# Adaptive Quantum State Tomography with Active Learning

Hannah Lange  Department of Physics and Arnold Sommerfeld Center for
Theoretical Physics (ASC), Ludwig-Maximilians-Universität München,
Theresienstr. 37, München D-80333, Germany  Munich Center for Quantum Science
and Technology (MCQST), Schellingstr. 4, D-80799 München, Germany  Matjaž
Kebrič  Department of Physics and Arnold Sommerfeld Center for Theoretical
Physics (ASC), Ludwig-Maximilians-Universität München, Theresienstr. 37,
München D-80333, Germany  Munich Center for Quantum Science and Technology
(MCQST), Schellingstr. 4, D-80799 München, Germany  Maximilian Buser
Department of Physics and Arnold Sommerfeld Center for Theoretical Physics
(ASC), Ludwig-Maximilians-Universität München, Theresienstr. 37, München
D-80333, Germany  Munich Center for Quantum Science and Technology (MCQST),
Schellingstr. 4, D-80799 München, Germany  Ulrich Schollwöck  Department of
Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC), Ludwig-
Maximilians-Universität München, Theresienstr. 37, München D-80333, Germany
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4,
D-80799 München, Germany  Fabian Grusdt  Department of Physics and Arnold
Sommerfeld Center for Theoretical Physics (ASC), Ludwig-Maximilians-
Universität München, Theresienstr. 37, München D-80333, Germany  Munich Center
for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 München,
Germany  Annabelle Bohrdt  ITAMP, Harvard-Smithsonian Center for Astrophysics,
Cambridge, MA 02138, USA  Department of Physics, Harvard University,
Cambridge, MA 02138, USA

###### Abstract

Recently, tremendous progress has been made in the field of quantum science
and technologies: different platforms for quantum simulation as well as
quantum computing, ranging from superconducting qubits to neutral atoms, are
starting to reach unprecedentedly large systems. In order to benchmark these
systems and gain physical insights, the need for efficient tools to
characterize quantum states arises. The exponential growth of the Hilbert
space with system size renders a full reconstruction of the quantum state
prohibitively demanding in terms of the number of necessary measurements. Here
we propose and implement an efficient and scalable scheme for quantum state
tomography using active learning. Based on a few initial measurements, the
active learning protocol proposes the next measurement basis, designed to
yield the maximum information gain. For a fixed total number of measurements
and basis configurations, our algorithm maximizes the information one can
obtain about the quantum state under consideration. We apply the active
learning quantum state tomography scheme to reconstruct different multi-qubit
states with varying degree of entanglement as well as to ground states of the
XXZ model in 1D and a kinetically constrained spin chain. In all cases, we
obtain a significantly improved reconstruction as compared to a reconstruction
based on the exact same number of measurements, but with randomly chosen basis
configurations. Our scheme is highly relevant to gain physical insights in
quantum many-body systems as well as for benchmarking and characterization of
quantum devices, and paves the way for scalable adaptive protocols to probe,
prepare, and manipulate quantum systems.

Since the turn of this century, the characterization of quantum states is of
high importance: It is needed for assessing the performance of quantum
algorithms for quantum computers, certifying the quality of experimental
quantum hardware, and enables the investigation of complex quantum systems [1,
2, 3, 4]. Hence, Quantum State Tomography (QST), which is the process of
reconstructing quantum states from measurements, is of high interest for both
experimental and theoretical research in the field of quantum physics and
quantum computing. The very nature of quantum states makes the state
reconstruction a complicated task and most available techniques, e.g. linear
inversion or the maximum likelihood method [5, 6], undergo an exponential
scaling of the number of samples needed for the reconstruction with the size
of the quantum system. This is due to the number of parameters containing the
information about the state, which increases exponentially with system size.
Therefore the application of conventional tomography methods is mostly limited
to only a few qubits [7].

A relatively new approach is neural network quantum state tomography. It is
based on the assumption that most physical states typically have a certain
degree of structure (e.g. symmetries) and can hence be described by a reduced
number of parameters. This enables the reconstruction of quantum states from
less measurement data and thus tackles the main shortcoming of the
aforementioned methods [8, 9, 10]. It has been shown that these neural network
based approaches allow the reconstruction of highly entangled states with more
than a hundred qubits, for example by using a restricted Boltzmann machine
(RBM) [11, 12]. Furthermore, machine learning based tomography is able to
access observables which cannot be directly inferred from an experiment itself
[13].

Here, we show that it is possible to drastically further reduce the number of
required measurement samples by combining restricted Boltzmann machines with
an adaptive active learning scheme as shown in Fig. 1. Active learning (AL)
machine learning models are able to interact with their environment. In the
case of the QST scheme we propose here, this interaction comprises of
requesting samples which are most informative for the further learning process
[14]. Active learning has been applied successfully, e.g. in improving
classification tasks [15, 16] or speech recognition [17], in computational
physics to speed up calculations of multidimensional functions [18], in
condensed matter theory to map out a phase diagram [19], and in quantum
experiment design [20]. Here, we use AL in the sense that our model decides
actively which measurement configuration to consider for consecutive
measurements in QST to improve the training most efficiently. The set of
measurements is hence adaptively optimized during the training, which has
already been shown to bring advantages for full quantum state tomography of
very small systems [21, 22, 23, 24]. Existing protocols, however, are often
not scalable to larger systems for various reasons, for example because either
the reconstruction of the quantum state or the procedure to obtain the next
measurement basis become numerically intractable. Another obstruction is the
optimal measurement basis itself: some adaptive protocols are based on
performing measurements along the direction of the current reconstruction,
which can be highly entangled many-body states, and thus extremely challenging
to implement. Our protocol overcomes these challenges by combining efficient
state tomography based on neural networks with an active learning scheme
relying on single qubit rotations only. We show that active learning can
highly reduce the number of samples needed for the reconstruction compared to
a passive procedure relying on RBMs only. We investigate quantum states
consisting of up to $19$ qubits or spin-$\frac{1}{2}$ particles which are
either generated on IBM quantum devices or simulated using the density-matrix
renormalization group [25]. Extensions of our scheme to larger system sizes
and other settings, such as fermionic systems or soft-core bosons, are
straightforward.

Figure 1: Quantum state tomography (QST) by active learning. We show our
active learning cycle: In the first stage a reference measurement basis is
chosen which is most suitable for reconstructing the state (grey). Then, based
on the already measured data, the active learner requests specific highly
informative samples (blue). These are used to train a RBM representing the
reconstructed quantum state.

