Quantum control of continuous systems via nonharmonic potential modulation
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Quantum control of continuous systems via nonharmonic potential modulation Piotr T. Grochowski1,2,3, Hannes Pichler1,2, Cindy A. Regal4,5, and Oriol Romero-Isart1,2,6,7 1Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria 2Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria 3Department of Optics, Palacký University, 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic 4JILA, National Institute of Standards and Technology and University of Colorado, Boulder, Colorado 80309, USA 5Department of Physics, University of Colorado, Boulder, Colorado 80309, USA 6ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 7ICREA, Passeig Lluis Companys 23, 08010 Barcelona, Spain We present a theoretical proposal for preparing and manipulating a state of a single continuous-variable degree of freedom confined to a nonharmonic potential. By utilizing optimally controlled modulation of the potential’s position and depth, we demonstrate the generation of non-Gaussian states, including Fock, Gottesman-Kitaev-Preskill, multilegged-cat, and cubic-phase states, as well as the implementation of arbitrary unitaries within a selected two-level subspace. Additionally, we propose protocols for single-shot orthogonal state discrimination, algorithmic cooling, and correcting for nonlinear evolution. We analyze the robustness of this control scheme against noise. Since all the presented protocols rely solely on the precise modulation of the effective nonharmonic potential landscape, they are relevant to several experiments with continuous-variable systems, including the motion of a single particle in an optical tweezer or lattice, or current in circuit quantum electrodynamics. 1 Introduction The preparation of a continuous-variable system in a non-Gaussian quantum state is of paramount importance in various aspects of quantum science. This ranges from fundamental tests of quantum mechanics [1–5], through the design of quantum sensors [6– 9], to quantum information processing [10–18]. The generation of non-Gaussian states requires a nonlinear resource, often introduced through coupling to an auxiliary degree of freedom, e.g., a two-level system [5,12,15,19–22]. On the other hand, some continuous-variable systems already possess intrinsic nonharmonicity in the potential of a canonical variable (see Fig. 1). Notable examples include the moPiotr T. Grochowski: [email protected] Figure 1: (a) Examples of continuous-variable nonlinear systems that can be optimally controlled without the need for auxiliary systems—single atoms in optical tweezers and fluxtunable transmons. (b) Time-evolved probability density P(x, τ) = |ψ(x, τ)|2, during the state preparation protocol with the snapshots of Wigner functions W(x, p)at the beginning and the end of the protocol. The potential is Gaussian and its optimally controlled position u(τ)is depicted via a solid black line. The top panel shows a comparison between a weak and long sinusoidal drive resonant with the groundfirst excited state transition and an optimized, much faster control. The bottom panel presents an optimal control leading to the GKP state. (c) Excited states occupation numbers |cn(τ)|2=|⟨ψn(x)|ψ(x, τ)⟩|2for ψ0(x)→ψ1(x)Fock excitation protocols of (b)top. The left panel corresponds to the slow Rabi flop, while the right one shows the fast optimal control. Utilizing more states within the nonharmonic potential accelerates and makes the control more versatile. tion of a particle in a trap of finite depth [23,24] and the current in an electric circuit with a Josephson junction [25]. These nonharmonicities in the potential Accepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 1 arXiv:2311.16819v4 [quant-ph] 5 Aug 2025
are typically used to define a qubit within continuousvariable systems [26–30]. In contrast, we investigate whether this intrinsic nonlinearity suffices to implement quantum protocols beyond the two-dimensional subspace. Since it is weaker than the nonlinearity provided by auxiliary two-level systems, we aim to assess how far optimal control can compensate for this limitation. More specifically, we develop protocols to generate a plethora of states, including Fock, GottesmanKitaev-Preskill (GKP) [10], multi-legged-cat [1,31], and cubic-phase [10,32], using optimal control [33–35] of the potential’s position and depth. Furthermore, we employ this control mechanism to design protocols that implement arbitrary unitaries in selected subspaces, enable single-shot orthogonal state discrimination [36,37], spatial state transfer, correct for nonGaussian evolution, and perform algorithmic cooling [38,39]. These protocols could be implemented with single atoms in optical tweezers [40–44] and lattices [23,45] to extend the motional states available in protocols with itinerant particles, e.g., fermionic quantum processors [18]; or with flux-tunable transmons [25,28,46] for minimally invasive state manipulation [cf. Fig. 1(a)]. Previous works, including control of Bose-Einstein condensates [47–56], fast atom transport [45,57,58], ion shuttling [59], cat state creation through two-photon driving [60], and lattice interferometry [61] have demonstrated similar concepts. However, the role of nonharmonic potentials in quantum information processing remains underexplored, despite their natural relevance—particularly for exploiting, for example, the motion of single atoms in scalable optical tweezer arrays [39]. The preparation of continuous-variable modes in quantum non-Gaussian states and their subsequent manipulation have been an object of increasing interest across several platforms, including single neutral atoms [44], trapped ions [62–65], superconducting circuits [66–68], propagating light at the telecommunication wavelength [69,70], etc. The control scheme we present offers a realization of a large variety of relevant tasks and can be applied to a generic continuous-variable platform that involves nonharmonicity. Our protocols take advantage of the controllability of the external potential landscape, naturally offered by, e.g., optical [39,44,45], electric [24], and magnetic [71–73] potentials, and their hybrid combinations [62–65,74]. With such schemes, a high information capacity of bosonic degree of freedom is activated via preparation and control of high-energy and high-quality quantum non-Gaussian states. Our protocols are particularly interesting and useful for systems without well-controlled access to nonlinear resources, e.g., internal degrees of freedom or auxiliary superconducting circuits. However, they can also be combined with such couplings for simultaneous control of both bosonic and spin degrees of freedom, opening new possibilities for, e.g., the use of hyperentanglement [75]. The paper is structured as follows. Section 2introduces details about the physical model and optimization techniques, while Section 3presents several protocols that can be achieved through potential modulation, including state preparation, correcting for non-Gaussian evolution, implementation of unitaries, single-shot measurements, cooling, and spatial state transfer. Further, Section 4analyzes the feasibility of our proposal, addressing the speed limits of the protocols, their robustness against decoherence, and how couplings to other modes can be incorporated into optimization. Section 5concludes the paper with a finishing discussion and an outlook. 2 Control of potential landscape In the following Section, we present two exemplary realizations of one-dimensional continuous-variable systems for which our proposal can be applied—a single neutral atom trapped via optical forces, either in a tweezer or in a lattice, and a flux-tunable transmon. In the Subsection 2.1, we provide details of the involved Hamiltonians in these platforms, introduce a universal description of our control scheme, and address how much nonharmonicity is needed for the presented protocols. The Subsection 2.2 is dedicated to details on the control optimization of our scheme. 2.1 Hamiltonians for different systems As a starting point, we focus on a one-dimensional continuous-variable system described by two conjugate quadrature operators, which may correspond to an arbitrary realization of a single mode, e.g., the motion of a single atom in an optical tweezer [44,76,77] or a lattice [45,78], or phase and charge operators in a flux-tunable transmon [25,79,80]. In the former case, coherent dynamics is driven by a Hamiltonian, ˆ H=ˆ P2 2m+ [1 + a(t)] Vhˆ X−U(t)i,(1) where mis the mass of an atom, ˆ Xis the position operator, ˆ Pis the momentum operator, tis time, and functions a(t)and U(t)control the depth and position of the potential. Here, the trapping potential V(ˆ X) is Gaussian for the tweezer, V(ˆ X) = V01−e−2ˆ X2 w2 0,(2) and a squared sine for the lattice, V(ˆ X) = V0sin22π λˆ X,(3) where w0is the tweezer waist, λis the optical wavelength, and V0is either the tweezer or the lattice Accepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 2
depth. For the transmon case, the Hamiltonian reads ˆ H= 4ECˆn2+EJcos πϕx(t) Φ0s1 + d2tan2πϕx(t) Φ0 ×1−cos ˆφ−dtan πϕx(t) Φ0.(4) Here, the phase difference operator is ˆφ=ˆφ1+ ˆφ2 2,(5) where ˆφi=2πˆ Φi Φ0denotes the phase difference across the ith Josephson junction, and Φ0=h/2eis the magnetic flux quantum. The charge number operator is defined as ˆn=ˆ Q 2e,(6) and the charge energy is EC h=e2 2hC ,(7) where Cis the total capacitance. The total Josephson energy is EJ=EJ1+EJ2,(8) with EJirepresenting the Josephson energy of the ith junction. The asymmetry between the two junctions is characterized by d=EJ2−EJ1 EJ2+EJ1 .(9) Finally, ϕx(t)denotes the time-dependent external magnetic flux threading the SQUID loop. In the leading order, each of these setups is harmonic with frequency ω, yielding natural time τ=ωt, canonical length and momentum scales, ˆx=ˆ X X0 =ˆn n0 =1 √2(ˆa†+ ˆa), ˆp=ˆ P P0 =ˆφ φ0 =1 √2i(ˆa†−ˆa),(10) where ˆa†creates a single excitation and [ˆx, ˆp] = i. For an atom trapped in a tweezer, we have X0= (ℏ2w2 0/4mV0)1/4, P0=ℏ/X0, ω=q4V0/mw2 0,(11) while for the optical lattice, X0= (ℏ2λ2/8π2mV0)1/4, P0=ℏ/X0, ω=p8π2V0/mλ2.