Wave-function engineering via conditional quantum teleportation with non-Gaussian entanglement resource
Warit Asavanant, Kan Takase, Kosuke Fukui, Mamoru Endo, Jun-ichi Yoshikawa, Akira Furusawa
WWave-function engineering via conditional quantum teleportation with non-Gaussianentanglement resource
Warit Asavanant, ∗ Kan Takase, Kosuke Fukui, Mamoru Endo, Jun-ichi Yoshikawa, and Akira Furusawa † Department of Applied Physics, School of Engineering,The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan (Dated: February 5, 2021)We propose and analyze a setup to tailor the wave functions of the quantum states. Our setupis based on the quantum teleportation circuit, but instead of the usual two-mode squeezed state,two-mode non-Gaussian entangled state is used. Using this setup, we can generate various classesof quantum states such as Schr¨odinger cat states, four-component cat states, superpositions of Fockstates, and cubic phase states. These results demonstrate the versatility of our system as a stategenerator and suggest that conditioning using homodyne measurements is an important tool in thegenerations of the non-Gaussian states in complementary to the photon number detection.
I. INTRODUCTION
Quantum information processing is widely researchedwith expectations of broad applications [1]. Amongmany candidates, continuous-variable (CV) optical sys-tems are currently one of the most promising platformsin terms of scalability. In CV optical systems, re-sources for measurement-based quantum computation(MBQC) [2, 3]—the cluster states—have been gener-ated in a scalable fashion using time-domain multiplexingmethod [4–7] and frequency-domain multiplexing method[8], and basic operations on time-domain-multiplexedcluster states have been recently demonstrated [9, 10].To achieve universal MBQC, however, non-Gaussian ele-ments have to be added to the cluster states [11, 12].A direct way to add the non-Gaussian elements isadding non-Guassian measurements to the cluster state.This, however, is experimentally difficult, especially inoptical systems, where direct non-Gaussian measure-ments are usually either probabilistic or require the non-available strong nonlinearity. A more viable option is touse ancillary non-Gaussian states (such as cubic phasestates) and realize non-Gaussian operations via gate tele-portation protocol [13, 14]. In addition, non-Gaussianstates also have many other applications. For example,a superposition of coherent states—commonly known asa Schr¨odinger cat state—have applications in quantumcomputation [15–18], quantum communication [19, 20],and quantum error correction [21–23]. Another non-Gaussian state called Gottesman-Kitaev-Preskill (GKP)qubit is currently the most promising logical qubit forfault-tolerant CV quantum computation [14, 24–29].Despite their varieties, optical generations of non-Gaussian states are mostly based on the same idea: thequantum state is first represented in the photon num-ber basis (Fock basis), i.e. | ψ (cid:105) = (cid:80) ∞ n =0 c n | n (cid:105) , and thetarget state is obtained by truncating the superpositionbelow a certain maximum photon number, | ψ target (cid:105) ≈ ∗ [email protected] † [email protected] (cid:80) n max n =0 c n | n (cid:105) . Then, we can find a setup that consistsof squeezed light sources, linear optics, and photon num-ber detectors that herald | ψ target (cid:105) when particular resultsare detected by the photon number detectors. Based onthis idea, generations of various non-Gaussian states havebeen explored [18, 22, 29–41]. In this approach, however,the generation system is highly dependent on the tar-get states and the generation systems tend to becomemore complex when the maximum photon number in-creases. Also, although these generations are formulatedbased on Fock basis, for CV quantum states, phase spacerepresentation in quadrature basis is also a natural rep-resentation. As the examples of previous researches inthis direction, generation of non-Gaussian state by theimplementation of the quadrature operator [42], gener-ation and amplification of Schr¨odinger cat states usingconditional homodyne measurement [43, 44], and high-rate Schr¨odinger cat state generation formulated usingthe wave function picture [45] have been studied. Evenso, the non-Gaussian state generation using the quadra-ture basis has remained largely unexplored.In this paper, we present a methodology to tailorthe wave functions of the quantum states using quan-tum teleportation circuit and non-Gaussian entangle-ment resource. The idea of our protocol is based ongate teleportation protocol [13, 14], where we have de-signed the non-Gaussian two-mode resource states to beequivalent to EPR state—a resource state for CV quan-tum teleportation—with quadrature operators (such asˆ x ) acting on one of its mode in the ideal limit. Asthe quadrature operators and their polynomials are non-unitary, we need to use the conditional quantum telepor-tation instead of the conventional unconditional quantumteleportation in the state preparation using our proto-col. In this case, the measurement results of the Bellmeasurements herald the generated states. In addition,this teleportation-based architecture possesses high affin-ity with the time-domain-multiplexing method [46], mak-ing it possible to realize our scheme in a scalable fashion.Our results demonstrate a programmable and scalablenon-Gaussian quantum state generator that utilizes thequadrature basis. a r X i v : . [ qu a n t - ph ] F e b The paper is structured as follows. In Sec. II, we definethe basic notations. In Sec. III, we analyze our method-ology. In Sec. IV, we show the actual generations of var-ious non-Gaussian states: Schr¨odinger cat states, four-component cat states, superpositions of Fock states, andcubic phase states. We discuss success rate and realisticimplementation of our architecture in Sec. V. Finally, weconclude our paper in Sec. VI.
