# Reconstructing Quantum States Using Basis-Enhanced Born Machines

Abigail McClain Gomez  Department of Physics, Harvard University
Cambridge, Massachusetts 02138, USA  Kadijeh Najafi  IBM Quantum, IBM T.J.
Watson Research Center
Yorktown Heights, NY 10598 USA  Susanne F. Yelin  Department of Physics,
Harvard University
Cambridge, Massachusetts 02138, USA

###### Abstract

Rapid improvement in quantum hardware has opened the door to complex problems,
but the precise characterization of quantum systems itself remains a
challenge. To address this obstacle, novel tomography schemes have been
developed that employ generative machine learning models, enabling quantum
state reconstruction from limited classical data. In particular, quantum-
inspired Born machines provide a natural way to encode measured data into a
model of a quantum state. Born machines have shown great success in learning
from classical data; however, the full potential of a Born machine in learning
from quantum measurement has thus far been unrealized. To this end, we devise
a complex-valued basis-enhanced Born machine and show that it can reconstruct
pure quantum states using projective measurements from only two Pauli
measurement bases. We implement the basis-enhanced Born machine to learn the
ground states across the phase diagram of a 1D chain of Rydberg atoms,
reconstructing quantum states deep in ordered phases and even at critical
points with quantum fidelities surpassing 99%. The model accurately predicts
quantum correlations and different observables, and system sizes as large as
37 qubits are considered. Quantum states across the phase diagram of a 1D XY
spin chain are also successfully reconstructed using this scheme. Our method
only requires simple Pauli measurements with a sample complexity that scales
quadratically with system size, making it amenable to experimental
implementation.

††preprint: APS/123-QED

##  I Introduction

Recent progress towards the creation of highly controllable and programmable
quantum systems has made it increasingly possible to tackle hard problems in
the realm of fundamental science and real world application [1, 2, 3, 4, 5, 6,
7]. However, characterization of a target quantum state is limited by the
information loss inherent to projective measurements and the noise that
pervades today’s quantum hardware. Quantum state tomography (QST) refers to
the task of reconstructing an unknown quantum state $\rho$ from measurement
for characterization. Traditional methods of QST suffer from exponential
scaling in system size $N$, limiting their applicability to small sizes
($N<10$) which have been outgrown by modern quantum systems [8, 9, 10]. Beyond
the concern of exponential scaling with $N$, the presence of quantum
correlations in a target quantum state can present another serious obstacle to
classical methods of QST.

A full state reconstruction is not necessary for many practical applications.
Using randomized measurement tools, it is possible to bypass a complete model
for $\rho$ and still estimate functions of $\rho$, thereby tremendously
reducing the number of experiments and amount of classical post processing
required, and potentially side-stepping the issue of exponential scaling [11,
12, 13, 14, 15]. Classical shadow tomography, which involves constructing a
succinct classical representation of the state from measurements, has been
successful in the efficient estimation of quantum properties and in
classification tasks that involve local observables [14, 15, 16, 17, 18]. The
scheme has been proven to asymptotically reach the scaling limit imposed by
quantum information theory in terms of number of measurements required, but
globally entangling gates are necessary to enjoy less-than-exponential scaling
for the task of fidelity estimation [16]. To tackle tasks like fidelity
estimation, alternative recent tomography schemes employ classical generative
machine learning (ML) tools like restricted Boltzmann machines (RBMs),
recurrent neural networks (RNNs) and attention-based tomography (AQT) [19, 20,
21, 22, 23, 24, 25]. These ML approaches learn directly from raw data but
generally require random measurements from an informationally complete (IC)
set of positive operator valued measures (POVMs), which unfavorably scales as
$4^{N}$ for an $N$-qubit system and is not required for general pure state
reconstruction [26, 27, 28].

Born machines are quantum-inspired generative models for machine learning that
are based on the probabilistic nature of quantum mechanics [29, 23, 30, 31,
32, 33, 34, 35]. The goal of a Born machine is to find a set of probability
amplitudes such that the associated Born probabilities would represent the
total probability distribution of the target data, mirroring the underlying
objective of maximum-likelihood approaches to QST. In a simple application of
a Born machine, a single probability distribution is learned by providing
training data sampled from the target distribution (single-basis learning)
[29, 36, 35]. Tensor networks are the primary Born machine ansatz for encoding
these probability amplitudes, as their structure can innately capture quantum
correlations [29, 37, 31]. This is advantageous for quantum applications as it
is generally known that quantum correlations can make the probability
distributions obtained from a quantum state more difficult to learn
classically, yet by the same token are one of the main prospects for quantum
advantage [2, 38, 39, 40]. In fact, it is possible to harness quantum
correlations in generative modeling, thereby increasing the model’s
expressivity and transporting the algorithm into the realm of quantum machine
learning [41, 42, 30].

In this work, we extend the capability of the Born machine to accept data from
different measurement bases (basis-enhancement) and thereby efficiently
achieve pure quantum state reconstruction. Furthermore, in contrast to using a
randomized measurement toolbox or an IC set of POVMs, we show that by using a
priori knowledge of the operators contained in the Hamiltonian, it is possible
to accurately train a tensor network-based Born machine using measurements
from only two distinct bases. More specifically, we use Born machines to
produce generative models of quantum states across ordered phases and close to
critical points of a one-dimensional Rydberg chain. This system is
characterized by strong Rydberg-Rydberg interactions which produce a
controllable platform for quantum computation [43, 44, 45]. The system also
exhibits unusual non-ergodic dynamics for special states known as quantum
scars [46, 47]. Beyond these features, the 1D Rydberg chain manifests a
nontrivial ground state phase diagram [48, 49], with multiple crystalline
phases that emerge out of a disordered phase for different regions of system
parameters. We further explore the critical points at the boundary of ordered
phases to test the Born machine’s ability to capture long range ordering. We
numerically show that a basis-enhanced Born machine (BEBM) can successfully
predict the quantum correlations at the critical point and can learn an
unknown quantum state to a quantum fidelity surpassing 99% across different
phases for system sizes up to $N=37$, outperforming the original Born machine.
The 1D anisotropic XY chain is also briefly investigated, and it is found that
the basis-enhanced Born machine can capture states from the oscillatory region
of the phase diagram where previous applications of the single-basis Born
machine were unsuccessful [35].

((a))

((b))

((c))

Figure 1: 1(a) Schematic picture of the ground state phase diagram for a 1D
Rydberg system, indicating three emergent ordered phases as function of
detuning and Rydberg radius. The critical points (red stars) and points of
interest in ordered phases (black circles) are highlighted. 1(b) Tensor
diagram illustrating how data native to the model can be accepted for
learning. 1(c) Tensor diagram illustrating how the model can be rotated into
another basis to accept data in a non-native basis. Figure 2: Schematic
diagram of the training procedure for a basis-enhanced Born machine with data
from the $z$\- and $x$-bases. First, a chain of tensors forming the MPS ansatz
are randomly initialized. The model is then iteratively trained by feeding a
batch of measurement data into the model, calculating the loss function, and
updating the model parameters through stochastic gradient descent optimizer
such as Adam. After training, the model is sampled to produce new
configurations of the system.

##  II The 1D Rydberg Model

The focus of this work is a 1D chain of Rydberg atoms. A single Rydberg atom
can be treated as a two level system with ground state $\ket{g_{i}}$ and
Rydberg excited state $\ket{r_{i}}$. Two excited Rydberg atoms in close
proximity to each other will interact via the Van der Waals interaction, which
must be included in the Hamiltonian of a chain of Rydberg atoms:

$\frac{\mathcal{H}}{\hbar}=\sum_{i}\frac{\Omega_{i}}{2}\sigma_{x}^{i}-\sum_{i}\Delta_{i}n_{i}+\sum_{i<j}V_{ij}n_{i}n_{j}.$
(1)

Here, $\sigma_{x}^{i}=\ket{g_{i}}\bra{r_{i}}+\ket{r_{i}}\bra{g_{i}}$,
$n_{i}=\ket{r_{i}}\bra{r_{i}}$, $V_{ij}=C_{6}/r^{6}_{ij}$ represents the Van
der Waals interaction, and $\Omega_{i}$, $\Delta_{i}$ represent the Rabi
frequency and detuning at the $i$th site. Note that in terms of Pauli
operators, $n_{i}=\frac{1}{2}((\sigma_{z}^{i})^{2}-\sigma_{z}^{i})$. For
simplicity, throughout this work we consider a homogeneous Rabi frequency and
detuning ($\Omega_{i}=\Omega$, $\Delta_{i}=\Delta$), as well as an equidistant
chain geometry where all atoms are separated from their neighbors by a
distance $a$.

Due to the Van der Waals interaction, an ensemble of Rydberg atoms resonantly
coupled to a single Rabi drive can only sustain one Rydberg excitation within
the Rydberg radius ($R_{b}$), which is defined by equating the relevant energy
scales of the system ($\Omega=V_{ij}(r=R_{b})$). This blockade mechanism leads
to ordered phases in the ground state phase diagram of a 1D Rydberg chain
(Fig. 1(a)) [44]. The ordered phases arise out of disorder as the detuning is
increased, and higher order phases can be reached by decreasing the inter-
atomic distance.

