# Inferring interpretable dynamical generators of local quantum observables
from projective measurements through machine learning

Giovanni Cemin  Institut für Theoretische Physik, Universität Tübingen, Auf
der Morgenstelle 14, 72076 Tübingen, Germany  Francesco Carnazza  Institut für
Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076
Tübingen, Germany  Sabine Andergassen  Institute for Solid State Physics and
Institute of Information Systems Engineering, Vienna University of Technology,
1040 Vienna, Austria  Georg Martius  Max Planck Institute for Intelligent
Systems, Max-Planck-Ring 4, 72076 Tübingen, Germany  Wilhelm Schickard
Institut für Informatik, Maria-von-Linden-Straße 6 72076 Tübingen  Federico
Carollo  Institut für Theoretische Physik, Universität Tübingen, Auf der
Morgenstelle 14, 72076 Tübingen, Germany  Igor Lesanovsky  Institut für
Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076
Tübingen, Germany  School of Physics and Astronomy and Centre for the
Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, The
University of Nottingham, Nottingham, NG7 2RD, United Kingdom

###### Abstract

To characterize the dynamical behavior of many-body quantum systems, one is
usually interested in the evolution of so-called order-parameters rather than
in characterizing the full quantum state. In many situations, these quantities
coincide with the expectation value of local observables, such as the
magnetization or the particle density. In experiment, however, these
expectation values can only be obtained with a finite degree of accuracy due
to the effects of the projection noise. Here, we utilize a machine-learning
approach to infer the dynamical generator governing the evolution of local
observables in a many-body system from noisy data. To benchmark our method, we
consider a variant of the quantum Ising model and generate synthetic
experimental data, containing the results of $N$ projective measurements at
$M$ sampling points in time, using the time-evolving block-decimation
algorithm. As we show, across a wide range of parameters the dynamical
generator of local observables can be approximated by a Markovian quantum
master equation. Our method is not only useful for extracting effective
dynamical generators from many-body systems, but may also be applied for
inferring decoherence mechanisms of quantum simulation and computing
platforms.

Introduction.—

Figure 1: System and measurement protocol. (a) The system is composed of the
two-spin subsystem S, which is embedded in the environment E consisting of
$L-2$ spins. The Hamiltonian describing the system [cf. Eq. (1)] features a
driving field with Rabi frequency $\Omega$ and nearest-neighbor interactions
with coupling strength $V$. (b) Synthetic data generation using the time-
evolving block-decimation algorithm. The time evolution operator is
approximated using Trotterization: $U_{t}=e^{-iHt}\approx\prod e^{-iH\delta
t}$. The time evolution is then obtained by repeatedly applying $e^{-iH\delta
t}$. (c) Sketch of the measurement protocol. The gray dashed line represents
the exact dynamics of the observable. For each trajectory, $M$ random points
are selected in the time window $[0,\Omega T]$ (here $\Omega T=10$). For each
point in time, we measure $N$ times in a suitable basis; here $z$/$y$-basis
for the first/second spin. The inset shows an example histogram of measurement
outcomes: $\ket{10}$ means the measurement on the first spin along $z$-basis
has outcome $1$, and the measurement on the second spin along $y$-basis has
outcome $0$. From the histogram, the expectation value depicted as orange dots
are obtained. (d) The synthetic data is fed to the ML algorithm, namely a
single layer perceptron, that learns the dynamical generator $\mathcal{L}$.
The algorithm is validated by comparison to the exact data, where the error
$\epsilon(N,M)$ is calculated for different values of $N$ and $M$. The ML
method is interpretable and can be used to “read out” the underlying dynamical
processes.

Reconstructing Hamiltonian operators or dynamical generators from physical
properties of a quantum system is a problem of current interest. For instance,
inverse methods can be applied to identify quantum Hamiltonians associated
with a given ground state [1] and interacting many-body theories can be
obtained from the knowledge of correlation functions [2, 3]. In many settings,
one is merely interested in reconstructing effective equations of motion for a
subsystem S embedded in a larger “environment” E, as it happens, for open
quantum systems [4, 5, 6, 7, 8]. Furthermore, in the framework of quantum
simulation [9, 10, 11, 12, 13, 14, 15, 16, 17], it is very important to
understand the (effective) dynamical equations under which artificial quantum
systems actually evolve and by how much these differ from the desired ones
[18, 19]. This is relevant for improving state-of-the-art hardware [20, 21]
and the identification of noise models. Another interesting instance concerns
the evolution of order parameters, often constructed from local observables,
such as the particle density or the magnetization.

Machine learning (ML) approaches appear to be particularly suited for this
task [22]. For instance, quantum process tomography with generative
adversarial methods [23], neural networks [24], and recurrent neural networks
[25, 26] have been developed. These approaches are very promising but have two
main drawbacks: they require a great number of measurements, and they treat ML
algorithms as black boxes, thus lacking in physical interpretation. Simpler
methods are capable of learning Hamiltonians from fewer local measurements
[27, 28, 29, 30, 31, 32, 33], yet they typically rely on a a priori ansatz for
the functional form of the Hamiltonian or of the dissipation. A more general
approach is to fit an open quantum system (OQS) dynamics by learning the
Nakajima-Zwanzig equation [34, 35] through transfer tensor techniques [36, 37,
38] or by learning convolutionless master equations [39]. However, these
approaches require a full state tomography at different time steps, which is
prohibitive to achieve in experiments. Ultimately, current methods thus either
rely on an ad hoc ansatz, or demand data which is not experimentally
accessible, or lack in physical interpretability (which is actually becoming
highly desirable [40]).

In this work, we show how to use ML methods to infer the effective dynamical
generator of a subsystem from a finite set of local measurements at randomly
selected times, which inevitably produce noisy data due to projection noise.
To illustrate our approach, we consider a many-body spin system [cf. Fig.
1(a)], which is ubiquitous in the context of experiments with trapped ion or
Rydberg atom quantum simulators [21, 41, 42]. By using synthetic
(experimental) data generated by tensor-network based algorithms we infer a
physically consistent Markovian dynamical generator [43, 44] governing the
evolution of a small subsystem. Our method, which works reliably across a wide
range of parameters – even in some instances outside the weak coupling limit –
yields interpretable results which may be used to infer noise models on
quantum simulators or to study thermalization dynamics in many-body systems.

Setting.— The system we consider is a $1D$ quantum spin chain consisting of
$L$ spins arranged on a circular lattice, as depicted in Fig. 1(a). The chain
is partitioned into a subsystem S, here formed by two adjacent spins, and the
environment E, that is, the remainder of the spin chain. We assume the whole
system to evolve unitarily, through the many-body Hamiltonian

$H_{S+E}=\frac{\Omega}{2}\sum_{i=1}^{L}\sigma_{i}^{x}+V\bigg{(}\sum_{i=1}^{L-1}n_{i}n_{i+1}+n_{L}n_{1}\bigg{)}\,.$
(1)

The first term in the equation above describes a transverse “laser” field,
while the second one accounts for nearest-neighbor interactions. Here,
$\sigma_{i}^{\alpha}$ denotes the $\alpha$ Pauli matrix for the $i^{th}$ spin
and we have defined the projector $n=\frac{1+\sigma^{z}}{2}$. The above
Hamiltonian is of practical interest for experiments with Rydberg atoms [42]
and essentially encodes an Ising model in the presence of transverse and
longitudinal fields. We simulate the time evolution of the whole system by
means of the time-evolving block-decimation (TEBD) algorithm [see Fig. 1(b)].

In our setting, the information on the state of S is obtained by a finite
number, $N$, of projective measurements, taken at randomly selected times
$t_{1},...\,t_{M}$ [see Fig. 1(c)]. From this noisy data we want to infer the
open quantum dynamics of the reduced state $\rho_{\mathrm{S}}(t)$ of subsystem
S. Formally, this dynamics is obtained as the partial trace of the evolution
of the full many-body state, i.e., $\rho_{\mathrm{S}}(t)={\rm
Tr}_{\mathrm{E}}\left(U_{t}\rho_{\mathrm{S+E}}U_{t}^{\dagger}\right)$, where
$U_{t}=e^{-iHt}$, $\rho_{\mathrm{S+E}}$ is the initial state of the system and
${\rm Tr}_{E}$ denotes the trace over the environment degrees of freedom. In
general, such a dynamics is rather involved and may show non-Markovian effects
or it may be nonlinear for generic initial states $\rho_{\mathrm{S+E}}$ [45].
Here, we restrict ourselves to learning a Markovian dynamics for
$\rho_{\mathrm{S}}(t)$, but more general approaches are certainly possible
[43]. The goal is then to identify the time-independent generator
$\mathcal{L}$, yielding the Markovian quantum master equation evolution [45,
46, 47],

$\dot{\rho}_{S}(t)=\mathcal{L}[\rho_{S}(t)]$ (2)

that optimally describes the dynamics of S. This simple form has the advantage
that it is straightforwardly interpretable, i.e., it allows to read off the
Hamiltonian and decoherence processes (see further below).

