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1

Robust preparation of Wigner-negative states

with optimized SNAP-displacement sequences

Marina Kudra,1, ∗ Mikael Kervinen,1 Ingrid Strandberg,1 Shahnawaz Ahmed,1 Marco

Scigliuzzo,1 Amr Osman,1 Daniel Perez Lozano,1 Giulia Ferrini,1 Jonas Bylander,1 Anton

Frisk Kockum,1 Fernando Quijandrıa,1, † Per Delsing,1, ‡ and Simone Gasparinetti1, §

1Department of Microtechnology and Nanoscience,

Chalmers University of Technology, 412 96 Gothenburg, Sweden

(Dated: November 16, 2021)

Hosting non-classical states of light in three-dimensional microwave cavities has emerged as apromising paradigm for continuous-variable quantum information processing. Here we experimen-tally demonstrate high-fidelity generation of a range of Wigner-negative states useful for quantumcomputation, such as Schrodinger-cat states, binomial states, Gottesman-Kitaev-Preskill (GKP)states, as well as cubic phase states. The latter states have been long sought after in quantumoptics and were never achieved experimentally before. We use a sequence of interleaved selectivenumber-dependent arbitrary phase (SNAP) gates and displacements. We optimize the state prepa-ration in two steps. First we use a gradient-descent algorithm to optimize the parameters of theSNAP and displacement gates. Then we optimize the envelope of the pulses implementing theSNAP gates. Our results show that this way of creating highly non-classical states in a harmonicoscillator is robust to fluctuations of the system parameters such as the qubit frequency and thedispersive shift.

I. INTRODUCTION

A promising approach to realize a quantum com-puter is to encode the quantum information into bosonicmodes [1, 2]. This approach has several advantages com-pared to a quantum computer based on two-level sys-tems [3]. First, the information is redundantly encodedin the infinite-dimensional Hilbert space of a harmonicoscillator. Choosing the encoding wisely, paired with asingle dominant error channel, single-photon loss, makesquantum error correction possible in a hardware-efficientway [4–6]. Moreover, non-Clifford gates can be imple-mented more efficiently on this platform [2, 7, 8].The basic building blocks of the bosonic quantum com-

puter are nonclassical states generated in a harmonic os-cillator. The nonclassicality of the states is convenientlycharacterised by the negativity of their Wigner function,which is a necessary resource for quantum computationaladvantage [9, 10]. One way to generate these states isto resonantly exchange excitations between a frequency-tunable ancillary qubit and the oscillator [11–13]. Thisapproach works well for states containing up to a fewphotons, thanks to fast swaps enabled by resonant in-teraction, but does not scale well beyond that, since oneneeds to climb the Fock-state ladder in a sequential man-ner to reach higher photon numbers.Another approach is to use optimal control pulses in a

system comprising a fixed-frequency qubit and an oscil-

∗ kudra@chalmers.se† Present address: Quantum Machines Unit, Okinawa Institute

of Science and Technology Graduate University, Onna-son, Oki-

nawa 904-0495, Japan.‡ per.delsing@chalmers.se§ simoneg@chalmers.se

lator in the strong dispersive regime [14]. This approachworks well, yielding high-fidelity states, because it ad-dresses all the relevant transitions at once. However, itrelies on very accurate estimation of the parameters ofthe model Hamiltonian, and fine-tuning of the drive pa-rameters. In addition, it works like a “black box” inwhich the system dynamics at intermediate times can-not be used to gain insight into the mechanisms thatlimit the fidelity of the state preparation. By contrast,a gate based approach using displacement and selectivenumber-dependent arbitrary phase (SNAP) gates [15, 16]offers a more transparent alternative. Not only are thesetwo gates enough for universal control of the harmonicoscillator [16], but they can also, in theory, be applied inan efficient way to greatly reduce the number of requiredgates [17]. A potential downside is that the standardimplementation of the SNAP gates limits the achievablefidelity of the target states, because the duration of thepulses must be significantly longer than the inverse of thedispersive interaction between the qubit and the oscilla-tor [15], which makes these gates slow and as a resultmore prone to decoherence effects.