This article is organized as follows: we start with a brief summary of
existing QST schemes by restricted Boltzmann machines, which constitute a
building block of our improved active learning QST. We then introduce our
active learning algorithm in section 2. In section 3, we define and motivate
the different quantum states considered in this work, in particular multi-
qubit states with variable degrees of entanglement and ground states of a one-
dimensional XXZ model and a kinetically constrained spin chain with a hidden
U(1) symmetry. We present the results of our active learning scheme for
quantum state reconstruction in section 4 and conclude in section 5.

##  1 Restricted Boltzmann Machines

Since our AL scheme also includes the training of a committee of restricted
Boltzmann machines, see Fig. 1, we will introduce the latter with a focus on
their application to QST in this section. The idea of our AL scheme is
independent of the specific representation of the reconstructed state and can
in principle be applied in combination with any quantum state tomography
scheme, such as other neural network architectures [26, 27, 28, 29, 30] as
well as matrix-product state based state reconstruction [31, 32].

Here, we use the implementation of RBM quantum state tomography by Beach et
al. [33] in the form of a python package called QuCumber. In this
implementation, the target many-body quantum state is represented in terms of
an artificial neural network (RBM) wave function

$\displaystyle\psi_{\mathrm{RBM}}(\bm{x})=\psi_{\lambda,\mu}(\bm{x})=\sqrt{\frac{p_{\lambda}(\bm{x})}{Z_{\lambda}}}\mathrm{e}^{i\theta_{\mu}(\bm{x})/2},$
(1)

(see appendix A), where $p_{\lambda}$ and $p_{\mu}=\mathrm{e}^{\theta_{\mu}}$
are probability distributions of two RBMs with weights $\lambda$ and $\mu$
which define the amplitude and the phase of the reconstructed state. Here,
$\bm{x}$ labels a general set of basis states.

A RBM consist of layers of so-called visible and hidden nodes $v_{i}$ and
$h_{j}$ with bias weights $b_{i}$ and $c_{j}$. The visible nodes correspond to
the input data (in our case the measurements of the target state). Both layers
are connected and each connection is weighted by parameters $W_{ij}$. To
reconstruct a state and represent it as a RBM, the network parameters are
learned from a set of measurements. To this end, the weights $\bm{b},\,\bm{c}$
and $\bm{W}$ are adjusted such that the probability distribution of the RBM
$p_{(\lambda,\mu)}=|\psi_{\mathrm{RBM}}(\bm{x})|^{2}$ become as close to the
distribution of target state measurements $q(\bm{x})$ in the set of
measurements $\mathcal{D}$ as possible (The empirical distribution $q(\bm{x})$
is defined as follows: if a possible outcome $\bm{v}$ is contained
$N_{\bm{v}}$ times in the set of $N_{\rm tot}$ existing measurements
$\mathcal{D}$, then $q(\bm{v})=N_{\bm{v}}/N_{\rm tot}$. If
$\bm{x}\notin\mathcal{D}$ has not been measured, $q(\bm{x})=0$.). This is
achieved by minimizing the so-called Kullback-Leibler divergence,

$\displaystyle
KL(q,p_{(\lambda,\mu)})=\sum_{\bm{x}\in\mathcal{D}}q(\bm{x})\mathrm{log}\frac{q(\bm{x})}{p_{(\lambda,\mu)}(\bm{x})},$
(2)

which quantifies how close the reconstructed distribution $p_{(\lambda,\mu)}$
is to the measured distribution $q$.

While the Kullback-Leibler divergence can be determined without prior
knowledge of the target state, a commonly used way of benchmarking of a QST
scheme is a direct comparison with the target state. In order to do so, a
useful measure for the performance of the reconstruction is the fidelity

$\displaystyle
f=|\langle\psi_{\mathrm{RBM}}|\psi_{\mathrm{target}}\rangle|^{2},$ (3)

which represents the square of the overlap of the RBM state representation and
the target state [35]. Note that the fidelity can only be evaluated if the
target state vector is explicitly known. For multi-qubit systems, $f$ is often
re-scaled to $\tilde{f}=f^{\frac{1}{N}}$, with $N$ the number of qubits, to
account for the exponential size of the underlying Hilbertspace. Another way
to benchmark the performance of the reconstruction scheme is to compare
observables like the density or correlators of the target and reconstructed
states.

To obtain information about both the amplitude and phase of the target state,
different types of measurements are required:

  * (i)

Samples from a reference measurement configuration are needed to learn the
amplitudes from the frequencies of measurements.

  * (ii)

Samples from measurements in further, different basis configurations are
required to extract information about the phase. Each measurement in a
different basis configuration corresponds to a rotation of one or several
qubits into the $x$, $y$ or $z$ basis individually (see Appendix A).

##  2 Active Learning Algorithm

Depending on the properties of the quantum state under consideration, there
are different configurations which contain relevant information and are hence
most useful for the reconstruction [22]. For most states it is difficult and
time-consuming to obtain the optimal set of measurement configurations and
amount of samples which leads to a good reconstruction (see Appendix B.1).
Here, we apply AL to choose the most informative measurement configurations
during the learning process. Hereby, no a-priori information about the type of
state or its properties is used. Instead, the configurations are chosen such
that the amount of information contained in a finite number of measurements
from this configuration is larger than for other configurations. We show that
by requesting this specific, highly informative data and feeding it into the
RBM, the total number of samples can be reduced while the accuracy of the
machine learning model is increased compared to a procedure which uses RBMs
only.

A schematic overview of our active learning algorithm is provided in Fig. 1.
It consists of the following steps:

  * 1.)

First, a few samples in three different measurement configurations are
generated: without rotations (denoted by $zz\dots z$), with all qubits rotated
to the $x$ axis ($xx\dots x$) and, third, with all qubits rotated to the $y$
axis ($yy\dots y$).

  * 2.)

The reference basis for the reconstruction of the quantum state’s amplitude is
selected: To this end, measurements from 1.) are fed into $n_{\mathrm{RBM}}$
RBMs. In general, the reference basis is chosen to be the basis where the
$n_{\mathrm{RBM}}$ RBMs agree most on the wave function’s state vectors. More
precisely, the configuration with the lowest variance of the RBM state
vector’s entries is chosen. Depending on the size of the Hilbert space, this
can be achieved by directly comparing the state vectors, or by sampling from
the $n_{\mathrm{RBM}}$ RBMs and comparing the resulting distributions.

  * 3.)

Third, we train $n_{\mathrm{RBM}}$ different RBMs on the data in the pool of
samples using the open source python package QuCumber [33].

  * 4.)

The active learner requests measurements in a configuration which improve the
reconstruction most efficiently. Here we apply the so called query-by-
committee strategy and request samples from those configurations for which the
RBMs disagree most: This is decided based on the variances of the amplitudes
and phases between different RBMs.