Data generation.— We simulate the time evolution of the system for times
$t\in[0,\Omega T]$ by means of matrix product states and the time-evolving
block-decimation algorithm [48, 49, 50] [see Fig. 1(b)], which allows us to
study systems of up to $50$ spins. We generate $30$ trajectories obtained by
initializing the system in state $\psi=\bigotimes_{k=1}^{L}\ket{0}$, with
$\sigma^{z}\ket{0}=-\ket{0}$ and perturbing the subsystem S through a random
two-spin unitary $\hat{U}_{\mathrm{rand}}$ distributed with the Haar measure.
As the system evolves in time, after each time-step $\Omega dt=0.01$, we
calculate the expectation value of all the $15$ independent observables
$\\{\mathds{1}_{1}\otimes\sigma_{2}^{x},\mathds{1}_{1}\otimes\sigma_{2}^{y},...,\sigma_{1}^{x}\otimes\sigma_{2}^{y},...,\sigma_{1}^{z}\otimes\sigma_{2}^{z}\\}/2$
of the subsystem S, which uniquely identify the reduced subsystem state. With
the reduced state at hand, we can emulate experimental measurements of the
above local observables. For each trajectory, we select $M$ random times in
the time-window $[0,\Omega T]$ and, for each of these points in time, we
perform $N$ measurements in all the relevant basis in order to produce a noisy
estimate of the reduced state. Such a procedure is sketched in Fig. 1(c), in
which the histogram depicts the counting of $N=10$ measurement outcomes for a
single time-point, whereas the orange dots represent the experimental
expectation value.

ML architecture and training.— The generator $\mathcal{L}$ defining the
quantum master equation (2) can be parametrized as
$\mathcal{L}=\mathcal{H}+\mathcal{D}$, where

$\displaystyle\mathcal{H}[\cdot]$
$\displaystyle=-i[H,\cdot]\,,\quad\mbox{with}\quad
H=\sum_{i=2}^{d^{2}}\theta^{H}_{i}F_{i}\,,$ (3)
$\displaystyle\mathcal{D}[\cdot]$
$\displaystyle=\frac{1}{2}\sum_{i,j=2}^{d^{2}}c_{ij}([F_{i},\cdot
F^{\dagger}_{j}]+[F_{i}\cdot,F^{\dagger}_{j}])\,.$

Here, we have introduced the Hermitian orthonormal basis
$\\{F_{i}\\}_{i=1}^{16}$, for the operators of the subsystem S. Note that we
choose $F_{1}$ to be proportional to the identity operator, specifically
$F_{1}=\mathds{1}/\sqrt{d}$. Moreover, we write the Hamiltonian in terms of
the “fields” $\theta^{H}_{i}$, and the dissipative contribution $\mathcal{D}$
in a non-diagonal form, fully specified by the so-called Kossakowski matrix
$c_{ij}$. The latter must be positive semi-definite in order for the open
quantum dynamics to be completely positive. This constraint can be “hard
coded” by setting $c=Z^{\dagger}Z$, for a complex matrix
$Z=\theta^{X}+i\theta^{Y}$, with $\theta^{X}$ and $\theta^{Y}$ being real-
valued.

In the same spirit, we decompose the reduced state $\rho_{S}$ on the basis
$\\{F_{i}\\}_{i=1}^{16}$ as [51]

$\rho_{\mathrm{S}}=\frac{\mathds{1}}{d}+\sum_{i=2}^{16}F_{i}v_{i}\,,$ (4)

which defines the coherence vector $v_{i}=\Tr(F_{i}\rho_{\mathrm{S}})$. Notice
the condition $\Tr(\rho_{\mathrm{S}})=1$ implies $v_{1}=\frac{1}{\sqrt{d}}$,
that we take outside the sum. The coherence vector encodes the full
information about the density matrix and provides a representation of it as a
vector. This is quite convenient, from a numerical point of view, as it allows
us to write the action of the generator on states as the action of the matrix
$\mathbf{L}$ on coherence vectors. Therefore, in this representation, the
quantum master equation (2) becomes

$\frac{d\mathbf{v}(t)}{dt}=\mathbf{L}\mathbf{v}(t)=(\mathbf{H}+\mathbf{D})\mathbf{v}(t)\,,$
(5)

where $\mathbf{H}_{ij}=-\Tr(\mathcal{H}[F_{i}]F_{j})$ and
$\mathbf{D}_{ij}=\Tr(\mathcal{D}[F_{i}]F_{j})$ are real-valued matrices. As
explicitly shown in the Supplemental Material (see Ref. [52]), the matrix
$\mathbf{H}=\mathbf{H}(\theta^{H}_{i})$ depends linearly on the parameters
$\theta^{H}_{i}$, while the matrix
$\mathbf{D}=\mathbf{D}(\theta^{X}_{ij},\theta^{Y}_{ij})$ depends quadratically
on $\theta^{X}_{ij}$ and $\theta^{Y}_{ij}$.

We build a simple neural network [53], here called Lindblad dynamics
approximator (LDA), as the exponential of the matrices $\mathbf{H}$ and
$\mathbf{D}$,

$\mathbf{M}(\theta,t)=e^{t[\mathbf{H}(\theta^{H}_{i})+\mathbf{D}(\theta^{X}_{ij},\theta^{Y}_{ij})]}\,,$
(6)

that is the structure of the Lindblad time propagator.

We train the LDA to learn the Lindblad representation $\mathbf{L}$ from
(synthetic) experimental data. In the training procedure, we feed the LDA with
the initial conditions $\mathbf{v}_{\mathrm{in}}=\mathbf{v}(0)$ and the time
of the measurement $t$, and optimize the parameters
$\theta=\\{\theta^{H}_{i},\theta^{X}_{ij},\theta^{Y}_{ij}\\}$ such that
$\mathbf{v}_{\mathrm{out}}\simeq\mathbf{v}(t)=\mathbf{M}(\theta,t)\mathbf{v}_{\mathrm{in}}$.
Training over a finite time $t$ is crucial when working with experimental
data. Indeed, training the LDA to propagate the coherence vector only over an
infinitesimal time-step $dt$ [44, 43], i.e.,
$\mathbf{v}_{\mathrm{in}}=\mathbf{v}(t)$ and
$\mathbf{v}_{\mathrm{out}}=\mathbf{v}(t+dt)$, is bound to fail as soon as the
noise is larger than the variation of the coherence vector.

The loss used for the optimization of the LDA parameters is the mean-squared
error function ($\mathbb{E}_{D}$ indicates average over the dataset)

$\mathrm{MSE}(\theta)=\mathbb{E}_{D}\Bigl{[}||M(\theta,\tau)\mathbf{v}(0)-\mathbf{v}(\tau)||^{2}\Bigl{]}\,.$
(7)

We additionally consider a regularization term on the parameters penalizing
nonzero elements of $\theta$. The rationale behind this term is twofold:
first, it yields a more stable training procedure, especially for small
training datasets; second, it keeps the learned generator as simple as
possible for enhanced interpretability. The total loss function is thus

$\mathrm{Loss}(\theta)=\mathrm{MSE}(\theta)+\alpha_{11}(||\theta^{X}||_{1}+||\theta^{Y}||_{1})+\alpha_{12}||\theta^{H}||_{1}\,,$
(8)

where $\|O\|_{1}=\sum_{i,j}|O_{ij}|$. The NN parameters $\theta$ are optimized
by means of Adam optimization algorithm [54], with a scheduled learning rate,
where it decays by a factor of $0.05$ at 100th and 50th epochs from the final
one.

After the training is completed, to test the correctness of the learned
generator, we produce $r$ new exact trajectories and compare them with our
prediction. To have a quantitative measure of the performance of the ML
algorithm, we compute the following error function

$\epsilon(N,M):=\frac{1}{r}\sum^{r}_{i=1}\frac{1}{T}\int^{T}_{0}\frac{\norm{\rho^{i}_{\mathrm{ML}}(t)-\rho^{i}_{\mathrm{S}}(t)}_{2}^{2}}{\norm{\rho^{i}_{\mathrm{S}}(t)}_{2}^{2}}\differential
t\,,$ (9)

where $\rho^{i}_{\mathrm{ML}}(t)$ is the prediction for the state of the
subsystem obtained from our ML algorithm, $\rho^{i}_{\mathrm{S}}(t)$
represents the synthetic data for a given choice of $N$ and $M$, and
$\norm{O}_{2}^{2}={\rm Tr}\left(O^{\dagger}O\right)$111The code for the
generation of the artificial data and the training of the ML algorithm is made
available at https://github.com/giovannicemin/lindblad_dynamics_approximator.
.