II. NOTATIONS
In CV systems, the quadrature operators ˆ x and ˆ p sat-isfy commutation relation: [ˆ x, ˆ p ] = i , where we use (cid:126) = 1.The quadrature operators are related to the annihilationand creation operators viaˆ x = 1 √ a + ˆ a † ) , (1)ˆ p = 1 i √ a − ˆ a † ) , (2)with [ˆ a, ˆ a † ] = 1, and the effect of the annihilation andthe creation operators on the Fock basis | n (cid:105) areˆ a | n (cid:105) = √ n | n − (cid:105) , (3)ˆ a † | n (cid:105) = √ n + 1 | n + 1 (cid:105) . (4)For a quantum states | ψ (cid:105) , the wave function in x and p are given by ψ ( x ) = (cid:104) x | ψ (cid:105) , (5)˜ ψ ( p ) = (cid:104) p | ψ (cid:105) , (6)respectively. When quadrature operators act on thequantum states, they transform the wave functions intoa new (unnormalized) wave function given by (cid:104) x | ˆ x | ψ (cid:105) = xψ ( x ) , (7) (cid:104) x | ˆ p | ψ (cid:105) = − i dd x ψ ( x ) , (8) (cid:104) p | ˆ x | ψ (cid:105) = i dd p ˜ ψ ( p ) , (9) (cid:104) p | ˆ p | ψ (cid:105) = p ˜ ψ ( p ) . (10)In the quantum teleportation circuit, the ideal two-mode resources are called EPR states and their unnor-malized form are given by | EPR (cid:105) = (cid:90) d x | x (cid:105) | x (cid:105) = ∞ (cid:88) n =0 | n (cid:105) | n (cid:105) . (11)Note that unless stated otherwise, we assume that theranges of the integrations are always from −∞ to ∞ . Inthe physical setting, two-mode squeezed state (TMSS) isused instead of the EPR state and it can be represented as | TMSS (cid:105) = (cid:90) d x d x Ψ TMSS ( x , x ) | x (cid:105) | x (cid:105) = (cid:112) − η ∞ (cid:88) n =0 η n | n (cid:105) | n (cid:105) , (12)withΨ TMSS ( x , x ) = exp (cid:34) − e r (cid:18) x − x √ (cid:19) (cid:35) × exp (cid:34) − e − r (cid:18) x + x √ (cid:19) (cid:35) (13) η = tanh r, (14)where r is the squeezing parameter of the initial squeezedvacua used in the generation of TMSS and is assumed tobe initially equal for both modes. We can see that in thelimit of the infinite squeezing, the first term approaches δ ( x − x ), while the second term is a Gaussian envelopethat becomes broader as r increases.TMSS can be generated by mixing two orthogonalsqueezed lights on a beam splitter: | TMSS (cid:105) = ˆ B ˆ S ( − r ) ˆ S ( r ) | (cid:105) | (cid:105) . (15) S ( r ) := exp (cid:2) r (ˆ a − ˆ a † ) (cid:3) is a squeezing operator where r > r <
0) corresponds to squeezing in ˆ x (ˆ p ) quadra-ture. ˆ B is an operator of 50:50 beamsplitter interactionwhich transforms the annihilation operator of the twomodes as ˆ B † (cid:18) ˆ a ˆ a (cid:19) ˆ B = 1 √ (cid:18) −
11 1 (cid:19) (cid:18) ˆ a ˆ a (cid:19) . (16) III. PROPOSED SETUP
In this section, we analyze the proposed setup. Thegeneralized form of our setup is shown in Fig. 1. Thissetup consists of photon-subtracted two-mode entangle-ment and conditioning quantum teleportation with thatresource. Let us look at each component.
A. Photon-subtracted TMSS
We consider what happens when photon subtractionsare combined with TMSS. The main reason we considerthis type of state is that the quadrature operators can bewritten as superpositions between the annihilation andcreation operators. Therefore, as we will shortly show,by implementing the photon subtraction before the 50:50beamsplitter, we can easily implement the superpositionof the annihilation and the creation operator, i.e. quadra-ture operators, on one of the modes of the TMSS. First, C ond i t i on i ng Non-Gaussian entangled state DisplacementBell measurement
FIG. 1. Schematic diagram of the proposed setup. Our setupis a quantum teleportation circuit where the usual two-modesqueezed state (TMSS) are replaced with the states whichwe will call non-Gaussian entangled state (NGES). One ofthe possible generation methods used in the analysis in thispaper is that by photon subtractions on the squeezed statesused in the generation of the TMSS. The success of the proto-col is conditioned by certain measurement results at the Bellmeasurements. let us consider subtraction of k photons from one of themode of the TMSS. This state can be written as(ˆ a ) k | TMSS (cid:105) = ∞ (cid:88) n = k η n (cid:115) n !( n − k )! | n − k (cid:105) | n (cid:105) . (17)Note that we will omit the normalization factor for thesimplicity. The above equation can be equivalently writ-ten as(ˆ a ) k | TMSS (cid:105) = (ˆ a † ) k ∞ (cid:88) n = k η n | n − k (cid:105) | n − k (cid:105) = ( η ˆ a † ) k | TMSS (cid:105) . (18)This means that photon subtraction in a single mode canbe considered as photon addition in another mode withadditional factor η .Now let us return to the non-Gaussian entanglement inFig. 