Deterministic nonlinear phase gates induced by a single qubit
aa r X i v : . [ qu a n t - ph ] J un Deterministic nonlinear phase gates induced by a single qubit
Kimin Park*, Petr Marek and Radim Filip
Department of Optics, Palack´y University, 17. listopadu 1192/12, 77146 Olomouc, Czech Republic ∗ (Dated: October 8, 2018)We propose deterministic realizations of nonlinear phase gates by repeating non-commuting Rabiinteractions feasible between a harmonic oscillator and only a single two-level ancillary qubit. Weshow explicitly that the key nonclassical features of the states after the ideal cubic phase gate and thequartic phase gate are reproduced faithfully by the engineered operators. This theoretical proposalcompletes the universal set of operators in continuous variable quantum computation. Quantum linear harmonic oscillators, systems withstates defined in an infinite-dimensional Fock space, areat the center of continuous variables (CV) quantum in-formation processing [1]. Physical realizations of quan-tum harmonic oscillators employ a traveling light [2] andcollective modes of atomic spins [3], motion modes oftrapped ions [4], and cavity [5] or circuit quantum elec-trodynamics [6]. In order to fully control these systemsas a quantum computation platform we need the abil-ity to implement an arbitrary nonlinear operation. Theprincipal requirement is reduced to an access to nonlin-earity of the third order - the cubic nonlinearity - andthen use it for implementation of nonlinearities of higherorder [7]. However, obtaining even the cubic nonlinearityis not straightforward. The evolution in cubic potentialis inherently instable, even in the overdamped regime [8].Naturally appearing nonlinear media do not come withsufficient strength and purity [9] and the evolution op-erator needs to be crafted by manipulating individualquanta of the harmonic oscillator. These approaches relyon nonlinear conditional measurements [10–12], deter-ministic feed-forward aided by nonlinear ancillary states[13–16], or deterministic control over a two-level systemcoupled to the oscillator [17, 18].These methods are specialized to different CV systemsand resources. By far the most effective approaches inquantum optics and quantum electrodynamics (QED) ex-ploit the coupling between the harmonic oscillator andan associated two level system, which is the standardscenario in trapped ions and circuit QED. This efficiencyoriginates from the inherent nonlinearity of the two-levelsystem, which can impart nonclassical behavior to theoscillator simply by “being there” [19]. There are twodistinct approaches towards realizing the nonlinear op-erations, either in a single involved step [18], or by in-cremental construction from weak elementary interac-tions [12, 17]. In the following, we will focus on themethod of elementary gates [17], directly extended to-wards continuous-variable phase gates by designing a dif-ferent coupling. Such gates offer the ability to addressthe wave-like features of quantum systems and can beemployed for simulation of particles in nonlinear poten- ∗ Electronic address: [email protected] tials [20].In this Letter we propose deterministic methods for re-alizing a unitary nonlinear quadrature phase gate of anarbitrary order for quantum harmonic oscillators. Thegates realized by our method gradually approach unitaryevolutions, which take advantage of the so-called Rabi in-teraction between the harmonic oscillator and a coupledtwo-level system. They consist of ordinary elementarybuilding blocks simply repeated to achieve nonlinearitiesof arbitrary strength in the limit of infinite resources.We analyze the performance of these methods under theassumption of finite resources.The continuous-variable phase gates are representedby unitary operators ˆ U m = exp[ iχ m ˆ X m ], where the ar-bitrary integer m denotes the order of the operation.