(12) For the flux-tunable transmon, the canonical length and momentum scales read n0= (8EC/EJ)1/4, φ0= 1/n0, ω=p8ECEJ/ℏ.(13) In the canonical variables ˆx,ˆp, each of the Hamiltonians (1),(4)can be rewritten as ˆ H ℏω=1 2ˆp2+ [1 + a(τ)] v[ˆx−u(τ)] ,(14) where v=V/ℏωis the dimensionless potential. The control of the system is performed through optimal modulation of position, u(τ), and depth, a(τ), of the potential. While for an atom in an optical tweezer, they are independent controls realized through, e.g., acousto-optic modulator [77], for flux-tunable transmons, they are constrained through a single control function, the intensity of the external flux [79], a(τ) = s1+4pEC/2EJu2(τ) 1+4pEC/2EJu2(τ)d−2−1, u(τ) = d 2ηtan πϕx(τ) Φ0 .(15) Such modulations have been realized in various other experimental platforms [45,54,55,61,81], where the choice of either position or depth control depends on feasibility within a specific setup. For example, Paul traps allow for both position and depth control via varying current through electrodes generating the electric field [54], while in optical lattices, phase control allows very precise position control [45]. As for optical tweezers, the use of multiple radiofrequency tones makes a powerful knob for the time-dependent control of both position and depth [44]. In general, we consider the shape of the potential to be symmetric in x, and we find it useful to express it as v(x) = 1 2η2v0(ηx)≈1 2x2−1 6η2x4(16) when expanded up to the second order in the small parameter η. Note that, as the potential v(x)is beyond quadratic in x, the ensuing dynamics is nonlinear, i.e., it does not preserve the Gaussianity of the states. For the examples considered in this work, the tweezer potential then reads as v0(x) = 3 21−exp −2 3x2,(17) while for the lattice it is v0(x) = sin2x, (18) and for the flux-tunable transmon, v0(x) = 1 2[1 −cos (2x)] (19) Accepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 3
Note that we have defined the units so that the functional forms of these potentials match up to the second leading order, allowing for a unified discussion of nonharmonicity. Here, ηis the measure of the nonharmonicity of the potential, comparing the canonical length to the characteristic length scale of the potential, and translates directly into an energy level splitting, ω21 −ω10 ω=−1 2η2,(20) where ωij is a transition frequency between ith and jth levels. For instance, for an atom in a tweezer, it is η=√3X0 w0 ;(21) for an optical lattice, it is a Lamb-Dicke parameter η=2πX0 λ;(22) for the flux-tunable transmon, it is a monotonic function of transmon anharmonicity α, α ω=rEC 8EJ =1 2η2;(23) and for Kerr oscillators, it can be associated with Kerr nonlinearity κ∼ωη2/24 [82]. The typical values of physical parameters for the examples introduced in this section lead to nonharmonicities no larger than η∼0.5(see Tab. 1) and can be tuned down at least an order of magnitude depending on the specific setup. Hence, here we will analyze such a parameter regime. Nevertheless, many other experimental platforms are characterized by much lower values, such as trapped ions [24,83] or levitated mechanical oscillators [84,85]. There, lower nonharmonicity is associated with either larger potential length scales or larger masses of trapped objects. In such cases, the additional enhancement of nonharmonicity is needed, e.g., through state excitation [85–87], to generate nonGaussian states. Optically trapped atom Superconducting circuit m101-102uEC/h MHz-GHz w0,λ102-103nm EJ101-104EC V0/kBµK-mK d0-1 Table 1: Typical physical parameters for atoms held in optical tweezers or optical lattices and flux-tunable transmons. 2.2 Control optimization Before presenting specific protocols, let us discuss possible methods of designing controls u(τ)and a(τ). Within each of the subsequent protocols, we aim to achieve some specific goal—be it state preparation, unitary implementation, or others—that can be quantified through a reward function that ought to be maximized. A set of methods for designing timevarying controls for the maximization of such a reward function is collectively called quantum optimal control (QOC) [33,34,88]. Introduced in the eighties and since then broadly developed, QOC has become one of the main quantum control tools, among other optimization approaches, such as adiabatic passages [89], shortcuts to adiabaticity [90], and composite pulses [91]. Various techniques have been established to design the control pulses, relying on both gradient-free and gradient-based approaches. The latter assume the ability to differentiate the reward function and include, among others, GRAPE (gradient ascent pulse engineering)- [92] and Krotov-like [93,94] techniques for optimizing controls that are piecewise constant in time. On the other hand, the former rely only on the evaluation of the reward