Quantum Motional State Tomography with Non-Quadratic Potentials and Neural Networks

TL;DR

This paper introduces a method combining non-quadratic potentials and neural networks to reconstruct unknown quantum motional states efficiently, with potential experimental applications.

Contribution

It presents a novel approach using complex quantum dynamics and neural networks for motional state tomography in non-quadratic potentials.

Findings

Neural networks can accurately reconstruct quantum states from position measurements.

The method is feasible despite decoherence and potential uncertainties.

Efficient state reconstruction is possible with measurements at different times.

Abstract

We propose to use the complex quantum dynamics of a massive particle in a non-quadratic potential to reconstruct an initial unknown motional quantum state. We theoretically show that the reconstruction can be efficiently done by measuring the mean value and the variance of the position quantum operator at different instances of time in a quartic potential. We train a neural network to successfully solve this hard regression problem. We discuss the experimental feasibility of the method by analyzing the impact of decoherence and uncertainties in the potential.

Tables

Table 1. Table 1: Summary of the neural network architecture and hyperparameters.

network type	feed-forward, densely connected
neurons	input layer: $2N$ ($N$ the number of points in each trajectory) hidden layers: $800$, $800$, $400$, $200$ output layer: $2d^2$ ($d$ the dimension of the initial quantum state)
activation functions	‘sigmoid’, for the final layer ‘tanh’
optimizer	adam (with keras default values)
loss function	custom mean-squared error
end of training	early stopping, with a patience of $500$ epochs (maximal $10000$ epochs)
batch size	512
number of training states	10000
number of validation states	10000

Equations

$$\hat{H}=\frac{{\hat{P}}^2}{2m}+\lambda {\hat{X}}^4=\hslash {\omega }_0\left(\frac{{\hat{p}}^2}{4}+\frac{{\hat{x}}^4}{{\alpha }^4}\right).$$

$$\dot{\hat{\rho }}=-\frac{i}{\hslash }[\hat{H},\hat{\rho }]-\Gamma [\hat{x},[\hat{x},\hat{\rho }]]=\hat{L}\hat{\rho }.$$

$$1-F=1-\text{tr}\left[\sqrt{\sqrt{\hat{\eta }}{\hat{\eta }}_{\text{est}}\sqrt{\hat{\eta }}}\right].$$

$$⟨\dot{\hat{x}}⟩$$

$$⟨\dot{\hat{p}}⟩$$

$${\dot{G}}^{a,b}=$$

$$-8\frac{a}{{\alpha }^4}\left[3⟨\hat{x}{⟩}^2{G}^{a-1,b+1}+3⟨\hat{x}⟩G^{a-1,b+2}+G^{a-1,b+3}\right]$$

$$+8\frac{a}{{\alpha }^4}(a-1)(a-2)\left[⟨\hat{x}⟩G^{a-3,b}+G^{a-3,b+1}\right]$$

Full text

I Introduction

One of the most fascinating possibilities in quantum physics is to prepare the motional degrees of freedom of a massive particle in a quantum state. The non-classical features of such a state can be demonstrated by reconstructing its quantum density-matrix operator and showing that its associated Wigner function has negative values. Such an endeavor has been successfully achieved with ions, see Ref. Leibfried et al. (2003) and references therein. Today, the field of quantum nano- and micromechanics aims to do the same with objects much more massive Aspelmeyer et al. (2014), for instance nano- and micro-particles, which contain billions of atoms Romero-Isart et al. (2010); Chang et al. (2010). Such an exciting goal has many challenges, and a crucial one is the faithful reconstruction (also called quantum tomography) of the quantum motional state.