    * a)

If the RBMs disagree more on the amplitudes than on the phases of the
reconstructed states (i.e. the variance of the amplitudes is larger than the
variance of the phases of $\psi_{\mathrm{RBM}}$ for different RBMs), the
active learner requests measurements from the reference basis.

    * b)

Else, the probability distributions
$p_{(\lambda_{i},\mu_{i})}(\bm{x})=|\psi_{\mathrm{RBM}i}(\bm{x})|^{2}$ with
$\psi_{\mathrm{RBM}i}$ rotated to $n$ different configurations for all
$i=1,\dots,n_{\mathrm{RBM}}$ RBMs (each RBM characterized by its set of
parameters $\lambda_{i},\,\mu_{i}$) is used to select the next measurement
basis. Here, we calculate the RBM probability distributions in up to $n\leq
2^{N}$ ($N$: system size or number of qubits) different configurations. Then,
the active learner evaluates for which measurement configuration the RBMs
disagree most on the probability distributions $p_{(\lambda_{i},\mu_{i})}$. To
this end, for each measurement configuration the variance of the probabilities
for each measurement outcome between the different RBMs is evaluated and
summed up. The configuration with the highest variance is considered to be the
most controversial between all RBMs and is selected as the next measurement
configuration.

  * 5.)

Samples in the requested configuration are measured and steps 3.) and 4.) are
repeated until the accuracy of the model reaches a predefined standard, or the
predefined maximum number of measurements is reached.

Furthermore, our AL-QST scheme automatically requests samples from the two
other configurations beside the reference configuration mentioned in 1.) if
the reference configuration is requested two times in a row. For example, if
the reference configuration is the $zz\dots z$ configuration and measurements
from this configuration are requested two times in a row, samples from the
$xx\dots x$ and $yy\dots y$ configurations will be added in the next learning
cycles. This is particularly relevant for rotationally invariant systems like
the Heisenberg chain.

##  3 Considered States

Before we present results by our AL-QST scheme, we provide an overview of the
quantum states used to benchmark our method. We chose two types of states:
first a set of generic qubit states with variable degree of entanglement,
which provide direct insights into the performance of our scheme. Second, we
consider many-body ground states of one-dimensional spin Hamiltonians as an
illustration of our method for quantum simulators.

###  3.1 Qubit states and IBM’s Quantum cloud

We investigate the reconstruction of states which are generated on real
quantum devices and classical simulators of the IBM Quantum platform [36]. IBM
provides 21 quantum systems based on superconducting qubits which can be
accessed via a cloud. Systems with up to five qubits are accessible for non-
internal users with a free account. Using Qiskit it is possible to implement
quantum algorithms on these quantum systems [37].

Here, we consider the reconstruction of four target states:

  * •

Greenberger–Horne–Zeilinger (GHZ) states,

$\displaystyle|\mathrm{GHZ}\rangle=\frac{1}{\sqrt{2}}\left(|0\dots
0\rangle+|1\dots 1\rangle\right),$ (4)

  * •

GHZ states with a complex phase,

$\displaystyle|\mathrm{GHZ}_{\varphi}\rangle=\frac{1}{\sqrt{2}}\left(|0\dots
0\rangle+i|1\dots 1\rangle\right),$ (5)

  * •

polarized product states with all qubits set to one (spin chains with spins
pointing in $z$ direction),

$\displaystyle|z\mathrm{-spins}\rangle=|1\dots 1\rangle,$ (6)

  * •

and states with a state vector with equal amplitudes for all components. They
correspond to spin chains with all spins pointing in $x$ direction, e.g. for
two qubits

$\displaystyle|x\mathrm{-spins}\rangle=\frac{1}{2}|00\rangle+\frac{1}{2}|01\rangle+\frac{1}{2}|10\rangle+\frac{1}{2}|11\rangle.$
(7)

Using the IBM platform we investigate the reconstruction of states with five
qubits. Tools for rotating and measuring the quantum systems are provided by
Qiskit. An example for measuring a two-qubit system in the $xy$ basis (first
qubit rotated to the $x$ axis, second qubit rotated to $y$) is shown in
Appendix A. In the following sections we will use the same notation, where
e.g. $xx\dots x$ denotes a system with all qubits rotated from the $z$ to the
$x$ axis.

###  3.2 XXZ and Heisenberg states

As a second example, we use our AL-QST scheme to reconstruct ground states of
the XXZ Hamiltonian

$\displaystyle\hat{\mathcal{H}}=\sum_{\langle
i,j\rangle}J(\hat{S}_{i}^{x}\hat{S}_{j}^{x}+\hat{S}_{i}^{y}\hat{S}_{j}^{y})+J(1+\Delta)\hat{S}_{i}^{z}\hat{S}_{j}^{z}\,,$
(8)

where $\hat{S}_{i(j)}^{\mu}$ denotes the $\mu\in\\{x,y,z\\}$ component of the
spin-1/2 operator at site $i(j)=1,\dots,L$. This model encloses ground states
with a strong polarization in $z$ direction for large $|\Delta|$ (broken
$\mathbb{Z}_{2}$ symmetry) to critical states without long-range order for
$-2J\leq\Delta\leq 0$. For $\Delta=0$ the ground states are SU(2) invariant
(Heisenberg antiferromagnet). In what follows we will compute the ground
states of Eq. (8) using a matrix product state representation with the SyTen
package [38, 39].

###  3.3 Kinetically constrained spin chain

As an additional illustrative example, we apply AL-QST to reconstruct ground
states of a kinetically constrained one-dimensional spin chain (KCS) model. It
is described by the following Hamiltonian [40, 41, 42]:

$\displaystyle\hat{\mathcal{H}}=t\sum_{j=2}^{L-1}$
$\displaystyle\left(4\hat{S}^{x}_{j-1}\hat{S}^{x}_{j+1}-1\right)\hat{S}^{z}_{j}$
$\displaystyle-h\sum_{j=1}^{L}2\hat{S}^{x}_{j}+\mu\sum_{j=2}^{L}\hat{S}^{x}_{j-1}\hat{S}^{x}_{j},$
(9)

where $\hat{S}^{\mu}_{j}$ denotes a spin-$1/2$ operator ($\mu=x,y,z$) on site
$j=1,...,L$. This model has a very interesting interpretation, where the spin
domain walls in the $x$-basis correspond to particles on a dual lattice. More
precisely, model (9) can be mapped to a one-dimensional $\mathbb{Z}_{2}$
lattice gauge theory with matter [41], see Appendix C.