In this work, we experimentally demonstrate high-fidelity generation of a variety of Wigner-negative statesin a microwave cavity by applying up to three interleaveddisplacement and SNAP gates. We show that statesused in error-correction protocols, such as the Fock, bino-mial [6, 18], and cat states [5, 19], can be prepared withfidelities around 0.96. We also show that complex quan-tum states enabling universal continuous-variable quan-tum computing when supplied to Gaussian circuits suchas the Gottesman-Kitaev-Preskill (GKP) state [7, 20–22]and the cubic phase state [7, 23–25] can be generated. Inparticular, the latter state has to our knowledge not beensuccessfully experimentally implemented before [26, 27].

2

FIG. 1. (a) Schematic representation of the pulse sequenceused to prepare and characterize the states. C, Q and R standfor cavity, qubit and readout resonator respectively. Dαi

are

the displacement and S(~θi) are the SNAP gates. Ryπ/2 are

the cavity state independent π/2 pulses. They are separatedin time by 1/2χ, where χ is the dispersive shift between thequbit and the cavity. M is the measurement. (b) There aretwo steps of optimization that we perform to find the desireddisplacement and SNAP gates. (c) Drawing of the stub geom-etry cavity coupled to the transmon qubit. (d) Enlarged chipcontaining the stripline readout resonator and the transmonqubit.

In contrast to previous work [14, 15], we optimize thepulse sequences with a two-step approach. First we op-timize the parameters that define the applied SNAP anddisplacement gates using a variation of an algorithm pro-posed by Fosel et al. [17]. Then we numerically optimizethe envelope of individual SNAP pulses to decrease thepulse length by about a factor of 8, which results in afivefold decrease in infidelity. Moreover, we show thatoptimizing the SNAP pulses makes these gates more ro-bust to fluctuations or erroneous estimation of systemparameters such dispersive shift and the qubit frequency,compared to the standard SNAP gates [15].

II. METHODS

The experiments are conducted on a circuit quan-tum electrodynamics platform. The system consists ofa three-dimensional (3D) cavity storage resonator, a pla-nar stripline readout resonator, and a superconductingtransmon qubit [28]. A 3D cavity in a coaxial λ/4 stubgeometry [29] is machined into a block of high-purity

aluminum. The cavity is first chemically etched to re-move surface damage from the machining and then subse-quently annealed [30]. A single-junction transmon qubitand the micro-strip readout resonator are fabricated ona silicon chip, which is then inserted into the 3D cavity(Fig. 1(c)-(d)). It is measured inside a dilution refriger-ator with a base temperature below 10mK.The device is controlled with two input lines: one for

driving the cavity and one for driving the readout res-onator and the qubit. The measurement and controlpulses are generated and read out with the microwavetransceiver platform Vivace [31]. Pulsed waveforms aresampled at a 2GSa/s rate and up-converted to microwavefrequencies by mixing them with continuous-wave carriertones. The signals are sent to the device through a seriesof attenuators and filters. The readout signal is ampli-fied by a chain having a Josephson travelling-wave am-plifier [32] as the first amplifier, down-converted to anintermediate frequency, and acquired in the time domainat a sampling rate of 2GSa/s by Vivace.The relevant part of the qubit-storage cavity system is

described by the Hamiltonian [15]

H = ωca†a− Kc

2(a†)2a2 + ωqb

†b

−χa†ab†b− χ′(a†)2a2b†b. (1)

Here ωc and ωq are the resonance frequencies of the cav-ity and the transmon qubit, respectively, Kc is the Kerrnonlinearity of the cavity, χ is the dispersive shift be-tween the cavity and the transmon qubit, χ′ is the firstorder correction to the dispersive shift, a† (a) is the cre-ation (annihilation) operator of the cavity field, and, sim-ilarly, b† (b) is the raising (lowering) operator of the qubit.When doing optimization or simulation we only considerthe lowest two energy levels of the qubit. The full Hamil-tonian, its parameters and the coherence properties of thesystem are listed in Appendix A.To prepare an arbitrary state in the cavity, we apply an

