution operators that we consider in this paperare QW of the form :

|ψ(t+ ε/l)〉 = WR |ψ(t)〉 (1)

with l = 2 for the square grid and l = 3 for the triangular grid.Here, R stands for the synchronous anti-clockwise rotationof all tiles. Notice that, wherever there is no missing tile, thesimultaneous rotations of the two tiles glued at x preciselycoincides with the implementation of a partial shift Tk,ε alonga direction uk :

Tk,ε

(ψ+(x)ψ−(x)

)=

(ψ+(x + ukε)ψ−(x− ukε)

).

Moreover,W stands for the synchronous application of a 2×2unitary W (x) on the spins ψ(x). This unitary depends on xonly in a very simple way which we now clarify. First of all,if it so happens that a tile is missing at x, then the spinor ψ(x)is incomplete, and so W (x) = I . Second of all, if there is nomissing tile, then W (x) = Wk, where the k is that correspon-ding to the partial translation direction uk occurring at x.It follows that, in the case of a full grid, any given walker willundergo Tk,ε and then Wk for k = 0 . . . l − 1 successively,amounting to

|ψ(t+ ε)〉 = Wl−1Tl−1,ε . . .W0T0,ε |ψ(t)〉= ΠkWkTk,ε |ψ(t)〉 . (2)

The way we choose these Wk is so that QW is Dirac QW,meaning that

ΠkWkTk,ε ≈ exp(iεHD)

as we neglect the second order terms in ε, with HD the DiracHamiltonian in natural ~ = c = 1 units, i.e.

HD = pxσx + pyσy +mσz.

Therefore, on the full grid, these QW simulate the Dirac Equa-tion, more and more closely as ε goes to zero.

Square grid. Let us consider the unit vectors along the x–axis and y–axis, namely {ux,uy} and use them to specify thedirections of the translations Tx,ε and Ty,ε. Eq. (2) then reads :

U = W+Ty,εW−Tx,ε (3)

where W± = exp(iσxθ±) with θ± = ±(π4 ± εm) and m a isreal constant, namely the mass. In the formal limit for ε→ 0,Eq.(3) recovers the Dirac Hamiltonian in (2 + 1)–spacetime.Iterations of the walk observationally converge towards solu-tions of the Dirac Eq., as was proven in full rigor in [5], whichmotivated the above choice of U on the square grid in the fol-lowing.

3

Triangular grid. For the Triangular grid let us considerthe unit vectors {u0,u1,u2}, as in Fig. 2 and defined by

uk = cos

(2kπ

3

)ux + sin

(2kπ

3

)uy for k = 0, 1, 2.

and use them to specify the directions of the translations Ti,ε.Eq. (2) then reads :

e−iεHD = WT2,εWT1,εWT0,ε (4)

with W = eiπ3 e−i

α2 σye−i

π3 σzei

α2 σye

−iε 3√5mσz the coin ope-

rator, which turns out not to depend on the direction uk. In [4]it has been proved in detail by some of the authors how thisparticular choice also leads, in the continuum limit, to the Di-rac Hamiltonian in (2 + 1)–spacetime, which again motivatedus to adopt the above W on the triangular grid in presence ofdefects.

Defects. A sector of a crystal lattice may be inaccessible,e.g. due to surface defects such as the vacancy of an atom(e.g. Schottky point defect) and others. These affect the phy-sical and chemical properties of the material, including elec-trical resistivity or conductivity [19]. Here we model these de-fects in the simplest possible way : locally, a small number ofsquares or triangles are missing, thereby breaking the transla-tional invariance of the lattice. In other words, the walker isforbidden access to a ball B of unit radius, as in Fig. 1.This isdone by reflecting those signals that reach the boundary ∂B ofthe ball, simply by letting W = I on the facets around ∂B.