ˆ X = (ˆ a + ˆ a † ) / √ a and ˆ a † denote the ladder opera-tors. The two-level system, or a qubit, is described by aBloch sphere with the SU[2] algebra represented by Paulimatrices σ i with i = x, y, z satisfying a set of commuta-tion relations [ σ i , σ j ] = 2 iǫ ijk σ k with a Levi-Civita sym-bol ǫ ijk . The main ingredient is a sequence of individualRabi interactions between the harmonic oscillator andtwo level systems, already feasible at the various plat-forms [21, 22]. A coupling referred to as the Rabi interac-tion [23] is represented by a unitary operator exp[ itσ i ˆ X ],where t is its effective strength. In trapped ionic sys-tem, this interaction can occur between a motional de-gree of freedom and the atomic internal two-level statesvia bichromatic laser driving red and blue motional side-bands in Lamb-Dicke regime [21]. In circuit QED system,this type of interaction arises between a superconductingqubit such as a flux or a transmon qubit and a waveguideresonator in the ultrastrong-coupling regime [22].Rabi gates with different orientation of σ i do not com-mute and we can exploit this non-commutative behaviorto elaborate important ingredients before the proposalof phase gates. With the help of the Baker-Campbell-Hausdorff formula [24], new combined operators to beused as building blocks are derived as:ˆ M ( ˆ T , ˆ T ) = exp[ − i ˆ T σ y ] exp[ i ˆ T σ z ] exp[ i ˆ T σ y ]= exp[ i ˆ T (cos[2 ˆ T ] σ z − sin[2 ˆ T ] σ x )] , ˆ M ( ˆ T , ˆ T ) = exp[ − i ˆ T σ y ] exp[ i ˆ T σ x ] exp[ i ˆ T σ y ]= exp[ i ˆ T (cos[2 ˆ T ] σ x + sin[2 ˆ T ] σ z )] , (1) |(cid:2)(cid:3) (cid:1)(cid:2) (cid:1) (cid:3) (cid:2) (cid:4) (cid:2) (cid:3) (cid:3) (cid:4) (cid:4) (cid:5)(cid:2) (cid:0) (cid:3) (cid:5) (cid:4) … (cid:1) (cid:6) (cid:2) (cid:7) (cid:3) (cid:8) (cid:4) (cid:1)(cid:2) (cid:9) (cid:3) (cid:10) (cid:4) (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:3)(cid:1)(cid:5)(cid:3) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:4)(cid:8)(cid:1)(cid:4)(cid:9)(cid:5)(cid:10)(cid:11)(cid:12) (cid:1)(cid:4)(cid:9)(cid:5)(cid:10)(cid:11) (cid:1) … (cid:4) (cid:11) (cid:5)(cid:6) (cid:7) (cid:6) (cid:12) (cid:8) (cid:4) (cid:13) (cid:5)(cid:6) (cid:7) (cid:6) (cid:14) (cid:8) (cid:1)(cid:6)(cid:3) (cid:2) (cid:15) (cid:2) (cid:16) (cid:1) (cid:1),(cid:3) (cid:7)(cid:13)(cid:14)(cid:2) (cid:3)(cid:3)/(cid:5)(cid:6) (cid:7) (cid:17) (cid:7) (cid:18) (cid:5)(cid:7) (cid:19) (cid:7) (cid:20) (cid:7) (cid:21) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5) (cid:15)(cid:3)(cid:16)(cid:13)(cid:17)(cid:17)(cid:6)(cid:7)(cid:4)(cid:8) (cid:2) … FIG. 1: (Color online) A deterministic setup to achieve anonlinear phase gate. (a) A five-element circuit model wherenon-commuting Rabi interactions at different strengths arearranged together with a correction operator. An ancillaryqubit (yellow elements) is prepared in a ground state at eachround, and R rounds are performed for a high-strength non-linear gate. (b) A trapped ion implementation in a Paultrap. Here the continuous system is the motional state ofthe ion, and the two-level system is its internal energy state.The preparation and measurement of the two-level systemare performed by control pulses. Rabi interaction is realizedby a bichromatic laser driving. (c) A circuit QED imple-mentation. The flux qubit is made of a superposition of theclockwise and counter-clockwise currents, and is interactingvia Josephson junction with the superconducting microwavecoplanar waveguide resonator. External magnetic flux Φ con-trols the interaction. (d) In trapped ionic system, the phaseof the bichromatic laser beam φ + = ( φ r + φ b + π ) / φ r,b is controlled in time segments alter-natingly corresponding to the assigned Pauli operators for agiven strength [21]. In circuit QED system, exactly the samecontrol sequence can be obtained by controlling the phase ofthe external driving fields φ d [25]. where the harmonic oscillator operators ˆ T , introducedfor a shorthand notation commute with all the other op-erators in the formula. When the qubit is prepared inthe ground state | g i of an eigenstate of σ z , subject toa sequence of two combined operators ˆ M , , and subse-quently projected