interleaved sequence of n+1 displacement operations andn SNAP gates [Fig. 1(a)]. Crucially, to implement thissequence, we introduce a two-step optimization routine[Fig. 1(b)]. The first optimization step takes the targetstate as an input and computes (see Appendix D) the am-plitudes of the displacement gates (α1,α2,...,αn+1) and

phases of the SNAP gates (~θ1,~θ2,...,~θn), for a fixed num-ber n of SNAP gates. The ith SNAP gate is parametrized

by the vector ~θi = (θi,0, θi,1, ..., θi,m) and amounts tosimultaneously applying the phase θi,j to the jth Fockstate, with j = 0, 1, . . . ,m. In this work, we have usedn = 2, 3 and m = 14, 17. In the second step, we calculatethe optimal pulse envelopes to implement the required

SNAP gates S(~θi), based on the measured parameters ofthe Hamiltonian in Eq. (1), using the optimization toolBoulder Opal [33]. The tool solves for the unitary dy-namics of the system in the presence of a driving term ofthe form Hd = γ(t)b + h.c., where γ(t) = I(t) + iQ(t) isthe complex control amplitude to be optimized.

3

SNAP gates have been previously implemented by ap-plying a superposition of m pulses centred at frequenciesωq+jχ, with the jth pulse rotating the qubit conditionedon the cavity being in Fock state j [15]. However, usingthat approach, the minimum duration of the SNAP gateis many times longer than 1/χ, limited by the require-ment that each pulse should condition the dynamics ona single Fock state. In our approach, the optimal controlpulses reduce the duration of the SNAP gates by about 8times, from 4µs down to 500ns. At the same time, opti-mal control pulses correct for the phases acquired duringthe gate duration that result from the small Kerr nonlin-earity of the cavity mode, Kc. Details of the calibrationof the Hamiltonian parameters and the cavity and qubitRabi rates are given in Appendix A. We realize displace-ment operations, D(α), by resonantly driving the cavitywith pulses with a sine-squared envelope, and a total du-ration of 50 ns, and a calibrated amplitude proportionalto α.Following the state preparation sequence, we perform

direct Wigner tomography of the prepared state [15]. Todo so, we apply a displacement of varying complex am-plitude α followed by a Ramsey measurement at fixedtime delay 1/2χ, which returns the photon parity of thecavity mode as the outcome of the qubit measurement.The Wigner function is then obtained as:

W (α) = 2πTr[D†(α)ρD(α)Π], (2)

with Π being the parity operator. We reconstruct thedensity matrix of the target state from the Wigner func-tion using a recently developed neural-network-based ap-proach [34, 35] which reconstructs the most likely den-sity matrix of the state based on the measured Wignertomography data.

III. RESULTS

To showcase our ability to prepare arbitrary statesof the cavity using displacement-SNAP gates, we pre-pare Wigner-negative states belonging to different classesof quantum states. We prepare the Fock state |2〉,the binomial state (|0〉 + |4〉)/

√2, and the cat state

(|α〉 − | − α〉)/N , where α =√2 and N is a normal-

ization constant [Fig. 2(a)-(c)]. All three states have anaverage photon number of two and are prepared usingtwo SNAP gates and three displacements. We also cre-ate a Gottesman-Kitaev-Preskill (GKP) state by apply-ing three SNAPs and four displacements [Fig. 2(d)] andthe cubic phase state by applying three SNAPs and threedisplacements [Fig. 2(e)] . The precise definitions of thetargeted GKP and cubic phase states are given in Ap-pendix E.The fidelities of the prepared states are listed

in Table I. along with the number of applied dis-placement and SNAP gates. The theoretical fidelityis obtained by numerically solving a master equation

TABLE I. State fidelities for the generated states shown inFigs. 2 and 3.