Benchmarking the algorithm.—

Figure 2: Benchmark. (a) Open system used to benchmark the ML algorithm,
described in terms of a Hamiltonian and jump operators [cf. Eq. (10)]. The
Hamiltonian consists of a driving ($\Omega$) and an interaction ($V$). Jump
operators describe decay (with rate $\gamma$) and dephasing (with rate
$\kappa$). (b) The error $\epsilon(N,M)$ as in Eq. (9) plotted for different
values of $N$ and $M$. The error is averaged over $r=10$ out-of-sample
trajectories with a doubled time-window $[0,2\Omega T]$ (see main text for
details). (c) Dynamical curves for two selected observables and different
combinations of $N$ and $M$. From the plot one can appreciate how the
predicted dynamics is already quite reliable for smaller values of $N$ and
$M$. The parameters are $V=0.5\Omega$, $\gamma=0.01\Omega$ and
$\kappa=0.05\Omega$.

Before training the algorithm on data for the many-body model in Eq. (1), we
benchmark its ability to learn a Lindblad generator within a well-controlled
setting. To this end we consider a two-spin Lindblad generator, which in its
diagonal form, is specified by the following Hamiltonian and jump operators
[cf. Fig. 2(a)]

$\displaystyle H$
$\displaystyle=\frac{\Omega}{2}(\sigma_{1}^{x}+\sigma_{2}^{x})+Vn_{1}n_{2}\,,$
$\displaystyle J_{1}$ $\displaystyle=\sqrt{\gamma}\sigma_{1}^{-}\,,\qquad
J_{2}=\sqrt{\gamma}\sigma_{2}^{-}\,,$ (10) $\displaystyle J_{3}$
$\displaystyle=\sqrt{\kappa}n_{1}\,,\qquad J_{4}=\sqrt{\kappa}n_{2}\,.$

The jump operators effectively describe the effects of the environment on the
subsystem. In particular, $J_{1},J_{2}$ encode decay from $\ket{1}$ to
$\ket{0}$ while $J_{3},J_{4}$ encode dephasing. The data generation procedure
follows the protocol described above, with two exceptions: the initial
conditions are given by a random density matrix $\rho_{\mathrm{rand}}$, and
the data for testing have a doubled time-window $[0,2\Omega T]$. In this way,
we can also test the ability of the algorithm to extrapolate to unseen times.
Since the network is, in principle, able to perfectly learn a Markovian
Lindblad generator, we expect the extrapolation to be accurate. The results
are reported in Fig. 2 (b-c). The color map [panel (b)] shows the error,
$\epsilon(N,M)$, averaged over $r=10$ test trajectories, for 100 different
combinations of $N$ and $M$. First, we observe the overall trend of the error
to decrease as $N\times M$ increases. However, the plot is not symmetric with
respect to the line $M=N$. In fact, for higher values of $M$ (and fixed
$N\times M$), the ML algorithm has a more stable training, and hence a better
performance. This is due to the fact that the choice of the $M$ time-points is
random, and a small value of $M$ has a high probability of yielding data which
is not representative of the dynamics. On the other hand, higher values of $M$
yields more representative data of the trajectory, despite relatively small
values of $N$. For high values of $N$ and $M$, as expected, the error becomes
small. The ML algorithm can thus successfully learn a Lindbladian, with a
precision that approximately depends on the product $N\times M$.

Many-body setting.—

Figure 3: Learning the subsystem dynamics. Error $\epsilon(N,M)$ as in Eq. (9)
for different values of $N$ and $M$, for two different combinations of the
parameters. The error is calculated over $r=10$ trajectories with
$t\in[0,\Omega T]$. (a) Data for the case $V=0.1\Omega$. Here, the dynamics is
slow and the ML algorithm learns an effective $\mathcal{L}$ already for small
values of $N$ and $M$. (b) Data for $V=2\Omega$. In this case, two different
regimes can be distinguished neatly: either the model does not learn the
dynamics (blue squares) or it learns it (yellow squares). In both cases, the
errors are due to the non-Markovianity assumed for the learned model. The
plots beneath the color maps reports the time evolution of an observable for
the two different models and different sets of $(N,M)$.

Having established the capability of the ML algorithm to exactly learn
Lindblad generators from experimental data, we now address the many-body
scenario described by Eq. (1). In this case, the reduced dynamics of the state
$\rho_{\rm S}(t)$ can feature non-Markovian effects. The ML algorithm will
thus, by construction, learn the “optimal” Markovian description of the
system. Whether this description will accurately describe the subsystem or not
depends on the relevance of non-Markovian effects.

In Fig. 3(a), we show results for weak interactions $V=0.1\Omega$. In this
case, the ML algorithm learns a dynamical generator which faithfully
reproduces the time evolution of the coherence vector. This suggests that the
subsystem dynamics is, in this weak-coupling regime, essentially Markovian.
Notably, the Lindblad generator can inferred even when $N$ and $M$ are small.
As already observed during the benchmarking, the training procedure is less
stable for the models in the bottom-right corner of Fig. 3(a), compared to the
top left, confirming that larger $M$ values are better than larger $N$ values
at fixed $N\times M$.

In Fig. 3(b) we show instead results for strong interactions, $V=2\Omega$.
Quite surprisingly, also in this case, the ML algorithm learns an effective
Lindbladian which reproduces very well the subsystem dynamics. Also in this
case, the latter has a Markovian character. A possible explanation for this is
that strong interactions lead to a faster decay of time correlations in the
environment, which thus renders the subsystem dynamics Markovian. Due to the
faster oscillations in this regime, more sampling points in time are needed
than the weakly interacting case, Fig. 3(a), for a same accuracy. In both
cases, for small $N\times M$, while the model cannot recover the exact
dynamics, it nonetheless provides an average over the fast oscillations [cf.
Figs. 3(a)-3(b)]. The data for the whole coherence vector are reported in the
supplemental material [52], where we also show additional results on the case
$V=0.5\Omega$. In the latter case, the error is higher, due to non-Markovian
effects which appear to be non-negligible [56] in this intermediate regime.

Conclusions.— We have presented a simple ML algorithm able to learn a
physically consistent and interpretable dynamical generator starting from
(synthetic) experimental data. We have shown that it can yield faithful
results for both weak and strong interactions. A wider range of systems could
be included by relaxing the assumed Markovianity of the learned generator,
namely by allowing it to be time-dependent $\mathcal{L}(t)$. Some approaches
already exist (see, e.g., Refs. [39, 36]), but they require an enormous amount
of data and lack physical interpretability.

Our ML method is interpretable, hence it can be used to “read out” the
underlying dynamical processes (see Supplemental Material [52]). In fact, the
learned matrix $\mathbf{L}$ gives direct access to the parameters
$\theta^{H}_{i},\theta^{X}_{ij},\theta^{Y}_{ij}$, which represent the
Hamiltonian and the jump operators of the subsystem S. In the future, it would
be interesting to understand whether feeding the ML algorithm with a sampling
of the full coherence vector is necessary or whether a bona fide dynamics can
still be learned when leaving out information about certain observables.

Acknowledgments.— We acknowledge financial support from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy—EXC-Number 2064/1-Project Number 390727645, under Project
No. 449905436 and through the Research Unit FOR 5413/1, Grant No. 465199066.
This project has also received funding from the European Union’s Horizon
Europe research and innovation program under Grant Agreement No. 101046968
(BRISQ). FC is indebted to the Baden-Württemberg Stiftung for the financial
support of this research project by the Eliteprogramme for Postdocs.