Throughout this work, a Rabi frequency $\Omega$ of 2 MHz and a Van der Waals
coefficient $C_{6}$ of 5.4e-24 m6/s were used for all computations, thereby
fixing the Rydberg radius; these numerical values are taken from [44]. A more
precise phase map is available in the SI Appendix (Fig. S1), produced
numerically using ITensor’s density-matrix renormalization group (DMRG)
algorithm [50, 51, 52, 53]. The long-range interaction in the Hamiltonian was
truncated in calculations to include at most the fifth nearest neighbor. In
order to converge on a solution in the ordered side of the phase diagram,
calculations were carried out from left to right along a particular $R_{b}/a$
line. Each new solution was used as the initial guess for a slightly larger
$\Delta/\Omega$, a procedure that closely resembles an experimental adiabatic
approach. We further treat the results of the DMRG algorithm as ground truth
for the purposes of quantifying how well a model can learn a particular
quantum state.

Points of interest across the phase diagram were chosen along lines of fixed
$R_{b}/a$ in each ordered phase. Specifically, the critical point at the
boundary of the ordered phase and one point deep within the ordered phase were
selected for further study; these points are highlighted in Fig. 1(a). The
atomic separation associated with each ordered phase was taken from [44], with
the exception of $Z_{4}$. A slightly smaller atomic separation is used in this
work for the $Z_{4}$ phase to avoid crossing the $Z_{3}$-$Z_{4}$ phase
boundary at large detuning $\Delta/\Omega$. For the $Z_{2}$, $Z_{3}$, and
$Z_{4}$ ordered phases, the critical point along a fixed $R_{b}/a$ line was
determined by examining three relevant parameters: the bipartite entanglement
entropy, the magnetization, and the first excited energy gap [54]. More
details can be found in the SI Appendix.

##  III The Generative Born Machine

Generative models are a central paradigm in unsupervised learning, aiming to
capture the total joint probability distribution of the data. This is in
contrast to discriminative models, which try to estimate the conditional
probabilities often for classification tasks [55]. Once training is performed,
a generative model can be used to produce new data according to the modeled
probability distribution – hence its name. Inspired by the probabilistic
nature of quantum mechanics, Born Machines model the quantum wave function
$\Psi(\mathbf{v})$ (or more generally, the probability amplitude distribution)
rather than directly modeling the target distribution
$\mathbbm{P}(\mathbf{v})$. The two quantities are related by Born’s rule:

$\hat{\mathbbm{P}}(\mathbf{v})=\frac{\lvert\hat{\Psi}(\mathbf{v})\rvert^{2}}{\sum_{\mathbf{v}\in\mathcal{V}}{\lvert\hat{\Psi}(\mathbf{v})\rvert^{2}}}.$
(2)

To differentiate the model from the target quantum state $\Psi(\mathbf{v})$,
we will denote our trained model with a hat: $\hat{\Psi}(\mathbf{v})$.

While representing a generic probability distribution requires a number of
parameters exponential in the system size, it turns out one can harness the
underlying structure in natural data to approximate $\mathbbm{P}(\mathbf{v})$
with only a polynomial number of parameters. This has been particularly
successful in probabilistic graphical models (PGM) [33, 56]. By a similar
analogy, a tensor network can provide an efficient representation of quantum
many body states by applying low rank decomposition [57, 58]. A particular
factorization that has shown great success in representing a wide category of
quantum states is the matrix product state (MPS), which is a one-dimensional
tensor network (TN). An MPS is an efficient way of classically storing
information from a high-dimensional (Hilbert) space with a lower-dimensional
representation, when the data satisfies the area law for entanglement [59, 60,
61]. For a one-dimensional chain of atoms such as our system of interest, the
mapping is well-established, and the number of parameters scales polynomially
with system size [62, 63, 64, 65]. Taking advantage of these facts, we use an
MPS as the model ansatz with one tensor for each atomic site:

${\hat{\Psi}}=\sum\limits_{D_{1}=1}^{D}\cdots\sum\limits_{D_{N-1}=1}^{D}{M_{1,i_{1}}^{D_{1}}\,M_{2,i_{2}}^{D_{1},D_{2}}\,\cdots\,M_{N,i_{N}}^{D_{N-1}}}$
(3)

where each tensor has one spin index $i$ and is connected to neighboring
tensors via bond dimensions $D$. The tensor elements themselves are the model
parameters that are optimized during training. For simplicity, we use the same
bond dimension $D$ between each tensor and select $D$ prior to learning.

We train the parameters through the minimization of the negative log
likelihood (NLL), defined below:

$\mathcal{L}=-\frac{1}{\lvert\mathcal{T}\rvert}\sum_{\mathbf{v}\in\mathcal{T}}\ln\hat{\mathbbm{P}}(\mathbf{v}).$
(4)

In the above expression, $\hat{\mathbbm{P}}(\mathbf{v})$ is the probability
distribution of our model, defined in Eq. 2, and $\mathcal{T}$ represents the
set of training data. Minimization of the NLL will maximize the model’s
probability of elements $\mathbf{v}$ that appear in the training data. If the
probability distribution $\hat{\mathbbm{P}}(\mathbf{v})$ perfectly captures
the probability distribution of the training data, then the NLL will approach
the Shannon entropy of the training data
$S(\mathcal{T})=-\sum_{\mathbf{v}}\mathcal{T}(\mathbf{v})\ln(\mathcal{T}(\mathbf{v}))$,
where $\mathcal{T}(\mathbf{v})$ represents the probability of configuration
$\mathbf{v}$ within the data $\mathcal{T}$. To accurately capturing the
quantum state of interest, it is essential to determine the required size of
$\mathcal{T}$. We used a Monte Carlo based-approach focusing on both local and
non-local observables: magnetization and Renyi entropy. The details of this
calculation can be found in the SI Appendix.

###  III.1 Basis Enhancement

In a traditional application of a Born machine, the model remains in a fixed
basis during learning, and all training data is provided in the same fixed
basis [29, 35]. We refer to this method of training as learning from a single
basis (SB). Note that feeding single-basis data into the model is as simple as
projecting a particular configuration $\mathbf{v}$ onto the spin indices of
$\hat{\Psi}$. In the language of tensor diagrams, if blue circles represent 1
in the computational basis (or equivalently represent a Rydberg excitation)
and red circles represent 0, then feeding a snapshot in the $Z_{2}$ phase into
$\hat{\Psi}$ could be represented as Fig. 1(b). Although having complex model
parameters has been proven to increase the model expressivity, data from a
single basis cannot constrain the relative phases of the probability
amplitudes, so purely real parameters are sufficient for SB learning [42].

((a))

((b))

((c))

((d))

((e))

((f))

Figure 3: Fidelity comparison for all combinations of learning basis
($x$-axis) and model parameters (real or complex) for: 3(a) the $Z_{2}$
critical point, 3(b) the $Z_{3}$ critical point, 3(c) the $Z_{4}$ critical
point, 3(d) within the $Z_{2}$ phase, 3(e) within the $Z_{3}$ phase, and 3(f)
within the $Z_{4}$ phase. The fidelity of successful choice ($xz$ learning
with complex model parameters) is provided, with error given by the standard
deviation of the $F$ over the final epoch of training.

SB learning can be successful in capturing classical data sets, but it is
found wanting for the task of faithful quantum state reconstruction. A Born
machine can be modified to allow multi-basis learning that captures more
complex data sets; this was alluded to in [23, 35], and a version named the
Quantum Circuit Born Machine (QCBM) was recently implemented to learn the
classical MNIST data set in [66]. Multi-basis quantum machine learning
algorithms usually involve random measurements from an IC set of POVMs, but
this is not required for the reconstruction of pure states, which are uniquely
defined with fewer parameters. Quantum systems are subject to the curse of
dimensionality as $2^{N}$ probability amplitudes are required to define a pure
quantum state of $N$ qubits. These amplitudes are generally complex, so
$2\times 2^{N}$ parameters are required (neglecting the normalization
constraint and the redundance of global phase) – see [26, 27, 28] for more
rigorous discussions. Global measurements of the system in a single basis will
only provide $2^{N}$ potential outcomes; however, the $2\times 2^{N}$
constraints necessary to fully define a pure quantum state can be supplied by
providing training data from two distinct bases. We chose to maintain the
$z$-basis as the initial reference basis for $\hat{\Psi}$. To feed training
data from a different basis into the model, $\hat{\Psi}$ must be rotated into
the appropriate basis. The MPS architecture can easily be rotated to a non-
native basis to accept or generate data in a different basis; this is
demonstrated for the case of feeding $x$-basis training data by the
application of Hadamard gates $H$ in Fig. 1(c). The loss function must be
modified to reflect the basis-enhanced training data. For the case of $z$\-
and $x$-basis data, $\mathcal{L}$ would take the form:

$\mathcal{L}=-\frac{1}{\lvert\mathcal{T}_{z}\rvert}\sum_{\mathbf{v}_{z}\in\mathcal{T}_{z}}\ln\hat{\mathbbm{P}}_{z}(\mathbf{v}_{z})-\frac{1}{\lvert\mathcal{T}_{x}\rvert}\sum_{\mathbf{v}_{x}\in\mathcal{T}_{x}}\ln\hat{\mathbbm{P}}_{x}(\mathbf{v}_{x}).$
(5)

In this expression, $\hat{\mathbbm{P}}_{z}$ is the familiar probability
distribution defined by the model $\ket{\hat{\Psi}}$, and
$\hat{\mathbbm{P}}_{x}$ is the probability distribution defined by the model
in the $x$-basis which can be accessed through the application of Hadamard
gates ($H^{\otimes n}\ket{\hat{\Psi}}$). In this work, the size of the
$x$-basis and $z$-basis data sets are chosen to be the same
($\lvert\mathcal{T}_{z}\rvert=\lvert\mathcal{T}_{x}\rvert$) in order to avoid
bias toward one particular basis. Note that the theoretical lower bound of the
updated loss is now the sum of the Shannon entropies of the $z$\- and
$x$-basis training data. We refer to a Born machine that is trained from two
measurement bases as a basis-enhanced Born machine (BEBM). A schematic
overview of the training process for a BEBM can be found in Fig. 2.