State (#Disp,#SNAP) Experiment Theory

Fock |2〉 (3,2) 0.996 0.972

Binomial (3,2) 0.937 0.972

Cat (3,2) 0.963 0.966

GKP (4,3) 0.942 0.932

Cubic phase (3,3) 0.717 0.922

|Ψ1〉 = Dα1|0〉 (1,0) 0.994 0.999

|Ψ2〉 = S(~θ1)|Ψ1〉 (1,1) 0.959 0.987

|Ψ3〉 = Dα2|Ψ2〉 (2,1) 0.946 0.986

|Ψ4〉 = S(~θ2)|Ψ3〉 (2,2) 0.921 0.974

(see Appendix C) which takes into account the measuredHamiltonian parameters [Eq. (1)], the applied SNAP anddisplacement gates, and the average measured decoher-ence and dephasing of both the qubit and the cavity. Thefidelities of the states with two photons on average (Fock,binomial, and cat) are limited by the coherence param-eters of our system (to about 0.97). The fidelity of theFock state |2〉 exceeds the predicted fidelity by 0.02, whilethe experimentally obtained binomial state has fidelity0.03 lower than theoretical prediction. We ascribe thesediscrepancies to fluctuations in the coherence time of thequbit [36], not captured by our theoretical model. TheGKP state, which has 4 photons on average, was createdby three SNAP gates that add phases to all Fock statesup to Fock number 17. The corresponding displacement-SNAP sequence would create a GKP state with fidelity0.99 in the ideal case, but with loss in our system thatnumber drops to 0.93. This is in good agreement with theexperimentally obtained 0.94. The cubic phase state hasfidelity 0.985 in the ideal case, when using three SNAPsand three displacements. After taking loss into account,the simulated fidelity is 0.92. The achieved experimentalfidelity is 0.71.

An advantage of the SNAP-displacement approach tostate preparation is that the contribution of each indi-vidual step of the sequence to the final fidelity can beindependently analyzed by taking a Wigner tomographyof intermediate states after each step. This allows us toidentify weak links in the sequence and optimize the gatesindividually. Such a series of “Wigner snapshots” takenafter each step of the sequence to prepare the binomialstate (|0〉+ |4〉)/

√2 is presented in Fig. 3. The fidelities

of intermediate states monotonically decrease as the se-quence progresses Table I. SNAP gates, being 10 timeslonger than the displacement gates, are responsible formost of the infidelity.

We also probe the sensitivity of the state preparationsequence to different calibration parameters that are fedto our pulse optimizer (Fig. 4). To do so, we preparethe two-photon Fock state with three displacements andtwo SNAPs [same as Fig. 2(a)], and measure the Wigner

4

FIG. 2. Wigner functions of the states created by optimized SNAP displacement gates. In the top row is the experimentaldata and in the bottom row the target states. From left to right: (a) Two-photon Fock state |2〉, (b) binomial state |0〉 + |4〉,and (c) an odd cat state |α〉 − | − α〉, with α =

√2. These three states have 2 photons on average and were created with two

SNAPs and three displacements. Next, (d) the GKP state with four photons on average and (e) the cubic phase state. GKPstate was prepared with three SNAPs and four displacements and cubic phase state was prepared with three SNAPs and threedisplacements. Fidelities are listed in Table I.

FIG. 3. Wigner function after each gate in the sequence to create a binomial state |0〉 + |4〉: (top) experimental data and(bottom) target states. The fidelities after each gate are listed in Table I. The parameters of each SNAP and displacementgate are given in Table A2.

function at the origin, W (0), as a proxy for its prepara-tion fidelity. This state has even parity; hence the valueof the Wigner function at the origin has the maximumvalue the Wigner function can take (2/π). Thus a lowervalue of the Wigner function must be associated with adecrease in fidelity, as confirm by simulations.