onto the ground state by a suitablemeasurement, the resulting operator is expressed asˆ O ,s ≡ h g | ˆ M ( ˆ T , ˆ T ) ˆ M ( − ˆ T , ˆ T ) | g i ≈ exp[ i ˆ T cos[2 ˆ T ]] , ˆ O ,s ≡ h g | ˆ M ( ˆ T , ˆ T ) ˆ M ( − ˆ T , − ˆ T ) | g i ≈ exp[ i ˆ T sin[2 ˆ T ]] . (2)Here the approximation holds when the eigenvalues of theoperators ˆ T and ˆ T relevant for the input states are allsmall enough [10]. In our case, ˆ T = t ˆ X and ˆ T = t ˆ X ,and the condition requires the states to be localized inX-representation. For a small t and t the gate is prac-tically applicable to an arbitrary state. The importanceof these new operators ˆ O ,s and ˆ O ,s lies in the observa- tion that they correspond to interaction Hamiltonians nolonger linear in ˆ T i . They instead contain trigonometricfunctions of ˆ T i , or infinite-order polynomials containingall orders of non-linear terms. These terms can be ex-ploited in realization of the nonlinear operations of anarbitrary order.Let us first begin with the realization of the cubic gaterepresented by the unitary operator ˆ U = exp[ iχ ˆ X ]with the target strength χ . We can start from (2) withRabi interactions arranged as in Fig. 1(a) with the sub-stution ˆ T = t ˆ X and ˆ T = t ˆ X asˆ O ,s ≈ h g | exp[ i t ˆ X cos[2 t ˆ X ] σ z ] | g i = exp[ i t ˆ X cos[2 t ˆ X ]] . (3)The last form can be expanded into the Taylor series ofonly two terms as exp[ i t ˆ X (cid:16) − (2 t ˆ X ) / (cid:17) ] for smallvalues of t . This operator represents a desired cubicgate, but with an residual displacement ˆ D ( − t ) whereˆ D ( z ) = exp( − iz ˆ X ). This displacement, however, can becompensated by an inverse correction displacement op-eration ˆ G c = ˆ D (2 t ) which can be achieved with anotherRabi interaction. A weak cubic gate can therefore beobtained as a sequence of operations ˆ G c ˆ O ,s . This oper-ator is close to a unitary operator and reliably approxi-mates the desired operation ˆ U with the target strength χ = 4 t t for small Rabi strengths t , t ≪ i t ˆ X cos[2 t ˆ X ] σ z ] in(3) has | g i as its eigenstate, beneficial in experimental im-plementation because the success probability of the oper-ation approaches one. The final projective measurementon ancilla is thus of a minor contribution, and we maychoose to ignore the measurement outcomes and achievea deterministic operation represented by a trace preserv-ing map Γ[ ρ ] = ˆ G c ( ˆ O ,s ρ ˆ O † ,s + ˆ O f ρ ˆ O † f ) ˆ G † c , (4)where ˆ O f = h e | ˆ M ( t ˆ X, t ˆ X ) ˆ M ( − t ˆ X, t ˆ X ) | g i = − sin ( t ˆ X ) sin(4 t ˆ X ) represents the failure operatorvanishing for weak Rabi strengths t , t ≪
1. In orderto enhance the total strength of the nonlinearity we mayapply the weak gate multiple times. After R repetitions,the output state ρ re = Γ R [ ρ ] produced by the cubic gatefeels the effective strength χ = 4 t t R . We note thattwo free parameters exist among t , t , and R to achievea target operator at a fixed strength χ . These free pa-rameters can be used to optimize the performance of thetarget gate. The cancellation by the correction displace-ment operator ˆ G c can equivalently be done altogether atonce after all the repetitions of rounds instead at eachround, due to the commutativity of ˆ G c with ˆ O ,s and ˆ O f to simplify the experimental setup.Instead of using five elementary Rabi interactions asin (3), different numbers of Rabi interactions can beused. Weak cubic gate can be also obtained by usingthree elementary Rabi interactions M ( t ˆ X, t ˆ X ) | g i , ortwo Rabi interactions M ( t ˆ X, t ) | g i in place of ˆ O ,s in (4). We can also effectively increase the numberof Rabi interactions in each round by reducing thefrequency of re-initialization of the ancillary state as( ˆ M ( t ˆ X, t ˆ X ) ˆ M ( − t ˆ X, t ˆ X )) k | g i in place of ˆ O ,s forintegers k >
1. The performance of these approachesis, however, worse for equal numbers of elementary Rabiinteractions (see supplemental material). We shall there-fore consider only the best operation (3) for the followinganalysis.
