Realizing Grover Search in Nonlinear Optical CrystalsMurphy Yuezhen Niu1 2 ∗

1 Research Laboratory of Electronics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge,Massachusetts 02139, USA

2 Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139,USA

Abstract.We demonstrate the first theoretical model of a generalized Grover search in nonlinear quantum optics through

spontaneous parametric down-conversion. The quantum search algorithm enables us to overcome the low conversionefficiency limit of conventional phase matching processes. Extending our method to multi-photon levels, a scalableGrover search can be realized with novel fabrication designs.

Keywords: quantum algorithm, quantum optics

1 IntroductionWith the advancement of quantum computation, more and

more quantum algorithms can be realized in experimental[5][2][1] and commercial protocols that outperform classicalcounterparts for specific problems. Despite the lack of knowl-edge of whether such quantum improvement exist in general,quantum search algorithms stand out unarguably as evidenceof quantum advantage: it provably provides quadratic speedupover classical search for an unsorted database. Nonetheless,the majority of quantum search models are realized in the bulkwith nuclear magnetic resonance[4], linear optics[6][5], cav-ity quantum electrodynamics or Rydberg atom ensembles[8].Scalable realizations have only been proposed in trapped ionatom systems[7]. As a subroutine to other exponentially fastquantum algorithm [29], a scalable quantum search realizationbecomes necessary. In this work, we demonstrate how to uti-lize nonlinear optical crystals to implement scalable quantumsearch algorithms.

The all-optical linear quantum computing (LOQC) is oneof the most promising candidates to realize quantum com-putation. Due to fast-advancing laser technology and newphoton detectors, the efficiency of LOQC gates has beensteadily improving[11][12][13][14]. With measurement in-duced nonlinearity, LOQC is sufficient for universal quantumcomputation[3]. However, without pre-compiling, the size ofelements grows exponentially with the qubit number[5].

Nonlinear quantum optics is used to generate entanglementsources and pure single photon states to further reduce the cir-cuit size of LOQC. For example, cluster states are highly en-tangled photon states used in one-way quantum computationwith only linear components. A new lower bound has beenproven on the entanglement resource required in the prepa-ration of the cluster state for universal quantum gates[10].Moreover, current experiments on cluster states use nonlinearprocesses such as spontaneous parametric down-conversion(SPDC) in beta-barium borate crystal (BBO)[9] to preparecluster states.

The downside of conventional SPDC is its low effi-ciency. Take periodically poled potassium titanium phos-phate(PPKTP) crystal, for example: a 100mW pump light

∗yzniu@mit.edu

can be converted to around 10pW degenerate entangled pho-ton pairs in type II SPDC, which gives a mere 10−10 conver-sion efficiency. In order to generate a usable entanglementsource, a strong input pump is required[15], and is assumedto be of constant power before and after the SPDC. Such anon-depleted pump assumption can only be justified in semi-classical theory.

However, in order to directly use SPDC in quantum mea-surement of LOQC outcome or to realize quantum logic gate,a complete quantum theory is needed. We review the single-mode quantum SPDC in Section 2, and in Section 3 we pro-pose the new Grover search model using nonlinear crystal de-sign at two-photon level, improving the quantum efficiency ofSPDC to unity. We further generalize the algorithm to fourpump photon case and propose the fabrication design in Sec-tion 3.3 . We validate the effective use of the proposed Groversearch in multi-photon level in Section 3.4.

2 Quantum Theory for SPDCGenerally, a strong pump field intensity is required to in-

duce nonlinearity in optical crystals. Even so, single-pump-photon SPDC has been successfully realized in experiment[16]. Both theoretical[17][18][19] and experimental studiesat the few-photon level [20] [21] [22] have exhibited greatpotential for more scalable quantum computing architectureusing nonlinear quantum optics.

SPDC occurs when pump photons enter quadratic nonlinearoptical crystals that lack centrosymmetry, and trigger elec-trons in the crystal to oscillate, absorbing the pump photonswhile spontaneously emitting pairs of lower energy photons.The photons of each down-converted pair are entangled intime and energy, and possess the same polarization for typeI SPDC, and orthogonal polarization for type II SPDC. For amono-chromatic plane wave pump input, the frequency sumof the converted photon pair equals the pump frequency. TheHamiltonian of this nonlinear process can be described in aninteraction picture as [18]

H = i~A(as†ai†ap − ap†asai) (1)

arX

iv:1

603.