Motivated by the fact that complex numbers are required to properly describe
quantum systems, we also extend our BEBM to employ complex valued tensors [67,
68]. Not only do complex parameters increase the expressibility of the model
[42], they also greatly increase the learnability by expanding the number of
possible solutions that differ only by global phase. Having a multitude of
solutions is particularly advantageous for basis-enhanced learning, as this
involves optimizing multiple potentially competing probability distributions
with a single model. In our computations, we found that when parameters are
restricted to be real, the success of the BEBM is heavily initialization-
dependent; however, permitting the parameters to be complex – providing a
multitude of solutions that are identical in practice – allows the model
optimization to evade initialization-dependent behavior.

##  IV Learning across quantum ordered phases via BEBM

###  IV.1 1D Rydberg Chain

((a))

((b))

Figure 4: 4(a) Overlaid probability distributions in the $z$\- and $x$-bases
of the DMRG results and: the real-valued, single-basis ($z$) Born machine
(left); the complex-valued, basis-enhanced ($xz$) Born machine (right). A
smaller chain ($N=9$) was used for figure clarity, and each histogram bin
contains two configurations. The classical fidelities $C_{i}$ are provided for
each basis $i$. 4(b) Overlaid probability amplitudes of the DMRG and complex-
valued basis-enhanced Born machine for $xz$ learning of the $Z_{2}$ critical
point.

As a first example, we examine the power of the BEBM in reconstructing the
ground states across different ordered phases and at critical points of a one
dimensional Rydberg system. By varying the values of the detuning parameter
$\Delta$ and the inter-atomic separation $a$, one can cross multiple quantum
phase transitions into ordered phases with underlying emergent $Z_{2}$,
$Z_{3}$ and $Z_{4}$ symmetries [44]. We tasked a variety of basis-enhanced and
single-basis Born machines with faithfully reconstructing the quantum state at
each point of interest indicated by the dark circles and red stars in the
ordered phases and at critical points as shown in Fig. 1(a), for Rydberg chain
of lengths ranging from $N=5$ to $N=37$. The procedure begins by sampling the
MPS output of the DMRG algorithm in one or two bases at the parameters of a
point of interest, simulating measurement of the physical quantum system and
forming the training data set, $\mathcal{T}$. A Born machine is initialized
with $N$ tensors (or $2N$ tensors for complex models) and some fixed bond
dimension $D$; bond dimensions ranging from $D=2$ to $D=16$ were used for
analysis. During training, the elements of the tensor network model are
updated through gradient descent of the NLL (Eq. 4 or a form of Eq. 5), using
PyTorch’s Adam optimizer [69]. After training, the model can efficiently
generate new samples [70, 71]. The resulting probability distribution can be
compared to that of the training data with a classical metric called classical
fidelity, $C$:

$C(\mathbbm{P},\mathbbm{Q})=\bigg{(}\sum_{\mathbf{v}}\sqrt{\mathbbm{P}(\mathbf{v})\mathbbm{Q}(\mathbf{v})}\bigg{)}^{2},$
(6)

where $\mathbbm{P}(\mathbf{v})$ ($\mathbbm{Q}(\mathbf{v})$) represents the
discrete probabilities contained by probability distribution $\mathbbm{P}$
($\mathbbm{Q}$). The classical fidelity between probability distributions
$\mathbbm{P}$ and $\mathbbm{Q}$ is bounded by 0 and 1, and will only equal 1
when $\mathbbm{P}$ and $\mathbbm{Q}$ are identical.

Basis |  $\mathbbm{R}$ / $\mathbbm{C}$ | $C_{x}$ | $C_{y}$ | $C_{z}$ | $\mathcal{L}-S$ | $F$
---|---|---|---|---|---|---
$x$ | $\mathbbm{R}$ | 0.9433 | 0.5588 | 0.1240 | 0.1317 | 0.0000
$\mathbbm{C}$ | 0.9619 | 0.4263 | 0.0549 | 0.0720 | 0.0022
$y$ | $\mathbbm{R}$ | 0.0035 | 0.9068 | 0.4881 | 0.1477 | 0.0000
$\mathbbm{C}$ | 0.0022 | 0.9063 | 0.2243 | 0.1485 | 0.0000
$z$ | $\mathbbm{R}$ | 0.0034 | 0.9049 | 0.9855 | 0.0346 | 0.0000
$\mathbbm{C}$ | 0.0073 | 0.8268 | 0.9852 | 0.0340 | 0.0000
$xy$ | $\mathbbm{R}$ | 0.9249 | 0.8544 | 0.0346 | 0.5459 | 0.0035
$\mathbbm{C}$ | 0.9567 | 0.9003 | 0.0184 | 0.2598 | 0.0003
$xz$ | $\mathbbm{R}$ | 0.6384 | 0.6009 | 0.5499 | 2.6052 | 0.0004
$\mathbbm{C}$ | 0.9644 | 0.9027 | 0.9859 | 0.0966 | 0.9831
$yz$ | $\mathbbm{R}$ | 0.0039 | 0.9079 | 0.9852 | 0.1805 | 0.0000
$\mathbbm{C}$ | 0.0033 | 0.9078 | 0.9856 | 0.1831 | 0.0000
Table 1: Classical fidelities in the $x$-, $y$-, and $z$-basis between the
training data $\mathcal{T}$ and data sampled from the trained model
$\hat{\Psi}$ for real and complex model parameters. The “Basis” column
indicates from which basis(es) the training data originated. The rightmost
columns report final loss $\mathcal{L}$ minus the Shannon entropy of
$\mathcal{T}$ (summed over appropriate bases, when applicable) and quantum
fidelity $F(\Psi,\hat{\Psi})$. These results are for the $Z_{2}$ critical
point, with a fixed bond dimension of $D=4$.

The efficacy of the trained model itself can be evaluated through the quantum
fidelity. Because the ground truth quantum system is simply an MPS calculated
using the DMRG algorithm, the quantum fidelity between our model and the state
of the quantum system can be calculated exactly:

$F=\lvert\langle\Psi|\hat{\Psi}\rangle\rvert^{2}$ (7)

Practically, this calculation amounts to contracting the spin indices of the
model with those of the DMRG output state and taking the modulus square. If
real experimental data was our only resource, a second full state tomography
would be required for this calculation, but for now this simple approach
enabled by DMRG provides a further test of our model and gives insight into
when and why it is possible to learn a quantum state.

Table 1 provides the classical fidelities between samples taken from the
trained model and samples taken from the DMRG algorithm output. All three
orthogonal Pauli bases are provided for verification, although only data from
the basis(es) indicated in the “Basis” column was used for training. Notice
that while there is usually high classical fidelity in the basis(es) used for
training, suggesting that all Born machines are successful in a classical
sense, only in the highlighted training scenario are all fidelities large –
even the quantum fidelity. For a more visual comparison, Fig. 3 provides the
quantum fidelity after training each combination of single-basis and basis-
enhanced Born machines with real and complex model parameters, for each point
of interest across the phase diagram. Notably, one specific choice – the
basis-enhanced Born machine learning from $xz$ data with complex model
parameters – is successful at every point, while all other learning scenarios
fail. The success / failure of different training choices is amplified at the
critical points, making up the top row of subfigures in Fig. 3.

Although Fig. 3 provides insight into which training choice can enable quantum
state learning, the quantum fidelity metric supplies little information
regarding what happens when learning fails. To examine the cases of failure
more closely, the left panel of Fig. 4(a) provides the overlaid probability
distributions at the $Z_{2}$ critical point for single-basis learning –
specifically, $\mathcal{T}_{z}$ was used with real-valued model parameters.
Notice that although there is very good overlap between the data sets in the
$z$-basis, a large overlap is not guaranteed in the other bases; this is
particularly apparent in the $x$-basis. The Born machine is given no
information about the correct relative phases of the quantum state, as phase
information cannot be extracted from measurement data in a single basis – such
a data set is purely classical, even if the data is acquired from a quantum
system. In contrast, the right panel of Fig. 4(a) highlights the success of
the complex-valued basis-enhanced Born machine for the judicious basis choice
of $xz$. Each sampled probability distribution has significant overlap with
the training data, and the quantum fidelity surpasses 98% with a modest bond
dimension of $D=4$.