Standard strategies to perform quantum motional state tomography Vanner et al. (2015) are to couple the motion of the particle to a few-level system Wallentowitz and Vogel (1995); Singh and Meystre (2010), to transfer the mechanical state to a cavity electromagnetic mode whose state can be reconstructed with homodyne tomography Parkins and Kimble (1999), to apply coherent displacements and phonon number measurements on the motional degree of freedom Wallentowitz and Vogel (1996); Banaszek and Wódkiewicz (1996); Poyatos et al. (1996), as done with ions see, e.g., Ref. Leibfried et al. (1996), or to measure the full position distribution function (i.e. having access to all moments) at different instances of time in a harmonic potential Vanner et al. (2013); Romero-Isart et al. (2011a). In this article, we propose an alternative approach based on exploiting two distinctive features of levitated particles: (i) their low level of motional decoherence and (ii) the possibility to engineer the potential of the particle, in particular to let the particle coherently evolve in a non-quadratic potential. We show that by solely measuring the mean value and the variance of the position of the particle (i.e. the first two moments only) as a function of time during the evolution in a non-quadratic potential, one can efficiently reconstruct the initial unknown quantum motional state (e.g. a given non-Gaussian state). Such reconstruction is a hard quantum regression problem that, as we show below, is ideally suited for neural networks.

This article is organized as follows: In Sec. II we introduce the physical scenario and argue that the time evolution of the mean value and variance of the particle’s position should allow us to reconstruct the initial state. In Sec. III we present our results: First, in Sec. 1, we explain the overall protocol and how we use and evaluate the neural network for quantum state tomography. Then, in Sec. 2, we show quantum state tomography in the absence of decoherence. In Sec. 3 we discuss the effects of decoherence on the achieved fidelity and in Sec. 4 we consider realistic scenarios where we take experimental limitations into account. In Sec. IV we summarize our results. Additional information can be found in the appendices: First, in Appendix A, we discuss an approximative analytical approach and how it becomes unfeasible in the regime relevant for quantum state tomography. Finally, in Appendix B, we present technical details about the architecture and training of the neural networks.

II Model

Let us consider the one-dimensional motion of a particle of mass $m$ in a quartic potential such that its coherent dynamics is described by the Hamiltonian

[TABLE]

Here, $\hat{X}=\hat{x}x_{0}$ and $\hat{P}=\hat{p}p_{0}$, with $[{\hat{X}},{\hat{P}}]=\text{i}\hbar$, are the position and momentum, and $\lambda$ the strength of the quartic potential. We have extracted units using $x_{0}=\hbar/(2p_{0})=[\hbar/(2m\omega_{0})]^{1/2}$ and defined the inverse quarticity parameter $\alpha=[\hbar\omega_{0}/(\lambda x_{0}^{4})]^{1/4}$, with the motivation that we will consider initial motional quantum states assumed to be prepared in a harmonic potential of frequency $\omega_{0}$. Additionally, we consider a standard source of decoherence for levitated particles Romero-Isart (2011); Joos et al. (2003); Schlosshauer (2007) that is of position localization type (e.g. due to recoil heating or a fluctuating white-noise force). In total, the evolution of the density-matrix operator of the motional state is described by the master equation

[TABLE]

Here, $\Gamma$ is the decoherence rate.

At $t=0$, the particle is assumed to be in an unknown motional quantum state $\hat{\eta}\equiv\hat{\rho}(0)$ that we aim to reconstruct. The system evolves according to Eq. (2) and the state at a later time $t\geq 0$ is given by $\hat{\rho}(t)=\exp(\hat{L}t)\hat{\eta}$. The position $\hat{X}$ is sufficiently measured at different instances of time to retrieve the mean value and its variance, that is, to obtain the dimensionless trajectories $u_{1}(t)\equiv\langle\hat{x}(t)\rangle-\langle\hat{x}(0)\rangle=\textrm{tr}[{\hat{x}({\hat{\rho}(t)-\hat{\eta}})}]$ and $u_{2}(t)\equiv\langle{\hat{x}^{2}(t)}\rangle-\langle{\hat{x}(t)}\rangle^{2}=\textrm{tr}[{\hat{x}^{2}\hat{\rho}(t)}]-({\textrm{tr}[{\hat{x}\hat{\rho}}]})^{2}$. Note that, as defined, $u_{1}(0)=0$, and hence one does not need to assume an absolute position measurement. In Appendix 2, we show examples of the trajectories $u_{1}(t)$ and $u_{2}(t)$ numerically calculated by solving Eq. (2) on a truncated Hilbert space using the python toolbox QuTiP Johansson et al. (2012, 2013). The question addressed in this article is the following: Can the information provided by $u_{1}(t)$ and $u_{2}(t)$ be used to reconstruct the unknown quantum motional state $\hat{\eta}$?