0685

7v1

[qu

ant-

ph]

22

Mar

201

6

where as† and ai† denote the photon creation operator forthe down-converted photons named as signal and idler by an-nihilating one pump photon denoted by the annihilation op-erator ap, and coefficient A ∝ χ is proportional to the non-linear electric susceptibility tensor[17]. At low light limit,given the monochromatic pump state represented on Fock ba-sis |Ψp〉 =

∑n cn|Ψn〉, the wave function evolves according

to the Schrodinger equation:

i~d|Ψp〉dt

= H|Ψp〉. (2)

For n-photon subspace of the pump state, the solu-tions to the Schrodinger equation can be expanded on thetri-partite n + 1 dimensional Fock bases |ns〉|ni〉|np〉 ∈{|0〉|0〉|n〉, |1〉|1〉|n− 1〉, |2〉|2〉|n− 2〉, ..., |n〉|n〉|0〉}, withthe first two photon number states representing signal andidler and the last one representing the pump. These n + 1orthogonal bases range between no conversion to completeconversion from pump photons into the signal and idler pho-ton pairs. A state in this n-pump-photon subspace can be ex-pressed as:

|Ψn〉 =

n∑k=0

fnk |k〉|k〉|n− k〉. (3)

Inserting above into the Schrodinger equation of Hamilto-nian Eq. (1) we find

(4)dfkdt

= −A ·√nfnk+1, k = 0;

dfkdt

= A · [k√n− k + 1fnk−1 − (k + 1)

√n− kfnk+1],

k = 1, 2, ...n− 1;

dfkdt

= A · nfnn , k = n.

Re-expressing the above equations in matrix representationwith ~V = {fn0 , fn1 , ...fnn } and (n+ 1)× (n+ 1) tri-diagonalmatrix M, we obtain

~V = M~V (5)

M = A

0, −

√n 0 0 · · ·√

n 0 −2√n− 1 0 · · ·

0 2√n− 1 0 −3

√n− 2 · · ·

... · · · 0 −n0 · · · n 0

For arbitrary dimension n, the numerical eigenvalues of M

give the frequency components of the wave function ampli-tude, and the differential problem is solved given initial con-ditions fn0 = 1, fnk = 0, k 6= 0.

2.1 Odd-number-pump-photon solutionsThe simplest non-trivial solution to Eq.(5) is in single-

pump-photon subspace:

df10 (t)

dt= −Af12 (t) (6)

df11 (t)

dt= Af10 (t) (7)

f10 (t) = f10 (0) cos(At+ φ0) (8)f11 (t) = f10 (0) sin(At+ φ0) (9)

where a pump photon can be converted to a pair of signaland idler photons completely. The unity conversion here isno longer true at multi-photon level. For three-pump-photoncase, the wave function solutions are:

f30 (t) = C[B1 cos(ω1At+ φ1) +B2 cos(ω2At+ φ2)] (10)

f31 (t) =C√

3[B1ω1 sin(ω1At+ φ1) +B2ω2 sin(ω2At+ φ2)]

f32 (t) = 2√

6C[cos(ω1At+ φ1)− cos(ω2At+ φ2)]

f33 (t) = 6√

6C[sin(ω1At+ φ1)

B2− sin(ω2At+ φ2)

B1]

with eigenvalues of the 4×4 matrix M determining the angu-lar frequency ω1 =

√10−

√73, ω2 =

√10 +

√73, B1 =√

73 + 7, B2 =√

73 − 7, and initial conditions f30 (0) =〈Ψ|3〉, f3k = 0, k 6= 0 determining the phase and scale pa-rameters φ1 = φ2 = 0, C = f30 .

Geometrically, the non-periodic solution for three-pump-photon Schrodinger equation is due to the two irrational val-ued angular frequency components ω1, ω2. However, as timegoes to infinity, all possible solutions have the amplitude offully converted state f33 (t) upper bound by 0.64 which givesthe highest efficiency of complete conversion to signal andidler photon pairs to only 40%. This is a dramatic decrease inefficiency by adding just two pump photons. The result shedslights on the connection yet to be discovered between quan-tum and semi-classical theories.

Figure 1: The wave function trajectory for the three-pump-photon evolution under quadratic interaction Hamiltonian.The amplitude for the completly converted photon state|3〉|3〉|0〉 is denoted as f33 (t),, that for the pump photon state|0〉|0〉|3〉 without conversion is denoted as f30 (t), and all othercomponents are projected to the third axis f⊥.

