Deterministic nonlinear phase gates induced by a single qubit

We propose deterministic realizations of nonlinear phase gates by repeating non-commuting Rabi interactions feasible between a harmonic oscillator and {\em only} a single two-level ancillary qubit. We show explicitly that the key nonclassical features of the states after the ideal cubic phase gate and the quartic phase gate are reproduced faithfully by the engineered operators. This theoretical proposal completes the universal set of operators in continuous variable quantum computation.

These blocks can be repeated to create unitary interaction of arbitrary strength. We analyze the performance of the method under the assumption of finite resources. For the analysis of robustness we employ a realistic model of imperfections such as dephasing of the two-level system and damping of the oscillator.
The CV quadrature phase gates, represented by unitary operators apply a phase to states in the respective quadrature basis and the integer m denotes the order of the operation. X a a 2 = + (ˆˆ) † is a quadrature operator of the harmonic oscillator, and â and â † denote the ladder operators. This operation generates a phase space displacement for m = 1 and a Gaussian squeezing with a phase space rotation for m = 2. For m 3, the phase gates deterministically generate nonlinearly squeezed states with Wigner functions exhibiting nontrivial oscillations [20], which can serve as sources of highly nonlinear effects [17]. The main ingredient of the nonlinear gate proposed here is a sequence of Rabi interactions between the harmonic oscillator and a two-level quantum system-a qubit. Qubits can be described by the SU [2] algebra represented by Pauli matrices i ŝ with i = x, y, z satisfying a set of commutation relations , 2i Levi-Civita symbol ò ijk . The main element, the Rabi interaction [29][30][31] between the qubit and the harmonic oscillator, is represented by a unitary operator t X exp i i s [ˆˆ], where t is its effective strength. The Rabi interaction is a nonlinear elementary interaction where a two-level system directly couples to the continuous X-quadrature of an oscillator. The actual experimental realization depends on the choice of the platform. Figure 1(a) shows the overall logic-circuit-like model. A circuit for a single round realizing a weak nonlinear operation (1) consists of a series of five non-commuting Rabi interactions of the oscillator coupling with the ancillary qubit prepared in the ground state, and a Gaussian correction operator. Nonlinear phase gates with higher strengths are then obtained by R repetitions of the single elementary rounds. The qubit can be re-initialized to the ground state after each round and re-used for the new one. For trapped ion implementation illustrated in figure 1(b), the required interaction can occur between a motional degree of freedom and the atomic internal two-level states via bichromatic laser driving red and blue motional sidebands in the Lamb-Dicke regime [29]. The preparation and measurement of the two-level system are performed by control pulses. The phase of the bichromatic laser beam f + = (f r +f b +π)/2 for the phase of each beam r b , f is controlled in time segments alternatingly corresponding to the assigned Pauli operators for a given strength [29]. In circuit QED system, illustrated in figure 1(c), the Rabi interaction arises between a superconducting qubit such as a flux or a transmon qubit and a waveguide resonator in the ultrastrong-coupling regime [30]. The flux qubit is made of a superposition of the clockwise and counter-clockwise currents, and is interacting via Josephson junction with the superconducting microwave coplanar waveguide resonator. Here external magnetic flux Φ controls the interaction. In circuit QED system, exactly the same control sequence can be obtained by controlling the phase of the external driving fields d f [32] as in figure 1(d).
Rabi gates with different i ŝ operators do not commute. A sequence of different non-commuting operators then implements a new operation, which can be highly nonlinear. To explicitly see that, let us employ the Figure 1. A deterministic setup to achieve a nonlinear phase gate. (a) A five-element circuit model: non-commuting Rabi interactions at different strengths are arranged together with a correction operator. The same circuit is sequentially repeated without any other optimization. An ancillary qubit (yellow elements) is initialized into a ground state at each round, and R rounds are performed for a high strength nonlinear gate. (b) A trapped ion implementation in a Paul trap: the cubic nonlinearity is applied to the motional state of the ion, and the two-level system is its internal energy state. The preparation and measurement of the two-level system are performed by control pulses. Rabi interaction is realized by a bichromatic laser driving. (c) A circuit QED implementation: the flux qubit is made of a superposition of the clockwise and counter-clockwise currents, and is interacting via Josephson junction with the superconducting microwave coplanar waveguide resonator. (c) reprinted by permission from Macmillan Publishers Limited, Nature Physics, [30]. External magnetic flux Φ controls the interaction. (d) For the trapped ions, the phase of the bichromatic laser beam f + =(f r +f b +π)/2 for the phase of each beam f r,b is controlled in time segments alternatingly corresponding to the assigned Pauli operators for a given strength [29]. In circuit QED system, exactly the same control sequence can be obtained by controlling the phase of the external driving fields f d [32].