## References

  * Chertkov and Clark [2018] E. Chertkov and B. K. Clark, Computational inverse method for constructing spaces of quantum models from wave functions, Phys. Rev. X 8, 031029 (2018).
  * Zache  _et al._ [2020] T. V. Zache, T. Schweigler, S. Erne, J. Schmiedmayer, and J. Berges, Extracting the field theory description of a quantum many-body system from experimental data, Phys. Rev. X 10, 011020 (2020).
  * Merger  _et al._ [2023] C. Merger, A. René, K. Fischer, P. Bouss, S. Nestler, D. Dahmen, C. Honerkamp, and M. Helias, Learning interacting theories from data (2023), arXiv:2304.00599 .
  * Liu  _et al._ [2011] B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems, Nat. Phys. 7, 931 (2011).
  * Navarrete-Benlloch  _et al._ [2011] C. Navarrete-Benlloch, I. de Vega, D. Porras, and J. I. Cirac, Simulating quantum-optical phenomena with cold atoms in optical lattices, New J. Phys. 13, 023024 (2011).
  * Ma  _et al._ [2012] J. Ma, Z. Sun, X. Wang, and F. Nori, Entanglement dynamics of two qubits in a common bath, Phys. Rev. A 85, 062323 (2012).
  * Everest  _et al._ [2017] B. Everest, I. Lesanovsky, J. P. Garrahan, and E. Levi, Role of interactions in a dissipative many-body localized system, Phys. Rev. B 95, 024310 (2017).
  * Hoeppe  _et al._ [2012] U. Hoeppe, C. Wolff, J. Küchenmeister, J. Niegemann, M. Drescher, H. Benner, and K. Busch, Direct observation of non-Markovian radiation dynamics in 3D bulk photonic crystals, Phys. Rev. Lett. 108, 043603 (2012).
  * Georgescu  _et al._ [2014] I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Rev. Mod. Phys. 86, 153 (2014).
  * Weimer  _et al._ [2010] H. Weimer, M. Müller, I. Lesanovsky, P. Zoller, and H. P. Büchler, A Rydberg quantum simulator, Nat. Phys. 6, 382 (2010).
  * Friedenauer  _et al._ [2008] A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Schätz, Simulating a quantum magnet with trapped ions, Nat. Phys. 4, 757 (2008).
  * Cirac and Zoller [2012] J. I. Cirac and P. Zoller, Goals and opportunities in quantum simulation, Nat. Phys. 8, 264 (2012).
  * Blatt and Roos [2012] R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. Phys. 8, 277 (2012).
  * Babbush  _et al._ [2023] R. Babbush, W. J. Huggins, D. W. Berry, S. F. Ung, A. Zhao, D. R. Reichman, H. Neven, A. D. Baczewski, and J. Lee, Quantum simulation of exact electron dynamics can be more efficient than classical mean-field methods (2023), arXiv:2301.01203 .
  * Jin  _et al._ [2023] S. Jin, N. Liu, X. Li, and Y. Yu, Quantum simulation for quantum dynamics with artificial boundary conditions (2023), arXiv:2304.00667 .
  * Zhang  _et al._ [2023] X. Zhang, E. Kim, D. K. Mark, S. Choi, and O. Painter, A superconducting quantum simulator based on a photonic-bandgap metamaterial, Science 379, 278 (2023).
  * Schäfer  _et al._ [2020] F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Tools for quantum simulation with ultracold atoms in optical lattices, Nat. Rev. Phys. 2, 411 (2020).
  * Carrasco  _et al._ [2021] J. Carrasco, A. Elben, C. Kokail, B. Kraus, and P. Zoller, Theoretical and experimental perspectives of quantum verification, PRX Quantum 2, 010102 (2021).
  * Hauke  _et al._ [2012] P. Hauke, F. M. Cucchietti, L. Tagliacozzo, I. Deutsch, and M. Lewenstein, Can one trust quantum simulators?, Rep. Prog. Phys. 75, 082401 (2012).
  * Altman  _et al._ [2021] E. Altman, K. R. Brown, G. Carleo, L. D. Carr, E. Demler, C. Chin, B. DeMarco, S. E. Economou, M. A. Eriksson, K.-M. C. Fu, M. Greiner, K. R. Hazzard, R. G. Hulet, A. J. Kollár, B. L. Lev, M. D. Lukin, R. Ma, X. Mi, S. Misra, C. Monroe, K. Murch, Z. Nazario, K.-K. Ni, A. C. Potter, P. Roushan, M. Saffman, M. Schleier-Smith, I. Siddiqi, R. Simmonds, M. Singh, I. Spielman, K. Temme, D. S. Weiss, J. Vučković, V. Vuletić, J. Ye, and M. Zwierlein, Quantum simulators: Architectures and opportunities, PRX Quantum 2, 017003 (2021).
  * Morgado and Whitlock [2021] M. Morgado and S. Whitlock, Quantum simulation and computing with Rydberg-interacting qubits, AVS Quantum Sci. 3, 10.1116/5.0036562 (2021), 023501.
  * Gebhart  _et al._ [2023] V. Gebhart, R. Santagati, A. A. Gentile, E. M. Gauger, D. Craig, N. Ares, L. Banchi, F. Marquardt, L. Pezzè, and C. Bonato, Learning quantum systems, Nat. Rev. Phys. 5, 141 (2023).
  * Braccia  _et al._ [2022] P. Braccia, L. Banchi, and F. Caruso, Quantum noise sensing by generating fake noise, Phys. Rev. Appl. 17, 024002 (2022).
  * Han  _et al._ [2021] C.-D. Han, B. Glaz, M. Haile, and Y.-C. Lai, Tomography of time-dependent quantum Hamiltonians with machine learning, Phys. Rev. A 104, 062404 (2021).
  * Mohseni  _et al._ [2022] N. Mohseni, T. Fösel, L. Guo, C. Navarrete-Benlloch, and F. Marquardt, Deep learning of quantum many-body dynamics via random driving, Quantum 6, 714 (2022).
  * Genois  _et al._ [2021] É. Genois, J. A. Gross, A. Di Paolo, N. J. Stevenson, G. Koolstra, A. Hashim, I. Siddiqi, and A. Blais, Quantum-tailored machine-learning characterization of a superconducting qubit, PRX Quantum 2, 040355 (2021).
  * Wilde  _et al._ [2022] F. Wilde, A. Kshetrimayum, I. Roth, D. Hangleiter, R. Sweke, and J. Eisert, Scalably learning quantum many-body Hamiltonians from dynamical data (2022), arXiv:2209.14328 .
  * Bairey  _et al._ [2019] E. Bairey, I. Arad, and N. H. Lindner, Learning a local Hamiltonian from local measurements, Phys. Rev. Lett. 122, 020504 (2019).
  * Di Franco  _et al._ [2009] C. Di Franco, M. Paternostro, and M. Kim, Hamiltonian tomography in an access-limited setting without state initialization, Phys. Rev. Lett. 102, 187203 (2009).
  * Cole  _et al._ [2005] J. H. Cole, S. G. Schirmer, A. D. Greentree, C. J. Wellard, D. K. Oi, and L. C. Hollenberg, Identifying an experimental two-state Hamiltonian to arbitrary accuracy, Phys. Rev. A 71, 062312 (2005).
  * Devitt  _et al._ [2006] S. J. Devitt, J. H. Cole, and L. C. Hollenberg, Scheme for direct measurement of a general two-qubit Hamiltonian, Phys. Rev. A 73, 052317 (2006).
  * Wang  _et al._ [2017] J. Wang, S. Paesani, R. Santagati, S. Knauer, A. A. Gentile, N. Wiebe, M. Petruzzella, J. L. O’Brien, J. G. Rarity, A. Laing, _et al._ , Experimental quantum Hamiltonian learning, Nat. Phys. 13, 551 (2017).
  * Gentile  _et al._ [2021] A. A. Gentile, B. Flynn, S. Knauer, N. Wiebe, S. Paesani, C. E. Granade, J. G. Rarity, R. Santagati, and A. Laing, Learning models of quantum systems from experiments, Nat. Phys. 17, 837 (2021).
  * Nakajima [1958] S. Nakajima, On quantum theory of transport phenomena: steady diffusion, Prog. Theor. Phys. 20, 948 (1958).
  * Zwanzig [1960] R. Zwanzig, Ensemble method in the theory of irreversibility, J. Chem. Phys. 33, 1338 (1960).
  * Cerrillo and Cao [2014] J. Cerrillo and J. Cao, Non-Markovian dynamical maps: numerical processing of open quantum trajectories, Phys. Rev. Lett. 112, 110401 (2014).
  * Gelzinis  _et al._ [2017] A. Gelzinis, E. Rybakovas, and L. Valkunas, Applicability of transfer tensor method for open quantum system dynamics, J. Chem. Phys. 147, 234108 (2017).
  * Pollock and Modi [2018] F. A. Pollock and K. Modi, Tomographically reconstructed master equations for any open quantum dynamics, Quantum 2, 76 (2018).
  * Banchi  _et al._ [2018] L. Banchi, E. Grant, A. Rocchetto, and S. Severini, Modelling non-Markovian quantum processes with recurrent neural networks, New J. Phys. 20, 123030 (2018).
  * Link  _et al._ [2023] M. Link, K. Gao, A. Kell, M. Breyer, D. Eberz, B. Rauf, and M. Köhl, Machine learning the phase diagram of a strongly interacting fermi gas, Phys. Rev. Lett. 130, 203401 (2023).
  * Wu  _et al._ [2021] X. Wu, X. Liang, Y. Tian, F. Yang, C. Chen, Y.-C. Liu, M. K. Tey, and L. You, A concise review of Rydberg atom based quantum computation and quantum simulation*, Chin. Phys. B 30, 020305 (2021).
  * Saffman  _et al._ [2010] M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010).
  * Mazza  _et al._ [2021] P. P. Mazza, D. Zietlow, F. Carollo, S. Andergassen, G. Martius, and I. Lesanovsky, Machine learning time-local generators of open quantum dynamics, Phys. Rev. Res. 3, 023084 (2021).
  * Carnazza  _et al._ [2022] F. Carnazza, F. Carollo, D. Zietlow, S. Andergassen, G. Martius, and I. Lesanovsky, Inferring Markovian quantum master equations of few-body observables in interacting spin chains, New J. Phys. 24, 073033 (2022).
  * Breuer and Petruccione [2002] H. P. Breuer and F. Petruccione,  _The theory of open quantum systems_ (Oxford University Press, Great Clarendon Street, 2002).
  * Gorini  _et al._ [1976] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys. 17, 821 (1976).
  * Lindblad [1976] G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  * Vidal [2004] G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett. 93, 040502 (2004).
  * Zwolak and Vidal [2004] M. Zwolak and G. Vidal, Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm, Phys. Rev. Lett. 93, 207205 (2004).
  * Gray [2018] J. Gray, quimb: a python library for quantum information and many-body calculations, Journal of Open Source Software 3, 819 (2018).
  * Byrd and Khaneja [2003] M. S. Byrd and N. Khaneja, Characterization of the positivity of the density matrix in terms of the coherence vector representation, Phys. Rev. A 68, 062322 (2003).
  * [52] See Supplemental Material for details.
  * Imambi  _et al._ [2021] S. Imambi, K. B. Prakash, and G. Kanagachidambaresan, Pytorch, Programming with TensorFlow: Solution for Edge Computing Applications , 87 (2021).
  * Kingma and Ba [2014] D. P. Kingma and J. Ba, Adam: A method for stochastic optimization (2014), arXiv:1412.6980 .
  * Note [1] The code for the generation of the artificial data and the training of the ML algorithm is made available at https://github.com/giovannicemin/lindblad_dynamics_approximator.
  * Roos  _et al._ [2020] J. Roos, J. I. Cirac, and M. C. Bañuls, Markovianity of an emitter coupled to a structured spin-chain bath, Phys. Rev. A 101, 042114 (2020).