2.2 Even-number-pump-photon solutionsTake photon number two eigenstate |0〉|0〉|2〉 as pump in-

put, the solution to the Schrodinger equations Eq. (5) is

f20 (t) = B2

3

√3

1 + 2m2(m+

1

2cos(Aω1t+ φ0)](11)

f21 (t) = B1√

1 + 2m2sin(Aω1t+ φ0)

f22 (t) = B

√6

3√

1 + 2m2[m− cos(Aω1t+ φ0)]

where ω1 =√

6 is the imaginary part of the eigenvalue ofthe matrix M in Eq. (6) and B,m, φ0 are determined by theinitial conditions of the input state. Given the amplitude ofbi-photon pump state asf20 (0) = 〈Ψp|2〉, f2k = 0, k 6= 0,we have B = f20 (0),m = 1, φo = 0. This solution geo-metrically corresponds to a circle centered at the wave vector|Ψ〉 = 1√

3|2〉|2〉|0〉 +

√2√3|0〉|0〉|2〉, shown in Fig. 2 as the

blue line. Changing initial conditions corresponds to differentvalue of m, geometrically represented by the shrinking andthe translation of the circle along the vector |Ψ〉.

The maximum complete conversion rate into signalsand idlers for two-pump-photon input is evaluated bymaxt |f22 (t)|2 = ( 2

√2

3 )2 ∼ 88% which sits in between the

single photon and three photon efficiency.An interesting observation is that only the even number

pump photon subspace have a constant factor m as a free pa-rameter that depends on the initial conditions. The reason be-hind this is the parity of the differential equation: in the odd-number-pump-photon subspace fnn (t) and fn0 (t) are oppositein their time reversal symmetry, meaning the fully convertedstate has a wave function that has a sign change for t → −t ,while the all pump photons state fn0 (t) is invariant under timereversal; in even number pump photons subspace, however,fnn (t) and fn0 (t) share the same parity and are both invari-ant under time reversal, allowing a constant term m as a freeparameter. This feature is proven to be helpful for realizingGrover search to be discussed in the next section.

Figure 2: The wave function trajectory for the two-pump-photon evolution under quadratic interaction Hamiltonian.The amplitude for the completed converted photon state|2〉|2〉|0〉 is denoted as zx, that for the pump photon state|0〉|0〉|2〉 without conversion is denoted as x and half con-verted state amplitude f21 (t) is represented as y. We takeB = 1 as an example, five circles are solutions of differentinitial conditions with m = 0, 0.4, 1, 2, 4 denoted in colororange, green, blue, red and purple circles, lying on the 3Dsphere centered at x = y = 0 of radius B..

Four-pump-photon subspace solution to Eq.(5) have twoangular frequencies similar to three-pump-photon solution Eq.(10). Here we focus on the amplitude of input pump state|0〉|0〉|4〉, the fully converted signal idler pairs |4〉|4〉|0〉, andsum of all other orthogonal state, denoted as x, z, y. The so-lutions to the Schrodinger equations expressed in this basisare

x = f40 (t) = −C 1

17/41 + m2

1168992

[B1 cos(ω1At+ φ1) +B2 cos(ω2At+ φ2) +m] (12)

z = f44 (t) = 6√

6C1

17/41 + m2

1168992

[ω2 cos(ω1At+ φ1)− ω1 cos(ω2At+ φ2)− m

24] (13)

y =√

1− x2 − z2,

where B1 = 17√

297 + 261, B2 = 17√

297 − 261, ω1 =√25−

√297, ω2 =

√25 +

√297, and m,φ1, φ2 are deter-

mined by the initial conditions of the input state. Given theamplitude of bi-photon pump state asf40 (0) = 〈Ψp|4〉, f2k =0, k 6= 0, we have C = f40 (0)/(82

√297),m = 1, φ1 = φ2 =

0. And the highest complete conversion efficiency is 74%which is lower than the two-pump-photon case but higher thanthe three-pump-photon case. We show the time evolution ofthe solution of given initial conditions in Fig.3. Choosing dif-ferent initial conditions form a saddle shape region in threedimensional space spanned by x, y, z defined in Eq. 12.13,which is shown in Fig. 4.

Figure 3: The wave function trajectory for the four-pump-photon evolution under quadratic interaction Hamiltonian. Werepresent the amplitude of unconverted pump state |0〉|0〉|4〉,the fully converted signal idler pairs |4〉|4〉|0〉, and sum of allother orthogonal state as x, z, y. The red thick line denotes theevolution t ∈ [0, π]. And the green sheet covers the evolutionof time t ∈ [0, 200π].