We also examine the actual probability amplitudes predicted by the basis-
enhanced model, to satisfy any remaining skeptics of the model’s success. In
Fig. 4(b), the probability amplitudes of the DMRG and basis-enhanced results
are provided for each basis state comprising the Hilbert space in the $z$-,
$x$-, and $y$-bases. Although a point for point comparison is visually
unreasonable, it is evident – especially for the linear $z$\- and $x$-bases
results – that the structure of basis-enhanced amplitudes closely matches that
of DMRG with a different global phase, which amounts to a rotation of the
amplitudes in the complex plane.

Figure 5: Learning correlation function as a function of site distance at
critical points across different ordered phases of 1D Rydberg system. For
comparison, the results obtained from the BEBM has overlaid with the DMRG
results indicating the power of BEBM in capturing quantum correlations at
critical point.

It is known that long-range ordering at and near quantum critical points leads
to a diverging correlation length. This has been reported to be the primary
reason of failure for learning the quantum state close to a critical point
[72, 73]. As a further test of the complex-valued BEBM, we examine the
learnabilty of the correlation function close to critical point. The
correlation function is defined as:

$G(r)=\sum_{i}\frac{\langle n_{i}n_{i+r}\rangle-\langle n_{i}\rangle\langle
n_{i+r}\rangle}{N_{p}},$ (8)

in this expression, $n_{i}$ is the Rydberg state projector at site $i$ as
defined in the Hamiltonian, and the normalization $N_{p}$ is the number of
pairs of sites included in the sum. $G(r)$ is a measure of Rydberg density
correlations separated by a number of sites $r$. There is generally good
agreement between $G(r)$ as calculated from the trained model and from the
DMRG results (Fig. 5), indicating the success of the BEBM at capturing quantum
correlations at and close to critical point.

###  IV.2 1D Anisotropic XY Chain

To further explore the success and necessity of the basis-enhancement, we
applied a Born machine to the anisotropic XY chain with a transverse magnetic
field, first attempted in [35]. The system is described by the Hamiltonian:

$\
\mathcal{H}=-J\sum_{i=1}^{N-1}\Big{(}\frac{1+\gamma}{4}\sigma_{i}^{x}\sigma_{i+1}^{x}+\frac{1-\gamma}{4}\sigma_{i}^{y}\sigma_{i+1}^{y}\Big{)}-\frac{h}{2}\sum_{i=1}^{N}\sigma_{i}^{z},$
(9)

where $J$ is the spin-spin coupling, $\gamma$ captures the relative strength
of interaction along $x$ and $y$, and $h$ is the external magnetic field
coupled to spins along the $z$-axis; we fixed $J=1$ as in [35]. Although this
system is exactly solvable, it spans multiple interesting phases. In [35], it
was demonstrated that the single-basis Born machine can capture the ground
states of the anisotropic XY chain at the Ising critical point ($\gamma=h=1$),
within the ordered phase ($\gamma>1;h<1$), and within the disordered phase
($\gamma>1;h>1$) with quantum fidelities surpassing 99% for $z$-basis training
data and real model parameters. However, in the oscillatory region
($h^{2}+\gamma^{2}<1$) where the system’s correlation function exhibits
oscillatory behavior, the single-basis Born machine was unsuccessful.

Figure 6: Comparison of quantum fidelity for the real-valued, single-basis
($z$) Born machine and the complex-valued, basis-enhanced ($xz$) Born machine
when learning various points from the ground state phase diagram of the 1D
anisotropic XY chain. Ising critical point: $h=1.0,\gamma=1.0$; ordered:
$h=0.5,\gamma=1.5$; disordered: $h=2.0,\gamma=2.0$; oscillatory:
$h=0.5,\gamma=0.5$.

In this work, we applied the single-basis and BEBM to the XY ground states
considered by [35]; the results are presented in Fig. 6. At fixed bond
dimension, the BEBM produces models with quantum fidelities comparable to
those of the single-basis Born machine for the states that could be captured
via SB learning. Additionally, the BEBM successfully captured the point inside
the oscillatory region that could not be learned via SB learning, highlighting
the robustness of basis-enhancement.

###  IV.3 Basis Enhanced Born Machine Under the Hood

In order to claim knowledge of a full quantum state, it is important to know
the relative phases between the probability amplitudes of basis states. Phase
information is lost when the probability amplitudes are converted to
probabilities, but reconstruction of the lost relative phase information is
achievable if one possesses knowledge of the state’s probability distribution
in multiple bases. This reconstruction is possible because the relative phases
in one basis will determine the probability amplitudes of the state – and
hence the probability distribution – in a different basis.

A tomographically complete set on the order of $2^{2N}$ measurements is
theoretically required to provide enough information to reconstruct an
arbitrary quantum state; however, this is an upper bound. Most quantum states
of physical interest such as pure states, ground states of local Hamiltonians,
and states with particular symmetries can be specified by a small number of
parameters and thus for such quantum states, fewer measurements are required
(a tomographically incomplete set) for reconstruction[8]. The numerical work
presented here illustrates that only measurements in the $z$-basis and
$x$-basis are sufficient to reconstruct the ground state of the Rydberg chain.
This number of measurements ($2\times 2^{N}=16384$ for $N=13$) is much smaller
than what would be considered tomographically complete ($4^{N}=67108864$ for
$N=13$). For the XY chain, it was demonstrated that only $z$-basis data is
required for state reconstruction in many cases, an even more extreme example
of the sufficiency of an incomplete measurement set.

To illuminate why such limited information is often sufficient for the XY
chain, here we investigate the probability amplitudes of the ground states and
consider model initialization. First, both the Rydberg chain and the XY chain
feature real-valued probability amplitudes in the computational basis. This
phenomenon can be explained simply by the fact that each system’s Hamiltonian
is purely real; hence, each Hamiltonian is symmetric rather than being merely
Hermitian. Any eigenstate of these Hamiltonians can be written to have purely
real probability amplitudes for a particular global phase. In addition to
being real, the probability amplitudes of the XY chain ground states are real
and non-negative – see Fig. 16 in the SI Appendix. This is a feature of some
local spin Hamiltonians known as stoquastic Hamiltonians, which are
characterized by real and non-positive off-diagonal matrix elements in the
standard basis [74]. For such Hamiltonians, it is proven that for any
$\beta\geq 0$, the Gibbs density matrix $\rho=e^{-\beta H}/\,{\rm
Tr}\,e^{-\beta H}$ has non-negative matrix elements in the standard basis
which simply implies that
$|\psi(\textbf{v})\rangle=\sum_{\textbf{v}}\alpha_{\textbf{v}}|\textbf{v}\rangle$
where $\alpha\geq 0$. Thus, the ground state can be fully described by a
single probability distribution:
$\alpha_{\textbf{v}}=\sqrt{\mathbbm{P}(\textbf{v})}$. Moreover, our choice to
initialize the real-valued model with random values between zero and one also
produces a state with purely positive probability amplitudes. The initial
state is therefore close in parameter space to the target ground state of any
stoquastic Hamiltonian, and the model arrives at the correct relative phase
choice with only $z$-basis training data. Considering the positive nature of
the probability amplitudes together with our model initialization explains why
single-basis learning can be sufficient for reconstructing the ground states
of the XY Hamiltonian in the ordered and disordered phases, as well as at the
Ising critical point. We will comment on the failure of single basis learning
in the oscillatory phase in the final section.

Theoretically, quantum state reconstruction of an arbitrary pure quantum state
is possible from $2\times 2^{N}$ measurements for any basis choice; however,
not all probability distributions are created equal when it comes to training
a Born machine. It is known that training a Born machine via stochastic
gradient descent (SGD) can fail to capture non-local information within a
distribution as these methods are based on local updates [71]. We
heuristically found that probability distributions which are spread across a
large fraction of the basis states are generally more difficult – or even
impossible – to learn with traditional stochastic gradient descent
optimization methods. We believe that this is the primary reason learning
fails for certain basis choices. The degree to which a probability
distribution is spread across the $2^{N}$ configurations can be quantified by
its Shannon entropy, which is linked to learning in that at the global minimum
of the loss function, $\mathcal{L}$ approaches the Shannon entropy of the
training data. A larger Shannon entropy will result in a shallower
optimization landscape.