FIG. 2: (Color online) Dependence of functionalities of deter-ministic cubic gates on the number of rounds R . (a) Fidelity F vs R , (b) fidelity of change F ⊥ vs R , (c) purity P vs R atvarious strengths χ = 0 . . . R is increasedfor all strength χ , and (d) purity of the orthogonal state P ⊥ vs R at various strengths χ = 0 . , . , .
3. All propertiesof the output states are enhanced as R is increased for allstrength χ . Let us now evaluate the performance of the engineeredcubic gate. We start by applying the operation to aninitial pure quantum state ρ = | ψ i h ψ | and verifyhow close the resulting state is to its ideal form. Thiscan be quantified by the fidelity F = Tr[ ρ id ρ re ], where ρ id = ˆ U ρ ˆ U † is the ideal state and ρ re = Γ R [ ρ ] is therealized state. In Fig. 2 we can see that with increasingnumber of repetitions corresponding to a high resourcelimit R → ∞ , the fidelity is approaching one for an arbi-trary value of desired cubic interaction strength for thevacuum state | i . However, for small values of interac-tion strength, the fidelity has a weakness as a figure ofmerit, because it draws dominant contribution from thepresence of the initial quantum state ρ [13, 14]. There-fore, the closeness of the generated state ρ re to the idealstate ρ id after the initial state component is removed fromboth states can be a complementary measure of perfor-mance without the weakness. This mathematical removalcan be achieved when a density matrix ρ is projectedonto a subspace orthogonal to the initial pure state ρ as ρ ⊥ = (ˆ − ρ ) ρ (ˆ − ρ ) / Tr[ ρ (ˆ − ρ )]. The fidelity af-ter the initial state removal between the target state and the simulated state F ⊥ = Tr[ ρ ⊥ id ρ ⊥ re ] will be denoted asthe fidelity of change, and is shown in Fig. 2 for varioustarget strengths. We again observe that F ⊥ → P [ ρ re ] = Tr[ ρ ] whichtakes a value 1 for a pure state. The purity is reduced bythe presence of the failure operators ˆ O f in (4), but Fig. 2demonstrates that this influence can be indeed vanishingas well in the high resource limit. The purity of the stateafter initial state removal P [ ρ ⊥ ] also approaches 1 as anauxiliary proof of the closeness of the operators. FIG. 3: (Color online) Cross sections of Wigner functions at x = 0 for χ = 0 . , . , . , . R =18 , , ,
72, respectively. Fidelities with the target states areshown as insets.
In addition to quantitative figures of merit, we can alsolook for qualitative ones to check the nonclassical aspectsof the engineered states. Wigner functions describe quan-tum states in quadrature phase space analogous to theclassical probability densities. Nonclassical states pro-duced by quantum nonlinearities exhibit peculiar prop-erties, one of which is the presence of negative values.Each nonclassical state has a specific pattern of thesenonclassical regions, and we can therefore check whetherthe states generated by the approximate cubic gate ex-hibit analogous patterns as the ideal cases. When thevacuum is the initial state, the ideal cubic states ˆ U | i exhibit negative fringes in the area given by p < x = 0, where x, p are the eigenvalues of the quadratureoperators ˆ X and ˆ P = (ˆ a − ˆ a † ) / √ i . In Fig 3, we haveplotted Wigner function values W ( x = 0 , p ) for severalcubic states at different strengths and the correspond-ing number of repetitions. We see that the two Wignerfunctions overlap nearly indistinguishably, and the ap-proximate gate closely follows the ideal scenario, whichconfirms the gradual reproduction of the nonlinear dy-namics.To test the validity of our scheme for arbitrary in-put states, we have considered a set of coherent states FIG. 4: (a) The region for which fidelity to the ideal state islarger than 0 .
99 for a cubic phase gate. With a relatively lownumber of repetition, a high strength cubic gate is achievedfor broad amplitudes of coherent states. The area of the re-gions can be expanded to an indefinite χ → ∞ when a highresource is accessible R → ∞ . (b) The region for which the fi-delity of change F ⊥ to the ideal state is larger than 0 .
95 for thesame generated operators. The fidelity of change is smallerthan the full fidelity, but we still can observe the expansionwhen a larger R is used. | α i = exp( −√ iα ˆ P ) | i with α > α and χ where the fidelity with the ideal cubic gate islarger than a chosen threshold F > .