Figure 4: The wave function for the four-pump-photon evo-lution over time period of t ∈ [0, 2π]. for different inputstate. We represent the amplitude of unconverted pump state|0〉|0〉|4〉, the fully converted signal idler pairs |4〉|4〉|0〉, andsum of all other orthogonal state as x, z, y. The green re-gion covers solution given different initial conditions m ∈[−2000, 2000].

3 Grover SearchGrover search[23] is the first quantum search algorithm that

find the marked item in an unsorted data set of size N us-ing time O(

√N). In comparison, it takes classical algorithm

time O(√N) . Grover search is proven to achieve the opti-

mal efficiency among all quantum search algorithms[24]. It isalso well known that quantum search in general is polynomialfaster than classical search due to the degree two quantum am-plitude amplification advantage over classical probability am-plification in search routines.

Given the input state as an equal superposition of all possi-ble states |Ψ〉 containing all the items to be searched, Groversearch repeatedly evokes two iteration steps: I. apply thequantum oracle O, adding a π phase shift for the search itemand leave other items unchanged; II. reflect in the plane de-fined by a uniform superposition of the search items |α〉 and auniform superposition of all unwanted items |β〉 around vector|Ψ〉. For a single marked item in a data set of size N, repeatingthe iteration π

4

√N times and measuring the input state finds

the marked item with probability close to unity.The three requirements for the conventional Grover search

include: an equal superposition input state |Ψ〉, quantum or-acle implementation, and a rotation defined by 2|Ψ〉〈Ψ| − I .Surprisingly,[25] later showed that almost any unitary trans-formationUαβ on the 2 dimensional plane spanned by |α〉, |β〉can be used for Grover iteration, given the ability to invertthe amplitude of a single basis state and that the amplitude|Uαβ | << 1. Moreover, [26] further relaxed the initial statepreparation in Grover search to be arbitrary complex ampli-tude state, where the search running time depends on the meanand variance of the initial state distribution. Now we show be-

low that the quantum SPDC together with linear optical dis-persion are enough to perform generalized Grover search.

In lieu of conventional qubit basis, we conduct Groversearch in quantum SPDC process on Fock basis. The goalis thus to find the state having the largest photon numberof signals and idlers given a fixed Fock state pump input.Represented in the basis we defined in Eq. (3), the Groversearch protocol defined below finds the fully converted state|n〉|n〉|0〉 of n pairs of signal and idler and a vacuum pumpamong n+ 1 data set within O(

√n).

In order for the Grover search to work, search item sizeM should be smaller than half of the total data set size N :M < N/2.More specifically, in n-pump-photon subspace Eq.(3), the searched item is unique |n〉|n〉|0〉 and the search spacehas n+1 elements in total. This requires that the pump photonnumber n to be larger than one: 1 < (n+ 1)/2→ n > 1. Wetherefore discuss two-pump-photon subspace Grover searchfor a start, and then extend the protocol to four photons. Gen-eralization to multi-photon case is discussed in the end.

3.1 Grover search iterationI. State preparation. Send a n-photon Fock state of pump

to the nonlinear crystal and wait for time t0, in the subspacespanned by {|0〉|0〉|n〉, |1〉|1〉|n− 1〉, ..., |n〉|n〉|0〉, we obtaina super position of all n+ 1 possible Fock states:

|Ψ0〉 = fn0 (t0)|0〉|0〉|n〉+ fn−11 (t0)|1〉|1〉|n− 1〉+...+ fnn (t0)|n〉|n〉|0〉,

which gives a complex amplitude state analyzed in [26].

II. Phase shift eiπ . The pump photon of frequency ωpdown-convert into signal and idler photons of smaller fre-quency ω1, ω2 obeying the energy conservation ωp = ω1+ω2.Photons of different frequency traveling through a linear dis-persion optics of length L gain different phases eik(ωi)L,where the wave number k(ωi) is determined by the refractiveindex of the given frequency as k(ωi) = ωi

c n(ωi). Sendingthe state into linear optical material of length L1 such thatei[k(ω1)+k(ω2)−k(ωp)]L1 = −1 fulfills the phase shift for thesearch item.