For each system considered, it was observed that careful selection of
measurement bases can improve state reconstruction. In the Rydberg chain
results presented in the previous section, the BEBM was successful only when
$z$\- and $x$-basis training data was employed. When $y$-basis training data
was substituted for either, the Born machine was unable to capture the quantum
state (see Table 1 and Fig. 6). The importance of basis choice may in part be
explained by the operators that appear in the Hamiltonian, as the Rydberg
Hamiltonian only involves $\sigma_{z}$ and $\sigma_{x}$ operators. This leads
to preferential configurations for the ground state in these bases, which in
turn produces more structured probability distributions that are easier to
learn. To test this hypothesis, the Rydberg Hamiltonian was modified by
replacing the $\sigma_{x}$ operator with a $\sigma_{y}$ operator. We found
that in this scenario, $z$ and $y$ was the only training data basis
combination that resulted in successful learning. Similarly, for the
anisotropic XY chain, a second point from the oscillatory region with
$\gamma<0$ was tested. A negative $\gamma$ results the the $\sigma^{y}$
operators appearing in the Hamiltonian to be weighted more heavily than the
$\sigma^{x}$ operators. We found that for this ground state, $yz$ training
data was successful for learning while $xz$ training data failed, in agreement
with the hypothesis. See the SI Appendix for more information. These results
stress the importance of using a priori knowledge about the Hamiltonian to
select the training bases. When this knowledge is not available, an active
learning strategy like that employed by [75] could be adapted for Born
machines.

Figure 7: Largest two eigenvalues of the entanglement spectrum for the reduced
density matrix of half of the chain, calculated after each epoch for a variety
of bond dimension $D$. These results are averaged over 100 trials of learning
the $Z_{2}$ critical point ($N=13$), for a complex BEBM learning from $x$\-
and $z$-basis data. The ribbon indicates $\pm$ one standard deviation.

Finally, to further investigate the efficiency and robustness of the enhanced-
Born Machine, we examined how the entanglement spectrum of the BEBM changes
during training at $Z_{2}$ critical point. Starting from our model MPS, we
first obtain the reduced density matrix for half of the quantum spin chain
($N=13$) by tracing out the complementary sites: $\rho_{A}={\rm
Tr}_{B}(\rho\ln\rho)$, where $\rho=|\hat{\psi}\rangle\langle\hat{\psi}|$ is
the density matrix of the full system. Then the entanglement spectrum can be
obtained by diagnalizing $\rho_{A}$. The largest two eigenvalues are plotted
in Fig. 7, as all others were vanishingly small. The plot demonstrates that
even for modest bond dimensions of $D=2,4,6$, the model can successfully
encode the system’s entanglement, even at a critical point. In fact, it is
even more beneficial to choose a small bond dimension, as over-
parameterization can lead to larger oscillations in the spectrum – see the
wider ribbon for the red $D=6$ curve in Fig. 7.

###  IV.4 Scaling Trends

To better understand the sample complexity of the BEBM and its scaling with
system size $N$, we investigated the accuracy of the BEBM in learning the
$Z_{2}$ critical point as a function of sampling size ($|\mathcal{T}|$) for
system sizes ranging from 5 to 37 and fixed bond dimension $D=4$. To contrast
the complexity of sampling at the critical point, we compare the results with
those within the ordered phase. The results are provided in Fig. 8. Overall,
the scaling law $1-F\sim c\times N\times|\mathcal{T}|^{-1/2}$ was uncovered,
where $c$ is a constant that depends on the complexity of the state under
consideration. We found that within the $Z_{2}$ phase, this constant is
approximately 0.096, while at the critical point, it increases to 0.28. For
fixed infidelity error $\epsilon=1-F$, this scaling implies that the number of
required training samples $|\mathcal{T}|$ is quadratic in the number of qubits
$N$. Note that this is for fixed bond dimension, such that the overall number
of model parameters $N_{p}$ scales only linearly in $N$. Similar scaling has
been observed for RBM tomography schemes, except the RBM tomography schemes
require a quadratic number of parameters with a linear number of training
samples to achieve a fixed error $\epsilon$ for Ising ground states, swapping
the BEBM scaling for $N_{p}$ and $|\mathcal{T}|$ [76, 20].

((a))

((b))

((c))

Figure 8: 8(a): Infidelity after training basis-enhanced born machines within
the $Z_{2}$ phase for a variety of system sizes $N$ as a function of training
data size $|\mathcal{T}|=|\mathcal{T}_{x}|+|\mathcal{T}_{z}|$. For consistency
across different $N$, the value of $\Delta/\Omega$ was selected at fixed $a$
such that the probability of being in the ordered phase is $\sim 75\%$. Each
point is the average of 10 trials, and the ribbon indicates $\pm 1$ standard
deviation of the trials. 8(b): Infidelity after training basis-enhanced born
machines at the $Z_{2}$ critical point for a variety of system sizes $N$.
8(c): The left panels display the linear relationship between infidelity and
$|\mathcal{T}|^{-1/2}$. On the right, the slopes of the linear fits from the
left panels are plotted against system size $N$, revealing another linear
relationship.

##  V Discussion and Conclusion

The characterization of quantum many body states lies at the heart of quantum
computing and quantum information. This task becomes even more crucial as we
rapidly progress toward quantum systems of larger size. In this paper, we have
presented an alternative machine learning method that harnesses both the power
of generative models and the probabilistic nature of quantum mechanics for
pure quantum state tomography. The scheme learns from a minimal number of
probability distributions obtained from easily accessible quantum measurements
such as single Pauli measurements.

While describing a generic quantum many body state requires an exponential
number of parameters, quantum states that occur in many practical situations
possess features such as locality and symmetries which lead to a reduction in
the required parameters [58]. In particular, inspired by the fact that the
internal architecture of a tensor network provides an efficient method for
targeting states that are concentrated in some part of the Hilbert space, we
have utilized the Born machine for learning across different quantum ordered
phases. We have further shown that extending the tensor-based Born machine to
complex values and encoding data from non-native measurement bases (basis-
enhancement) not only leads to an increase in model expressibilty, but also
allows one to access information about the quantum relative phases, which is
essential for full reconstruction of non-trivial pure quantum states. While
our model can be generalized to any tensor network architecture, we have
utilized the one dimensional matrix product state as a model ansatz for our
basis-enhanced Born machine, and we have shown that it is capable of learning
across the emergent $Z_{2}$, $Z_{3}$ and $Z_{4}$ ordered phases of a Rydberg
system with fidelities surpassing 99%. Surprisingly, our scheme is even
capable of reconstructing quantum states at and close to critical points,
which tend to be much harder to learn due to the existence of long range
correlations.

If enough samples are gathered, information from a standard basis such as $z$
would theoretically be sufficient to reconstruct quantum states that have real
and non-nonegative amplitudes (which generally occurs for the ground states of
stoquastic Hamiltonians [74]). While our numerical results for the XY
Hamiltonian confirm that learning only from the $z$-basis is sufficient to
reconstruct the ground state in many parts of the phase diagram, in the
oscillatory phase, the $z$-basis probability distribution is spread across a
larger fraction of the $2^{N}$ possible configurations $\mathbf{v}$. This
complex structure poses an obstacle to learning the single-basis probability
distribution, but it is still possible to reconstruct such ground states via
basis-enhancement.

More precisely, the presence of barren plateaus in the optimization landscape
may explain the single-basis Born machine’s failure in learning the ground
state of the XY chain in the oscillatory region. We found that the probability
amplitudes of this particular state were qualitatively identical to the other
XY chain ground states we examined, so one might expect the single-basis Born
machine to capture this state as it did for the other points. We believe that
the large Shannon entropy associated with the $z$-basis data for this state
may lead to a barren plateau – a region in parameter space where the gradient
vanishes, halting optimization [77]. Barren plateaus have been observed in
tensor network training scenarios, and it is likely that the global nature of
the NLL (Eq. (4)) contributes to their presence in the Born machine’s training
landscape [78]. Even so, the basis-enhanced Born machine can capture the
oscillatory ground state of the XY chain. Providing the $x$-basis training
data – which has much lower Shannon entropy – in addition to the $z$-basis
data improves optimization while still providing the necessary state
information. Using complex parameters greatly increases the number of
identical solutions up to an overall phase, such that locating one of the
global minimums is much more likely regardless of model initialization.

We have shown that complete ground state reconstruction is achievable using
only two Pauli measurement bases if care is taken to select these bases using
a priori knowledge of which operators appear in the Hamiltonian. However, when
no a priori knowledge about the state or Hamiltonian is available, it may
still be possible to use the BEBM scheme to achieve quantum state
reconstruction. The naive approach would be to include data from three
orthogonal bases ($x$, $y$, and $z$) in training, although this may include
redundant information. Another approach would involve using information about
the training data sets’ Shannon entropies for basis selection, either prior to
learning or adaptively as in [75]. A few global bases that are experimentally
easily accessible could be considered as candidates, and the Shannon entropy
of each could be estimated from a small amount of data. The training bases
should be selected such the net Shannon entropy is small, thereby increasing
the information provided to the model but also improving the trainability of
the Born machine. If only one basis possesses a small Shannon entropy, a
secondary basis could be selected that is rotated only partially ($<\pi/2$)
from the first. We also note that more measurement bases and either a matrix
product operator architecture or a purification scheme such as that presented
in [79] will generally be necessary for mixed state reconstruction, but
further study is required. Moreover, methods beyond stochastic gradient
descent for optimization of tensor networks should be explored to relax the
importance of training data basis choice.