We choose the system in Eq. (9) due to several non-trivial properties for
which we can check in the reconstructed state. These include: (i) that the
underlying excitations are domain walls of the spins, extending beyond one
site – a fact that needs to be captured by a reliable QST scheme; (ii) the
Hamiltonian features a hidden U(1) symmetry describing the conservation of the
total number of domain walls – their number is controlled by the chemical
potential $\mu$; (iii) the model hosts gapless Luttinger liquids with
significant amount of non-local entanglement, presenting a general challenge
for any QST scheme; and (iv) by tuning the effective field from $h=0$ to
$h\neq 0$, a confinement-deconfinement transition exists where the nature of
constituents of the Luttinger liquid changes, as indicated by a change of the
Fermi momentum $k_{\rm F}$ and reflected in the period of Friedel oscillations
[41, 42].

Below we compute the ground states of Eq. (9) with the density-matrix
renormalization group (DMRG) using the SyTen package [38, 39]. We obtain a
matrix-product state (MPS) representation of the ground states, from which
efficient snapshot sampling is possible [43], including in variable bases
[44]. This provides us with the data required to run and benchmark the AL-QST
algorithm.

To probe how close the reconstructed state is to the actual ground state, we
compute the following observables. We start with the local domain-wall
density,

$\hat{n}(j)=\frac{1}{2}\left(1-4\hat{S}^{x}_{j}\hat{S}^{x}_{j+1}\right).$ (10)

This also immediately leads us to the conserved total system density
$\hat{n}^{\rm tot}=\frac{1}{L-1}\sum_{j=1}^{L-1}\hat{n}(j)$. In practice, we
find it convenient to define a vector $\bm{n}$ of local densities with the
following entries,

$\bm{n}=\bigl{(}\langle\hat{n}_{1}\rangle,\langle\hat{n}_{2}\rangle,...,\langle\hat{n}_{L}\rangle\bigr{)}^{T}.$
(11)

In order to probe the confinement of domain walls we consider their non-local
equal-time Green’s function measured relative to the center $L/2$ (for $L$
odd: $L/2+1/2$) of the chain (see Appendix C for more details):

$c(d)=\left\langle\frac{1}{2}\left(1-4\hat{S}^{x}_{L/2}\hat{S}^{x}_{L/2+1}\right)\left(\prod_{L/2<j\leq
L/2+d}2\hat{S}^{z}_{j}\right)\right.\\\
\times\left.\frac{1}{2}\left(1-4\hat{S}^{x}_{L/2+d}\hat{S}^{x}_{L/2+1+d}\right)\right\rangle.$
(12)

We highlight the following key properties of this function: (a) for distance
$d=0$, the density is recovered, $c(0)=\langle\hat{n}(L/2)\rangle$; (b) the
decay of $c(d)$ allows to distinguish between confined (exponential decay) and
deconfined (power-law decay) regimes.

##  4 Results

Figure 2: Learning curve: re-scaled fidelities (top), Kullback-Leibler
divergence (middle) and number of samples within the learning cycles (bottom)
with active learning for a GHZ state with phase structure and consisting of
$5$ qubits. Here, $4$ different RBMs are used in step 4.) of the active
learning cycle (blue lines). At the end, $107$ samples were used. The results
can be compared to a learning cycle without active learning (green). The green
band corresponds to the entire span of results without AL. The active learner
decided to choose the $zz\dots z$ basis as reference basis in step 2.) and
requested measurements from $2$ additional configurations in $7$ queries (see
epoch s $1000$ to $8000$ and step 4.) of the AL-QST learning scheme).

The active learning curve for an exemplary state, namely a GHZ state with
phase structure consisting of $5$ qubits and generated with the IBM quantum
simulator, is shown in Fig. 2. The results for AL tomography are compared to
QST without active learning (denoted baseline) by using the pure RBM
reconstruction of QuCumber, the same number of samples as for AL and equally
many, but randomly chosen measurement configurations. Note that we consider
the number of samples and configurations at the end of the AL scheme for the
baseline reconstruction. Hence, the results for AL and baseline can differ
already at the beginning of the training. The pool of samples which is used
for the RBM training is the same for all RBMs (both for AL and baseline,
respectively).

Fig. 2 shows the learning curve for the AL scheme as presented in Fig. 1.
After the selection of the reference configuration ($zzzzz$, not shown in Fig.
2) the training starts with $100$ samples drawn from the reference basis,
which are fed into the QuCumber state reconstruction as implemented by Beach
et al. [33] (see step 3.). In the subsequent learning sequence up to epoch
$1000$ the re-scaled fidelity $f^{1/N}$ increases ($1-f^{1/N}$ decreases) and
the Kullback-Leibler divergence $KL$ decreases. However, the samples measured
in step 1.) of the active learning routine do not contain enough information
to decrease $1-f^{1/N}$ below a threshold value of $10\,\%$ for all RBMs (see
first section in Fig. 2). The lack of information is mostly due to the fact
that no information on the phase structure of the state under consideration is
contained in the measurements from the $zzzzz$ configuration.

After that, more and more samples are added to the pool of measurements which
are requested by the learner and the RBM learning process is re-started (with
the same initialisation of the RBMs as before). For the state reconstruction
presented in Fig. 1, the learner requests a measurement from the $xzzxx$
configuration in the first $6$ queries (see epochs $1000$ to $6000$), and from
$zyyyz$ in the last learning cycle starting in epoch $7000$. Note that samples
from these measurement configurations contain not only information on the
amplitude, but also on the phase structure of the state under investigation.
Step by step, these samples are added to the learning process. This process is
stopped when a specified re-scaled fidelity (here
$f^{1/N}_{\mathrm{stop}}=90\,\%$) is reached or the number of posed queries
exceeds a maximal value $N_{\mathrm{query}}^{\mathrm{max}}$ (in this paper we
use $N_{\mathrm{query}}^{\mathrm{max}}=30$). The final fidelity in this
example is $f^{1/N}=95.0\pm 1.5\,\%$ (average over the four RBMs used for step
3.)) using only $107$ samples and three different configurations. Furthermore,
in Fig. 2 it can be observed that there are regimes in the learning procedure
where the difference between reconstructed and target state stagnates or even
grows when adding the requested samples. This is not unintentional within our
learning scheme: If a new sample is requested that contains completely
different information (i.e. when adding the first sample from the $xzzxx$
configuration to a pool of $100$ samples from the $zz\dots z$ configuration in
epoch $1000$) the learner may get confused in the first place. However, when
requesting more and more samples afterwards, the same piece of information can
become a valuable contribution to a good reconstruction.

A learning procedure without AL with the same number of measurements and
equally many, but randomly chosen configurations ends with
$f^{1/N}_{\mathrm{baseline}}=75\pm 11\,\%$ (average over four RBMs, all of
them trained with the same samples), indicated by the green dotted line in
Fig. 2. Hence, the quality of the reconstruction was improved strongly by the
use of active learning. Furthermore, the span of outcomes for different RBMs
is significantly decreased compared to a learning procedure without active
learning (visualized by the green band in Fig. 2), which makes the results
much more reliable than without AL.