SUPPLEMENTAL MATERIAL

Inferring interpretable dynamical generators of local quantum observables from
projective measurements through machine learning

Giovanni Cemin,1 Francesco Carnazza,1 Sabine Andergassen,2

Georg Martius,3,4 Federico Carollo,1 and Igor Lesanovsky1,5

1Institut für Theoretische Physik, Universität Tübingen,
Auf der Morgenstelle 14, 72076 Tübingen, Germany
2Institute for Solid State Physics and Institute of Information Systems
Engineering, Vienna University of Technology, 1040 Vienna, Austria
3Max Planck Institute for Intelligent Systems, Max-Planck-Ring 4, 72076
Tübingen, Germany

4 Fachbereich Informatik, Universität Tübingen
5School of Physics and Astronomy and Centre for the Mathematics
and Theoretical Physics of Quantum Non-Equilibrium Systems,
The University of Nottingham, Nottingham, NG7 2RD, United Kingdom

## I. Matrix representation of $\mathbf{L}$

We here report a brief derivation of the explicit matrix elements appearing in
eq. (5). The stating point is the coherence vector $v_{i}=\Tr(F_{i}\rho_{S})$.
The time derivative is simply given by

$\frac{d}{dt}v_{i}(t)=\Tr\bigg(F_{i}\frac{d}{dt}\rho_{S}(t)\bigg{missing})=\Tr(F_{i}\mathcal{L}[\rho_{S}(t)])=\Tr(\mathcal{L}^{*}[F_{i}]\rho_{S}(t))$
(S1)

where in the second equality we used eq. (2). In the last equality
$\mathcal{L}^{*}$ is the dual map of $\mathcal{L}$, i.e. the one evolving
observables instead of states. By expanding the dual map
$\mathcal{L}^{*}[F_{i}]=\sum_{j=1}^{d^{2}}\Tr(\mathcal{L}^{*}[F_{i}]F_{j})F_{j}$,
and substituting into eq. (S1), one obtains

$\frac{d}{dt}v_{i}(t)=\sum_{j=1}^{d^{2}}\Tr(\mathcal{L}^{*}[F_{i}]F_{j})\Tr(F_{j}\rho_{S})=[\mathbf{Lv}(t)]_{i}\,,$
(S2)

where we defined the matrix $\mathbf{L}$ as
$\mathbf{L}_{ij}\equiv\Tr(\mathcal{L}^{*}[F_{i}]F_{j})$. By exploiting the
cyclic property of the trace, one can explicitly calculate
$\mathcal{L}^{*}[F_{i}]$ starting from the known action of
$\mathcal{L}[\rho]$. Explicitly we obtain

$\Tr(\mathcal{L}^{*}[F_{k}]F_{l})=\Tr\bigg(i[H,F_{k}]F_{l}+\frac{1}{2}\sum_{i,j=2}^{d^{2}}c_{ij}(F^{\dagger}_{j}[F_{k},F_{i}]F_{l}+[F^{\dagger}_{j},F_{k}]F_{i}F_{l})\bigg{missing})\,.$
(S3)

Comparing this result to eq. (5) we obtain

$\mathbf{H}_{kl}=i\Tr([H,F_{k}]F_{l})\,,\qquad\mathbf{D}_{kl}=\frac{1}{2}\sum_{i,j=2}^{d^{2}}c_{ij}\Tr(F^{\dagger}_{j}[F_{k},F_{i}]F_{l}+[F^{\dagger}_{j},F_{k}]F_{i}F_{l})\,.$
(S4)

To simplify this expressions we use the structure constants $d_{ijk}$ and
$f_{ijk}$, defined from the commutation and anticommutaton relations as

$\displaystyle d_{ijk}=\frac{1}{4}\Tr(\\{F_{i},F_{j}\\}F_{k})\,,$ (S5)
$\displaystyle f_{ijk}=-\frac{i}{4}\Tr([F_{i},F_{j}]F_{k})\,.$ (S6)

By substituting those definitions into eq. (S4), we obtain the following
expressions for the matrix elements of $\mathbf{H}$

$\displaystyle\mathbf{H}_{ij}$
$\displaystyle=-4\sum_{k=2}^{d^{2}}f_{ijk}\theta^{H}_{k}\,,\quad$
$\displaystyle i,j\in\\{2,...,d^{2}\\}$ (S7) $\displaystyle\mathbf{H}_{i1}$
$\displaystyle=\mathbf{H}_{1i}=0\,,\quad$ $\displaystyle
i\in\\{1,...,d^{2}\\}$

where $\theta^{H}_{k}$ provides the expansion of the Hamiltonian over the
basis $F_{i}$, namely $H=\sum_{i=2}^{i=d^{2}}F_{i}\theta^{H}_{i}$. The matrix
elements of $\mathbf{D}$ reads

$\displaystyle\mathbf{D}_{mn}$
$\displaystyle=-8\sum_{i,j,k=2}^{d^{2}}f_{mik}(f_{njk}\real(c)_{ij}+d_{njk}\imaginary(c)_{ij})\,,$
$\displaystyle m,n\in\\{2,...,d^{2}\\}$ (S8) $\displaystyle\mathbf{D}_{m1}$
$\displaystyle=2\sum_{i,j=2}^{d^{2}}f_{imj}\imaginary(c)_{ij}\,,\quad\mathbf{D}_{1m}=0\,,$
$\displaystyle m\in\\{1,...,d^{2}\\}$

where $c_{ij}$ is the so-called Kossakowski matrix, which we parametrized as
$c=Z^{\dagger}Z$ with $Z=\theta^{X}+i\theta^{Y}$ and $\theta^{X}$,$\theta^{Y}$
real matrices.

Equations (S7) and (S8) are the matrix representation of the Lindblad operator
$\mathcal{L}$ in terms of the vector $\omega_{i}$ and the Kossakowski matrix
$c_{ij}$.

## II. Further results

In the main text, we investigate the behavior of the ML framework applied to
spin chains having $V=0.1\Omega,2\Omega$. Here we report the case of a spin
chain having $V=0.5\Omega$. The results are shownin Figure S1. The graph
exhibits a similar behavior to the cases analyzed in the main text. However,
there is a notable distinction: the model saturates at a higher error. This
discrepancy arises from the stronger coupling between the system and the bath,
rendering non-Markovian effects non-negligible.

Figure S1: ML algorithm’s performance on open system dynamics. The figure
reports the error $\epsilon(N,M)$ as in (9) plotted for different values of
$N$ and $M$, and for the $V=0.5\Omega$ spin chain.

## III. Additional plots and physical “read out”

For completeness, we report here the plots of all $15$ non-trivial components
of the coherence vector $v_{i}$. The plots represent the exact time evolution
(black solid line) and the dynamics predicted by the ML algorithm (red dashed
line).

Figure S2: Full plot of the ML algorithm’s performance. In figure the $15$
non-trivial components of the coherence vector $v_{i}$. The data refers to a
spin chain with $V=0.1\Omega$, and $N=20,M=20$. The plots represent the exact
time evolution (black solid line) and the dynamics predicted by the ML
algorithm (red dashed line). In this case, the overall dynamics is roughly
captured. Figure S3: Full plot of the ML algorithm’s performance. In figure
the $15$ non-trivial components of the coherence vector $v_{i}$. The data
refers to a spin chain with the same parameters as in Fig. S2, but $N=2,M=2$.
The plots represent the exact time evolution (black solid line) and the
dynamics predicted by the ML algorithm (red dashed line). In this case, the
overall dynamics is well captured.