The basis-enhanced tensor-based Born machine is a useful tool for quantum
state reconstruction in the current age of noisy devices and limited
experimentally accessible measurements. The scheme could additionally be used
to compute various non-local quantities that are hard to measure, including
von Neumann entanglement entropy, higher correlation functions, or even local
observables in a particular basis [80, 7]. While in this paper we have focused
on reconstructing quantum states across phases with well-defined symmetries,
the BEBM could be applied to learning less trivial phases, such as the
recently realized topological phases or spin liquids [81]. Applying the BEBM
to other non-local states like the GHZ state or cluster states that are
challenging to learn using a single basis Born machine would be another
important future direction [71].

###### Acknowledgements.

K.N would likes to thank Xun Gao, Mohammad Ali Rajabpour, Anatoli Dymarsky and
Dolev Bluvstein for their helpful discussions. A.M.G acknowledges support from
the NSF through the Graduate Research Fellowships Program, as well as support
through the Theodore H. Ashford Fellowships in the Sciences. S.F.Y would like
to acknowledge funding by the NSF and DOE.

## References

  * Ball  _et al._ [2021] P. Ball _et al._ , First quantum computer to pack 100 qubits enters crowded race, Nature 599, 542 (2021).
  * Arute  _et al._ [2019] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrà, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  * Pednault  _et al._ [2019] E. Pednault, J. A. Gunnels, G. Nannicini, L. Horesh, and R. Wisnieff, Leveraging secondary storage to simulate deep 54-qubit sycamore circuits (2019).
  * Zhong  _et al._ [2020] H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, P. Hu, X.-Y. Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang, L. Gan, G. Yang, L. You, Z. Wang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, Quantum computational advantage using photons, Science 370, 1460 (2020), https://www.science.org/doi/pdf/10.1126/science.abe8770 .
  * Gong  _et al._ [2021] M. Gong, S. Wang, C. Zha, M.-C. Chen, H.-L. Huang, Y. Wu, Q. Zhu, Y. Zhao, S. Li, S. Guo, H. Qian, Y. Ye, F. Chen, C. Ying, J. Yu, D. Fan, D. Wu, H. Su, H. Deng, H. Rong, K. Zhang, S. Cao, J. Lin, Y. Xu, L. Sun, C. Guo, N. Li, F. Liang, V. M. Bastidas, K. Nemoto, W. J. Munro, Y.-H. Huo, C.-Y. Lu, C.-Z. Peng, X. Zhu, and J.-W. Pan, Quantum walks on a programmable two-dimensional 62-qubit superconducting processor, Science 372, 948 (2021), https://www.science.org/doi/pdf/10.1126/science.abg7812 .
  * Wu  _et al._ [2021] Y. Wu, W.-S. Bao, S. Cao, F. Chen, M.-C. Chen, X. Chen, T.-H. Chung, H. Deng, Y. Du, D. Fan, _et al._ , Strong quantum computational advantage using a superconducting quantum processor, Physical review letters 127, 180501 (2021).
  * Ebadi  _et al._ [2021] S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, _et al._ , Quantum phases of matter on a 256-atom programmable quantum simulator, Nature 595, 227 (2021).
  * Flammia  _et al._ [2012] S. T. Flammia, D. Gross, Y.-K. Liu, and J. Eisert, Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators, New J. Phys. 14, 095022 (2012).
  * Haah  _et al._ [2016] J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu, Sample-optimal tomography of quantum states, in  _Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing_, STOC ’16 (Association for Computing Machinery, New York, NY, USA, 2016) p. 913–925.
  * O’Donnell and Wright [2016] R. O’Donnell and J. Wright, Efficient quantum tomography, in  _Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing_, STOC ’16 (Association for Computing Machinery, New York, NY, USA, 2016) p. 899–912.
  * Elben  _et al._ [2022] A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The randomized measurement toolbox (2022).
  * van Enk and Beenakker [2012] S. J. van Enk and C. W. J. Beenakker, Measuring $\mathrm{Tr}{\rho}^{n}$ on single copies of $\rho$ using random measurements, Phys. Rev. Lett. 108, 110503 (2012).
  * Elben  _et al._ [2018] A. Elben, B. Vermersch, M. Dalmonte, J. I. Cirac, and P. Zoller, Rényi entropies from random quenches in atomic hubbard and spin models, Phys. Rev. Lett. 120, 050406 (2018).
  * Aaronson [2018] S. Aaronson, Shadow tomography of quantum states, in  _Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing_, STOC 2018 (Association for Computing Machinery, New York, NY, USA, 2018) p. 325–338.
  * Aaronson and Rothblum [2019] S. Aaronson and G. N. Rothblum, Gentle measurement of quantum states and differential privacy, in  _Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing_, STOC 2019 (Association for Computing Machinery, New York, NY, USA, 2019) p. 322–333.
  * Huang  _et al._ [2020] H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nature Physics 16, 1050 (2020).
  * Huang  _et al._ [2021] H.-Y. Huang, R. Kueng, G. Torlai, V. V. Albert, and J. Preskill, Provably efficient machine learning for quantum many-body problems, arXiv:2106.12627 [quant-ph] (2021).
  * Hu  _et al._ [2022] H.-Y. Hu, R. LaRose, Y.-Z. You, E. Rieffel, and Z. Wang, Logical shadow tomography: Efficient estimation of error-mitigated observables, arXiv:2203.07263 [quant-ph] (2022).
  * Carleo and Troyer [2017] G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  * Torlai  _et al._ [2019] G. Torlai, B. Timar, E. P. L. van Nieuwenburg, H. Levine, A. Omran, A. Keesling, H. Bernien, M. Greiner, V. Vuletić, M. D. Lukin, R. G. Melko, and M. Endres, Integrating neural networks with a quantum simulator for state reconstruction, Physical Review Letters 123, 230504 (2019), arXiv: 1904.08441.
  * Torlai  _et al._ [2018] G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, and G. Carleo, Neural-network quantum state tomography, Nature Physics 14, 447 (2018).
  * Carrasquilla  _et al._ [2019] J. Carrasquilla, G. Torlai, R. G. Melko, and L. Aolita, Reconstructing quantum states with generative models, Nature Machine Intelligence 1, 155 (2019).
  * Wang  _et al._ [2020] J. Wang, Z.-Y. Han, S.-B. Wang, Z. Li, L.-Z. Mu, H. Fan, and L. Wang, Scalable quantum tomography with fidelity estimation, Physical Review A 101, 032321 (2020).
  * Carrasquilla and Torlai [2021] J. Carrasquilla and G. Torlai, How to use neural networks to investigate quantum many-body physics, PRX Quantum 2, 040201 (2021).
  * Cha  _et al._ [2022] P. Cha, P. Ginsparg, F. Wu, J. Carrasquilla, P. L. McMahon, and E.-A. Kim, Attention-based quantum tomography, Machine Learning: Science and Technology 3, 01LT01 (2022).
  * Finkelstein [2004] J. Finkelstein, Pure-state informationally complete and “really” complete measurements, Physical Review A 70, 052107 (2004).
  * Flammia  _et al._ [2005] S. T. Flammia, A. Silberfarb, and C. M. Caves, Minimal Informationally Complete Measurements for Pure States, Foundations of Physics 35, 1985 (2005).
  * Heinosaari  _et al._ [2013] T. Heinosaari, L. Mazzarella, and M. M. Wolf, Quantum Tomography under Prior Information, Communications in Mathematical Physics 318, 355 (2013).
  * Han  _et al._ [2018] Z.-Y. Han, J. Wang, H. Fan, L. Wang, and P. Zhang, Unsupervised Generative Modeling Using Matrix Product States, Physical Review X 8, 031012 (2018), publisher: American Physical Society.
  * Coyle  _et al._ [2020a] B. Coyle, D. Mills, V. Danos, and E. Kashefi, The born supremacy: quantum advantage and training of an ising born machine, Npj Quantum Inf. 6 (2020a).
  * Wall  _et al._ [2021] M. L. Wall, M. R. Abernathy, and G. Quiroz, Generative machine learning with tensor networks: Benchmarks on near-term quantum computers, Physical Review Research 3, 10.1103/physrevresearch.3.023010 (2021).
  * Zang  _et al._ [2022] W. Zang, X. Gao, S. F. Yelin, and K. Najafi, Many body localized hidden born machines (2022), In Preparation .
  * Cheng  _et al._ [2018] S. Cheng, J. Chen, and L. Wang, Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines, Entropy 20, 583 (2018).
  * Coyle  _et al._ [2020b] B. Coyle, D. Mills, V. Danos, and E. Kashefi, The Born supremacy: quantum advantage and training of an Ising Born machine, npj Quantum Information 6, 60 (2020b).
  * McClain Gomez  _et al._ [2021] A. McClain Gomez, S. F. Yelin, and K. Najafi, Born Machines for Periodic and Open XY Quantum Spin Chains, arXiv:2112.05326 [quant-ph] (2021), arXiv: 2112.05326.
  * Stoudenmire and Schwab [2016] E. Stoudenmire and D. J. Schwab, Supervised Learning with Tensor Networks, in  _Advances in Neural Information Processing Systems_, Vol. 29, edited by D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, and R. Garnett (Curran Associates, Inc., 2016).