In the following sections, the results for the active learning procedure as
explained in section 2 will be presented for states on IBM’s quantum devices
and states generated with DMRG. The number of samples $N_{\mathrm{tot}}$,
number of queries $N_{\mathrm{queries}}$, number of samples per query
$N_{\mathrm{per\,query}}$ and number of different configurations
$N_{\mathrm{config}}$ are presented in Tab. 1 and 2. Based on our experience
that for a good reconstruction of the amplitude in most cases more information
about the reference configuration is needed (compared to the reconstruction of
the phase based on locally rotated configurations), we request more samples
than $N_{\mathrm{per\,query}}$ (if not stated differently $3\cdot
N_{\mathrm{per\,query}}$) if the reference configuration is requested.

###  4.1 Tomography results for quantum states on IBM’s Quantum cloud

Figure 3: Final re-scaled fidelities for the states defined in Eqs. (4), (5),
(6) and (7) with $5$ qubits generated on classical quantum simulators (left)
and real quantum devices (right). An exemplary learning curve can be found in
Fig. 4. Figure 4: Examplary learning curves for the GHZ state (see Eq. (4))
with $5$ qubits generated on a classical quantum simulator by IBM.

In this section the results for states defined in Eqs. (4) to (7) generated on
IBM’s classical quantum simulator, which is designed to mimic the execution of
an actual device, and on real quantum devices are presented (for more details
see appendix B.1). We consider a system size of $5$ qubits. The reconstruction
results can be found in Fig. 3. Here the training is stopped as soon as
$f^{1/N}_{\mathrm{stop}}=90\,\%$ is reached for the classical quantum
simulators. For the real devices, $f^{1/N}_{\mathrm{stop}}=80\,\%$ is used
(except for the GHZ state with $f^{1/N}_{\mathrm{stop}}=90\,\%$ and the
$\mathrm{GHZ}_{\varphi}$ state with $f^{1/N}_{\mathrm{stop}}=85\,\%$). The
number of samples, queries and configurations at the end of the training is
presented in Tab. 1.

When reconstructing the GHZ state with AL, the active learner selects the
$zzzzz$ basis as the reference basis. The fidelity of the GHZ classical
quantum simulator states is improved by around $25\,\%$ from $f^{1/N}=70.49\pm
0.10\,\%$ without AL to $f^{1/N}=95.667\pm 0.001\,\%$. For real quantum
devices the quality of the reconstruction improves as well, by around $22\,\%$
from $f^{1/N}=68.6\pm 8.3\,\%$ without AL to $f^{1/N}=90.93\pm 0.16\,\%$.
Moreover, the standard deviation of the results for different RBMs is lowered
by up to two order of magnitude for the classical quantum simulators and
quantum devices, which makes the results more robust when using AL. This
improvement is achieved by requesting samples from only two additional
measurement configurations.

In Fig. 4 the learning curve for the GHZ state is shown. When using AL the
fidelity and $KL$ divergence are decreased strongly at the end of the
training. One can see that in the first learning cycle, which uses $60$
samples drawn from the $zzzzz$ reference basis, the information contained in
these samples is not enough to decrease the fidelity below the threshold value
of $f^{1/N}_{\mathrm{stop}}=90\,\%$ for most RBMs. By adding only one sample
of the $yzzzy$ configuration to the pool of samples in the first query, the
RBMs can not make sense of the phase information that is added and the
fidelity increases compared to the first learning step. However, one can see
that the RBMs still adapt to the new information since the $KL$ is decreased
to a comparable amount as in the first part of the learning. In the second
query one sample from $xxyxy$ is added. Together with the other samples, the
information contained in the measurements is enough to adapt the phase of the
reconstructed wave function such that the fidelity increases above the
threshold value.

Figure 5: Examplary learning curves for the GHZ state with phase (see Eq. (5))
with $5$ qubits generated on a quantum device of IBM.

For the GHZ state with phase structure the AL-QST scheme also requests
additional measurements in 2 (5) additional measurement configurations for
states generated with the classical simulator (real quantum device). In
contrast to the GHZ state without phase structure, much more information is
contained in these configurations, namely the phase difference of $|0\dots
0\rangle$ and $|1\dots 1\rangle$. Our scheme is able to capture the importance
of measurements from other configurations than the reference configuration and
requests many of the configurations several times until the the learning is
stopped after 7 (25) queries. The AL-QST scheme improves the reconstruction
results to a fidelity of $f^{1/N}=95.0\pm 1.5\,\%$ ($87.5\pm 4.3\,\%$)
compared to the baseline with $75\pm 11\,\%$ ($61\pm 13\,\%$) for additional
measurements from randomly selected configurations. The learning curves for
states generated with the classical simulator and on a real quantum device are
shown in Figures 2 and 5. In both cases it can be seen that adding samples
which contain more information on the phase structure can confuse the learner
at first place (see epochs $1000$ to $2000$ in both figures), but when enough
information on the phase is added, the reconstruction improves again. For the
state generated with the classical simulator (see Fig. 2) this happens by
adding samples from the $xzzxx$ and $xyyyx$ configurations. For the states
generated on the real devices five additional measurement configurations are
needed.

For the state with all spins pointing upwards (all qubits having value one),
the reference basis containing the most valuable information about the state
is the $zzzzz$ basis. When using AL, this configuration is selected and the
results are extremely good already in the first cycle of AL, with a fidelity
of $f^{1/N}=99\,\%$ for simulator and real device. The QuCumber reconstruction
without AL coincidentally uses the $zzzzz$ reference basis by default and
hence coincidentally the perfect reference basis for the reconstruction of
this state. Therefore, no difference between AL compared to the baseline can
be observed.