The reconstruction of a quantum motional state $\hat{\eta}$ could be performed should one have full knowledge of the mean values $\langle\hat{x}\rangle$ and $\langle\hat{p}\rangle$ and all the moments defined by $G^{a,b}=\langle(\hat{p}-\langle{\hat{p}}\rangle)^{a}(\hat{x}-\langle{\hat{x}}\rangle)^{b}\rangle_{\text{Weyl}}$, where $\langle\cdot\rangle_{\text{Weyl}}$ denotes the mean value of the Weyl-ordered product of operators calculated for the state $\hat{\eta}$, and $a,b$ are non-negative integers. As we show in detail in Appendix A based on Refs. Ballentine and McRae (1998); Brizuela (2014a, b), the trajectories $u_{1}(t)$ and $u_{2}(t)$ depend, for $t$ larger than a given critical time, on basically all the moments $G^{a,b}$ of the state at $t=0$. This is a manifestation of the non-linear quantum dynamics induced by the quartic potential and has two consequences. First, it shows that, indeed, the trajectories $u_{1}(t)$ and $u_{2}(t)$ should provide sufficient information to reconstruct $\hat{\eta}$. This is in contrast to a harmonic potential where $u_{1}(t)$ and $u_{2}(t)$ would depend at most on quadratic moments. Second, it shows that in a quartic potential, consequently, it is not possible to correctly approximate $u_{1}(t)$ and $u_{2}(t)$ as a function of a finite set of initial moments and, hence, the regression problem of deriving $\hat{\eta}$ based on $u_{1}(t)$ and $u_{2}(t)$ is a hard problem that, to our knowledge, cannot be solved with analytical tools. Nevertheless, this problem is very well suited to a neural network trained by supervised learning. The neural network will not require us to input how exactly the initial moments affect the trajectories. Instead, the neural network will, based on the training examples, find by itself an internal representation of the underlying regression problem. We remark that such a setting, inferring the quantum state from the time evolution of observables, is very different from recent works using neural networks for quantum state tomography of systems of many qubits Torlai et al. (2018); Xu and Xu (2018); Xin et al. (2018); Quek et al. (2018), or for filtering experimental data before performing quantum state tomography Palmieri et al. (2019).

III Results

In this section we present our results on quantum state tomography based solely on trajectories in non-quadratic potentials. In Sec. 1 we explain the protocol and how we obtain our results using neural networks. In Sec. 2 we investigate the ideal decoherence-free scenario, while in 3 we include decoherence and give an estimate of the thereby introduced necessary conditions. Eventually, in Sec. 4, we study realistic scenarios including experimental limitations.

1 Protocol

Let us now introduce the overall procedure: We propose to train the neural network on simulated data and then use the trained network to deduce the initial quantum state from experimentally measured trajectories $u_{1}(t)$ and $u_{2}(t)$. Experimentally these trajectories could be obtained by repeatedly re-preparing a particle in the same initial state and then evolving it (in the absence of measurements) up to a time $t_{1}$ when the position is measured, for instance, via optical position detection Tebbenjohanns et al. (2019, 2018). Averaging over the many repetitions, this reveals the expectation value and variance of the position at this time $t_{1}$. Consecutively, this procedure is repeated evolving the particle (in the absence of a measurement) to a later time $t_{1}+\delta t$ in order to measure the next point of the trajectory. This is repeated for all points of the trajectory.