1. The non-Gaussian entangled state | NGES( k, l ) (cid:105) is given by | NGES( k, l ) (cid:105) = ˆ B (ˆ a ) k (ˆ a ) l ˆ S ( − r ) ˆ S ( r ) | (cid:105) | (cid:105) . (19)Using the beamsplitter transformation, this state can betransformed into | NGES( k, l ) (cid:105) = (cid:18) ˆ a + ˆ a √ (cid:19) k (cid:18) ˆ a − ˆ a √ (cid:19) l | TMSS (cid:105) . (20)Using Eq. (18), we get | NGES( k, l ) (cid:105) = ˆ f k,l ( η ) | TMSS (cid:105) , (21)where ˆ f k,l ( η ) ≡ √ k + l k + l (cid:88) j =0 c k,lj ˆ a k + l − j ( η ˆ a † ) j (22) and c k,lj is a coefficient of the polynomial( a + b ) k ( a − b ) l = k + l (cid:88) j =0 c k,lj a j b k + l − j . (23)Therefore, by implementing photon subtractions onthe initial squeezed states, we can implement a poly-nomial of annihilation and creation operators on one ofthe modes of the TMSS. This is because the subtractionbefore the beamsplitter is equivalent to the superposi-tion between the photon subtraction on each mode of theTMSS, which is also equivalent superposition of the pho-ton subtraction and addition on one of the modes. In thatsense, the physical intuition of our method is that quan-tum teleportation using this non-Gaussian entanglementis roughly the same as the implementation of the polyno-mial f kl (ˆ a , η ˆ a † ) on the input state. The implementationof the coherent superposition of the annihilation and thecreation operators on a quantum state has been studiedin a different context [47]. There are also effects from themeasurement results of the bell measurements which willbe discussed in Sec. III B.Although Eq. (21) and Eq. (22) give complete charac-terization of NGES, it will be more convenient to defineˆ f k,l ( η ) via a recursive formula. From Eq. (21), we canwrite down the following recursive formulaˆ f k,l = 1 √ a ˆ f k − ,l + 1 √ η ˆ f k − ,l ˆ a † , (24)ˆ f k,l = − √ a ˆ f k,l − + 1 √ η ˆ f k,l − ˆ a † , (25)with ˆ f , = ˆ I and we dropped the index “2” and thedependence on η of ˆ f k,l ( η ). This can be further writtenusing quadrature operators asˆ f k,l = 12 (cid:16) ˆ x ˆ f k − ,l + η ˆ f k − ,l ˆ x (cid:17) + i (cid:16) ˆ p ˆ f k − ,l − η ˆ f k − ,l ˆ p (cid:17) , (26)ˆ f k,l = − (cid:16) ˆ x ˆ f k,l − − η ˆ f k,l − ˆ x (cid:17) − i (cid:16) ˆ p ˆ f k,l − + η ˆ f k,l − ˆ p (cid:17) . (27)In the limit of η →
1, we can show that ˆ f k, is the poly-nomial of solely ˆ x and ˆ f ,l is the polynomial of solely ˆ p .This can be done as follows. Let us put ˆ g k,l ≡ lim η → ˆ f k,l .Then, ˆ g k,l = 12 { ˆ x, ˆ g k − ,l } + i p, ˆ g k − ,l ] , (28)ˆ g k,l = −
12 [ˆ x, ˆ g k,l − ] − i { ˆ p, ˆ g k,l − } . (29)As we can easily show that ˆ g , = ˆ x and ˆ g , = − i ˆ p , usingthe recursive formula of the Hermite polynomial, we canwrite down ˆ g k, = 1(2 i ) k H k ( i ˆ x ) , (30)ˆ g ,l = 1( − l H l ( i ˆ p ) , (31)where H n ( · ) is the Hermite polynomials of the order n .The reason that both ˆ g k, and ˆ g ,l are polynomial in ˆ x or ˆ p with real coefficients is because we are utilizing thephase information and the splitting ratio of the beam-splitter, whereas general arbitrary superposition of theannihilation and creation operators do not necessarily re-sult in the real-coefficient polynomial in the quadratureoperators.As an example of ˆ f k,l , we list a few of them here. Notethat ˆ f k,l are always at most the polynomial of the order k + l in ˆ x and ˆ p .ˆ f , = 1 + η x + i − η p (32)ˆ f , = − − η x − i η p (33)ˆ f , = − − η x − ˆ p ) − i η x ˆ p + ˆ p ˆ x ) (34)ˆ f , = (cid:20) η (cid:21) ˆ x + (cid:20) i − η (cid:21) ˆ p + 14 ( η − η ) (35) B. Output states
Now that we have established that our two-mode re-source is equivalent to coherent superposition of photonsubtraction and photon addition acting on one of themodes of the TMSS, we look at the conditional quantumteleportation part. When we implement Bell measure-ment, we are projecting the mode “in” and “1” onto thedisplaced EPR states via the projection operator:ˆΠ( m x , m p ) =ˆ D x, in ( m x ) ˆ D p, ( m p ) | EPR (cid:105) in , (cid:104) EPR | ˆ D † x, in ( m x ) ˆ D † p, ( m p ) , (36)where the displacement operators ˆ D x ( m x ) and ˆ D p ( m p )transform quadrature operators as ˆ D † x ( m x )ˆ x ˆ D x ( m x ) =ˆ x + m x and ˆ D † p ( m p )ˆ p ˆ D p ( m p ) = ˆ p + m p . Note that m x and m p are not directly the measurement results of thetwo homodyne measurements but are related to that bya factor of √