All bases except the pure pump photon state |0〉|0〉|n〉 gainsa minus sign under this transformation, which is not exactlyequivalent to the first part of the Grover iteration. However,this difference does not affect the result of iteration since thephase shift here will land the state into a new orbit that sharesdifferent initial conditions (denoted as different colored circlein Fig.2). The rotation that bring the state closer to the searchitem can then be carried out by evolving the state on the neworbit of bigger radius in the step discussed below.

III. Rotation towards marked state. After the phase shiftin step II, the wave function intersects another solution tothe Schrodinger equation of a different initial conditions andevolves under a new orbit. Send the photon state back to thenonlinear crystal of length L2 = t2 · c. This effects in a rota-tion closer towards the marked item |n〉|n〉|0〉.

3.2 two-photonShown in section 2.2, two-pump-photon subspace solutions

to the Schrodinger equation lie on circles centered at the vec-tor z = 1√

2x, with x = f20 (t) denoting the two photon pump

amplitude and z = f22 (t) denoting the two-photon signal,two-photon idler and vacuum pump Fock state amplitude. Asthe result, the state |Ψ〉 = 1√

3|2〉|2〉|0〉 +

√2√3|0〉|0〉|2〉 can be

used for the reflection 2|Ψ〉〈Ψ| − I in the second step of theGrover iteration. Such reflection is easily realized by a rota-tion in three dimensional basis after evolving the state for atime ∆t = π/

√6 in SPDC process.

Now we prove below that in two-pump-photon subspacewith data set size n = 3 and search item size m = 1, oneiteration of the Grover search defined above enables us to findthe marked state |2〉|2〉|0〉 with unit probability.

1. Before the Grover search starts, the system is describedby two-pump photon Fock state represented as the deep bluedot in Fig. 5

|Ψ0〉 = |0〉|0〉|2〉 (14)

2. After t0 = 0.63/(ω1A) time the initial state preparationis complete, shown as the light blue dot in Fig. 5

|Ψ1〉 =2

3(1 +

1

2cos(0.63)|0〉|0〉|2〉 (15)

+1√3

sin(0.63)|1〉|1〉|1〉+

√2

3(1− cos(0.63))|2〉|2〉|0〉

3. Sending the photons into the linear optical material oflength L1 in step II, a minus sign is added for all basis exceptpure pump state which intersect another solution that has unityamplitude for z = f22 (t), shown as the purple dot in Fig. 5:

|Ψ2〉 =2

3(1 +

1

2cos(0.63)|0〉|0〉|2〉 (16)

− 1√3

sin(0.63)|1〉|1〉|1〉 −√

2

3(1− cos(0.63))|2〉|2〉|0〉

3.Sending the photons back into the nonlinear crystal oflength L2 = t2 · c, with t2 = π/(ω1A) − t0 the state evolveaccording to

|Ψ2〉 =

√2

3(1 + cos(Aω1(t+ t0))|0〉|0〉|2〉 (17)

−√

2√3

sin(Aω1(t+ t0))|1〉|1〉|1〉

−2

3(

1

m− cos(Aω1(t+ t0)))|2〉|2〉|0〉

which is effectively equivalent to swapping the amplitudebetween the all-pump Fock state and all-down-converted pho-tons Fock state, seen in Eq. (16). As a result f22 (t) reachesunity at time t2 = π/(ω1A)− t0 denoted as red dot in Fig. 5.This corresponds to a 100% conversion efficiency comparedto the original 88% efficiency.

Figure 5: The figure shows the Grover search process fortwo-pump-photon subspace. We represent the amplitude ofunconverted pump state |0〉|0〉|2〉, the fully converted signalidler pairs |2〉|2〉|0〉, and partially converted state |1〉|1〉|1〉as x, z, y. The wave function for the two-pump-photon evo-lution over time period of t ∈ [0, 2π] with different initialvalue m ∈ [−1, 1] are denoted by the Green surface. fordifferent input state. The green region covers solution givenm ∈ [−2000, 2000].

3.3 four-photonThe four-pump-photon solution is much more complicated

due to its lack of closed-orbit evolution of wave function.However, closed orbit is not necessary for Grover search it-eration: the unitary evolution corresponds to each iterationdoes not need to be the same. Following similar three stepsin two-photon Grover search model, we can find the markeditem |4〉|4〉|0〉 with 100% probability within t < 4π of oneiteration compared with the original efficiency of 74%.