For the state with all spins pointing in $x$ direction, the reconstruction
results are improved as well. Here, the $xxxxx$ basis is chosen and the
fidelity is increased by $14\,\%$ to $f^{1/N}=98.841\pm 0.022\,\%$ for
simulated states ($10\,\%$ to $f^{1/N}=85.28\pm 0.40\,\%$ for real quantum
devices). Furthermore, the variance is decreased by a factor of up to $10$. In
contrast to the GHZ state, where the improvement is achieved by increasing the
fidelity step by step with every query, for this system the underlying reason
for the improvement is the rotation of the reference configuration: When the
$x$-spins state is rotated from the $zz\dots z$ configuration to the $xx\dots
x$ configuration, the measurement distribution changes from equally
distributed peaks for all outcomes to a peaked distribution at $00\dots 0$.
Similarly to the state with all spins in $z$ direction, it is relatively easy
for the RBMs to learn this distribution.

| cl. quant. simulator | quant. device
---|---|---
| $N_{\mathrm{tot}}$ | $N_{\mathrm{queries}}$ | $N_{\mathrm{config}}$ | $N_{\mathrm{tot}}$ | $N_{\mathrm{queries}}$ | $N_{\mathrm{config}}$
GHZ | $62$ | $2$ | $3$ | $102$ | $2$ | $3$
$\mathrm{GHZ}_{\varphi}$ | $107$ | $7$ | $3$ | $450$ | $25$ | $6$
$x$-spins | $2$ | $0$ | $1$ | $2$ | $0$ | $1$
$z$-spins | $4$ | $0$ | $1$ | $4$ | $0$ | $1$
Table 1: Number of samples $N_{\mathrm{tot}}$, number of queries
$N_{\mathrm{queries}}$ selected by the active learner and total number of
configurations $N_{\mathrm{config}}$ at the end of the learning for the
reconstruction of the quantum states generated on IBM’s classical quantum
simulators and quantum devices. Here, the number of samples is
$N_{\mathrm{per\,query}}=1$ except for the $\mathrm{GHZ}_{\varphi}$ state on
the quantum device, where $N_{\mathrm{per\,query}}=10$. The reference
configuration selected by the AL is $zz\dots z$ for $z$-spins, GHZ and
$\mathrm{GHZ}_{\varphi}$ states, and $xx\dots x$ for the $x$-spins state. All
states are defined in Eqs. (4) to (7).

###  4.2 Tomography results for DMRG states

To investigate the AL reconstruction of many-body quantum states we can use
the matrix product state framework for representing quantum states, such as
for example ground states of the XXZ and the KCS models, and sampling in
different basis configurations. In this section, the AL results for these
states with $5$ to $19$ qubits are presented in Figs. 6 to 11, with the number
of samples and configurations used for the reconstruction from Tab. 2. For all
states a committee of four RBMs was used.

####  4.2.1 Reconstruction of XXZ and Heisenberg States

Figure 6: Reconstruction results for the ground states of the XXZ Hamiltonian
(8).

In figure 6 the reconstruction results for the ground states of the XXZ model
with $8$ spins and different $\Delta=-1,0,1,5$ are presented. Here, we focus
on the reconstruction of the correlator
$\langle\hat{S}_{i}^{\alpha}\hat{S}_{i+1}^{\alpha}\rangle$
($\alpha\in\\{x,y,z\\}$) for nearest neighbors and sum over all sites
$i=1,\dots,L-1$. The learning is stopped when the reconstructed correlator
$|\langle\hat{S}_{i}^{\alpha}\hat{S}_{i+1}^{\alpha}\rangle|$ is at least 2/3
of the target value and the correct sign is obtained. It can be seen that for
$\Delta=-1,0,1$ the AL-QST scheme performs significantly better than the
baseline, and for $\Delta=5$ equally well. Furthermore, we emphasize that the
RBM representation in general yields better reconstruction results for the
amplitudes than for the phases of the considered states, which has
consequences for the reconstruction of the XXZ states: Firstly, the RBM
reconstruction is not able to capture the SU(2) invariance of the $\Delta=0$
Heisenberg state, as can be seen e.g. also in Ref. [11]. Secondly, a better
reconstruction of quantities measured in the reference direction compared to
the orthogonal directions is obtained for all values of $\Delta$, e.g. the
reconstruction of the correlator
$\langle\hat{S}_{i}^{\alpha}\hat{S}_{i+1}^{\alpha}\rangle$ for $\Delta=0$ with
$zz\dots z$ being the reference configuration has a value closer to the actual
value for $\alpha=z$ than for $\alpha=x,\,y$ since the values for the latter
correlators are systematically underestimated by the RBM representation. A
similar tendency can be observed for small $\Delta$ like $\Delta=-1(1)$, where
the reference configurations are selected by the AL-QST scheme to be the
$xx\dots x$ ($zz\dots z$) configurations. Details on the AL-QST scheme can be
found in Tab. 2.

Figure 7: Learning curve for the ground state of the XXZ Hamiltonian (8) with
$\Delta=-1$. The values for the target states are represented by the black
lines.

The reconstruction curve for for $\Delta=-1$ is shown in Fig. 7. This state
has a small polarization in $z$ direction compared to $x$ and $y$ directions.
Our AL-QST scheme is able to capture this by selecting the $xx\dots x$
configuration as the reference configuration and not the $zz\dots z$
configuration. Consequently, as explained above, the reconstruction of the
correlator in the reference direction
$\langle\hat{S}_{i}^{x}\hat{S}_{i+1}^{x}\rangle$ is better than e.g.
$\langle\hat{S}_{i}^{z}\hat{S}_{i+1}^{z}\rangle$.
$\langle\hat{S}_{i}^{z}\hat{S}_{i+1}^{z}\rangle$ even has a wrong sign after
the first learning phase up to epoch $3000$. After that, the AL-QST scheme
requests the reference (i.e. $xx\dots x$) configuration in the first step (see
epochs $3000$ to $6000$), which does not yield a significant improvement of
the reconstruction of $\langle\hat{S}_{i}^{z}\hat{S}_{i+1}^{z}\rangle$. In the
next two phases up to epoch $1200$ the AL scheme as explained in section 2
selects the orthogonal configurations $zz\dots z$ and $yy\dots y$ (see epochs
$6000$ to $12000$, which yields strongly improved results of
$\langle\hat{S}_{i}^{z}\hat{S}_{i+1}^{z}\rangle$. In the last three queries
measurements from the $xx\dots x$, $xxzxxzzx$ and $xzyzzzzz$ configurations
improve the reconstruction of the correlators in all directions up to the
threshold values and the learning procedure is stopped.

####  4.2.2 Kinetically constrained spin chain model

In Figs. 8 to 11 the tomography results for the kinetically constrained spin
chain with $t=1$ and $h/t=0$, $\mu/t=0$ or respectively $h/t=1$, $\mu/t=1$ are
summarized. For these states no full state vectors are available and hence the
fidelity cannot be used to evaluate the quality of the reconstruction.
Instead, we calculate the density and the correlator from Eqs. (10) and (12)
of the reconstructed states as defined in Sec. 3 and compare them to the
values for the target states. We have used the stopping conditions
$\frac{|\bm{n}-\bm{n}_{\mathrm{target}}|}{|\bm{n}_{\mathrm{target}}|}\leq\tilde{n}_{\mathrm{stop}}=0.2$
and
$\frac{|\bm{c}-\bm{c}_{\mathrm{target}}|}{|\bm{c}_{\mathrm{target}}|}\leq\tilde{c}_{\mathrm{stop}}=0.2$.