2. We, however, use the projector operatorin Eq. (36) so that we do not have the factor √ ψ in ( x ) as a wave function of the in-put mode, the output after the conditional teleportationbefore the displacement operations can be written downas | ψ (cid:48) (cid:105) = ˆ D † p ( m p ) ˆ D † x ( m x ) (cid:90) (cid:90) d x d x (cid:48) e im p ( x − x (cid:48) ) ψ in ( x )Ψ TMSS ( x − m x , x (cid:48) − m x ) | x (cid:48) (cid:105) . (37)In the normal context, by displacing, we can recover theinput state and remove the dependence on the measure-ment results when Ψ TMSS ( x, x (cid:48) ) → δ ( x − x (cid:48) ).As we have previously stated, since the | NGES( k, l ) (cid:105) is equivalent to | TMSS (cid:105) acted on with ˆ f k,l , we can writedown the (unnormalized) state when | NGES( k, l ) (cid:105) is usedas | ψ (cid:48) (cid:105) = ˆ f k,l ( η ) ˆ D † p ( m p ) ˆ D † x ( m x ) (cid:90) (cid:90) d x d x (cid:48) e im p ( x − x (cid:48) ) ψ in ( x )Ψ TMSS ( x − m x , x (cid:48) − m x ) | x (cid:48) (cid:105) . (38)Therefore, the wave function ψ out ( x (cid:48) , m x , m p ) after dis-placement operations is ψ out ( x (cid:48) , m x , m p ) =ˆ h k,l ( η, m x , m p ) ψ cond ( x (cid:48) , m x , m p ) (39)where each part is defined as follows.ˆ h k,l ( η, m x , m p ) ≡ ˆ D p ( m p ) ˆ D x ( m x ) ˆ f k,l ( η ) ˆ D † p ( m p ) ˆ D † x ( m x ) (40) ψ cond ( x (cid:48) , m x , m p ) ≡ (cid:90) d x e im p ( x − x (cid:48) ) ψ in ( x )Ψ TMSS ( x − m x , x (cid:48) − m x ) (41)Let us consider the physical intuition of each partof ψ out ( x (cid:48) , m x , m p ). The conditional wave function ψ cond ( x (cid:48) , m x , m p ) consists of three parts: modulation dueto m p , input wave function ψ in ( x ), and convolution dueto the TMSS. We observe that even if we displace both x and x (cid:48) of the Ψ TMSS ( x, x (cid:48) ), the argument of the firstexponential term in Eq. (13) remains the same. Thus,the integral is evaluated around x ≈ x (cid:48) . This makes thedemodulation term with m p negligible as long as m p isnot too big and the squeezing level is sufficient. On theother hand, the argument of the second exponential termin Eq. (13) is now centered around m x . As such the effectdue to the finite squeezing of the TMSS is the Gaussianconvolution and a Gaussian envelope.Next, regarding the term ˆ h k,l ( η, m x , m p ), let us lookat a special case where k = 1, l = 0, and η →
1. Insuch case, this term becomes ˆ h k,l (1 , m x , m p ) = ˆ x − m x .Therefore, in the infinite squeezing limit, the wave func-tion of the output state is equal to the wave function ofthe input state multiplied by ( x − m x ).Using this setup, it is possible to tailor the wave func-tion of the input states. For example, if we consider | p = 0 (cid:105) as our input state and iteratively use this circuitfor the case where k = 1 and l = 0, then after n itera-tions, the wave function becomes ψ ( x ) = (cid:81) ni =1 ( x − m i ),where m i are the measurement results of the homodynedetectors. As we can see, ψ ( x ) is a n -th order poly-nomial with real roots. We could also have imaginaryroots by using, for example, ˆ f , ( η ). In general, we canuse this setup and realize arbitrary wave function withreal roots. Although we believe that there should exista modification of our setup to include arbitrary complexroots and complex coefficients, we leave the explorationsof such possibility to the future work. Note that in a re-alistic setup, we would have to approximate | p = 0 (cid:105) with p -squeezed states, which means that the wave functionwill be attenuated at large x . Moreover, the additionalGaussian envelope due to the finite squeezing and finite-width of the conditioning window m x and m p must betaken into an account.In the next section, we will show the simulation resultsof various quantum states that can be generated withour system. We will first assume that the conditioning isapplied with zero width. Afterward, the discussions onthe actual conditioning window and success probabilitywill be given. IV. SIMULATIONS OF THE GENERATEDSTATES
In most of the simulations, we will be modest and as-sume the squeezing parameter to be | r | ≤ .