Notice that, wherever we replace the coin W by identity,both Dirac walks reduce to just anti-clockwise rotation R asin Eq. (1), see Fig. 1. Still, the operators (3) and (4) may havedifferent topological properties around ∂B. For instance, thesquare grid QW has vanishing Chern number and trivial to-pological properties [20] for vanishing m, which can still be-come non-trivial from m > 0 [6]. The triangular walk on theother hand is always topologically non-trivial and has Chernnumber equal to one [21]. In the triangular case the positiveand negative component decouple respectively in the grey andthe white triangles, and may be thought of as inducing po-larized local topological currents of spin, called edge states[31]. According to [31], this phenomenon will be observedwhenever initial states have an overlap with ∂B, elsewhere thewalker does not localize and explores the lattice with ballis-tic speed. Thus, we expect these topological effects to becomesignificant in the triangular case and it is indeed the case.

Our conjecture is that, starting from a uniformly superposedwavefunction, the walker will, in finite time, localise aroundthe defect inO(

√N) steps, with probability inO(1/ log(N)),

with N the total number of squares/triangles. In the followingwe discuss the numerical evidence we have for such a conjec-ture.

GROVER SEARCH

Our numerical simulations over the square and triangulargrids are exactly in line with a series of results [1, 12, 30] sho-

0 100 200 300 400 500

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

FIGURE 3: Square grid periodic localization. Probability of beinglocalized around the center of the defect versus time. For m = 0 andN = 2500.

wing that 2D spatial search can be performed inO(√N) steps

with a probability of success in O(1/ logN). With O(logN)repetitions of the experiment one makes the success probabi-lity anO(1), yielding an overall complexity ofO(

√N logN).

Making use of quantum amplitude amplification [10], howe-ver, one just needs O(

√logN) repetitions of the experiment

in order to make the probability an O(1), yielding an overallcomplexity ofO(

√N logN). This bound is unlikely to be im-

proved, given the strong arguments given by [22, 25, 28].These works were not using Dirac QW, nor defects. Our aimhere is demonstrate that QW which recover the Dirac equa-tion, also perform a Grover search, as they propagate over thediscrete surface and localise around its defects. More concre-tely we proceed as follows : (i) Prepare, as the initial statethe wavefunction which is uniformly superposed over everysquare or triangle, and whose coin degree of freedom is alsothe uniformly superposed (|v+〉+ |v−〉)/

√2. Notice that am-

plitude inside the defect is zero ; (ii) Let the walker evolvewith time ; (iii) Quantify the number of steps t(N) before thewalker reaches its probability peak p(N) of being localizedin a ball of radius 2 around the center of the defect, namelythe peak recurrence time and estimate this probability peak,at fixed N ; (iv) Characterize t(N) and p(N), i.e. the way thepeak recurrence time and the probability peak depend uponthe total number of squares/triangles N .

We indeed observe that the probability of being foundaround the defect has a periodic behaviour, see in Fig. 3 forthe case of the square lattice : for instance with N = 2500sites, for m = 0, the peak recurrence time is at t ∼ 25, withmaximum probability is p ' 10−1. The dependencies in Nwere interpolated from the data set, as shown in Fig. 4.a. Weobserve that t(N) =

√N and p(N) ' 1/ logN asympto-

tically, with a prefactor that depends on m. In this masslesscase the interpolation p(N) ' 1/ logN works right-away, butwhen the mass gets larger, the curve remains longer along ap(N) ' 1/N trajectory, before it eventually enters its asymp-

4

totic p(N) ' 1/ logN regime. Moreover, as shown in Fig.4.b, in presence of more than one topological defects we theway the peak recurrence time and the probability depend uponN is the same. Notice that the prefactors do not depend on thenumber of defects but only depend onm, as shown in Fig. 4.a.