Graphical illustration for the Grover search iteration can beseen in Fig.6. z axis deontes the amplitude f44 (t) of the fullyconverted state |4〉|4〉|0〉, y axis denotes the sum of two otherbasis amplitude y =

√(f43 (t))2 + (f42 (t))2, and x axis de-

notes the amplitude f40 (t) of pure pump state shown in Fig. 8,same as that defined in Eq. (13)(12).

The green curve in Fig.6 denotes step I of the Groversearch, where input pure four-pump-photon Fock state evolvesfor time a t0 = 2π. The inversion of step II for the search itemcorresponds to the blue curve to the upper left quadrant. TheGrover step III corresponds to a unitary evolution along thenew orbit represented as the red curve that approach unity zusing t ∼ 2π. Notice that in the last step the state evolveson an orbit of eiπ phase different from the initial wave vec-tor in all components, which is still a solution of the sameSchrodinger equations.

-1.0 -0.5 0.5 1.0y

-1.0

-0.5

0.5

1.0

z

Figure 6: The figure shows the Grover search process for four-pump-photon subspace. The z and y axis are defined in Eq.(12)(13). The wave function for the two-pump-photon evolu-tion over time period of t ∈ [0, 2π] with different initial valuem ∈ [−1, 1] are denoted by the Green surface. The green linerepresents the step I wave function evolution, the blue linerepresnts the inversion in step II and red line represents thelast step of Grover search.

A fabrication of interlacing linear and nonlinear opticalcrystal will suffice to implement the Grover search we pro-posed here, see Fig. 7 for more detailed visualization.

III

IIIII

III

Figure 7: The figure shows the proposed fabrication interlac-ing the linear optical components(purple region) in betweennonlinear(light blue region) optical crystals to realize Groversearch iteration of step I- III as denoted in parallel with thecrystal.

One question is the robustness of such search algorithm.We need the fabrication between linear and nonlinear crystals,and both the crystal length and the boundary positions of thetwo suffer from fabrication inaccuracy when manifests in theinaccurate evolution phases. We analyze this by blurring thetiming of each step of the Grover search.

Figure 8: The figure shows the Grover search process for four-pump-photon subspace. The x, y, z axis are defined in Eq.(12)(13). The wave function for the two-pump-photon evo-lution over time period of t ∈ [0, 200π] with different initialvalue m ∈ [−800, 800]. Pureple area denotes the original in-put state, blue area denotes the state after step II of Groversearch, and red area denotes the wave functions for the laststep III.

As can be seen in Fig. 8, the measure of the intersectionbetween blue and red area is significant, implying a high toler-ance towards the actual timing of the initial state preparation.More rigorously, we prove the effectiveness of our Groversearch protocol in section 3.4 without requiring the fine-tunedexperimental parameters.

3.4 multi-photonThe secret behind the successful implementation of Grover

search in two and four pump photon subspace of SPDC quan-tum process lies in the parity of the Schrodinger Eq. 5: onlyin even number pump photon subspace, the completely con-verted state amplitude fnn (t) and the input all-pump-photonamplitude fn0 (t) share the time reversal symmetry and possessa constant free parameter m determined by the initial condi-tions. Since the input state is always all-pump-photon statefnk (0) = 0, k 6= 0, the only way to find the marked state|n〉|n〉|0〉 with close to unity probability is to be able to swapthe amplitude between fn0 (t) and fnn (t). This swapping, how-ever, is carried out in a continuous manner explained in thetwo steps of the Grover iteration. The step II change the ini-tial state to a slightly different orbit of a different m, step IIIevolves the photon state under the new orbit which has largermaximum amplitude for the fully converted state fnn (t). If itis not unity, one can perform step I again to rotate the wavevector to another orbit that has an even larger intersection withthe marked vector |n〉|n〉|0〉.

When the pump photon number n is relatively big, the ex-act Schrodinger Eq. 5 will have around n − 2 many differentfrequency components, and our geometrical insight as wellas the symmetry argument no longer help. However, we canreduce the problem to three dimensional SO(3) rotation byconsidering only the pure pump amplitude fn0 (t), the com-pletely converted state amplitude fn0n(t), and the square rootof all the other amplitude square sum, since the amplitudes ofdifferent Fock states are real in solution of Eq. (5). Denote

superposition of all state orthogonal to the marked state andto the input pump as y axis ~y =

∑n−1i=1

1√n−2 |i〉|i〉|n− i〉 , x

axis unit vector defined by the Fock state ~x = |0〉|0〉|n〉 and~z = |n〉|n〉|0〉 as z axis unit vector, same as that described inEq. (13)(12).