function itself. The examples are numerous, including NelderMead [95], evolution strategies [96], simulated annealing [97], and many others. Notably, recent years have brought a rapid surge in the use of machine learning approaches for quantum control problems [98]. The choice of approach relies on constraints given by a specific experimental setup, including how accurately and fast we can solve the dynamics and what the limitations of control pulses are, such as maximal Fourier bandwidth and intensity. We choose to utilize the dressed chopped random-basis technique (dCRAB) [99] that is based on the Nelder-Mead gradient-free method. Here, the control function is expanded into a Fourier basis with randomized frequencies and a high-frequency cutoff corresponding to approximations of experimentally accessible bandwidths, i.e., u(τ) = Np X k=1 akcos νkτ+bksin νkτ, (24) where akand bkare parameters to be optimized, νk are frequencies that are probabilistically drawn from the uniform distribution in the range [0, νmax][100], νmax is the frequency cutoff, and Npis the number of frequency components. We have chosen this particular gradient-free method as we consider several different reward functions, it naturally implements a frequency cutoff for the control function, and there are available well-developed open-source packages, including the Quantum Optimal Control Suite [35], which we utilize for the optimization. However, depending on the specifics of the experimental implementation, our results can be achieved with an alternative approach, such as, e.g., GRAPE. Generally speaking, controllability of the system described by Eq. (1)depends sensitively on the shape and depth of the potential. For shallow potentials, the finite number of bound states can limit the set Accepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 4
of achievable state transformations. However, in the case of sufficiently deep or unbounded potentials— such as the quartic potential—full controllability can be established under certain control schemes [101]. The chosen potential landscape is then crucial in determining the accessible Hilbert space and, consequently, the effectiveness of optimal control protocols. For all the simulations, the split-step method [102] was used to simulate the dynamics on a spatial grid. In the case of one-dimensional calculations, throughout optimization runs, the numerical parameters read: number of spatial grid points Nx= 28, where the grid spanned x∈[−13,13], and number of time grid points Nτ= 500. After optimizing the pulses, the dynamics was run and further optimized with increased accuracy to make sure the results are converged. The optimization has been performed with QuoCS library version 0.0.44 [35]. The number of Fourier components used for dCRAB varied from 20 to 50, depending on a particular protocol, and was randomized from a uniform distribution on a bandwidth that also varied between different protocols. The choice of these optimization parameters was fine-tuned for each of the protocols, however, it was not exclusive— different choices usually yielded similar values of specific reward functions, but with different total optimization iterations. Note that further optimization of all the protocols towards lower infidelities is possible and would involve further fine-tuning of optimization parameters. 3 Protocols In the following section, we discuss several protocols that can be implemented via a nonharmonic potential modulation. It includes a versatile Gaussian (Sec. 3.2) and non-Gaussian state preparation, both in a single- (Sec. 3.1) and double-well (Sec. 3.5) potential landscape, as well as the implementation of unitaries (Sec. 3.4), single-shot measurements (Sec. 3.6), cooling (Sec. 3.7), and correcting state evolution for nonlinear evolution due to nonharmonicities (Sec. 3.3). 3.1 State preparation As the first type of protocol, we consider state preparation. The initial state of the system is assumed to be the ground state ψ0(x)of the nonharmonic potential v(x), well approximated by the wave function ψ0(x)≈1 π1/4e−x2/2,(25) which is the ground state of the leading harmonic approximation, x2/2. For this protocol, we choose to control only the position of the potential u(τ), to maximize the fidelity F=|⟨ψ(x, τ =τmax)|ψT(x)⟩|2(26) Figure 2: Examples of a state preparation protocol involving Gaussian [(a), (c), (e), (g)] and cosine [(b), (d), (f), (h)] single-well potentials, with η= 0.25 and τmax/2π= 6, using only optimized position of the potential u(τ)in the case of the Gaussian potential. The protocols include second excited Fock [(a-b)], cat [(c-d), s= 9], cubic-phase [(e-f), κ= 2,r= 0.7], and GKP [(g-h), r= 0.6, (d1, d2, d3) = (−√4π, 0,√4π),(h1, h2, h3) = (1,2,1)] state preparation. Protocol fidelities Fare: (a) 99.8% (b) 99.8% (c) 92.7% (d) 93.4% (e) 97.2% (f) 94.9% (g) 99.8% (h) 99.8%. See the caption of Fig. 1for subplot and curve legend details. with the target state ψT(x). We consider the preparation of several states, including Fock, finite GKP, cubic-phase, and cat states with high fidelity in a one-dimensional geometry. As these states are nonGaussian, the role of nonharmonicity of the potential is evident—if it was quadratic, and dynamics linear, the Gaussianity of the state would be preserved durAccepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 5