0, whichcorresponds to about − . m x = ( m x, , m x, , . . . , m x,n ) and m p = ( m p, , m p, , . . . , m p,n ) as vectors showing the val-ues we condition the homodyne measurements in eachiteration with. A. Schr¨odinger cat states
Here, we will show that our system can be used togenerate cat states. A Schr¨odinger cat state | CAT , α, ±(cid:105) is given by | CAT , α, ±(cid:105) = N α, ± ( | α (cid:105) ± |− α (cid:105) ) , (42)where N α, ± is a normalization factor and we will assume α ∈ R for the simplicity. We call | CAT , α, + (cid:105) a pluscat state and | CAT , α, −(cid:105) a minus cat state. In general,the generated states in this section will be close to thesqueezed cat state ˆ S ( ξ ) | CAT , α, ±(cid:105) and we will considerthe fidelity of the generated states to the squeezed catstate with parameter ( ξ, α ).Let us first consider the infinite squeezing limit. If westart from a p -squeezed as our input and implement ˆ x n on it by repeating the circuit in Fig. 1 for n times, the(unnormalized) wave function in both x and p becomes ψ n ( x ) = x n exp (cid:18) − x e r (cid:19) , (43)˜ ψ n ( p ) = H n (cid:18) p √ e − r (cid:19) exp (cid:18) − p e − r (cid:19) . (44)This type of wave function is similar to those in Ref.[43, 45] and is known to be a good approximation to thecat state. When n is odd (even), the generated stateapproximates minus (plus) cat state. The wave function ψ n ( x ) has two extrema located at x ext = ±√ ne r (45)This means that the amplitude of the cat state willroughly scale with square root of the number of the it-erations and the initial scale of the cat state will be de-termined by the squeezing of the input. In the p quadra-ture, there is an oscillatory structure due to the Hermitepolynomial. Note that in addition to ˆ g , , we can alsoconsider using the NGES with more number of photonsubtracted. In such a case, since ˆ g k, ∝ H k ( i ˆ x ), we ex-pect that the increase in the amplitude will be amplifiedby roughly √ k , since the leading term when x becomeslarge is ˆ x k . Moreover, if we plot H k, ( ix ), we can see thatits values around x = 0 are very small and the functionrapidly increases at the large x . Such behavior is advan-tageous for cat state generations as we want the ψ ( x ) tohave two peaks that are far apart from each other.In some protocols such as generation of GKP qubitsusing cat states, it is more advantageous to use squeezedcat states, rather than normal cat states with large ampli-tude [23]. In the usual photon subtraction, a conventionalmethod to approximate cat state, since the annihilationoperators ˆ a are applied to the state, there can be no phaseinformation and the cat states are always generated withthe amplitude pointing out in the antisqueezing direc-tion. In our method, since we are effectively applyingquadrature operator ˆ x , it is possible to apply ˆ x to a x -squeezed states so that the amplitudes is in the squeezing xp xp xp xpxp xp xp xp -8 -4 0 4 8-101 -8 -4 0 4 8-101 -8 -4 0 4 8-101 -8 -4 0 4 8-101 (A)(B) -8 -4 0 4 8-101 -8 -4 0 4 8-101 -8 -4 0 4 8-101 -8 -4 0 4 8-101 FIG. 2. Simulation results of the wave functions and the Wigner functions of the generation of the Schr¨odinger cat states. Weuse k = 1 and l = 0 in this simulation. The squeezing parameters for the NGES are r tele = 1 . r of the initial squeezed states for (A) and (B) are − . .
0, respectively. Wave functions of the generated states (solidlines) and of the closest squeezed cat states ˆ S ( ξ ) | CAT , α, ±(cid:105) (dashed lines) are shown. The parameters α and ξ are written inthe lower-right corner of each subfigure. The numbers of the iterations are n = 1 , , , direction. The resulting wave functions after n iterationsin the infinite squeezing limit are ψ n ( x ) = x n exp (cid:18) − x e − r (cid:19) , (46)˜ ψ n ( p ) = H n (cid:18) p √ e r (cid:19) exp (cid:18) − p e r (cid:19) , (47)which are simply the squeezed version of Eqs. (43) and(44).Figure 2 shows the simulation results. We observe thatthe number of the interference fringes, which are charac-teristics of cat states, increases with each iteration. Con-trary to our expectation, however, even for the case wherewe used p -squeezed states as our inputs, the resultingstates are weakly squeezed in the x -direction. This isthe effects from both the finite squeezing of the initialsqueezed states and the finite squeezing in NGES. Eventhen, we observe that for the case where the initial stateis squeezed in ˆ x , the generated states is also squeezed in ˆ x without having to additionally squeeze the state. How-ever, the amplitude α tends to be larger for the caseswhere p -squeezed states are used. Even so, the fidelity to the squeezed cat states are over 0 .
995 for every subfigure.The procedure here can be repeated to achieve cat stateswith large amplitudes. Although we restrict ourselves tothe case where k = 1 and l = 0 in Fig. 2, by increas-ing k , we could reach the large-amplitude cat states withmuch fewer iterations. Figure 3 shows an example of this.Thus, depending on the number of the photon that wecan resolve, the number of the iterations can be adjustedproperly. B. Four-component cat states
In addition to the usual cat state, there is also animportant class of state called four-component cat state[22, 49, 50]. This type of state is a superposition of fourcoherent states, i.e. | ψ m (cid:105) ∝ | β (cid:105) +( − m |− β (cid:105) +( i ) m | iβ (cid:105) +( − i ) m |− iβ (cid:105) with β = | β | exp( iπ/ x and p , we should start with an input state with xp xp -8 -4 0 4 8-101 -8 -4 0 4 8-101 FIG. 3. Simulation results of the generation of theSchr¨odinger cat states for the case where k = 3 and l = 0.The squeezing parameters for the NGES are r tele = 1 . r of the initial squeezed states are − . . xp xp -8 -4 0 4 8-101 -8 -4 0 4 8-101 FIG. 4. Four-component cat states generated with ˆ f , . Thesqueezing parameters for the NGES are r tele = 1 . such symmetry. After that there are two ways where wecan evolve the state into a four-component cat states.First, we can implement ˆ f , and ˆ f , alternatively withthe conditioning at quadrature values 0 at the homodynedetectors. Note that this is also equivalent to keep imple-menting ˆ f , but changing the measurement basis of theconditioning teleportation so that the teleported statesare rotated by 90 degrees [51]. Another method is touse ˆ f , which is symmetric in ˆ x and ˆ p in the infinitesqueezing limit. Note that we expect the operations tobe symmetric in both quadrature for all k = l .Figure 4 shows the simulation