Figure 8: Correlator over distance (top) and density (bottom) from Eqs. (12)
and (10) for the KCS model state with $h/t=0$ and $19$ qubits for target state
(black) and the reconstructed states using AL (blue) and the baseline (green).

Figure 9: Correlator over distance (top) and density (bottom) from Eqs. (12)
and (10) for the KCS model state with $h/t=1$, $\mu/t=1$ and $19$ qubits for
target state (black) and the reconstructed states using AL (blue) and the
baseline (green).

The target and reconstructed density $\bm{n}_{\mathrm{target}}$ and $\bm{n}$
have $N-1$ entries (for each possible domain wall between site $i$ and $i+1$).
The spatially resolved domain wall densities for the kinetically constrained
spin chain ground states with $h/t=0$, $\mu/t=0$ and $h/t=1$, $\mu/t=1$ are
shown in Figs. 8 and 9 (bottom) for a system with $19$ qubits. For $h/t=0$ and
$\mu/t=0$ the local target density is $n_{i}=0.5$. For $h/t=1$ and $\mu/t=1$,
the conserved total system density is equal to $n^{\rm tot}=8/18$ and one can
observe Friedel oscillations, where the oscillations are proportional to
$k_{f}=\pi n^{\rm tot}$ [41]. For both target KCS model ground states the
reconstructed states have a domain wall density which agrees with the target
density in terms of magnitude and general characteristics (i.e. oscillations)
when using AL. In contrast, the baseline without AL results in local densities
$n_{i}$ which are only of around half of the magnitude of the target
densities.

A similar tendency can be observed for the target and reconstructed
correlators $\bm{c}_{\mathrm{target}}$ and $\bm{c}$. They have
$\lfloor\frac{N}{2}\rfloor$ entries since we calculate the correlator over
distance $d$ for the site at the middle of the chain. Also here the baseline
reconstruction without AL yields values of $c(d)$ with a much smaller
magnitude than for the target state for all distances $d$ and both parameter
sets ($h/t=0$, $\mu/t=0$ and $h/t=1$, $\mu/t=1$). In contrast, when using
active learning the reconstructed correlator values are of the same magnitude
as for the target state and even follow local features (see bending of the
curve in Fig. 9). Moreover, the power-law for $h/t=0$ and exponential decay
for $h/t=1$ can be reconstructed when using AL, but not for the baseline (see
Appendix B.2).

To conclude, for the KCS model with $19$ qubits the reconstruction is improved
drastically when using AL compared to the baseline scenario. This conclusion
can also be drawn from the reconstruction of the KCS model states for other
system sizes shown in Figs. 10 and 11. For $h/t=0$ the change of reference
basis yields a reduction of the density differences between target and
reconstructed state by a factor $5$ for AL, from a relative value of around
$44\,\%$ to around $6\,\%$ when averaging over all system sizes. The
difference of correlators is decreased from $69\,\%$ to about $17\,\%$. For
$h/t=1$ the difference in densities is decreased from an absolute value of
around $61\,\%$ to less than $12\,\%$. The difference of target and
reconstructed correlators is decreased to around $16\,\%$ from $74\,\%$. For
both values of $h$ and all system sizes the active learner selected the
$xx\dots x$ configuration as reference basis. This is different from the
choice for the usual RBM procedure implemented in QuCumber which always uses
the $zz\dots z$ configuration as reference basis. For almost all sizes the
stopping condition was reached within the first learning cycle in step 3.)
(except for the $15$ qubit state ($h/t=0$), where another measurement in the
reference basis was requested). Hence, this change of the reference frame by
applying active learning already improves the results extremely, even without
a need for the further steps 4.) and 5.).

Figure 10: Relative difference of target and reconstructed densities (left)
and correlators (right) from Eqs. (10) and (12) of the KCS model with $h/t=0$.
Error bars correspond to the standard error of the mean.

Figure 11: Relative difference of target and reconstructed densities (left)
and correlators (right) from Eqs. (10) and (12) of the KCS model with $h/t=1$
and $\mu/t=1$. An exemplary learning curve for $19$ qubits is shown in the
Appendix B.2.

##  5 Summary and Outlook

XXZ |  $\Delta=-1$ |  $\Delta=0$ |  $\Delta=1$ |  $\Delta=5$
---|---|---|---|---
reference |  $xx\dots x$ |  $zz\dots z$ |  $zz\dots z$ |  $zz\dots z$
$N_{\mathrm{tot}}$ |  $2000$ |  $2730$ |  $1750$ |  $1550$
$N_{\mathrm{queries}}$ |  $7$ |  $16$ |  $6$ |  $30$
$N_{\mathrm{per\,query}}$ |  $50$ |  $10$ |  $50$ |  $50$
$N_{\mathrm{config}}$ |  $5$ |  $8$ |  $4$ |  $1$

KCS ($\frac{h}{t}=0$) |  $7$ qubits |  $9$ qubits |  $15$ qub. |  $19$ qub.
---|---|---|---|---
reference |  $xx\dots x$ |  $xx\dots x$ |  $xx\dots x$ |  $xx\dots x$
$N_{\mathrm{tot}}$ |  $200$ |  $200$ |  $202$ |  $500$
$N_{\mathrm{queries}}$ |  $0$ |  $0$ |  $1$ |  $0$
$N_{\mathrm{per\,query}}$ |  $-$ |  $-$ |  $2$ |  $-$
$N_{\mathrm{config}}$ |  $1$ |  $1$ |  $1$ |  $1$

KCS ($\frac{h}{t}=1$) |  $7$ qubits |  $9$ qubits |  $15$ qub. |  $19$ qub.
---|---|---|---|---
reference |  $xx\dots x$ |  $xx\dots x$ |  $xx\dots x$ |  $xx\dots x$
$N_{\mathrm{tot}}$ |  $200$ |  $200$ |  $400$ |  $500$
$N_{\mathrm{queries}}$ |  $0$ |  $0$ |  $0$ |  $0$
$N_{\mathrm{config}}$ |  $1$ |  $1$ |  $1$ |  $1$

Table 2: Reference configuration selected by the AL, number of samples
$N_{\mathrm{tot}}$, number of queries $N_{\mathrm{queries}}$, number of
samples per query $N_{\mathrm{per\,query}}$ and configurations
$N_{\mathrm{config}}$ selected by the active learner for the reconstruction of
the DMRG states: For the XXZ model and kinetically constrained spin (KCS)
model ground states.

In this work, we propose and implement an active learning scheme for adaptive
quantum state tomography. Assuming that the quantum state under consideration
has some internal structure, as it is typically the case for states of
interest, the active learning scheme uses the information available in the
measurement data taken already to propose the basis configuration for the next
measurement with the most possible information gain. We show that for a given
number of measurements, our scheme provides a significant improvement in the
reconstructed quantum state compared to a random choice of basis
configurations. For a quantum state without any structure, the active learning
scheme cannot provide an advantage, but will perform as good as a random
choice of configurations (baseline).