In Figure S2 and S3 we report the performance of the model trained on data
that refers to a spin chain with $V=0.1\Omega$. In Fig. S2 the synthetic
experimental data is generated with $N=20,M=20$. In this case, the ML
prediction is almost perfectly overlapped on the exact line, indicating the
correctness of the learned Lindbladian. In Fig. S3 the synthetic experimental
data is generated with $N=2,M=2$. In this case, the dynamics is roughly
captured, made exception for some observables e.g.,
$\sigma^{y}_{1}\sigma^{y}_{2},\sigma^{y}_{1}\sigma^{z}_{2},\sigma^{z}_{1}\sigma^{y}_{2}\text{
and }\sigma^{z}_{1}\sigma^{z}_{2},$.

In addition to the complete plots, we here report the learned expressions for
the Hamiltonian and jump operators. In this case, we take into consideration
the model trained over $N=20$ and $M=20$. For the Hamiltonian, we round up to
two decimal places to improve readability; it reads

$H=0.5(\sigma^{x}_{1}+\sigma^{x}_{2})+0.055(\sigma^{z}_{1}+\sigma^{z}_{2})+0.025\,\sigma^{z}_{1}\sigma^{z}_{2}-0.03(\sigma^{y}_{1}+\sigma^{y}_{2})+0.005(-\sigma^{z}_{1}\sigma^{x}_{2}-\sigma^{y}_{1}\sigma^{x}_{2}+\sigma^{z}_{1}\sigma^{y}_{2}-\sigma^{x}_{1}\sigma^{z}_{2}+\sigma^{x}_{1}\sigma^{y}_{2})\,.$

Regarding dissipation part, there is only one non-zero eigenvalue (rounded to
three decimal places) of the Kossakowski matrix $\gamma=0.013$. Regarding the
jump operator, rounded up to two decimal places, it reads

$\displaystyle J=$ $\displaystyle
0.245\,\sigma^{x}_{1}\sigma^{z}_{2}+(0.19-0.1i)\,\sigma_{2}^{y}+(-0.16+0.07i)\,\sigma^{x}_{1}\sigma^{y}_{2}+(0.055+0.145i)\,\sigma_{1}^{y}\sigma_{2}^{y}+(0.095-0.075i)\,\sigma_{1}^{y}+$
$\displaystyle+(-0.09-0.08i)\,\sigma_{1}^{y}\sigma_{2}^{x}-(0.05+0.105i)\,\sigma_{1}^{x}+(0.09-0.065i)\,\sigma_{1}^{x}\sigma_{2}^{x}+(0.03-0.1i)\,\sigma^{z}_{1}\sigma^{x}_{2}+(-0.09+0.02i)\,\sigma_{1}^{z}+$
$\displaystyle+(0.005+0.08i)\,\sigma_{1}^{z}\sigma_{2}^{x}+(-0.03+0.06i)\,\sigma_{1}^{z}\sigma_{2}^{z}+(0.02+0.035i)\,\sigma_{1}^{y}\sigma_{2}^{z}+0.005\,\sigma_{2}^{z}$

Figure S4: Full plot of the ML algorithm’s performance. In figure the $15$
non-trivial components of the coherence vector $v_{i}$. The data refers to a
spin chain with $V=2\Omega$, and $N=20,M=20$. The plots represent the exact
time evolution (black solid line) and the dynamics predicted by the ML
algorithm (red dashed line). In this case, the overall dynamics is porly
captured. Figure S5: Full plot of the ML algorithm’s performance. In figure
the $15$ non-trivial components of the coherence vector $v_{i}$. The data
refers to a spin chain with the same parameters as in Fig. S4, but $N=3,M=2$.
The plots represent the exact time evolution (black solid line) and the
dynamics predicted by the ML algorithm (red dashed line). In this case, the
overall dynamics is well captured.

In Figure S4 and S5 we report the performance of the model trained on data
that refers to a spin chain with $V=2\Omega$. In Fig. S4 the synthetic
experimental data is generated with $N=20,M=20$. In this case, the ML
prediction is almost perfectly overlapped on the exact line, indicating the
correctness of the learned Lindbladian. In Fig. S3 the synthetic experimental
data is generated with $N=3,M=2$. In this case, the dynamics is poorly
captured.


# Structure-aware Unsupervised Tagged-to-Cine MRI Synthesis with Self
Disentanglement

Xiaofeng Liu  Gordon Center for Medical Imaging, Massachusetts General
Hospital and Harvard Medical School, Boston, MA 02114 USA  Fangxu Xing  Gordon
Center for Medical Imaging, Massachusetts General Hospital and Harvard Medical
School, Boston, MA 02114 USA  Jerry L. Prince  Deportment of Electrical and
Computer Engineering, Johns Hopkins University, Baltimore, MD 21218 USA
Maureen Stone  Deportment of Neural and Pain Sciences, University of Maryland
School of Dentistry, Baltimore, MD 21201 USA  Georges El Fakhri  Gordon Center
for Medical Imaging, Massachusetts General Hospital and Harvard Medical
School, Boston, MA 02114 USA  Jonghye Woo  Gordon Center for Medical Imaging,
Massachusetts General Hospital and Harvard Medical School, Boston, MA 02114
USA

###### Abstract

Cycle reconstruction regularized adversarial training—e.g., CycleGAN,
DiscoGAN, and DualGAN—has been widely used for image style transfer with
unpaired training data. Several recent works, however, have shown that local
distortions are frequent, and structural consistency cannot be guaranteed.
Targeting this issue, prior works usually relied on additional segmentation or
consistent feature extraction steps that are task-specific. To counter this,
this work aims to learn a general add-on structural feature extractor, by
explicitly enforcing the structural alignment between an input and its
synthesized image. Specifically, we propose a novel input-output image patches
self-training scheme to achieve a disentanglement of underlying anatomical
structures and imaging modalities. The translator and structure encoder are
updated, following an alternating training protocol. In addition, the
information w.r.t. imaging modality can be eliminated with an asymmetric
adversarial game. We train, validate, and test our network on 1,768, 416, and
1,560 unpaired subject-independent slices of tagged and cine magnetic
resonance imaging from a total of twenty healthy subjects, respectively,
demonstrating superior performance over competing methods.

###### keywords:

Tagged MRI, Image synthesis, Unsupervised image translation, Anatomical
disentanglement.

##  1 Introduction

Cross-modality image-to-image translation plays a vital role in medical image
analysis. For example, tagged-to-cine magnetic resonance (MR) imaging (MRI)
synthesis can potentially be used to reduce the extra cine MRI acquisition
time and cost, without interfering with subsequent tasks, such as motion
analyses [1, 2]. To this end, previous works usually relied on the paired cine
and tagged MRI for training [1]. However, because of imaging artifacts or
because patients might not tolerate long enough in the scanner, one of the
modalities (e.g., cine-MRI) could be missing. In such circumstances, it is
necessary to relax the requirement of the paired training data.

A typical solution for unpaired image-to-image translation would be to use the
cycle constraint of the bi-directional mapping—e.g., CycleGAN [3],
DiscoGAN[4], and DualGAN[5]. Although prior works yielded visually realistic
results by means of a generative adversarial network (GAN) loss [6, 7, 8, 9,
10], the structural consistency was not enforced, thereby easily resulting in
local distortions in the synthesized images. We note that, in the tagged-to-
cine MRI synthesis task, keeping the anatomical structure is essential for the
subsequent tissue segmentation and motion analysis. In related developments, a
segmentation task was added for co-training [11]. However, it requires
additional segmentation labels to train an anatomical segmentation network
[12, 13, 14, 15]. In addition, Yang et al. [16] enforced the consistency of
the MIND feature between input CT and output MRI data. However, that work
cannot be applied to our tagged-to-cine MRI synthesis, since the contour
patterns cannot be the same between tagged- and cine-MRI due to tag patterns.

In this work, to address the aforementioned challenges, we propose to achieve
a structure-aware translator, by learning general structural features,
compared with task-specific or hand-crafted features, such as the contour-
based MIND feature [16]. This is achieved by an explicit disentanglement of
the anatomical structures (e.g., tongue shape) and imaging modalities via a
novel input-output image patches self-training protocol. Our framework can be
simply added on top of the conventional cycle-constrained GANs for structure-
preserving synthesis.

Figure 1: Illustration of our framework for the tagged-to-cine MRI synthesis,
which consists of a synthesis module and a structure extraction module.

##  2 Related work

Tagged MRI has been a crucial imaging modality for measuring tissue
deformation in moving organs [17]. Since it has intrinsically low anatomical
resolution, an additional matching set of cine MRI is typically acquired for
subsequent tissue segmentation, which added extra scanning time and cost. With
the development of GAN-based image generation methods [18, 19, 20, 21], recent
methods [1, 2] proposed to synthesize cine MRI from acquired tagged MRI. In
these methods, precisely co-registered paired tagged and cine MRI datasets
were required for training.