ing the evolution. In Fig. 1(b), we present an example of the excitation protocol, where we specialize to a Gaussian potential with η= 0.25. The target state is a finite GKP state, ψGKP(x) = N 3 X i=1 hier/2ψ0[er(x−di)] (27) with (h1, h2, h3) = (1,2,1),(d1, d2, d3) = (−√4π, 0,√4π), the squeezing parameter r= 0.6, the normalization constant N, and the whole protocol is set to take τmax/2π= 6. The presented protocol yields fidelity F ≈ 99.8%. The further examples of the state preparation are presented in Fig. 2, both for the Gaussian potential and for cosine potential with the constraint (15),η= 0.25 and d= 0.8. Specifically, we additionally show the second excited Fock state, ψ2(x) = 1 √2 1 π1/42x2−1e−x2/2,(28) the squeezed cubic-phase state, ψcub(x) = eiκx3er/2ψ0(erx),(29) and the cat state, ψcat(x) = 1 √2[ψ0(x+s/2) + ψ0(x−s/2)],(30) where κis the cubicity, ris the squeezing parameter, and sis the separation between the coherent states. The specific values of the state parameters and fidelities are shown in the caption. 3.2 Gaussian operations As we consider nonlinear dynamics, a relevant question is whether it is still possible to implement Gaussian operations within this framework. In our setting, this typically involves protocols that transiently generate non-Gaussian states during the evolution, but yield a final state that is Gaussian. A relevant example is the preparation of squeezed vacuum states, for which we demonstrate two distinct realizations. The first protocol employs only displacement modulation, while the second relies solely on depth modulation. Both cases are shown in Fig. 3and achieve high final fidelities. The second example represents a generalized version of the frequency-jump protocol, in which a sudden change in potential depth excites breathing in the wave function, resulting in squeezing. In our case [Fig. 3(b)], the achieved squeezing surpasses what would be expected from a single-frequency-jump protocol, which is fundamentally limited by the ratio of initial and final frequencies. Figure 3: Gaussian squeezing of the initial ground state via (a) displacement and (b) modulation. The potential is taken to be Gaussian with η= 0.15. Protocols fidelities read: (a) 99.8% (b) 99.6%. See the caption of Fig. 1for subplot and curve legend details. Figure 4: (a) Temporal evolution of a cat state in a perfectly harmonic potential. It rotates in the phase space and fully revives every trap period. (b) Evolution of a cat state in a tweezer characterized by η= 0.25. (c) Evolution of a cat state in a tweezer characterized by η= 0.25, however, with optimally controlled potential position and depth. (d) Fidelities between a cat state rotating in a perfectly harmonic potential and in Gaussian potentials with different nonharmonicities without (shades of green) and with (red) optimal control. See the caption of Fig. 1for subplot and curve legend details. 3.3 State stabilization Note that undriven evolution is nonlinear and, hence, the prepared state gets distorted during the evolution. In contrast, when evolved in the perfectly harmonic potential, the state undergoes rotation in the phase space with the trap period. In a realistic setup, such a state preservation in the rotating frame is deAccepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 6
stroyed by the nonharmonicity of the potential. The remedy lies in either decreasing the nonharmonicity or, alternatively, mimicking perfect phase space rotation through optimal control [60]. We present the latter method, which consists of two optimization steps. The first one involves using the average fidelity with respect to the perfectly rotating state during a multiple of the trap period τmax =n2πas a reward function, Frot =1 τmax Zdτ|⟨ψ(x, τ)|ψrot(x, τ)⟩|2,(31) where |ψrot(τ)⟩=e−iˆa†ˆaτ |ψ(0)⟩(32) is the initial state undergoing a phase-space rotation in a perfectly harmonic trap. After optimizing Frot, the next step involves maximizing the final fidelity with the initial state after a single trap period, F=|⟨ψ(x, τmax)|ψ(x, 0)⟩|2.(33) The second step guarantees high fidelity, should the protocol be applied consecutively many times, while the first one ensures that the state is not distorted at any time. In Fig. 4, we present an example of an optimally controlled cat state stabilization. The optimization is conducted for a single trap period, τmax = 2π, with both position and depth control of a Gaussian potential. Subsequently, the optimized control is applied six times consecutively over six trap periods and compared with the bare evolution in a Gaussian potential. The results showcase that the presented optimal stabilization of the state surpasses that of a Gaussian potential with a relatively low nonharmonicity η= 0.05, within this particular time scale. 