Now, we proceed to prove that iterations of step I-III given|0〉 as input state can bring the state closer and closer to themarked state |1〉. In this SO(3) representation, step II cor-responds to a rotation around the x axis for π Step I andIII can be represented as the SO(3) rotation around the axis~n = {sin(φ) cos(α), sin(φ) sin(α), cos(φ)} for a small angleθg denoted as R(~n, θg). For large-photon-number case, eachGrover search takes only small step in angular rotation, mean-ing θg << 1. At this limit, we approximate the time for ini-tial state preparation t0 and that for the second part of Groveriteration t2 to be the same, which contributes an angular rota-tion θg.Using the isomorphism between the universal cover ofSO(3) and SU(2), the rotation R(~n, θg) can be representedwith unitary U(θ, ~n) = e−iθ/2~n·~σ , where ~σ = {σx, σy, σz}are three Pauli operators eigenstate and the market state cor-responds to the negative eigenstate of σz . The initial statepreparation gives

|Ψ0〉 = R(θg, ~n(α))|x〉 (18)

= (cos(θg2

)− i sin(θg2

) cos(α))|0〉 − i sin(θ

2) sin(α)|1〉

u (1− θg2

cos(α))|0〉 − iθg2

sin(α)|1〉

Apply the step II of the Grover search, we obtain

|Ψ1〉 = σzU |0〉 (19)

u (1− θg2

cos(α))|0〉+ iθg2

sin(α)|1〉.

Apply the second part of the Grover search the state be-comes:

|Ψ2〉 = UσzU |0〉 (20)

u (1− iθg cos(α)− (θg2

)2 cos(2α))|0〉 − (θg2

)2 sin(2α)|1〉.

After N repetitions the amplitude of the marked state|1〉 is fnn = N(

θg2 )2 sin(2α), the probability of finding

the completely converted state approach unity when N ∼1

(θg2 )2 sin(2α)

. However,the value of rotation α is yet to be de-

termined for different input states. One convenient estimationgives a lower bound on α, which corresponds to equally ro-tating all n states α = 1√

n[24]. This proves our conjectured

polynomial improvement in search iterations of N ∼√n.

4 SummaryWe study the quantum theory of spontaneous parametric

down-conversion in nonlinear optical crystals, such as PP-KTP and BBO. Using these results, we propose a new Groversearch realization in nonlinear crystal that can improve theconversion efficiency at even pump photon subspace to unity.

We further generalize the protocol to multi-photon level. Forthe first time, we are able to conduct Grover search using lin-ear and nonlinear optical materials achieving scalability. Wepresent the simple fabrication design realizable in the near fu-ture. A modification of the current model into two dimen-sional wave guide might be used to implement Grover searchin spatial mode through random walk process[27][28].

5 AcknowledgementWe thank Franco N. C. Wong, Theodore Yoder, Guanghao.

Low and Elton Zhu for helpful discussions.

References[1] M. Steffen, et al., Experimental Implementation of an

Adiabatic Quantum Optimization Algorithm Phys. Rev.Lett. 90, 067903 2003

[2] Thomas Monz1,et al., Realization of a scalable Shor al-gorithm, Science 04 Mar 2016: Vol. 351, Issue 6277, pp.1068-107

[3] E Knill, R Laflamme, GJ Milburn,A scheme for efficientquantum computation with linear optics - Nature 409,46-52 (4 January 2001) -

[4] Isaac L. Chuang, Neil Gershenfeld, and Mark Kubinec,Experimental Implementation of Fast Quantum Search-ing, Phys. Rev. Lett. 80, 3408 Published 13 April 1998

[5] P. G. Kwiat et al., J. Mod. Opt. 47, 257, 2000

[6] P. Walther et al., Experimental one-way quantum com-puting, Nature, 434, 169, (2005).