  * Cheng  _et al._ [2019] S. Cheng, L. Wang, T. Xiang, and P. Zhang, Tree tensor networks for generative modeling, Phys. Rev. B 99, 155131 (2019).
  * Anders and Browne [2009] J. Anders and D. E. Browne, Computational Power of Correlations, Physical Review Letters 102, 050502 (2009).
  * Buhrman  _et al._ [2010] H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, Nonlocality and communication complexity, Reviews of Modern Physics 82, 665 (2010).
  * Liu  _et al._ [2021a] Y. Liu, W.-J. Li, X. Zhang, M. Lewenstein, G. Su, and S.-J. Ran, Entanglement-based feature extraction by tensor network machine learning, Frontiers in Applied Mathematics and Statistics 7, 10.3389/fams.2021.716044 (2021a).
  * Gao  _et al._ [2021] X. Gao, E. R. Anschuetz, S.-T. Wang, J. I. Cirac, and M. D. Lukin, Enhancing generative models via quantum correlations (2021), arXiv:2101.08354 [quant-ph] .
  * Glasser  _et al._ [2019] I. Glasser, R. Sweke, N. Pancotti, J. Eisert, and I. Cirac, Expressive power of tensor-network factorizations for probabilistic modeling, in  _33rd Conference on Neural Information Processing Systems_ (Vancouver, Canada, 2019) p. 13.
  * Jaksch  _et al._ [2000] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Fast Quantum Gates for Neutral Atoms, Physical Review Letters 85, 2208 (2000).
  * Bernien  _et al._ [2017] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature 551, 579 (2017).
  * Weimer  _et al._ [2010] H. Weimer, M. Müller, I. Lesanovsky, P. Zoller, and H. P. Büchler, A Rydberg quantum simulator, Nature Physics 6, 382 (2010).
  * Turner  _et al._ [2018] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Weak ergodicity breaking from quantum many-body scars, Nature Physics 14, 745 (2018).
  * Heller [1984] E. J. Heller, Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits, Physical Review Letters 53, 1515 (1984), publisher: American Physical Society.
  * Rader and Läuchli [2019] M. Rader and A. M. Läuchli, Floating Phases in One-Dimensional Rydberg Ising Chains, arXiv:1908.02068 [cond-mat, physics:quant-ph] (2019).
  * Keesling  _et al._ [2019] A. Keesling, A. Omran, H. Levine, H. Bernien, H. Pichler, S. Choi, R. Samajdar, S. Schwartz, P. Silvi, S. Sachdev, P. Zoller, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg simulator, Nature 568, 207 (2019).
  * Schollwöck [2011] U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011).
  * Fishman  _et al._ [2020] M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calculations, arXiv:2007.14822 [cond-mat, physics:physics] (2020).
  * White [1992] S. R. White, Density matrix formulation for quantum renormalization groups, Physical Review Letters 69, 2863 (1992).
  * Stoudenmire and White [2012] E. Stoudenmire and S. R. White, Studying Two-Dimensional Systems with the Density Matrix Renormalization Group, Annual Review of Condensed Matter Physics 3, 111 (2012).
  * Samajdar  _et al._ [2020] R. Samajdar, W. W. Ho, H. Pichler, M. D. Lukin, and S. Sachdev, Complex Density Wave Orders and Quantum Phase Transitions in a Model of Square-Lattice Rydberg Atom Arrays, Physical Review Letters 124, 103601 (2020).
  * Salakhutdinov [2015] R. Salakhutdinov, Learning deep generative models, Annual Review of Statistics and Its Application 2, 361 (2015), https://doi.org/10.1146/annurev-statistics-010814-020120 .
  * Lin  _et al._ [2017] H. W. Lin, M. Tegmark, and D. Rolnick, Why does deep and cheap learning work so well?, J. Stat. Phys. 168, 1223 (2017).
  * Orus [2013] R. Orus, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of Physics 349 (2013).
  * Poulin  _et al._ [2011] D. Poulin, A. Qarry, R. Somma, and F. Verstraete, Quantum simulation of time-dependent hamiltonians and the convenient illusion of hilbert space, Phys. Rev. Lett. 106, 170501 (2011).
  * Hastings [2007] M. B. Hastings, An area law for one-dimensional quantum systems, Journal of Statistical Mechanics: Theory and Experiment 2007, P08024 (2007).
  * Eisert  _et al._ [2010] J. Eisert, M. Cramer, and M. B. Plenio, Colloquium : Area laws for the entanglement entropy, Reviews of Modern Physics 82, 277 (2010).
  * Lu  _et al._ [2021] S. Lu, M. Kanász-Nagy, I. Kukuljan, and J. I. Cirac, Tensor networks and efficient descriptions of classical data, arXiv:2103.06872 [cond-mat, physics:quant-ph, stat] (2021).
  * Baumgratz  _et al._ [2013] T. Baumgratz, D. Gross, M. Cramer, and M. B. Plenio, Scalable Reconstruction of Density Matrices, Physical Review Letters 111, 020401 (2013).
  * Lanyon  _et al._ [2017] B. P. Lanyon, C. Maier, M. Holzäpfel, T. Baumgratz, C. Hempel, P. Jurcevic, I. Dhand, A. S. Buyskikh, A. J. Daley, M. Cramer, M. B. Plenio, R. Blatt, and C. F. Roos, Efficient tomography of a quantum many-body system, Nature Physics 13, 1158 (2017).
  * Perez-Garcia  _et al._ [2007] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Matrix product state representations, Quantum Info. Comput. 7, 401–430 (2007).
  * Huggins  _et al._ [2019] W. Huggins, P. Patil, B. Mitchell, K. B. Whaley, and E. M. Stoudenmire, Towards quantum machine learning with tensor networks, Quantum Science and Technology 4, 024001 (2019).
  * Rudolph  _et al._ [2021] M. S. Rudolph, N. B. Toussaint, A. Katabarwa, S. Johri, B. Peropadre, and A. Perdomo-Ortiz, Generation of High-Resolution Handwritten Digits with an Ion-Trap Quantum Computer, arXiv:2012.03924 [quant-ph] (2021).
  * Renou  _et al._ [2021] M.-O. Renou, D. Trillo, M. Weilenmann, T. P. Le, A. Tavakoli, N. Gisin, A. Acín, and M. Navascués, Quantum theory based on real numbers can be experimentally falsified, Nature 600, 625 (2021).
  * Chen  _et al._ [2022] M.-C. Chen, C. Wang, F.-M. Liu, J.-W. Wang, C. Ying, Z.-X. Shang, Y. Wu, M. Gong, H. Deng, F.-T. Liang, Q. Zhang, C.-Z. Peng, X. Zhu, A. Cabello, C.-Y. Lu, and J.-W. Pan, Ruling out real-valued standard formalism of quantum theory, Phys. Rev. Lett. 128, 040403 (2022).
  * Kingma and Ba [2017] D. P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization, arXiv:1412.6980 [cs] (2017).
  * Ferris and Vidal [2012] A. J. Ferris and G. Vidal, Perfect sampling with unitary tensor networks, Physical Review B 85, 165146 (2012), publisher: American Physical Society.
  * Najaﬁ  _et al._ [2020] K. Najaﬁ, M. E. Stoudenmire, S. Yelin, A. Azizi, X. Gao, M. D. Lukin, and M. Mohseni, Limitations of gradient-based Born Machines over tensor networks on learning quantum nonlocality, in  _34th Conference on Neural Information Processing Systems_ , First Workshop on Quantum Tensor Networks in Machine Learning (2020) p. 10.
  * Torlai and Melko [2016] G. Torlai and R. G. Melko, Learning thermodynamics with boltzmann machines, Phys. Rev. B 94, 165134 (2016).
  * Azizi  _et al._ [2020] A. Azizi, K. Najafi, and M. Mohseni, Learning phase transition in ising model with tensor-network born machines, in  _34th Conference on Neural Information Processing Systems_ , First Workshop on Quantum Tensor Networks in Machine Learning (2020).
  * Bravyi  _et al._ [2007] S. Bravyi, D. P. DiVincenzo, R. I. Oliveira, and B. M. Terhal, The Complexity of Stoquastic Local Hamiltonian Problems, arXiv:quant-ph/0606140 (2007), arXiv: quant-ph/0606140.
  * Lange  _et al._ [2022] H. Lange, M. Kebrič, M. Buser, U. Schollwöck, F. Grusdt, and A. Bohrdt, Adaptive Quantum State Tomography with Active Learning, arXiv:2203.15719 [cond-mat, physics:quant-ph] (2022).
  * Sehayek  _et al._ [2019] D. Sehayek, A. Golubeva, M. S. Albergo, B. Kulchytskyy, G. Torlai, and R. G. Melko, The learnability scaling of quantum states: restricted Boltzmann machines, Physical Review B 100, 195125 (2019).
  * McClean  _et al._ [2018] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature Communications 9, 4812 (2018).
  * Liu  _et al._ [2021b] Z. Liu, L.-W. Yu, L.-M. Duan, and D.-L. Deng, The Presence and Absence of Barren Plateaus in Tensor-network Based Machine Learning, arXiv:2108.08312 [cond-mat, physics:quant-ph] (2021b).
  * Torlai and Melko [2018] G. Torlai and R. G. Melko, Latent Space Purification via Neural Density Operators, Physical Review Letters 120, 240503 (2018).
  * Omran  _et al._ [2019] A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler, S. Choi, J. Cui, M. Rossignolo, P. Rembold, S. Montangero, T. Calarco, M. Endres, M. Greiner, V. Vuletić , and M. D. Lukin, Generation and manipulation of schrödinger cat states in rydberg atom arrays, Science 365, 570 (2019).
  * Semeghini  _et al._ [2021] G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, A. Omran, S. Sachdev, A. Vishwanath, M. Greiner, V. Vuletić , and M. D. Lukin, Probing topological spin liquids on a programmable quantum simulator, Science 374, 1242 (2021).