3.4 Unitaries The next type of protocol involves the implementation of a specific unitary within a selected subspace. For simplicity and relevance to several proposals and realizations of bosonic qubits [13–17,62,103,104], we analyze an example of a two-level subspace. This subspace can be spanned by any two orthogonal states ψ±(x), including two lowest-lying vibrational states ψ0(x)and ψ1(x)(Fock basis), two mutually displaced GKP states (GKP basis), four-, and two-legged-cat bases. For the protocol considered, the reward function is the subspace average gate fidelity [105] Fˆ U=1 6[Tr (ˆ Mˆ M†) + |Tr ˆ M|2],(34) where ˆ M=ˆ Pˆ UTˆ Uˆ P, (35) Figure 5: (a,b) Two orthogonal four-legged-cat states spanning a two-level subspace evolved in a positionand depthcontrolled single Gaussian well, realizing a σxoperation. (c) Cat state preparation using a double-well optimal control. The superscript L(R) signifies the state centered in the left(right) potential well. (d,e) State transfer between the wells—each of the vibrational states, ψ0(x)and ψ1(x), is transferred to the other well, without altering the relative phase. See the caption of Fig. 1for subplot and curve legend details. ˆ Pis a projector onto a subspace, ˆ UTis a target unitary, and ˆ Uis a unitary generated through Eq. (14). Taking advantage of the optimized position of the potential u(τ)and additional slight, optimized modulation of the depth a(τ)(to increase the fidelity of the protocol), we present examples of unitaries, e.g., σxor Hadamard operations, for the above-mentioned subspaces. In Fig. 5(a,b), we show an exemplary case of four-legged-cat basis and σxoperation performed with a Gaussian potential, where the basis states are given by ψ± 4LC(x) = N±[φβ(x) + φ−β(x)±φiβ(x)±φ−iβ(x)] , (36) respectively [106,107]. Here, the coherent state reads φβ(x) = 1 π1/4exph−(x−√2 Re β)2/2 + i√2xIm βi, (37) N±are the normalization constants, and we use β= 2. Again, the protocol is performed with high fidelity, Accepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 7
Fˆ U≈99%. In Fig. 6, we show additional examples for the Gaussian potential, including, among others, the GKP basis, where the basis is spanned by two mutually displaced GKP states, ψ±=ψGKP(x±d3/4),(38) where d3is defined as in (27)and the squeezing and displacement need to be enough for the states not to overlap. Further unitary implementations, but for the cosine potential with the constraint (15), are presented in the Appendix A. 3.5 Double-well potential landscape Up to now, we have analyzed quantum dynamics taking place in a single well of a potential landscape. Here, we bring our attention to the nonharmonic potential landscape case that involves two independently controlled potential wells. Such a realization is available for optical lattices, optical tweezers [42,77,108– 110], or various circuit quantum electrodynamics setups [111]. First, we address the state preparation and single-particle unitary implementation with two potential wells. We assume that we have two independently controlled wells at our disposal, such that the total potential reads v(x) = X i={1,2} 1 2η2v0[η(x−ui(τ))] (39) with two independent position control functions, ui(τ). Here, we assume that v0contains only a single well with some characteristic width. Controlling the relative distance between the wells amounts to changing the barrier height, which affects the coupling between the bound states in each of the wells. It can be understood as mode multiplexing [112], accelerated through the optimal control. In Fig. 5(c), we show a balanced cat state preparation utilizing two Gaussian potential wells, ψ0(x±s/2) →1 √2[ψ0(x±s/2) + iψ0(x∓s/2)], (40) where sis the separation between the wells. Note that the presented control is symmetric, u1(τ) = −u2(τ) = u(τ), and the instantaneous separation between the wells is constrained in the optimization. The fidelity of the presented protocol yields F ≈ 99%. The cat states produced in such a manner are pertinent to the fundamental tests of quantum mechanics, especially for massive objects. This protocol, along with a fast optimized transport [45], should allow for a fast macroscopic cat state creation and interferometric protocols. The double-well potential has also been shown to provide a platform for fault-tolerant quantum computing with so-called Kerr-cats [15]. There, the computational subspace is spanned by superpositions of Figure 6: Evolution of selected orthogonal states in a Gaussian potential characterized by η= 0.25 and with optimally controlled displacement u(τ)and intensity a(τ). The solid line shows u(τ)and the protocol lasts τmax/2π= 9. (a-b) σxunitary within a subspace spanned by ψ0and ψ1Fock states. The fidelity reads Fˆ U≈99.8%. (c-d) Hadamard unitary within a subspace spanned by GKP states with r= 0.7,(d1, d2, d3) = (−√6π, 0,√6π). The fidelity reads Fˆ U≈96.4%. (e-f) σyunitary within a subspace spanned by four-legged-cat states states with β= 2. The fidelity reads Fˆ U≈96.2%. See the caption of Fig. 1for subplot and curve legend details. the ground states of the respective wells, corresponding to the (almost) degenerate ground-state manifold of a full double-well potential, ψ± KC(x) = 1 √2[ψ0(x+s/2) ±ψ0(x−s/2)].