We now explain how the neural network is trained and tested. Initial states $\hat{\eta}$ are randomly sampled from a Hilbert Schmidt ensemble of density matrices of dimensions $d\times d$. That is, we assume that $\hat{\eta}$ is prepared in a harmonic potential of frequency $\omega_{0}$ with zero probability to contain more than $d-1$ excitations, an assumption motivated by experiments preparing non-Gaussian quantum states after the particle has been cooled near the ground state of a harmonic potential Romero-Isart et al. (2011a). While such a subspace is considered for the initial state $\hat{\eta}$, note that during the evolution the state $\hat{\rho}(t)=\exp(\hat{L}t)\hat{\eta}$ can populate a much larger space that is only limited by numerical restrictions in the integration of the master equation Eq. (2). With the input of the trajectories $u_{1}(t)$ and $u_{2}(t)$, the neural network reconstructs a density-matrix operator $\hat{\eta}_{\text{est}}$ of size $d\times d$ with infidelity

[TABLE]

We remark that, in practice, any prior knowledge about the initially prepared quantum state should be used to accordingly choose the subspace and sampling distribution of training states in order to optimize the performance. Here, we sample the trajectories with timesteps $\delta t=0.05/\omega_{0}$ and each data point is represented by a neuron in the input layer of the network. Throughout the article we used four hidden layers of $800$, $800$, $400$, and $200$ neurons and an output layer of $2d^{2}$ neurons, representing the real numbers defining $\hat{\eta}_{\text{est}}$. The output is interpreted as a complex matrix $M$ that, generally, does not strictly fulfill the conditions of a physical density-matrix operator (positive semi-definite with unit trace). Consequently, the reconstructed physical state is obtained via $\hat{\eta}_{\text{est}}={M^{\dagger}M}/\textrm{tr}[{M^{\dagger}M}]$  Xu and Xu (2018). The network is trained via supervised learning using the mean squared error as the loss function and a training set of $10000$ randomly drawn quantum states. All results shown in the figures are obtained from a validation set, i.e., another set of $10000$ random states that were not used during training. More details on the network architecture and training can be found in Appendix B.

2 Decoherence-free scenario

Let us first show the results obtained in the absence of decoherence using Eq. (2) with $\Gamma=0$. In Fig. 1a, we show the average infidelity on the validation set reached by a neural network given input trajectories of a certain length (denoted by the time $t$) for $d=2,3$ and $4$, and with a quartic potential defined by the inverse quarticity $\alpha=5$. In every data point a new network was trained on the specific trajectory length (defining the input layer size of the network) and initial state dimension $d$ (defining the output layer size). In all cases, the infidelity decreases significantly with trajectory length and eventually saturates (all fluctuations around the saturation level are not of physical origin, as explained in Appendix 3). The saturation occurs later for larger $d$, as an increasing number of $d^{2}-1$ independent moments need to be extracted from the trajectories in order to determine an arbitrary state of dimension $d$. The achieved infidelity also saturates on different levels depending on $d$, as we use the same number of training states ($10000$) despite the increasing size of the initial subspace. The achieved low infidelities demonstrate that the neural network can reconstruct the initial state from the trajectories with high accuracy, see Figs. 1b,c and caption for some examples. To show that the non-quadratic potential is indeed crucial, we also plot the performance of neural networks that were trained on trajectories in a harmonic potential (dotted lines in Fig. 1a) described by the Hamiltonian $\hat{H}=\hat{P}^{2}/(2m)+m\omega_{0}^{2}\hat{X}^{2}/2$. As expected, the non-quadratic potential outperforms the quadratic one, with the exception of the $d=2$ case, where the trajectories in the quadratic potential contain information about the only three moments that are required to fully determine the state.