results. We start withvacuum states and evolve the state in each iteration us-ing ˆ f , . As ˆ f , does not change the parity of the states, xp xpxp xp -8 -4 0 4 8-101 -8 -4 0 4 8-101 (A) (B) -8 -4 0 4 8-101 -8 -4 0 4 8-101 (C) (D) FIG. 5. Simulation of various superpositions of Fock statesup to three photons. The squeezing parameters for the NGESare r tele = 1 . | (cid:105) + | (cid:105) . (B) | (cid:105) + | (cid:105) . (C) | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) . (D) | (cid:105) + | (cid:105) .For all cases, we assume m p = and the m x for each statesare (-0.63), (-0.91, 0.93, 0.46), (-1,06, 0.13, 0.36), and (-1.27,0.13, 0.99), respectively. The fidelity to the target states are0.99, 0.97, ∼ ∼ the generated states are four-component cat states whosewave functions are even functions. Although we onlyshow the simulation results for the such cases, we canmake the four-component cat states with odd wave func-tions by adding, for example ˆ f , . C. Fock state superpositions
In the previous two examples regarding cat states, weassume that the results of the homodyne measurementsare 0. In this section, to illustrate the possible applica-tions of utilizing the other measurement results, we showhow our system can be used to generate qubits. In thesame way as a genuinely CV states can be approximatedin Fock basis by truncating the infinitely large Hilbertspaces to subspaces below certain photon number, ourmethod of tailoring wave function can also be used toapproximate superposition of Fock states. This is doneby attempting to shape the wave function to be close tothe wave function of the Fock state superpositions.For a Fock state superposition up to n max photons, theket vector becomes | ψ (cid:105) = n max (cid:88) n =0 c n | n (cid:105) . (48)Then, the corresponding wave function is | ψ (cid:105) = 1 π / n max (cid:88) n =0 c n √ n n ! H n ( x ) exp (cid:18) − x (cid:19) . (49)Therefore, for c n ∈ R , the wave function is a polyno-mial of at most n max order, multiplied with the wavefunction of the vacuum state. Therefore, we expect thatatmost n max iterations of our circuit on vacuum states isrequired to generate such states. Note that we could alsoeasily make the squeezed version of the Fock superposi-tion state, ˆ S ( r ) | ψ (cid:105) , by injecting squeezed states insteadof the vacuum states and scaling the conditioning of thehomodyne measurement results.As an example, we consider Fock superposition statesof the form | ψ F (cid:105) = c | (cid:105) + c | (cid:105) + c | (cid:105) + c | (cid:105) . This typeof arbitrary superposition of Fock states have been exper-imentally realized using TMSS, three avalanche photodi-odes, and three coherent beams for displacements [35].When c = c = 0, | ψ F (cid:105) reduces to the usual qubit.Figure 5 shows the simulation results. We observe thatthey have high fidelity to the target state and show howconditioning at other measurement results than 0 can beused to tailor the wave function. The general strategyhere is that we select initial m x so that they are close to the 0 points or the local minima of the wave functionsand then optimize the wave function to match the targetstates. Another remark is that, unlike most experimentwith Fock basis that usually implicitly assume low pumplimit and truncate multiphoton components, no trunca-tions in Fock basis are used and we work on the Hilbertspace using quadrature basis. D. Approximated cubic phase state
One of important classes of the nonclassical state is acubic phase state | CPS (cid:105) = exp( iγ ˆ x ) | p = 0 (cid:105) . This stateis an ancillary state for cubic phase gate which is a can-didate for non-Gaussian gates in CV quantum computa-tion [14, 52]. In the previous attempts to generate thisstate, truncation based on Fock state was used and thisstate was approximated to up to three photons [35]. Inthis section, we will show that it is possible to use ourmethod to realize cubic phase state.If we expand the ket vector of the cubic phase state,we get | CPS (cid:105) = (cid:20) iγ ˆ x + 12! ( iγ ˆ x ) + . . . (cid:21) | p = 0 (cid:105) . (50)As it was previously mentioned in Sec. III B, our methodcan realize wave function in x that has real roots. There-fore, we need to modified the above equation. By lookingat the wave function in p instead of x and approximating | p = 0 (cid:105) with p -squeezed states, we get˜ ψ | CPS (cid:105) ( p ) ≈ (cid:20) γ (cid:48) H (cid:18) p (cid:48) √ (cid:19) + γ (cid:48) H (cid:18) p (cid:48) √ (cid:19) + . . . (cid:21) exp (cid:18) − p (cid:48) (cid:19) , (51)with γ (cid:48) = γ/ ( √ e − ξ ) and p (cid:48) = p/e − ξ . In that sense,the parameter γ and the squeezing of the squeezed statesare in the scaling relationship. In the approximation, wewill assume that ξ = 0 for simplicity, as the squeezing canbe added after the state is generated when necessary.There is also another possible approximation. If werecall that the unnormalized wave function of the ideal | CPS (cid:105) is ψ | CPS (cid:105) ( x ) ∝ exp( iγ ˆ x ) , (52)the Fourier transform of the above function is [53]˜ ψ | CPS (cid:105) ( p ) ∝ Ai (cid:18) − p √ γ (cid:19) , (53)where Ai( · ) is the Airy function. As the Airy function is areal function, it can be approximated using our method-ology. However, there are two points that need consid-erations. First, as ˜ ψ | CPS (cid:105) ( p ) is mainly contained in the upper-region of the phase space (i.e. region with positive p ), it will be more advantageous to generate a displacedversion of this. Second, ideal Airy functions extend to theinfinity, meaning that we have to add another approxi-mation. One of possible ways to do so is considering aGaussian envelope over a displaced Airy function as ourapproximation, i.e.˜ ψ | CPS (cid:105) ( p ) ≈ exp (cid:18) − p e ξ (cid:19) Ai (cid:18) − ( p + p ) √ γ (cid:19) (54)as our target state which approaches ideal | CPS (cid:105) in thelimit of ξ → ∞ .Figure 6 shows the generated state using both approxi-mation that is targeted to this state. We observe that al-though both approximations yield different Wigner func-tions, they share similar traits: parabolic structure andoscillatory structure in p -direction. The first type of ap-proximation is actually equivalent to Fock basis trun- xp xpxp xp -8 -4 0 4 8-101 -8 -4 0 4 8-101 (A) (B) -8 -4 0 4 8-101 -8 -4 0 4 8-101 (C) (D) p pp p FIG. 6. Simulations of cubic phase state generations. Thetarget states for all subfigures assume γ = 0 . m p = and generation using ˆ f , . The squeezing parameters of theNGES for all subfigures are r tele = 1 .