(41) With optimal control, one can again realize arbitrary unitaries within this subspace, utilizing the subspace average gate fidelity (34). Examples of σx,σy, and Hadamard unitaries are presented in detail in Appendix B. Furthermore, independent control of two wells can also be used to perform a state-preserving transport between the wells [113]. Here, the initial state is prepared within a two-dimensional subspace spanned by two orthogonal states localized in the left Accepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 8
(L) well, ψL ±(x) = ψ±(x+s/2).(42) The aim of the protocol is to transfer such a state to a subspace localized in the right (R) well, ψR ±(x) = ψ±(x−s/2).(43) Then, if we introduce the following matrix, ˆ M′=ψL +(x, τmax)ψR +(x) ψL +(x, τmax)ψR −(x) ψL −(x, τmax)ψR +(x) ψL −(x, τmax)ψR −(x), (44) where ψL ±(x, τ)is evolved according to the potential (39), we can define the state transfer fidelity, Fst =1 6[Tr (ˆ M′ˆ M′†) + |Tr ˆ M′|2],(45) as the reward function. In Fig. 5(d,e), we show the example of a state of the two lowest vibrational levels of the left well transferred to the right one with Fst ≈ 99%. 3.6 Single-shot discrimination After presenting state preparation and unitary implementation in various nonharmonic potential landscapes, we move on to two protocols that perform single-shot discrimination between two orthogonal states ψ±(x)(see Fig. 7), providing an alternative to already existing methods involving auxiliary systems in, e.g., superconducting circuits [36]. The first one involves a single nonharmonic well and is performed through imprinting opposite momentum kicks onto each of the states via a potential displacement, ψ±(x)→ψ± k(x) = ψ±(x)e±ikx,(46) where kis chosen such that two phase-imprinted states are nonoverlapping in phase space. After the phase imprinting, a selective measurement takes place, which can be realized in, e.g., an optical tweezer setup through the subsequent release of the trap, in which time-of-flight evolution reveals spatially separated detection clicks for each of the states [44,114]. The reward function for such a momentum-kick protocol involves a sum of equally weighted fidelities, Fmk =1 2X j=±Dψj(x, τmax)ψj k(x)E 2(47) Note that in contrast to the unitary implementation, here the final relative phase between the states does not matter, as we aim only to distinguish the initial states. As an example, in Fig. 7ψ±(x)are taken to be the two lowest eigenstates of the well, and the protocol is shown in a schematic way. In Fig. 8, we show in detail specific realizations of this and other momentumkick protocols, utilizing a double-well Gaussian potential. They also involve other choices of orthogonal Figure 7: Two proposed discrimination protocols, based on a phase-space separation. (i) The opposite momentum kicks are imprinted onto each of the states. (ii) States are separated spatially via a second potential well. Figure 8: Examples of the implementation of momentum kick protocols via an optimally controlled single-well Gaussian potential with η= 0.25,τmax/2π= 6, and k= 2. (a-b) ψ+(x) = ψ0(x)and ψ−(x) = ψ1(x), (c-d) ψ±(x)=[ψ0(x)±ψ1(x)]/√2, and (e-f) ψ±(x) = [ψ0(x)±iψ1(x)]/√2. Momentum-kick fidelities read (a-b) Fmk ≈98.5%, (c-d) Fmk ≈99.4%, and (e-f) Fmk ≈99.5%. See the caption of Fig. 1for subplot and curve legend details. states for the discrimination, namely eigenstates of both σxand σyPauli matrices. Such discrimination procedures then correspond to σxand σymeasurements within a given two-level subspace. The second discrimination protocol involves a Accepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 9
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A Single-well two-level unitary implementations In this section, we present examples of the implementation of selected unitaries within a two-level subspace spanned by either Fock, GKP, or four-legged-cat states. Fig. 6shows results for a single-well Gaussian potential, while Fig. A1 concerns a cosine one. Figure A1: Evolution of selected orthogonal states in a cosine potential characterized by η= 0.2and with optimally controlled displacement u(τ)and intensity a(τ)that are constrained through the flux control of a flux-tunable transmon with d= 0.8. The solid line shows u(τ)and the protocol lasts τmax/2π= 9. (a-b) Hadamard unitary within a subspace spanned by |0⟩and |1⟩Fock states. The fidelity reads Fˆ U≈99.99%. (c-d) σyunitary within a subspace spanned by GKP states, ψ±=ψGKP(x±d3/4) with r= 0.7,(d1, d2, d3) = (−√6π, 0,√6π). The fidelity reads Fˆ U≈89.5%. (e-f) σxunitary within a subspace spanned by four-legged-cat states states with β= 2. The fidelity reads Fˆ U≈94.0%. See the caption of Fig. 1for subplot and curve legend details. B Double-well two-level unitary implementations In Fig. B1, we present examples of the implementation of selected unitaries within a two-level subspace spanned by Kerr-cat states, ψ±(x) = [ψ0(x+s/2) ± iψ0(x−s/2)]/√2with s= 9 via a double-well Gaussian potential with η= 0.25. Figure B1: Evolution of selected orthogonal Kerr-cat states in a double-well Gaussian potential with optimally controlled position displacements u1(τ)and u2(τ). The solid lines show u1(τ)and u2(τ)and the protocol lasts τmax/2π= 6. (a-b) σxunitary. The fidelity reads Fˆ U≈99.2%. (c-d) σxunitary. The fidelity reads Fˆ U≈90.9%. (e-f) Hadamard unitary. The fidelity reads Fˆ U≈95.9%. See the caption of Fig. 1for subplot and curve legend details. Accepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 19