The theory lines plotted in red in Fig. 4 are obtainedby simulating an open system given by the qubit andthe cavity with the measured average decay and dephas-

ing rates. The only fitting parameter is the offset in thex-axis (x0). The offsets tell us how far from the opti-mal calibration point our applied pulses are. The theorylines capture well the parabolic decrease in W (0) as thepulse parameters deviate from the optimum; however,they deviate from the experimental data by a constantfactor in Fig. 4(b) and (d). We attribute these devia-tions to fluctuations in the coherence parameters. Toprobe the sensitivity to the dispersive shift χ, we vary its

5

FIG. 4. Sensitivity of the displacement-SNAP sequence todifferent calibration parameters. Wigner function at the ori-gin for a two-photon Fock state vs (a) dispersive shift, χ, (b)modulation frequency of SNAP gates fSNAP, (c) amplitude ofthe displacement Adisp and (d) SNAP ASNAP pulses. Dots:experiment. Solid line: theory.

value fed to the optimizer by ±0.07 around its nominalvalue. For each value, we generate a new pulse sequence,execute it, and measureW (0) [Fig. 4(a)]. In addition, weexplore variations in the carrier frequency of the SNAPpulses fSNAP [Fig. 4(b)], the amplitude of the displace-ment pulses Adisp [Fig. 4(c)], and the amplitude of theSNAP pulses ASNAP [Fig. 4(d)].

We further analyze how the state preparation fi-delity of Fock state two depends on the parameters inFig. 4, for both our optimized SNAP and ”standard”SNAP gates (Fig. 5). We define a standard SNAP gateS(θ0, θ1, ..., θm) as a sum of m pulses at frequenciesωq − iχ, i ∈ {0, 1, ...,m}, where m is the maximum Focklevel the SNAP gate adds phase to. At each of the fre-quencies, two consecutive π pulses (2 µs long) with axesof rotation θi apart are sent [15]. The distance betweenthe dotted lines represents the parameter range whichis within 1% of the maximum fidelity. The fidelity ofthe state made by the standard SNAPs is 5 times moresensitive to calibration of dispersive shift χ [Fig. 5(a)]and 7 times more sensitive to fluctuations in qubit fre-quency [Fig. 5(b)]. This robustness to fluctuations inqubit frequency and dispersive shift is a feature of theoptimized SNAP pulses. The displacement amplitudecan be about ±0.06 from the optimal value for both theoptimized and standard SNAP pulses [Fig. 5(c)]. The dis-

FIG. 5. Theoretical comparison of the sensitivity of the opti-mized vs standard SNAP to the same calibration parametersas in Fig. 4. The distance between the dashed lines representsthe parameter span where the fidelity is within 1% of the max-imum. The optimized SNAP gates are identical to those inFig. 4, while the standard SNAP gates are a superposition of4µs long pulses centered at photon number dependent qubitfrequencies [15].

placement pulses are identical for both state preparationcases, so we would expect them to have similar sensitiv-ity to amplitude of the displacement pulses. Finally, theamplitude of the SNAP pulses can vary ±0.05 for opti-mized SNAP pulses and ±0.07 for standard SNAP pulses[Fig. 5(d)]. An important thing to note here is that the fi-delity close to the maximum has a parabolic dependenceon all of the calibration parameters. This allows us toindependently further fine-tune these parameters.

IV. DISCUSSION

We have shown that only a few optimized SNAP anddisplacement gates provide a well-controlled method forgenerating a range of complex, highly nonclassical statesrelevant for quantum computation. We tested the per-formance of this state preparation scheme by generat-ing Fock, binomial, cat, GKP, and cubic phase states,achieving the first experimental implementation of thelatter. We showed that the optimized SNAP gateshave parabolic dependence in fidelity with respect toperturbations of relevant calibration parameters closeto the maximum fidelity. Furthermore, this prepara-tion scheme is robust to fluctuations in both displace-

6

ment amplitude (∆Adisp/Adisp = ±0.06) and in theamplitude (∆ASNAP/ASNAP = ±0.05) and frequency(∆fSNAP/∆χ = ±0.04) of the SNAP drive. Compared tothe SNAP gates in Ref. [15], the optimized SNAP gatesare about 8 times shorter, 5 times less sensitive to the cal-ibration of dispersive shift χ, and 7 times less sensitiveto fluctuations of qubit frequency fq. Our results arerelevant for any implementation of continuous-variablequantum computing with qubit-oscillator systems in thestrong dispersive regime, including 3D microwave cavi-ties [29, 37], superconducting resonators [38], and acous-tic resonators coupled to superconducting qubits [39–42].