0. (A,B) Approximationusing Eq. (51) with ξ = 0 up to the first-order (A) and thesecond-order (B) in γ . The initial squeezing level of the inputsis r = 0 and − .
7, and the m x are (0.78, -1.51, 0.58) and (0.61,-1.15, -0.23, 0.60), respectively. (C,D) Approximation usingEq. (54). The squeezing of the envelope is ξ = 0 . p = 8and 9, respectively. The m x are (2.80, 1.39, -1.18, -0.08, 1.02)and (-1.73, 1.72, -0.68, 1.02, 0.08). The fidelities of each stateto its targeted approximated | CPS (cid:105) are ∼ .
00, 0.985, 0.978,and 0.962, respectively. Note that the state is rotated by 90degrees after the generation. cations when ξ = 0 is equivalent to truncation up tosix photons. On the other hand, the second type ofapproximation would roughly be equivalent to reducingthe weight of the multiphoton components, but not com-pletely truncating them in a sense that the Wigner func-tions of the Fock states with lower photon numbers tendto be localized near the origin of the phase space. Notethat a more rigorous approximation of the cubic phasestate is to look at its performance when it is used torealize the cubic phase gate [52, 54]. P r obab ili t y Threshold fidelity 0.50.60.70.80.9110 -5 -4 -3 -2 -1 FIG. 7. Success probabilities of the homodyne condition-ing for states generated with a single step of ˆ f , to havefidelities to the target states above the threshold fidelity.The initial states are ˆ S ( r ) | (cid:105) (circle), ˆ S ( − r ) | (cid:105) (diamond),ˆ S ( r ) | (cid:105) (square), and ˆ S ( − r ) | (cid:105) (triangle), with r = 1 . r tele = 1 .
0. The target states are the states generated when m x = m p = 0 for all input states. V. DISCUSSIONSA. Success rate and fidelity
In this section, we discuss success rate and fidelity. Upuntil this point, we have assumed for the simplicity thatthe window of the conditioning can be infinitesimal. Inreality, we have to have a finite window to have a finitesuccess rate. Since we are effectively tailoring by approx-imating them as polynomial and the measurement resultsof the homodyne detector determine the positions of theroots of the polynomials, the conditioning window sizeis determined by how the polynomials change when theroots are changed. In general, we would expect that fora wave function that is broadly distributed, the positionsof the roots do not greatly affect the overall polynomi-als, thus we can have a large conditioning window. Thismeans that, it tends to be more advantageous to tailorthe wave function of the antisqueezed version of the tar-get state.Figure 7 shows the probability of successfully generat-ing quantum states whose fidelity to the target state ishigher than a certain threshold. Indeed, the success prob-ability is higher as we lower the threshold fidelity. Wealso observe that, as expected, the success probability isindeed higher for a state that is broadly distributed in thequadrature x for the current case where the operator ˆ f , is applied to the initial state in the case where the con-ditioning window is infinitesimal. Interestingly, althoughwe are conditioning near m x = m p = 0, rather thansqueezed states whose wave function is a Gaussian func-tion centered at the origin, the squeezed single photon0 p -8 -4 0 4 8-1-0.500.51 FIG. 8. Effects due to the shifts of the measurement results.Blue line: The wave function of the approximated cubic phasestate in Fig. 6(B). Orange line: the wave function of the stategenerated when the measurement results m x is shifted by (0.1,0.1, 0.1, 0.1). states, which have zero probability of finding the quadra-ture value 0, have much higher success probability. Thisis due to the fact that when the squeezed single photonis interfered with one of the modes of the | NGES(1 , (cid:105) ,the probability of Bell measurement giving m x = m p = 0increases due to the interference of the non-Gaussian fea-tures of both states.For the case where the conditioning teleportation withNGES is implemented multiple times, there are addi-tional aspects that must be considered. First, the overallsuccess probability. To illustrate this aspect quantita-tively, let us consider the ideal case with k = 1 and l = 0 and infinite squeezing. Then, if the initial wavefunction is ψ ( x ), the unnormalized wave function af-ter one and two step of our circuit is ( x − m ) ψ ( x ) and( x − m (cid:48) )( x − m (cid:48)(cid:48) ) ψ ( x ) where m , m (cid:48) , and m (cid:48)(cid:48) are the resultsof the Bell measurement in x quadrature. If we want todo conditioning near the place where m = m (cid:48) = m (cid:48)(cid:48) = 0,then for the one-step case, the allowable range of m should be well below (cid:104) ˆ x (cid:105) / , where the mean (cid:104)·(cid:105) here istaken with respect to the initial state | ψ (cid:105) . On the otherhand, for the two-step case, if we restrict to the casewhere m (cid:48) = − m (cid:48)(cid:48) , it is easy to show that the allowablerange should be well below (cid:104) ˆ x (cid:105) / . As (cid:104) ˆ x (cid:105) / ≥ (cid:104) ˆ x (cid:105) / ,the allowable range of each measurement in the two-stepcase should be broader given that they have appropriaterelation. As such, this qualitative example suggests thatfor the multi-step case, rather than consider each step in-dividually, we should perform conditioning and heraldingon a set or a range of the measurement results.Another aspect we have to consider is the possibility ofincreasing the fidelity to the target state using Gaussianoperations. Figure 8 shows an example of such cases.Although the fidelity between the two wave functions is 0.87, it is obvious that the two are related via displace-ment operation. If the wave function is displaced by theamount of the shifts, the fidelity becomes 