C Selective stealing protocols In Fig. C1, we present examples of the implementation of selective stealing protocols via an optimally controlled double-well Gaussian potential with η= 0.25 and τmax/2π= 6. Figure C1: Spatial separation of orthogonal states ψ±(x+ s/2) →ψ±(x∓s/2) with s= 9 for (a-b) ψ+(x) = ψ0(x) and ψ−(x) = ψ1(x), (c-d) ψ±(x)=[ψ0(x)±ψ1(x)]/√2, and (e-f) ψ±(x) = [ψ0(x)±iψ1(x)]/√2. Fidelities read (ab) Fˆ U≈98.9%, (c-d) Fˆ U≈98.9%, and (e-f) Fˆ U≈97.4%. See the caption of Fig. 1for subplot and curve legend details. D Fourier transforms of control functions In this section, we present Fourier transforms of displacement control functions, Fu=Zu(τ) exp−i˜ω ωτ ,(62) for all the presented protocols. Specifically, Figs. D1,D-5, and D-9 pertain to the figures from the main text, while Figs. D-2,D-6,D-A1,D-B1,D-8,D-C1, and D-11 are associated to the Supplemental Materials. Fig. 1(b)top Fig. 1(b)center Fig. 1(b)bottom 2 4 6 8 10 Figure D-1: Fourier transforms of displacement control functions u(τ)from Fig. 1in the main text. The vertical axis is in arbitrary units, normalized to the highest frequency contribution. Accepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 20
Fig. 2(a) Fig. 2(b) Fig. 2(c) Fig. 2(d) Fig. 2(e) Fig. 2(f) Fig. 2(g) Fig. 2(h) 2 4 6 8 10 Figure D-2: Fourier transforms of displacement control functions u(τ)from Fig. 2. The vertical axis is in arbitrary units, normalized to the highest frequency contribution. Fig. 3(a) 2 4 6 8 10 Figure D-3: Fourier transform of displacement control function u(τ)from Fig. 3(a). The vertical axis is in arbitrary units, normalized to the highest frequency contribution. Fig. 5(a,b) Fig. 5(c) Fig. 5(d,e) Fig. 5(d,e) 2 4 6 8 10 Figure D-5: Fourier transforms of displacement control functions u(τ)from Fig. 5in the main text. The vertical axis is in arbitrary units, normalized to the highest frequency contribution. Fig. 6(a,b) Fig. 6(c,d) Fig. 6(e,f) 2 4 6 8 10 Figure D-6: Fourier transforms of displacement control functions u(τ)from Fig. 6. The vertical axis is in arbitrary units, normalized to the highest frequency contribution. Accepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 21
Fig. 8(a,b) Fig. 8(c,d) Fig. 8(e,f) 2 4 6 8 10 Figure D-8: Fourier transforms of displacement control functions u(τ)from Fig. 8. The vertical axis is in arbitrary units, normalized to the highest frequency contribution. Fig. 9 Fig. 9 2 4 6 8 10 Figure D-9: Fourier transforms of displacement control functions u(τ)from Fig. 9in the main text. The vertical axis is in arbitrary units, normalized to the highest frequency contribution. Fig. 11(a) Fig. 11(b) Fig. 11(c) 2 4 6 8 10 Figure D-11: Fourier transforms of displacement control functions u(τ)from Fig. 11. The vertical axis is in arbitrary units, normalized to the highest frequency contribution. Fig. A1(a,b) Fig. A1(c,d) Fig. A1(e,f) 2 4 6 8 10 Figure D-A1: Fourier transforms of displacement control functions u(τ)from Fig. A1. The vertical axis is in arbitrary units, normalized to the highest frequency contribution. Fig. B1(a,b) Fig. B1(a,b) Fig. B1(c,d) Fig. B1(c,d) Fig. B1(e,f) Fig. B1(e,f) 2 4 6 8 10 Figure D-B1: Fourier transforms of displacement control functions u(τ)from Fig. B1. The vertical axis is in arbitrary units, normalized to the highest frequency contribution. Accepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 22
Fig. C1(a,b) Fig. C1(a,b) Fig. C1(c,d) Fig. C1(c,d) Fig. C1(e,f) Fig. C1(e,f) 2 4 6 8 10 Figure D-C1: Fourier transforms of displacement control functions u(τ)from Fig. C1. The vertical axis is in arbitrary units, normalized to the highest frequency contribution. E Depth modulations Some of the protocols, presented in both the main text and the supplementary materials, utilized additional optimal control of the potential’s depth, a(τ). In Figs. E-5,E-6, and E-8, these control functions are shown. Fig. 5(a,b) -0.2 0.2 Fig. 5(d,e) 12345 -0.2 0.2 Figure E-5: Depth control functions a(τ)from Fig. 5. Fig. 6(a,b) -0.2 0.2 Fig. 6(c,d) -0.2 0.2 Fig. 6(e,f) 12345678 -0.2 0.2 Figure E-6: Depth control functions a(τ)from Fig. 6. Fig. 8(a,b) -0.2 0.2 Fig. 8(c,d) -0.2 0.2 Fig. 8(e,f) 12345 -0.2 0.2 Figure E-8: Depth control functions a(τ)from Fig. 8. Accepted in Quantum 2025-08-04, click title to verify. Published under CC-BY 4.0. 23