3 Limitations due to decoherence

Let us now show the impact of decoherence, which will limit the length of the trajectories that can be used for quantum state reconstruction since, eventually, all information about the initial state is lost. In Fig. 2, we plot the achieved infidelities using Eq. (2) for different values of $\Gamma$, an inverse quarticity $\alpha=5$, the same set of quantum states as sampled for $d=4$ in Fig. 1a, and with neural networks trained and validated using the simulated trajectories in the presences of decoherence. The inset shows the distribution of infidelities reached in the validation set (see the caption for details). If the decoherence is sufficiently small (dashed line) the reached infidelity does not differ significantly from the performance achievable in absence of decoherence (green line, same as in Fig. 1a), since the trajectories are only significantly altered by decoherence after all information necessary to reconstruct the initial state has already been extracted. In contrast, at larger decoherence rates (dashed-dotted and dotted line), the trajectories are altered much earlier and both the average performance at intermediate times and the final performance become worse. The reason is that there is neither enough information contained in a trajectory up to the time where decoherence acts, nor sufficient time for the neural network to infer all the moments determining the initial state before decoherence erases the initial state dependence.

The above discussion shows that in an experimental implementation of the proposed method the decoherence rate plays a limiting role. In the following we will show that eventually the ratio between decoherence and the strength of the non-quadratic potential is decisive. To this end, let us obtain a rough quantitative estimate of a necessary requirement. The initial state $\hat{\eta}$ is spatially confined in a length scale given by $(\textrm{tr}[\hat{\eta}\hat{X}^{2}])^{1/2}\sim x_{0}$ and has a kinetic energy of the order of $\hbar\omega_{0}$. During the evolution in the quartic potential, the initial state spreads as $(\textrm{tr}[\rho(t)\hat{X}^{2}])^{1/2}\sim x_{0}\omega_{0}t$ and the effect of the quartic potential is relevant when the potential energy is comparable to the initial kinetic energy, namely for $t^{\star}$ such that $\lambda(\textrm{tr}[\rho(t^{\star})\hat{X}^{2}])^{2}\sim\lambda(x_{0}\omega_{0}t^{\star})^{4}\approx\hbar\omega_{0}$. Using the definition of the dimensionless inverse quarticity $\alpha=[4m^{2}\omega_{0}^{3}/(\hbar\lambda)]^{1/4}$, one obtains $t^{\star}\approx\alpha/\omega_{0}$. Trajectories longer than $t^{\star}$ are required to be affected from the quartic potential, and the requirement that they are coherent demands that $1/\Gamma\gg t^{\star}$, or alternatively, $\alpha\ll\omega_{0}/\Gamma$. Thus, a stronger quartic potential (smaller inverse quarticity $\alpha$) helps to cope with decoherence. Such a rough estimate is a necessary but not a sufficient requirement, as can be seen in Fig. 2. Nevertheless, it provides a good reference for experimental implementations. For instance, let us assume that the quartic potential is engineered using a potential of the form $V_{G}(X)=-m\omega_{0}^{2}\sigma^{2}\exp[-X^{2}/\sigma^{2}]/2$ (e.g. generated by optical tweezers), which has been used to generate the quadratic potential, $V_{G}(X)\approx V_{G}(0)+m\omega_{0}^{2}X^{2}/2$. A (to leading order) pure quartic potential can then be obtained around $X=0$ by superimposing two such Gaussians $V(X)=V_{G}(X-\sigma/\sqrt{2})+V_{G}(X+\sigma/\sqrt{2})\approx 2V_{G}(0)+\lambda X^{4}$, where $\lambda=m\omega_{0}^{2}/[3\sqrt{e}\sigma^{2}]$, and, hence, $\alpha\approx 1.8\sqrt{\sigma/x_{0}}$. In this case, the necessary requirement reads $1.8\sqrt{\sigma/x_{0}}\ll\omega_{0}/\Gamma$. For ions, this condition is not challenging since $\omega_{0}/\Gamma\sim 10^{3}$ Leibfried et al. (2003) and $x_{0}\approx 10\leavevmode\nobreak\ \text{nm}$, and, hence, one requires potentials with $\sigma\ll 10^{3}\leavevmode\nobreak\ \mu\text{m}$. For optically levitated nanoparticles, where $\omega_{0}/\Gamma\sim 10^{2}$ Chang et al. (2010); Windey et al. (2019); Gonzalez-Ballestero et al. (2019), and $x_{0}\sim 10^{-12}\leavevmode\nobreak\ \text{m}$, the condition reads $\sigma\ll 10\leavevmode\nobreak\ \text{nm}$, which is not compatible with optical potentials where $\sigma$ is lower bounded by an optical wavelength. Therefore, levitated nanoparticles require either longer coherence times, achievable by evolution in the absence of recoil heating from laser light (quasi-electrostatic traps Home et al. (2011); Kuhlicke et al. (2014); Alda et al. (2016); Delord et al. (2018); Bykov et al. (2019), magnetic traps Romero-Isart et al. (2012); Slezak et al. (2018); Prat-Camps et al. (2017), or in free fall Romero-Isart et al. (2011b); Hebestreit et al. (2018) where the quartic potential is only applied after the state has sufficiently broadened) or the use of electromagnetic forces near surfaces Diehl et al. (2018); Magrini et al. (2018); Pino et al. (2018) such that $\sigma$ can be potentially smaller than an optical wavelength. Instead of aiming for stronger non-quadratic potentials or longer decoherence times one could also speed up the broadening of the initially prepared state by introducing an inverted harmonic potential Pino et al. (2018); Romero-Isart (2017) at the center of the quartic trap, that is, using a double-well potential.