Baker-Campbell-Hausdorff formula [33] to derive two new operators M T T  T  T  T  T  T  T   M T T  T  T  T  T  T  where the harmonic oscillator operators T t X 1 1 =ˆand T t X 2 2 =ˆintroduced as a shorthand notation commute with all the other operators in the formula. We can see that the exponents of the resulting unitary operators contain nonlinear functions of operator X . We can now define two operations acting on the state of the harmonic oscillator. In both cases the qubit, which is initially prepared in the ground state, interacts with the oscillator through different sequences of operators M 1,2 and is then projected to the ground state by a measurement. The resulting operators acting on the state of the oscillator can be expressed as Here the final approximation holds when the eigenvalues of the operators T 1 and T 2 relevant for the input states are all small enough (or equivalently when t 1 and t 2 are small) to allow merging the equations in (2) as in the Suzuki-Trotter technique [12]. We note that the full sequence consists of five elementary Rabi interactions instead of six because two interactions in the middle are indistinguishable and can be merged. The importance of these new operators O s

3,
and O s 4, lies in the observation that they correspond to interaction Hamiltonians no longer linear in T î . They instead contain trigonometric functions of T î which can be seen as infinite-order polynomials containing all even or odd orders of nonlinear terms. These terms can be exploited in realization of nonlinear operations of arbitrary orders. All these operators can be simply obtained when the ground state gñ | stays highly more probable than the excited state eñ | after the application of M 1 and M 2 . Below, particular examples will be discussed about the actual applicability of the scheme.
Let us first begin with the realization of the cubic phase gate represented by the unitary operator The cosine function in the argument can be approximated by a Taylor series of only two terms leading to t X t X exp i2 for small values of t 1 . The Trotterization condition also requires that t 1,2 are small and the input states of the harmonic oscillator are localized in quadrature representation. This operator O s 3, represents a desired cubic phase gate practically applicable to an arbitrary state, only with a residual displacement operator D t . This displacement can be compensated by an inverse correction displacement operation G D t 2 c 2 =ˆ( )which can be frequently implemented directly. It can also be achieved by an additional Rabi interaction with an independent ancilla in offline mode a as . Alternatively, the correction can be made with an additional Rabi interaction t X exp 2i c = for small Rabi strengths t 1 , t 2 =1 (see supplemental material available online at stacks.iop.org/NJP/20/053022/mmedia for the discussion of near-unitarity). This nearunitarity arises from the fact that the approximative operator t X t X exp i2 cos 2 | as its eigenstate, and therefore the oscillator is nearly decoupled from the qubit and the success probability of the implemented operation M T T M T T , , It is therefore viable to leave two-level system unmeasured and the operation will still approach the cubic phase gate. Moreover, we can easily recycle the qubit back to the ground state gñ | to be re-used in the next sequences. The measurement-free sequence = á -ñ = -|ˆ(ˆˆ)ˆ(ˆˆ)| (ˆ) (ˆ) corresponding to a conversion into the excited state eñ | of the two-level system, vanishes for weak Rabi strengths t 1 , t 2 =1, which allows to reach the desired gate deterministically. In order to enhance the total strength of the nonlinearity we may apply the weak gate multiple times. After R repetitions with cyclically re-initialized two-level system into the initial ground state, the output state We note that two free parameters exist among t 1 , t 2 , and R to achieve a target operator at a fixed strength χ 3 . These free parameters can be used for optimization of the performance of the designed gate. The correction displacement operator G ĉ can equivalently be done altogether at once after all the repetitions of rounds instead at each round, due to the commutativity of G ĉ with O s

3,
and O f to simplify the experimental setup.