Clearly, repeating the experiment an O(logN) number oftimes will make the probability of finding the defect as closeto 1 as desired, leading to an overall time complexity inO(√N logN). Again we could, instead, propose to use quan-

tum amplitude amplification [10] in order to bring the needednumber of repetitions down anO(

√logN), leading to an ove-

rall time complexity in O(√N logN). But it seems that this

would defeat the purpose of this paper to some extent : sinceour aim is to show that there is a ‘natural implementation’ ofthe Grover search, we must not rely on higher-level routinessuch as quantum amplitude amplification.

Over the triangular grid the Grover search is again at play.Indeed the data set of Fig. 5, confirms the results obtainedover the square grid : the peak recurrence time is againt(N) ' O(

√N), and its corresponding probability peak

is again p(N) ' O(1/ logN) for large N , again with aprefactor that depends on the mass. Again this leads to anoverall complexity of O(

√N logN), or O(

√N logN) using

amplitude amplification.

CONCLUSION

It is now common knowledge that Quantum Walks (QW)implement the Grover search, and that some QW mimic thepropagation of the free 1/2-spin fermion. Yet, could this meanthat these particles naturally implement the Grover search?Answering this question positively may be the path to a se-rious technological leap, whereby experimentalist would by-pass the need for a full-fledged scalable and error-correctingQuantum Computer, and take the shortcut of looking for ‘na-tural occurrences’ of the Grover search instead. So far, howe-ver, this idea has remained unexplored. The QW used to im-plement the Grover diffusion step were unrelated to the DiracQW used to simulate the 1/2-spin fermion, with the noticeableexception of [25]. More crucially, the Grover oracle step see-med like a rather artificial, involved controlled-phase, far fromsomething that could occur in nature. This contribution beginsto remedy both these objections.

We used Dirac QW over both the triangular and the squaregrid as the Grover diffusion step and, instead of alternatingthis with an extrinsic oracle step, we coded for the solutiondirectly inside the grid, by introducing a topological defect.We obtained strong numerical evidence showing that the Di-rac QW localize around the defect in O(

√N) steps with pro-

bability O(1/ logN), just like previous QW search would.Our next step is to use QW to locate not just a hole defect,but a particular QR code–like defect, amongst many possibleothers that could be present on the lattice. This would bring usone step closer to a natural implementation of an unstructured

0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16

10-4

10-3

10-2

10-1

0 200 400

0

100

200

300

400

0.08 0.1 0.12 0.14 0.16

0.02

0.03

0.04

2 defects

3 defects

4 defects

0 200 400

0

100

200

300

400

FIGURE 4: Square grid scalings. (Top) Probability peak of being lo-calized around one defect, versus the number of squares in the gridfor different value of the mass m. (Bottom) Probability peak of beinglocalized around two, three and four defects respectively, versus thenumber of squares in the grid for m = 0. The inset shows the peakrecurrence time.

database Grover search. Replacing the Grover oracle step bysurface defects seems way more practical in terms of experi-mental realizations, whatever the substrate, possibly even in abiological setting [26]. At a more abstract level, this suggestsusing QW to search, not just for ’good’ configurations withina space, but rather for topological properties of the configura-tion space itself.

Acknowledgements The authors acknowledge inspiringconversations with Fabrice Debbasch, that sparked the ideaof Grover searching for surface defects ; enlightening discus-sions on topological effects with Alberto Verga ; and usefulremarks on how to better present this work by Janos As-both, Tapabrata Ghosh, Apoorva Patel and the anonymousreferees. This work has been funded by the Pepiniere d’Ex-cellence 2018, AMIDEX fondation, project DiTiQuS and the

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0.115 0.12 0.125 0.13 0.135 0.14 0.145

10-4

10-3

10-2

30 40 50 60 70 80 900

100

200

300

400

m=0m=0.2m=0.4

FIGURE 5: Triangular grid scalings. Recurring probability peak ofbeing localized around the defect, versus the number of triangles inthe grid. The inset shows the peak recurrence time.

ID# 60609 grant from the John Templeton Foundation, aspart of the “The Quantum Information Structure of Spacetime(QISS)” Project.

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