4 Realistic scenarios

Regarding the experimental implementation of the method, it is also clear that a perfect quartic potential cannot be engineered. Related to the discussion above, let us assume that the two Gaussian potentials are not perfectly symmetric aligned, namely one has $V_{\epsilon}(X)=V_{G}(X-\sigma/\sqrt{2})+V_{G}(X+\epsilon+\sigma/\sqrt{2})$, where $\epsilon$ parametrizes the imperfection of the quartic potential. The form of such imperfect potentials is illustrated in the inset of Fig. 3 (dotted line for $\epsilon>0$, dashed line for $\epsilon<0$). In Fig. 3 we show the quantum state tomography performance of neural networks trained and tested on trajectories from the perturbed potential (black dotted line: $\epsilon/x_{0}=0.1$, black dashed line: $\epsilon/x_{0}=-0.1$). A similar overall performance can be achieved compared to the purely quartic potential $\epsilon=0$ (solid green line). Thus, the neural network finds an appropriate model to each scenario and the quantum state tomography does not crucially depend on the details of the non-quadratic potential, even in the presence of small linear and quadratic contributions.

So far, the training and testing scenario of the neural network were always the same. However, the experimental situation might not perfectly match the scenario used for training. For example, one could be ignorant of the exact form of the potential and hence use a neural network trained in slightly different potentials. The red lines in Fig. 3 show the reached average infidelity of neural networks that were trained on trajectories from the purely quartic potential ($\epsilon=0$) but that are then used to estimate the quantum state given trajectories from the perturbed potential (dotted and dashed again refer to $\epsilon/x_{0}=0.1$ and $\epsilon/x_{0}=-0.1$, respectively). At very short trajectory lengths the internal model of the trained neural network allows to reconstruct the quantum state with similar infidelity to the scenarios where training and validation situation had the same physical origin. If longer trajectories are used, the performance still improves, although the network is not able to retrieve as much information as in the ideal scenario. Given any specific accuracy goal, a numerical study would easily allow to estimate beforehand what size of experimental uncertainties a neural network trained with ignorance could bear.

IV Summary

In summary, we have proposed a method to perform quantum motional state tomography for levitated particles (e.g. ions, nanoparticles), based on inferring the initial state from the time evolution of a few moments in a non-quadratic potential. The reconstruction is efficiently done with a neural network. We have analyzed the impact of decoherence and potential imperfections. As a proof-of-principle, we have shown results for a quartic potential in which the mean value and variance of the position is measured. We emphasize, however, that the method is very general since a neural network allows to optimally adapt the quantum state tomography to any given physical scenario in an experiment by using training examples from the particular situation. For the case of levitated nanoparticles, ground-state cooling is closely approached Tebbenjohanns et al. (2019); Windey et al. (2019); Delić et al. (2019); Gonzalez-Ballestero et al. (2019); Tebbenjohanns et al. (2018); Jain et al. (2016); Fonseca et al. (2016); Millen et al. (2015); Asenbaum et al. (2013); Kiesel et al. (2013); Gieseler et al. (2012); Li et al. (2013), hence the development of quantum tomography schemes is not only important but timely. At the same time, implementing non-quadratic potentials is also a fantastic tool to prepare non-Gaussian states. We therefore hope that this work will further motivate experimentalists in the field of levitated nanoparticles to engineer non-quadratic potentials to bring and probe nanoparticles in the quantum regime.