It should be noted that the same nonlinear phase gates can be realized also with a simplified architecture, which does not use all the five five elementary Rabi interactions as is depicted in figure 1. In particular, weak cubic phase gate U 3 may be obtained by using three elementary Rabi interactions g M t X t X g , in (5). These simpler approaches can be useful for first tests of the cubic phase gate but generally perform worse than the five-gate sequence, which is why we will not consider them in the further analysis.
Let us now evaluate the performance of the engineered cubic phase gate by numerical simulations, which were based on the full evolution operators without any approximation or assumption. We start by applying the operation to an initial pure quantum state 0 0 0 r y y = ñá | |and verifying how close the resulting state is to its ideal form. The similarity of the states can be quantified by the fidelity F Tr  figure 2(a) we can see that the fidelity is approaching one for an arbitrary value of desired cubic interaction strength χ 3 by increasing number of repetitions R. This asymptotic behavior holds true for other states such as weak coherent states, while the actual fidelities and the number of Rabi interactions differ. However, for a small interaction strength χ 3 , the fidelity has a weakness as a figure of merit because it draws dominant contribution from the initial input state ρ 0 [15,16]. To avoid this weakness, let us consider a subspace orthogonal to the initial state ρ 0 and formally define the complementary density matrix as ], together with the regular fidelity, then faithfully evaluate the quality of the transformation. In our case, the complementary fidelity F ⊥ also approaches one for increasing R, as is shown in figure 2 for various target strengths. We note that this result also implies that the generated state ρ re becomes pure when R  ¥.
In addition to quantitative figures of merit, we can also look for qualitative ones to check the nonclassical aspects of the engineered states. Wigner functions describe quantum states in quadrature phase space analogous to the classical probability densities which can predict all features of the oscillator. Nonclassical states produced by quantum nonlinearities U m exhibit peculiar properties, one of which is the presence of multiple negative values. Each nonclassical state has a specific pattern of these nonclassical regions and we can therefore check whether the states generated by the approximate cubic phase gate exhibit the same patterns as the ideal cases. When the initial state is the vacuum, the ideal cubic states U 0 3 ñ | exhibit negative fringes in the area given by p 0 < around x=0 [17], where x and p are the eigenvalues of the quadrature operators X and P a a 2 i = -(ˆˆ) † . In figure 3, we have plotted cross sections of Wigner function W(x=0, p) for several cubic states at different strengths and the corresponding number of repetitions. We see that the two Wigner functions overlap nearly indistinguishably, and the approximate gate closely follows the ideal scenario, which confirms the gradual reproduction of the nonlinear dynamics.
To test the validity of our scheme for arbitrary input states, we have considered a set of coherent states represents the amplitude of the state. Coherent states are suitable for feasible experimental verification due to the easiness of generation in an experimental setting, while spanning the entire Hilbert space to form an overcomplete basis. Figure 4(a) shows the region over α and χ where the fidelity with the ideal cubic phase gate is larger than the chosen threshold F>0.99. We can see that even with a relatively low number of repetitions R, a high strength cubic phase gate is achieved with a reasonable fidelity for broad range of amplitudes of coherent states, while a higher number of rounds is necessary for the coherent  The region over various strengths χ 3 and amplitudes α of the input coherent states for which fidelity F of output states generated by implemented cubic phase gates to the ideal state is larger than 0.99. With a relatively low number of repetitions, a high strength cubic phase gate is achieved for broad amplitudes of coherent states. The area of the regions can again be expanded to an indefinite χ 3 when a high R is accessible. (b) The region for which the complementary fidelity F^to the ideal state is larger than 0.95 for the same generated operators. The complementary fidelity is smaller than the full fidelity, but we still can observe a full coverage as well when a larger R is used.
states to achieve the same fidelity as the vacuum state. Figure 4(b) shows, analogously, the area in which the complementary fidelity satisfies F 0.95 > . Both of these results imply that the approximate cubic phase gate scheme is applicable to a wide class of states and that increasing the number of repetitions can extend this range.