Cycle reconstruction for image style translation is an important technology
for the unpaired translation task [3, 4, 5]. However, local structures can
hardly be constrained, which thus leads to significant distortions [16]. To
enforce structural consistency, a recent work [16] incorporated the MIND
texture feature to extract an additional supervision signal for CT-to-MRI
translation. In addition, deformation invariant CycleGAN[22] was proposed to
alleviate large nonlinear deformations. Similarly, in that work, a structural
dissimilarity loss [23] was enforced to preserve local structural consistency.
Though low-level texture features, e.g., MIND feature, work efficiently for MR
to CT translation, those features cannot be used in our tagged to cine MRI
translation task, due to additional tag patterns present in tagged MRI.

##  3 METHODS

###  3.1 MRI Data Acquisition

All participants were speaking a word, “a souk,” along with a periodic
metronome-like sound during the MRI scan. MRI scanning was carried out on a
Siemens 3.0T TIM Trio system with a 12-channel head coil and a 4-channel neck
coil using a segmented gradient echo sequence [24]. The field of view was
240$\times$240 mm. In-plane resolution was 1.87$\times$1.87 mm and a slice
thickness was 6 mm. The image sequence was acquired at the rate of 26 fps. We
note that both cine and tagged MRI are in the same coordinate space. The
detailed collection protocol can be found in [1].

###  3.2 Our Proposed Network

In an unpaired setting, we have a group of tagged MR images $\\{x_{t}\\}$ and
a group of cine MR images $\\{x_{c}\\}$. The basic framework of CycleGAN [3]
is illustrated in Fig. 1 left, which has two bi-directional U-Net-based
translators $G_{TC}$: Tagged MRI$\rightarrow$Cine MRI and $G_{CT}$: Cine
MRI$\rightarrow$Tagged MRI, and two discriminators $D_{T}$ and $D_{C}$ for
tagged and cine MRI, respectively. In addition, the cycle constraint minimizes
the image-level reconstruction loss of $L_{1}^{I}$. Note that we only focus on
the tagged-to-cine MRI synthesis in this work. Specifically, the optimization
objectives can be formulated as:

$\displaystyle L_{1}^{I}=||G_{CT}(G_{TC}(x_{t}))-x_{t}||_{1},$ (1)
$\displaystyle
L_{D_{C}}=\mathbb{E}_{x_{c}}[\text{log}(D_{C}(x_{c}))]+\mathbb{E}_{x_{t}}[\text{log}(1-D_{C}(G_{TC}(x_{t}))],$
(2) $\displaystyle
L_{D_{T}}=\mathbb{E}_{x_{t}}[\text{log}(D_{T}(x_{t}))]+\mathbb{E}_{x_{c}}[\text{log}(1-D_{T}(G_{CT}(G_{TC}(x_{c})))],$
(3)

which are trained in a round-based manner in each iteration.

A recent study [11] demonstrated that with a bijective geometric
transformation $T$ with its inverse transformation $T^{-1}$, the cycle
constraint can still be satisfied with the translators of
$G_{TC}^{\prime}=G_{TC}T$ and $G_{CT}^{\prime}=G_{CT}T^{-1}$. Therefore, the
geometrical distortions were not punished at the training stage. In addition,
Yang et al. [16] extracted the MIND feature $M(x_{t})$ and $M(G_{TC}(x_{t}))$
of $x_{t}$ and $G_{TC}(x_{t})$ with a manually defined extractor $M$, and
minimized their reconstruction loss $||M(x_{t})-M(G_{TC}(x_{t}))||_{1}$.
Specifically, since the MIND feature mainly focuses on the boundaries, it it
not straightforward to apply the same MIND feature to tagged MRI due to tag
patterns. To improve upon the prior work [16], in the present work, we propose
to learn a general structure feature extractor $f$ as an alternative to $M$.
Our $f$ can be learned with a novel input-output image patches self-training
scheme to achieve a disentanglement of the structures and imaging modalities.

Inspired by the recent self-supervised learning [25], we simply split both the
input $x_{t}$ and the output $G_{TC}(x_{t})$ to a set of $3\times 3$ patches.
In each forward pass, we randomly choose two patches from a batch of 18
patches as our input. Accordingly, there are four possible combinations:

(a) From the same position of tagged MRI $x_{t}$ and generated cine MRI
$G_{TC}(x_{t})$.

(b) Both from tagged MRI $x_{t}$, with different positions.

(c) Both from generated cine MRI $G_{TC}(x_{t})$, with different positions.

(d) From different modalities and different positions.

Based on these four cases, we have two assumptions: (1) the patches in the
case of (a) should have consistent anatomical structures, and (2) the patch
pairs in these cases have three combinations from two modalities—i.e., both
from tagged MRI, both from cine MRI, and from tagged and cine MRI.

To remove the modality information from the embedding of $f$, we adopt a
classifier $C$ for checking the modality combination of the concatenated two
patch features. $C$ is trained to minimize a 3-class cross-entropy loss
$\mathcal{L}_{CE}$ of modality combination classification as the adversarial
classifier as in Pix2Pix [26]. There can be three possible binary
combinations, i.e., $\\{f(G_{TC}(x_{t})),f(x_{t})\\}$,
$\\{f(G_{TC}(x_{t})),f(G_{TC}(x_{t}))\\}$, and $\\{f(x_{t}),f(x_{t})\\}$. We
denote the true binary combination as $y\in\mathbb{R}^{3}$, which is a one-hot
vector. The combination classification loss can be formulated as:

$\displaystyle\mathcal{L}_{CE}=\sum_{i}^{3}-y_{i}\log C_{i}(\cdot),$ (4)

where $C_{i}(\cdot)$ is the classifier prediction for the i-th class. Our
classifier $C$ and feature extractor $f$ play an asymmetrical adversarial game
to encourage that $f$ eliminates the modality information [27]. Rather than
maximizing the cross-entropy loss, $f$ minimizes the KL-divergence of its
softmax prediction and a uniform distribution. Specifically, we minimize the
following loss:

$\displaystyle\mathcal{L}_{KL}=D_{KL}(C(\cdot)||\mathcal{U}).$ (5)

We note that the modality and position label of the sampled two patches are
known, which can be used for supervised training.

Figure 2: Comparison of different unpaired tagged-to-cine MR generation
methods, including the vanilla GAN (i.e., the half direction of CycleGAN),
CycleGAN [3], and our proposed method. * indicates the first attempt at
unpaired tagged-to-cine MR image synthesis.

Other than removing the modality information from the embedding of $f$, it is
necessary to preserve the anatomic information. In the case of (a), we would
expect the $\mathcal{L}_{1}^{f}$ loss between the embeddings of two patches
can be minimized.

$\displaystyle\mathcal{L}_{1}^{f}=||f(G_{TC}(x_{t}))-f(x_{t})||_{1},$ (6)

Similar to the self-training [2], we adopt an alternating training scheme,
which firstly fixes the structure extraction module to update $G_{TC}$ with
the $\mathcal{L}_{1}^{f}$ loss, and then fix $G_{TC}$ to update the structure
extraction module with the $\mathcal{L}_{1}^{f}$ loss.

After several iterations, the structure extraction module can be expected to
embed the anatomic information and filter the modality information out, and
the CycleGAN can be well regularized by the $\mathcal{L}_{1}$ loss—i.e., the
structural consistency constraint. We use $f$ to replace the manually defined
$M$ as in [16] to achieve a better structural consistency translation. In
testing, only the trained CycleGAN part is used for translation.

##  4 RESULTS

During training, we used unpaired 1,500 tagged MR images with horizontal tag
patterns and 1,500 cine MR images from a total of ten subjects. In addition,
two subjects (416 slice pairs) and eight subjects (1,560 slice pairs) are used
for hyper-parameter validation and evaluation, respectively.

For a fair comparison, we resized the tagged and cine MR images to
256$\times$256 and adopted the $G_{TC}$, $G_{CT}$ backbones from CycleGAN [3].
For the structure encoder $f$, we used five fully convolutional layers, which
resulted in features with the size of $32\times 32\times 128$. The classifier
$C$ has two convolutional layers and two fully-connected layers with a three-
dimensional output layer. We used the PyTorch deep learning toolbox for our
implementation. We also used an NVIDIA V100 GPU for the training, which took
about 6 hours for 200 epochs.

The synthesis results using GAN without the cycle constraint, CycleGAN [3],
and our proposed method are shown in Fig. 2. We can see that the CycleGAN
produced realistic images, while not being able to achieve structural
consistency. By contrast, our method was able to keep the position and shape
of the tongue consistent between the two modalities. The resulting images were
expected to have realistic appearances and to be structurally consistent with
its corresponding paired ground-truth $x_{c}$. For our quantitative
evaluation, we used four evaluation metrics, including mean L1 error,
structural similarity index measure (SSIM), peak signal-to-noise ratio (PSNR),
and inception score (IS) [1]. Table 1 lists numerical comparisons for the
eight testing subjects. The proposed framework outperformed the other two
comparison methods w.r.t. SSIM, PSNR, and IS. Of note, all of the compared
methods do not have the L1 minimization objective as in [1].