99. We can seethat even with a relatively low number of repetitions R ,a high strength cubic gate is achieved at a reasonable fi-delity for broad range of amplitudes of coherent states,while a higher number of rounds is necessary for the co-herent states to achieve a same fidelity for the vacuumstate. Fig. 4 (b) shows, analogously, the area in whichthe fidelity of change satisfies F ⊥ > .
95. Both of theseresults imply that the approximate cubic gate scheme isapplicable to a wide class of states and that increasingthe number of repetitions can extend this range.The basic principles applied to the cubic gate can beextended towards the fourth- and higher-order nonlinearquadrature phase gates. The fourth order quartic gateˆ U = exp[ iχ ˆ X ] can be realized by a sequence of ele-mentary Rabi interactionsˆ O ,s = h g | ˆ M ( t ˆ X, t ˆ X ) ˆ M ( − t ˆ X, − t ˆ X ) | g i≈ exp[ i t ˆ X sin[2 t ˆ X ]] . (5)As in the case of the cubic gate, the crucialstep lies in compensation of the lowest orderterm exp[ i t t ˆ X ] in the approximate expansionexp[ i t ˆ X (cid:16) t ˆ X − (2 t ˆ X ) / (cid:17) ] for small values of t .In this case, the correction step needs to be realizedby a squeezing operation G c = exp[ − i ζt t ˆ X ]. Thissqueezing operation can be similarly engineered fromnon-commuting Rabi interactions from ˆ O ,s with differ-ent parameters t ′ and t ′ . The squeezing parameter ζ can be optimized for the best performance. In Fig. 5, we FIG. 5: The cross section of the Wigner function at x = 1after a quartic gate of (a) χ = 0 . χ = 0 .
4. Wenotice that the overall oscillation is qualitatively analogous.The region for which (c)
F > .
99 and (d) F ⊥ > .
95 for aquartic gate. The area of the regions can again be expandedto an indefinite χ when a high R is accessible. perform the same analysis for Wigner functions, F and F ⊥ as for cubic gates. The simulation of the negativeregions in the Wigner function of the quartic phasegate again shows a qualitative similarity, with only asmall deviation. We can also achieve interaction with anarbitrary strength χ , even though the requirements arestricter than for the cubic gate.In general, the phase gates ˆ U m = exp[ iχ m ˆ X m ] for anarbitrary integer m can be similarly achieved by apply-ing (3) when m is odd, or (5) when m is even, and elim-inating all the undesirable lower order terms by alreadyacheived phase gates G c = ˆ U k We acknowledge Project GB14-36681G of the CzechScience Foundation. K.P. acknowledges support by theDevelopment Project of Faculty of Science, Palack´y Uni-versity. Authors thank L. Slodiˇcka for a helpful discus- sion. [1] C. Weedbrook, S. Pirandola, R. C.-Patr´on, N. J. Cerf, T.C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys.84, 621 (2012); S. L. Braunstein, and P. Van Loock, Rev.Mod. Phys., 77(2), 513 (2005); N. J. Cerf, G. Leuchs,and E. S. Polzik (eds.), Quantum Information with Con-tinuous Variables of Atoms and Light (Imperial CollegePress, London, 2007).[2] P. A. M. Dirac, The Quantum Theory of the Emissionand Absorption of Radiation , Proc. Royal Soc. (London)A114, pp. 243265 (1927).[3] T. Opatrny, arXiv:1702.03124 (2017); I. D. Leroux, M.H. Schleier-Smith, and V. Vulet´ıc, Phys. Rev. Lett. 104,073602 (2010).[4] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev.Mod. Phys. 75, 281 (2003).[5] A. Reiserer and G. Rempe, Rev. Mod. Phys. 87, 1379(2015).[6] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R.J. Schoelkopf, Phys. Rev. A 69, 062320 (2004); A. Wall-raff, D. I. Schuster, A. Blais, L. Frunzio, R.- S. Huang,J. Majer, S. Kumar, S. M. Girvin and R. J. Schoelkopf,Nature 431, 162 (2004).[7] S. Lloyd and S. L. Braunstein, Phys. Rev. Lett. 82, 1784(1999).[8] M. ˇSiler, P. J´akl, O. Brzobohat´y, A. Ryabov, R. Filip,and P. Zem´anek, Sci. Rep. 7, 1697 (2017).[9] L. Lugiato, F. Prati, and M. Brambilla,