The effects of imperfections often hinder the prospects of achieving high-order nonlinearities, as the quantumness is easily erased by the decoherence [34]. In our analysis we consider two possible sources of imperfections: dephasing of the ancillary qubit, and energy loss in the oscillator state. We model both sources of these dominant errors by discrete steps acting after each Rabi gate. The dephasing acts by damping the offdiagonal terms of the qubit density matrix by factor η=e −γt , where γ is a dimensionless phase-damping constant of the two-level system and t is the strength of the respective Rabi interaction. The energy loss is modeled by a beam-splitter-like coupling with transmissivity e s t s h = g between the oscillator and vacuum baths, where γ s is dimensionless energy-damping constant. These imperfections are assumed to apply after each elementary Rabi interaction. In figure 5, we show the maximal fidelity for the target strengths of cubic nonlinearity under different levels of noise. All free parameters (t 1 , t 2 , R) were optimized numerically for each target strength χ 3 . We can see that in these realistic situations the maximal fidelity is reduced but not critically from the fidelity at unity attainable under the ideal conditions. For comparison we also show the values for maximally incoherent benchmark corresponding to γ and γ s approaching infinity. We can see that even though the decoherence reduces the quality of the operation it does so gradually and if the imperfections are low enough to be tolerated. An important question is how the decoherence interacts with the gradual build-up of the nonlinearity, i.e. if the dampings suppress the creation of features associated with strong nonlinear features such as the negative fringes of the Wigner function. These features are related to higher Fock states and their multiple negative regions. We therefore analyze this aspect by investigating the oscillations of the generated Wigner functions at χ 3 = 0.4, with the free parameters (t 1 , t 2 , R) optimized for noiseless case. For clarity we have focused on the behavior of the second negative region which appears in the Wigner function cut W(x=0, p<0). We remind that the dephasing constants γ and loss constants γ s corresponding to overall factors η and η s are applied after each single Rabi interaction. For dephasing, the magnitudes of the negative regions are reduced and the first negative region survives for η>0.9 (corresponding to γ<0.098) while the second one for η>0.935 (γ<0.062). For the loss of oscillator energy, the first and the second negativities survive for η s >0.987 (γ s <0.0121553) and η s >0.990 (γ s <0.0093), respectively. We can also see that the energy loss in the oscillator affects the oscillations more significantly than the dephasing, even though the overall fidelities of the operations are comparable. Nevertheless, the oscillations to negative values of the Wigner function remain still observable. The cold trapped ion and circuit QED are therefore very good platforms for the experimental demonstrations, because their oscillators are very well isolated. This advantage has been demonstrated in the preparation of the large cat states [29,35].
The basic principles applied to the cubic phase gate can be extended towards the fourth-and higher-order nonlinear quadrature phase gates. The fourth-order quartic gate U X exp i 4 for small values of t 1 . In this case, the correction shows that the inherent nonlinear terms can be uncovered by canceling the lower order terms Several trigonometric functions in the exponents can be summed sequentially applying the trigonometric nonlinear gates. For example, the fifth order term required for the quintic gate U 5 can be obtained by combining three O 3 gates to obtain unitary with argument: [ˆ] of (a) χ 4 =0.2 and (b) χ 4 =0.4 on the vacuum. We notice that the overall oscillation is qualitatively analogous. The region for which (c) F>0.99 and (d) F 0.95 > for a quartic gate. The area of the regions can again be expanded to an indefinite χ when a high R is accessible. [ˆ] ( )ˆˆ( )ˆ( ) The errors by the higher-order are typically much smaller than the contribution from the dominant term. In the same way, all powers of X m can be realized by summing the trigonometric functions in (4) and (6). By a serial combination of U k , emulation of interaction operator in any nonlinear potential can be implemented on the trapped ion motion [29] or superconducting electric oscillator [35].
The proposed deterministic scheme thus asymptotically realize an arbitrary unitary nonlinear quadrature phase gate achievable with the current technology. This scheme requires a sequence of controllable Rabi interactions between a harmonic oscillator and a single ancillary two-level system [29,30], employing periodical re-initialization of the two-level system into its ground state. Each individual sequence of Rabi interactions realizes a weak nonlinear quadrature phase gate, and stronger gates can be obtained by repeating the elementary steps. This repetition allows building up the desired operations incrementally and monitoring the dynamic evolution of the systems, even under a modest level of noise. This type of phase gates provides essential elements for simulating arbitrary Hamiltonian dynamics. This approach without a set of specific ancillary states [24] allows closing the universal set of operators in CV quantum computation [1].