##  5 CONCLUSION

In this work, we proposed a novel input-output image patches self-training
scheme to achieve a disentanglement of the anatomical structures and imaging
modalities. The structure extraction module and the structure tagged-to-cine
MRI translator $G_{TC}$ were trained with an alternating training protocol.
Our translator was able to achieve the structure-aware translation as
demonstrated by the tagged-to-cine MRI synthesis task. Both qualitative and
quantitative evaluation results showed that our framework outperformed
CycleGAN for unpaired training. Our approach can be a simple add-on module to
CycleGAN, or the other cycle constrained translators—e.g., DiscoGAN and
DualGAN. Additionally, our method can be applied to other modality synthesis
tasks, including MRI-to-CT synthesis, which is subject to our future work.

Table 1: Numerical comparisons of four methods in testing across 1,560 slice
pairs. The best results are in bold.

Methods | L1 $\downarrow$ | SSIM $\uparrow$ | PSNR $\uparrow$ | IS $\uparrow$
---|---|---|---|---
GAN | 183.8$\pm$0.2 | 0.8863$\pm$0.0015 | 25.84$\pm$0.07 | 7.19$\pm$0.14
CycleGAN | 174.2$\pm$0.3 | 0.9014$\pm$0.0017 | 27.33$\pm$0.06 | 8.86$\pm$0.13
Proposed |  168.5$\pm$0.3 |  0.9237$\pm$0.0014 |  29.52$\pm$0.08 |  9.91$\pm$0.16

###### Acknowledgements.

This work is supported by NIH R01DC014717, R01DC018511, and R01CA133015.

## References

  * [1] Liu, X., Xing, F., Prince, J. L., Carass, A., Stone, M., El Fakhri, G., and Woo, J., “Dual-cycle constrained bijective vae-gan for tagged-to-cine magnetic resonance image synthesis,” in [2021 IEEE 18th International Symposium on Biomedical Imaging (ISBI) ], 1448–1452, IEEE (2021).
  * [2] Liu, X., Xing, F., Stone, M., Zhuo, J., Reese, T., Prince, J. L., El Fakhri, G., and Woo, J., “Generative self-training for cross-domain unsupervised tagged-to-cine mri synthesis,” in [International Conference on Medical Image Computing and Computer-Assisted Intervention ], 138–148, Springer (2021).
  * [3] Zhu, J.-Y., Park, T., Isola, P., and Efros, A. A., “Unpaired image-to-image translation using cycle-consistent adversarial networks,” in [Proceedings of the IEEE international conference on computer vision ], 2223–2232 (2017).
  * [4] Kim, T., Cha, M., Kim, H., Lee, J. K., and Kim, J., “Learning to discover cross-domain relations with generative adversarial networks,” in [International Conference on Machine Learning ], 1857–1865, PMLR (2017).
  * [5] Yi, Z., Zhang, H., Tan, P., and Gong, M., “Dualgan: Unsupervised dual learning for image-to-image translation,” in [Proceedings of the IEEE international conference on computer vision ], 2849–2857 (2017).
  * [6] Liu, X., Guo, Z., Li, S., Xing, F., You, J., Kuo, C.-C. J., El Fakhri, G., and Woo, J., “Adversarial unsupervised domain adaptation with conditional and label shift: Infer, align and iterate,” in [Proceedings of the IEEE/CVF International Conference on Computer Vision ], 10367–10376 (2021).
  * [7] Liu, X., Li, S., Ge, Y., Ye, P., You, J., and Lu, J., “Recursively conditional gaussian for ordinal unsupervised domain adaptation,” in [Proceedings of the IEEE/CVF International Conference on Computer Vision ], 764–773 (2021).
  * [8] Liu, X., Xing, F., El Fakhri, G., and Woo, J., “A unified conditional disentanglement framework for multimodal brain mr image translation,” in [2021 IEEE 18th International Symposium on Biomedical Imaging (ISBI) ], 10–14, IEEE (2021).
  * [9] Liu, X., Liu, X., Hu, B., Ji, W., Xing, F., Lu, J., You, J., Kuo, C.-C. J., Fakhri, G. E., and Woo, J., “Subtype-aware unsupervised domain adaptation for medical diagnosis,” AAAI (2021).
  * [10] He, G., Liu, X., Fan, F., and You, J., “Classification-aware semi-supervised domain adaptation,” in [Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops ], 964–965 (2020).
  * [11] Zhang, Z., Yang, L., and Zheng, Y., “Translating and segmenting multimodal medical volumes with cycle-and shape-consistency generative adversarial network,” in [Proceedings of the IEEE conference on computer vision and pattern Recognition ], 9242–9251 (2018).
  * [12] Liu, X., Han, Y., Bai, S., Ge, Y., Wang, T., Han, X., Li, S., You, J., and Lu, J., “Importance-aware semantic segmentation in self-driving with discrete wasserstein training,” in [Proceedings of the AAAI Conference on Artificial Intelligence ], 34(07), 11629–11636 (2020).
  * [13] Liu, X., Ji, W., You, J., Fakhri, G. E., and Woo, J., “Severity-aware semantic segmentation with reinforced wasserstein training,” in [Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition ], 12566–12575 (2020).
  * [14] Liu, X., Lu, Y., Liu, X., Bai, S., Li, S., and You, J., “Wasserstein loss with alternative reinforcement learning for severity-aware semantic segmentation,” IEEE Transactions on Intelligent Transportation Systems (2020).
  * [15] Liu, X., Xing, F., Gaggin, H. K., Wang, W., Kuo, C.-C. J., Fakhri, G. E., and Woo, J., “Segmentation of cardiac structures via successive subspace learning with saab transform from cine mri,” arXiv preprint arXiv:2107.10718 (2021).
  * [16] Yang, H., Sun, J., Carass, A., Zhao, C., Lee, J., Prince, J. L., and Xu, Z., “Unsupervised mr-to-ct synthesis using structure-constrained cyclegan,” IEEE transactions on medical imaging 39(12), 4249–4261 (2020).
  * [17] Xing, F., Liu, X., Stone, M., Wedeen, V. J., Prince, J. L., El Fakhri, G., and Woo, J., “Measuring strain in diffusion-weighted data using tagged magnetic resonance imaging,” in [SPIE Medical Imaging 2022: Image Processing ], (2022).
  * [18] Liu, X., Kumar, B. V., Jia, P., and You, J., “Hard negative generation for identity-disentangled facial expression recognition,” Pattern Recognition 88, 1–12 (2019).
  * [19] Liu, X., Che, T., Lu, Y., Yang, C., Li, S., and You, J., “Auto3d: Novel view synthesis through unsupervisely learned variational viewpoint and global 3d representation,” in [Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part IX 16 ], 52–71, Springer (2020).
  * [20] Liu, X., Jin, L., Han, X., and You, J., “Mutual information regularized identity-aware facial expression recognition in compressed video,” Pattern Recognition , 108105 (2021).
  * [21] He, G., Liu, X., Fan, F., and You, J., “Image2audio: Facilitating semi-supervised audio emotion recognition with facial expression image,” in [Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops ], 912–913 (2020).
  * [22] Wang, C., Macnaught, G., Papanastasiou, G., MacGillivray, T., and Newby, D., “Unsupervised learning for cross-domain medical image synthesis using deformation invariant cycle consistency networks,” in [International Workshop on Simulation and Synthesis in Medical Imaging ], 52–60, Springer (2018).
  * [23] Xiang, L., Li, Y., Lin, W., Wang, Q., and Shen, D., “Unpaired deep cross-modality synthesis with fast training,” in [Deep Learning in Medical Image Analysis and Multimodal Learning for Clinical Decision Support ], 155–164, Springer (2018).
  * [24] Xing, F., Woo, J., Lee, J., Murano, E. Z., Stone, M., and Prince, J. L., “Analysis of 3-D tongue motion from tagged and cine magnetic resonance images,” Journal of Speech, Language, and Hearing Research 59(3), 468–479 (2016).
  * [25] Chen, T., Zhai, X., Ritter, M., Lucic, M., and Houlsby, N., “Self-supervised gans via auxiliary rotation loss,” in [Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition ], 12154–12163 (2019).
  * [26] Isola, P., Zhu, J.-Y., Zhou, T., and Efros, A. A., “Image-to-image translation with conditional adversarial networks,” in [Proceedings of the IEEE conference on computer vision and pattern recognition ], 1125–1134 (2017).
  * [27] Liu, X., Chao, Y., You, J. J., Kuo, C.-C. J., and Vijayakumar, B., “Mutual information regularized feature-level frankenstein for discriminative recognition,” IEEE Transactions on Pattern Analysis and Machine Intelligence (2021).