0.95 which ismuch higher. The effects from the displacements dueto the shifts in the measurement results from the targetstate are more obvious when the wave function is oscilla-tory as shown in this example. In addition, we could alsoconsider a case where all the measurement results are justthe scaling of the target case. In such case, the gener-ated state would be roughly the squeezed or antisqueezedversion of the target state. Therefore, when consideringthe success probability and the fidelity, it is more advan-tageous to optimize the fidelity to the target state usingGaussian operations. This optimization is expected toincrease the success probability further.By combining these two aspects, it is expected thatthe success probability of our method can be further in-creased. As the calculations for the actual experimentalsetup would be highly dependent on the initial states andthe target states, and also the experimental imperfectionssuch as optical losses, we leave the detailed considerationas a future experimental work. B. Experimental feasibility
In addition to the probabilistic nature of the homo-dyne conditioning, we also need to consider how to realizeNGES. We will restrict our discussions to the realizationin the optical systems. One of the simplest implemen-tations to realize NGES is that via photon subtraction,which is a method widely used to approximate cat states.As photon subtraction is probabilistic, using the photonsubtraction as it is will limit the generation rate evenfurther. There are a few possibilities to overcome this.First, we could simply employ quantum memory as ourprotocol used NGES as resource states that can be gen-erated offline. Second, for k = 1 and l = 0 (or k = 0and l = 1), the squeezed states become squeezed singlephoton states. As there exists on-demand single photonsource based on architecture such as quantum dots [55]and deterministic squeezing of single photons have beenrealized [56], we could use such architecture for determin-istic generation. Third, as photon-subtracted squeezedstates are usually approximations of cat states [36], wecould also consider a possibility of replacing them withcat state sources. Recently, there is a proposal for asystem to generate Schr¨odinger cat state with high gen-eration rate [45] which might enable realistic realizationof the method in this paper. C. Time-domain-multiplexing and Non-Gaussiancluster states
Figure 9 shows a possible experimental realization ofour setup using the time-domain-multiplexing method.As our setup is based on sequential quantum teleporta-1 C ond i t i on i ng Non-Gaussianentangled state Bell measurementDelay line To displacementOptical switch
FIG. 9. A setup for iterative implementation of our protocolusing the time-domain multiplexing method. The NGES usedis for the implementation of ˆ f , and is realized by replacingone of the squeezed light source in the one-dimensional clusterstate setup with a squeezed single photon source. tion circuits, by using the time-domain multiplexing andreplacing the source of the TMSS with the NGES, it ispossible to realize a versatile and compact experimen-tal setup for generation of various non-Gaussian states,where the generated state can be programmed via thechoice of the homodyne conditioning. The optical switchis used to inject the initial quantum state or retrieve thequantum state after the implementation of our protocol.The usage of such optical switch on the quantum stateshas recently been demonstrated [57, 58].Also, as the setup in Fig. 9 resembles that of the one-dimensional cluster state generation [4, 5, 46], we couldalso interpret our protocol as a cluster state computationwhere the NGES is used instead of the usual Gaussian CVcluster states. Degaussification of Gaussian cluster statesand their properties have been studied recently [59–61]and a basic demonstration of the generation of such non-Gaussian cluster state has also been realized [62]. In thiscontext, our protocol here demonstrates a possible appli-cation of non-Gaussian entanglement resource and couldserve as a setup to study the combination of the non-Gaussian element to the time-domain-multiplexed cluster state. VI. CONCLUSION
In this paper, we present a methodology to generatenon-Gaussian states based on tailoring of the wave func-tion using the non-Gaussian entanglements and the con-ditional quantum teleportation. Our approach is a com-plementary approach to the Fock basis approach. Wedemonstrate the versatility of our method by showingthat, without modifying our system at all, we can gen-erate various quantum states with iterative conditionalquantum teleportation using the non-Gaussian entangle-ment resources. In addition to a system that can generateany states, we could also consider a system specialized fora certain state such as GKP states. Such a study has beendone in Ref. [29] where parameters of the optical systemsand photon number resolving detector are optimized inFock basis. As we have demonstrated that homodyneconditioning is also useful for state generation, it will beinteresting to see if we can also incorporate homodyneconditioning and realize a kind of hybridized state gen-erator that utilizes both Fock basis and wave functionpicture.
ACKNOWLEDGMENTS
This work was partly supported by JST [Moon-shot R&D][Grant No. JPMJMS2064], JSPS KAK-ENHI (Grant No. 18H05207, No. 18H01149, and No.20K15187), UTokyo Foundation, and donations fromNichia Corporation. K.T. acknowledges financial sup-ports from the Japan Society for the Promotion of Sci-ence (JSPS). The authors would like to thank TakahiroMitani for proofreading of the manuscript. [1] M. A. Nielsen and I. L. Chuang,
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