[7] K.-A. Brickman, et al., Implementation of Grovers quan-tum search algorithm in a scalable system Phys. Rev. A72, 050306(R) 2005

[8] J. Ahn, et al., Information Storage and RetrievalThrough Quantum Phase, Science 21 Jan 2000: 287,5452, 463-465

[9] Kai Chen, Che-Ming Li, Qiang Zhang, Yu-Ao Chen,Alexander Goebel, Shuai Chen, Alois Mair, and Jian-Wei Pan, Experimental Realization of One-Way Quan-tum Computing with Two-Photon Four-Qubit ClusterStates, Phys. Rev. Lett. 99, 120503 Published 17September 2007

[10] Hussain A. Zaidi, Chris Dawson, Peter van Loock, andTerry Rudolph, Near-deterministic creation of univer-sal cluster states with probabilistic Bell measurementsand three-qubit resource states, Phys. Rev. A 91, 042301Published 1 April 2015

[11] Pieter Kok, et al., Linear optical quantum computingwith photonic qubits Rev. Mod. Phys. 79, 135 Published24 January 2007

[12] Jeremy L. O’Brien, Optical Quantum Computing, Sci-ence 07 Dec 2007: Vol. 318, Issue 5856, pp. 1567-1570

[13] A. P. Lund, T. C. Ralph, and H. L. Haselgrove,Fault-Tolerant Linear Optical Quantum Computing withSmall-Amplitude Coherent States, Phys. Rev. Lett. 100,030503 2008

[14] Peter C. Humphreys, et al., Linear Optical QuantumComputing in a Single Spatial Mode, Phys. Rev. Lett.111, 150501 Published 9 October 2013

[15] P. Ben Dixon, Jeffrey H. Shapiro, and Franco N.C. Wong, Spectral engineering by Gaussian phase-matching for quantum photonics, (2013)

[16] H Hubel, et al., Direct generation of photon triplets us-ing cascaded photon-pair sources, Nature 466, 601603( 2010)

[17] Timothy E. Keller and Morton H. Rubin, Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse, Phys. Rev.A 56, 1534 Published 1 August 1997

[18] Podoshvedov, S.A., Noh, J., Kim, K., A full quantumtheory of parametric down conversion and its applica-tion to coincidence measurements, (2005) Journal of theKorean Physical Society, 47 (2), pp. 213-222.

[19] D. A. Antonosyan, A. S. Solntsev, A. A. Sukho-rukov, Single-photon spontaneous parametric down-conversion in quadratic nonlinear waveguide arrays,Optics Communications, Volume 327, 15 September2014, Pages 22-26

[20] S. Sensarn, G. Y. Yin, and S. E. Harris, Generationand Compression of Chirped Biphotons, Phys. Rev. Lett.104, 253602 2010

[21] N. K. Langford, et al., Efficient quantum computing us-ing coherent photon conversion, Nature 478, 360363(2011)

[22] T. Guerreiro, et al., Interaction of independent singlephotons based on integrated nonlinear optics, NatureCommunications 4, 2324 (2013)

[23] Lov K. Grover, A fast quantum mechanical algorithmfor database search, Proceedings of the twenty-eighthannual ACM symposium on Theory of computing, 212-219

[24] M. A. Nielsen, I. L. Chuang, Quantum computationand quantum information, Cambridge University Press.- 2010

[25] Lov K. Grover, Quantum Computers Can SearchRapidly by Using Almost Any Transformation, Phys.Rev. Lett. 80, 4329 1998

[26] E. Biham, et al., Grovers quantum search algorithm foran arbitrary initial amplitude distribution, Phys. Rev. A60, 2742 1999

[27] S. Aaronson and A. Ambainis, Quantum search of spa-tial regions, Foundations of Computer Science, 2003.Proceedings. 44th Annual IEEE Symposium on, 2003,pp. 200-209.

[28] Andrew M. Childs and Jeffrey Goldstone, Spatial searchby quantum walk, Phys. Rev. A 70, 022314 Published 23August 2004

[29] S. Lloyd, S. Garnerone, P. Zanardi, Quantum algorithmsfor topological and geometric analysis of data, NatureCommunications 7, 10138, (2016)

[30] A. Dosseva, L. Cincio, A. M. Branczyk, Shaping thespectrum of downconverted photons through optimizedcustom poling, arXiv:1410.7714, (2014 )

[31] M. Fejer, G. Magel, D. Jundt, and R. Byer, Quasi-phase-matched second harmonic generation: tuning and toler-

ances, IEEE J. Quant.Electron. 28, 26312654 (1992).

[32] R. S. Bennink, Optimal collinear Gaussian beams forspontaneous parametric down-conversion, Phys. Rev. A81, 053805 (2010) Engineered optical nonlinearity forquantum light sources

[33] A. M. Branczyk, A. Fedrizzi, T. M. Stace, T. C. Ralph-Optics Express, Engineered optical nonlinearity forquantum light sources, 19, 1, pp. 55-65 (2011)