##  Appendix A Supplementary Material

###  A.1 Characterization of the 1D Rydberg Phase Diagram

####  A.1.1 Locating a Critical Point

A numerical phase diagram for a 1D Rydberg chain of 13 atoms is provided in
Fig. 9(a). To produce this figure, the ground state at different $a$ and
$\Delta$ (for fixed $R_{b}$ and $\Omega$) was calculated using DMRG. The
double sum appearing in the Hamiltonian was truncated to include at most 5 of
the nearest neighbors. The probabilities indicated in the figure by the color
bar were found by calculating the overlap of the ground state MPS with the
corresponding ordered product state.

Three different observables were examined along fixed $R_{b}/a$ lines to
locate the critical points: the bipartite entanglement entropy $\mathrm{SvN}$,
the magnetization $M$, and the first excited energy gap $\Delta E$. The
bipartite entanglement entropy approaches a maximum near the critical point
due to diverging correlations. The slope of the magnetization will approach an
extremum at the critical point as the preferential ground state shifts from
all spins aligned to an ordered crystalline phase; this is less noticeable for
the higher order phases, as fewer spins are flipped within the ordered phase.
Finally, for our choice of $N=13$, the first excited energy gap will approach
a minimum at the critical point. The energy gap would vanish throughout the
ordered phase for the case of an infinite chain due to the degenerate cyclic
permutations of the ordered state; however, when the size of the unit cell
within the ordered phase and the length of a finite chain are congruent modulo
1, there is a preference for one specific configuration of the ordered phase
over any otherwise degenerate permutation. In this special configuration, the
atoms on either end of the chain (which have only one neighbor and therefore a
smaller interaction energy) are excited to the Rydberg state, resulting in a
lower energy for a finite chain. For a chain of 13 atoms, the $Z_{2}$,
$Z_{3}$, and $Z_{4}$ ordered phases all have units cells that are congruent
modulo 1 with the chain length. This breaks the degeneracy and the energy gap
increases after the phase boundary is crossed in Fig. 9(b) for this reason.

####  A.1.2 Monte Carlo Sampling

Prior to uncovering the scaling law discuessed in the main text, we used a
Monte Carlo approach to estimate the number of samples $|\mathcal{T}|$
required to accurately capture a quantum state of interest. Two observables –
magnetization ($M=(1/N)\sum_{i}\langle S_{z}^{i}\rangle$) and Renyi entropy
($H_{\alpha}(\mathbbm{P})=(1-\alpha)^{-1}\ln{\sum_{\mathbf{v}\in\mathcal{T}}\mathbbm{P}(\mathbf{v})^{\alpha}}$)
– were considered, as these can easily be quantified from a discrete number of
samples while providing insight into the quantum state.

The system was sampled 100,000 times in the computational basis, and these
samples were cumulatively used to estimate the expectation value of the
system’s magnetization and Renyi entropy every 1,000 samples, creating a
single “sampling trajectory.” This process was repeated 50 times. The number
of samples required to capture the state was estimated by determining when the
sampling trajectories converged within $\pm$1% of the true target observable,
which was calculated from the DMRG output. After reviewing the Monte Carlo
results (see Fig. 9(c)), 30,000 samples were used to form each training data
set.

###  A.2 Additional Results

####  A.2.1 1D Rydberg Chain

Additional results for points across the 1D Rydberg chain phase diagram are
provided in Fig. 10 \- 15.

####  A.2.2 1D Anisotropic XY Chain

More in-depth results across the 1D anisotropic XY chain phase diagram are
provided in Fig. 16.

####  A.2.3 Testing Basis Dependence

To test the idea that the basis-dependent success of basis-enhanced Born
machine can be partially explained by the most prominent bases of operators
appearing in the Hamiltonian, we constructed an altered Rydberg chain
Hamiltonian in which the $\sigma_{x}$ operator was replaced by a $\sigma_{y}$
operator. All combinations of training data basis(es) and model parameters
were tested for the $Z_{2}$ critical point of this altered Hamiltonian; the
results are provided in Fig. 17. The basis-enhanced Born machine is successful
for the choice of $yz$ learning and complex model parameters, in accordance
with the basis of the operators appearing in the altered Hamiltonian.

((a))

((b))

((c))

Figure 9: 9(a) Numerical phase diagram of a 1D Rydberg chain with 13 atoms.
Red stars indicate critical points between ordered phases and disorder. Black
dots indicate points in phase selected for study. 9(b) Observables examined
along fixed $R_{b}/a$ to determine critical point location. Located critical
points are indicated by the vertical black dot-dashed lines. 9(c)
Magnetization ($M$), first order Rényi entropy ($H_{1}$), and second order
Rényi entropy ($H_{2}$) results from 50 unique sampling trajectories.
Observables are calculated cumulatively from the samples over the course of
one trajectory.

((a))

((b))

Figure 10: 10(a) Probability amplitudes in the $z$-, $x$-, and $y$-bases
calculated from the DMRG MPS and from the trained basis-enhanced Born machine
at the $Z_{2}$ critical point. 10(b) Overlaid histograms of the probability
distributions in the $z$-, $x$-, and $y$-bases from the DMRG MPS and from the
trained basis-enhanced Born machine at the $Z_{2}$ critical point.

((a))

((b))

Figure 11: 11(a) Probability amplitudes in the $z$-, $x$-, and $y$-bases
calculated from the DMRG MPS and from the trained basis-enhanced Born machine
within the $Z_{2}$ ordered phase. 11(b) Overlaid histograms of the probability
distributions in the $z$-, $x$-, and $y$-bases from the DMRG MPS and from the
trained basis-enhanced Born machine within the $Z_{2}$ ordered phase.

((a))

((b))

Figure 12: 12(a) Probability amplitudes in the $z$-, $x$-, and $y$-bases
calculated from the DMRG MPS and from the trained basis-enhanced Born machine
at the $Z_{3}$ critical point. 12(b) Overlaid histograms of the probability
distributions in the $z$-, $x$-, and $y$-bases from the DMRG MPS and from the
trained basis-enhanced Born machine at the $Z_{3}$ critical point.

((a))

((b))

Figure 13: 13(a) Probability amplitudes in the $z$-, $x$-, and $y$-bases
calculated from the DMRG MPS and from the trained basis-enhanced Born machine
within the $Z_{3}$ ordered phase. 13(b) Overlaid histograms of the probability
distributions in the $z$-, $x$-, and $y$-bases from the DMRG MPS and from the
trained basis-enhanced Born machine within the $Z_{3}$ ordered phase.

((a))

((b))

Figure 14: 14(a) Probability amplitudes in the $z$-, $x$-, and $y$-bases
calculated from the DMRG MPS and from the trained basis-enhanced Born machine
at the $Z_{4}$ critical point. 14(b) Overlaid histograms of the probability
distributions in the $z$-, $x$-, and $y$-bases from the DMRG MPS and from the
trained basis-enhanced Born machine at the $Z_{4}$ critical point.

((a))

((b))

Figure 15: 15(a) Probability amplitudes in the $z$-, $x$-, and $y$-bases
calculated from the DMRG MPS and from the trained basis-enhanced Born machine
within the $Z_{4}$ ordered phase. 15(b) Overlaid histograms of the probability
distributions in the $z$-, $x$-, and $y$-bases from the DMRG MPS and from the
trained basis-enhanced Born machine within the $Z_{4}$ ordered phase.

((a))

((b))

((c))

Figure 16: 16(a) Probability amplitudes in the $z$-, $x$-, and $y$-bases
calculated from the DMRG MPS and from the trained basis-enhanced Born machine
at a point from the oscillatory region of the 1D anistropic XY phase diagram.
16(b) Probability amplitudes in the $z$-, $x$-, and $y$-bases calculated from
the DMRG MPS and from the trained basis-enhanced Born machine at the Ising
critical point of the 1D anistropic XY phase diagram. 16(c) Probability
amplitudes in the $z$-, $x$-, and $y$-bases calculated from the DMRG MPS and
from the trained basis-enhanced Born machine at a point from the ordered
region of the 1D anistropic XY phase diagram. Figure 17: Final quantum
fidelity after training a Born machine with different combinations of training
data basis (indicated by the $x$-axis) and model parameters (real or complex)
at the $Z_{2}$ critical point of the altered Rydberg Hamiltonian.