ACKNOWLEDGMENTS

We would like to thank Mats Myremark and LarsJonsson for machining the cavity, Gustav Grannsjo andJohan Blomberg for their help with the Vivace driver,and Mats Tholen and Riccardo Borgani for their helpwith the Vivace platform. We would further like to thankAndre Carvalho and Viktor Perunicic from Q-CTRL fortheir help with the optimization software Boulder Opal.The simulations and visualization of the quantum stateswere performed using QuTiP [43, 44], NumPy [45], andMatplotlib [46]. The automatic differentiation tools Ten-sorFlow [47] and Jax [48] were used in state reconstruc-tion and optimization. This work was supported by theKnut and Alice Wallenberg foundation via the Wallen-berg Centre for Quantum Technology (WACQT) and bythe Swedish Research Council. M. Kervinen acknowl-edges support from the Emil Aaltonen Foundation. Thechips were fabricated in Chalmers Myfab cleanroom. Weacknowledge IARPA and Lincoln Labs for providing theTWPA used in this experiment.

Appendix A: Hamiltonian parameters

TABLE A1. Parameter values for the Hamiltonian in Eq. (1)and measured coherence times.

Parameter Value (ω/2π) Parameter Valuefq 6.267388 GHz T1q 42 ± 4µsfc 4.428018 GHz T2q 32 ± 4µsfr 7.28592GHz T1c 248µsχqc 3.137616MHzχqr 1.7MHzKc 5.943 kHzαq 300MHzχ′qc 25.856 kHz

The system parameters are listed in Table A1. Thedifferent parameters such as the dispersive shift χ, thecorrection to the dispersive shift χ′, and the Kerr nonlin-earity of the cavity are measured following the calibrationexperiments in Ref. [49]. Besides the Hamiltonian param-

TABLE A2. Gate parameters used to create the states inFig. 2(a)-(c).

StateGate

parametersValues

Fockstate|2〉

α1, α2, α3 1.390, -0.494, 0.622

~θ1(2.049, -0.654, 1.130, -1.106,0, 0, 0, 0, 0, 0, 0, 0, 0 , 0 , 0)

~θ2(0.003, 1.592, 0, -0.869, 0, -0.234,

0.067, 0, 0, 0, 0, 0, 0, 0, 0)

Binomialstate

α1, α2, α3 1.304, -1.144, -0.291

~θ1(0.111, -1.535, 0.377, 0, -0.836,

-0.856, 0, 0, 0, 0, 0, 0)

~θ2(0.404, 0, 0, -0.777, -0.976, 2.083,

-1.087, 1.552, 0, 0, 0, 0)

Catstate

α1, α2, α3 1.373, -0.614, 0.529

~θ1(1.277, -0.302, -1.906, 0.093, -1.161,

0.463, -0.569, 0.286, 0, 0, 0, 0, 0, 0, 0)

~θ2(0.650, 2.109, 0, 0.770, 0.392, 0.272, 0,-0.132, -0.248, -0.251, -0.027, 0, 0, 0, 0)

eters from Eq. (1), the pulse optimization is constrainedby the maximum Rabi rate, the sampling frequency, andthe cut-off for the digital low-pass filter function. Thelow-pass cut-off is 40-80MHz depending on the targetstate which ensures the drive can be implemented on ourhardware without exciting unwanted transitions. Sincethe anharmonicity of the qubit αq is much larger than thefilter cut-off, we need to take only two qubit levels intoaccount in Eq. (1). The maximum Rabi rate is ensuredto be under 30MHz to be below the maximum outputamplitude of our instrument and to avoid any non-lineareffects of strong driving. Finally, the sampling frequencywas chosen to be 1GSa/s (i.e., 500 points for a 500nspulse).