Acknowledgements.

We acknowledge discussions with L. Novotny, F. Tebbenjohanns, and A. E. Rubio López. We thank the European Union Horizon 2020 Project FET-OPEN MaQSens (Grant No. 736943) for financial support. T.W. also acknowledges financial support from the Alexander von Humboldt foundation.

Appendix A Expansion in moments

In this appendix, we discuss an approximative approach to analytically describe the time evolution of the initial quantum state and demonstrate how it becomes unfeasible in the regime important for quantum state tomography. This approach is based on truncating the infinite system of coupled equations of motion of all moments.

To this end, we consider the motion of a particle in a quartic trap as described by the Hamiltonian (1). Using the results of Refs. Ballentine and McRae (1998); Brizuela (2014a, b), one can write the equations of motions of all moments:

[TABLE]

Recall that $G^{a,b}=\langle(\hat{p}-\langle{\hat{p}}\rangle)^{a}(\hat{x}-\langle{\hat{x}}\rangle)^{b}\rangle_{\text{Weyl}}$ denotes moments of the order $a+b$, with $\langle\cdot\rangle_{\text{Weyl}}$ the expectation value of the Weyl-ordered operators. This set of differential equations is exact but infinite. However, an approximation can be applied by only keeping all moments with combinations of $a$ and $b$ such that $a+b\leq N_{\text{t}}$, where $N_{\text{t}}$ is the truncation order. The resulting system of coupled, non-linear differential equations can then be solved numerically.

In Fig. 4 we illustrate the performance of this approximation using the Fock state $|{1}\rangle$ as the initial state, $\alpha=5$, and no decoherence $\Gamma=0$. The symmetry of a Fock state leads to vanishing first order moments, i.e. , $\langle{\hat{x}}\rangle=\langle{\hat{p}}\rangle=0$ for all times, and odd order moments do not significantly contribute to the motion (the results of truncating to an odd order are shown as black dashed lines and coincide with the respective solution of the next lower even truncation order). In Fig. 4a we also display the exact solution (blue line) which we obtain by numerically integrating the Schrödinger equation using QuTiP. We show approximations up to $N_{\text{t}}=14$. In Fig. 4b, we show the relative error between the full quantum solution $\langle{\hat{x}^{2}}\rangle_{\text{q}}$ and the truncated solution $\langle{\hat{x}^{2}}\rangle_{\text{t}}$. Increasing the truncation order increases the time where the error remains below a certain threshold only slightly. Even more importantly, the gain of accuracy by increasing the truncation order (by two) decreases. Therefore, truncating the number of moments used to simulate the non-linear dynamics is not a good approximation. This means that the trajectories depend, in general, on an unbounded number of initial moments, and hence provide sufficient information to reconstruct the initial quantum state. Such task for arbitrary quantum states cannot, however, be done analytically but with a neural network, as we show in this work.

Appendix B Neural network

In this appendix, we provide additional information on the neural networks that we used throughout this work. In particular, in Appendix 1, we summarize the network architecture and relevant hyperparameters. In Appendix 2 we discuss the input to the neural network and show example trajectories for the various scenarios discussed in the main text. Finally, in Appendix 3, we describe the technical aspects of training the neural network in detail and present a typical learning curve.

1 Details on the neural network

The neural networks used throughout this work were the same for all tasks, with only the input and output layer depending on the trajectory length and dimensionality of the specific problem respectively. In Table 1 we describe the network architecture and specify the relevant hyperparameters.