Appendix B: Wigner function reconstruction

In order to obtain the density matrix and to esti-mate fidelity of the target state, we reconstruct thedensity matrix from the Wigner tomography measure-ment using a recently developed neural-network-basedapproach [34, 35], which reconstructs the density matrixthat most faithfully reproduces the Wigner tomographyof the state. The fidelity is then given by

F (ρ, σ) =

(

Tr√√

ρσ√ρ

)2

, (B1)

where ρ and σ are the density matrices for the targetand the reconstruction. For the reconstructed densitymatrix, we use a Hilbert space size of 32. The maximumand minimum values of the experimental data are scaledwith a linear scaling to compensate for the loss of contrastin the Wigner measurement for states with high photonnumber [37].

7

Appendix C: Master equation simulations

To obtain the theoretical fidelities of our target states,we simulate the system dynamics using a master equa-tion with the known decoherence sources, and the ap-plied pulse waveforms. The Lindblad master equation isdefined as

∂

∂tρ(t) = −i[H(t), ρ(t)]+

(

1

T1cD[a] +

1

T1qD[b] +

1

TφD[b†b]

)

ρ(t), (C1)

where T1q and Tφ are the decay and the dephasing ofthe qubit, respectively, whereas T1c is the decay of thecavity mode. D is the Lindblad operator describing thedecoherent procesess in the system evolution.

Appendix D: Optimization using gradient descent

The parameters of the displacement and SNAP gatesthat are used to create the states in the main text aregiven in Table A2. These parameters were obtained byusing gradient-based optimization of the cost function:

L(αi, ..., ~θj , ...) = [1− F (Ψtarget,Ψ(αi, ..., ~θj , ...))]

+ λ∑

i,m

|θim|, (D1)

where Ψ(αi, ..., ~θj , ...) represents a variational state. Thisvariational state is prepared by applying displacementsand SNAP gates to the vacuum state. We use the fi-delity F (Ψ1,Ψ2) between the two states to constructour loss function. The LASSO regularization [50] termλ∑

i,m |θim| is used to keep the SNAP parameters sparse.

In Ref. [17], the authors demonstrated that only 3 to 4SNAP gates were sufficient for relevant experimental im-plementation — an improvement from the ∼ 50 gates inprevious approaches. In our work, we fix the number ofSNAP-displacement operations and then use gradient de-scent to minimize Eq. (D1) starting from a random setof displacement and SNAP parameters. Our approach

differs from that of Ref. [17] in that we directly optimizethe displacement and SNAP parameters using gradient-descent by assuming a fixed length for the gate sequence.However the method in Ref. [17] uses a two step processwhere in the first step, a heuristic is used to determine areasonable length for the gate sequence and find an ap-proximate initialization for the parameters. In the nextstep, the parameters are fine tuned with gradient-basedoptimization.Since in our experiment, we can only apply a limited

number of gates, we fixed the length of the sequencedirectly and only optimize the parameters akin to thefine-tuning step of Ref. [17]. We observed that usingappropriate hyperparameters and regularization of theloss function, we can directly obtain the gate parametersstarting from a random initialization.

Appendix E: GKP and cubic phase state

We consider the finite GKP state [35, 51–53] charac-terized by the real parameter σ ∈ [0, 1] and defined inthe Fock basis as,

|ψσ,µGKP

〉 =∑

α∈K(µ)

e−σ2|α|2e−iRe[α]Im[α]|α〉, (E1)

where |α〉 are coherent states with the complex ampli-tudes in a gridK(µ) =

√

π2 (2n1+µ))+i

√

π2n2, µ ∈ {0, 1}

represents the logical 0 or 1 encoding, and n1, n2 are in-tegers in [−8, 8]. We take the Fock space cutoff as 25with σ = 0.35 and µ = 0.The cubic phase state is defined as

∣

∣

∣ψγ,ζcubic

⟩

= eiγq3

S(ζ)|0〉, (E2)

where q = (a+ a†)/√2 is defined using the ladder opera-

tors a, a† and S(ζ) is the single-mode squeezing operator

S(ζ) = e1

2 (ζ∗a2−ζa†2). (E3)

We use a cubicity γ = −0.3 and a squeezing parameterζ = 0.5 to generate our cubic phase state. An additionaldisplacement of β = 1.5i is applied to move the state inorder to capture more features in the Wigner functionplot.

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