The active learning scheme is generally applicable to different quantum states
and devices, such as trapped ions, neutral atoms in optical tweezers, and
superconducting qubits, as shown here. With the increasing number of quantum
devices, the need for an efficient way to characterize the realized quantum
state arises. Applications range from the verification of quantum computing
devices, e.g. testing how faithfully a given quantum state can be prepared, to
probing exotic states of matter realized in (analog) quantum simulators, such
as the recently realized quantum spin liquid states [45, 46], where
measurements in different bases are necessary to characterize the quantum
state.

Apart from the implementation of our protocol in an interactive experimental
feedback loop, possible directions for future work include more advanced
schemes, e.g. more possible reference bases in the first step. Our active
learning scheme can furthermore be generalized to state representations other
than the restricted Boltzmann machines considered here, such as variational
autoencoders [26], recurrent [27] and convolutional neural networks [28],
generative adversarial networks [29], and transformer architectures [30].

Another exciting future direction is the combination of the active learning
scheme introduced here with the recently proposed classical shadows [47] by
extending the possible active learning actions to different unitary gates,
potentially involving two or more qubits.

_Acknowledgements.–_ We thank Tizian Blatz, Juan Felipe Carrasquilla, Anna
Dawid, Ehsan Khatami, Sam Mardazad, Sebastian Paeckel, Henning Schlömer, and
Jeffrey Thompson for fruitful discussions. This research was funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy – EXC-2111 – 390814868, by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programm (Grant Agreement no 948141) — ERC Starting Grant SimUcQuam and by the
NSF through a grant for the Institute for Theoretical Atomic, Molecular, and
Optical Physics at Harvard University and the Smithsonian Astrophysical
Observatory.

## References

  * Nimbe  _et al._ [2021] P. Nimbe, Weyori, and A. F. B. A.Adekoya, Quantum Information Processing 20 (2021), 10.1007/s11128-021-03021-3.
  * Bloch  _et al._ [2012] I. Bloch, J. Dalibard, and S. Nascimbène, Nature Physics 8 (2012), 10.1038/nphys2259.
  * Preskill [2018] J. Preskill, Quantum 2, 79 (2018).
  * 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, and authors, PRX Quantum 2, 017003 (2021).
  * Häffner  _et al._ [2005] H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-al kar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, Nature 438, 643 (2005).
  * Hradil [1997] Z. Hradil, Phys. Rev. A 55, R1561 (1997).
  * Coles  _et al._ [2018] P. J. Coles, S. J. Eidenbenz, S. Pakin, A. Adedoyin, J. Ambrosiano, P. M. Anisimov, W. Casper, G. Chennupati, C. Coffrin, H. N. Djidjev, D. Gunter, S. Karra, N. Lemons, S. Lin, A. Malyzhenkov, and authors, arXiv e-prints (2018), arXiv:1804.03719 .
  * Carrasquilla  _et al._ [2019] J. Carrasquilla, G. Torlai, R. G. Melko, and L. Aolita, Nature Machine Intelligence 1, 155 (2019).
  * Torlai and Melko [2020] G. Torlai and R. G. Melko, Annual Review of Condensed Matter Physics 11, 325 (2020), https://doi.org/10.1146/annurev-conmatphys-031119-050651 .
  * Carrasquilla and Torlai [2021] J. Carrasquilla and G. Torlai, PRX Quantum 2, 040201 (2021).
  * Torlai  _et al._ [2018] G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, and G. Carleo, Nature Phys 14, 447–450 (2018).
  * Melko  _et al._ [2019] R. G. Melko, G. Carleo, J. Carrasquilla, and J. I. Cirac, Nature Physics 15, 887 (2019).
  * Torlai  _et al._ [2019] G. Torlai, B. Timar, E. P. L. van Nieuwenburg, A. Levine, H.and Omran, A. Keesling, H. Bernien, M. Greiner, V. Vuletic, M. D. Lukin, R. G. Melko, and M. Endres, Phys. Rev. Lett. 123, 230504 (2019).
  * Settles [2009] B. Settles, Computer Sciences Technical Report (2009).
  * Greiner  _et al._ [2002] R. Greiner, A. J. Grove, and D. Roth, Artificial Intelligence 139, 137 (2002).
  * Tong and Chang [2001] S. Tong and E. Chang, Proc. of the 9th ACM Intern. Conf. on Multimedia , 107–118 (2001).
  * Tur  _et al._ [2005] G. Tur, D. Hakkani-Tür, and R. E. Schapire, Speech Communication 45, 171 (2005).
  * Yao  _et al._ [2020] J. Yao, Y. Wu, J. Koo, B. Yan, and H. Zhai, Phys. Rev. Research 2, 013287 (2020).
  * Ding  _et al._ [2022] Y. Ding, J. D. Martín-Guerrero, Y. Song, R. Magdalena-Benedicto, and X. Chen, Phys. Rev. Research 4, 013213 (2022).
  * Ding  _et al._ [2020] Y. Ding, J. D. Martín-Guerrero, M. Sanz, R. Magdalena-Benedicto, X. Chen, and E. Solano, Phys. Rev. Lett. 124, 140504 (2020).
  * Huszár and Houlsby [2012] F. Huszár and N. M. T. Houlsby, Phys. Rev. A 85, 052120 (2012).
  * Mahler  _et al._ [2013] D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, and A. M. Steinberg, Phys. Rev. Lett. 111, 183601 (2013).
  * Ferrie [2014] C. Ferrie, Phys. Rev. Lett. 113, 190404 (2014).
  * Straupe [2016] S. Straupe, Jetp Lett. 104, 510 (2016).
  * Schollwoeck [2011] U. Schollwoeck, Annals of Physics 326, 96 (2011).
  * Rocchetto  _et al._ [2018] A. Rocchetto, E. Grant, S. Strelchuk, G. Carleo, and S. Severini, npj Quantum Information 4, 28 (2018).
  * Morawetz  _et al._ [2021] S. Morawetz, I. J. S. De Vlugt, J. Carrasquilla, and R. G. Melko, Physical Review A 104 (2021), 10.1103/physreva.104.012401.
  * Schmale  _et al._ [2021] T. Schmale, M. Reh, and M. Gärttner, arXiv e-prints , 2109.13776 (2021), arXiv:2109.13776 [quant-ph] .
  * Ahmed  _et al._ [2021] S. Ahmed, C. Sánchez Muñoz, F. Nori, and A. F. Kockum, Phys. Rev. Lett. 127, 140502 (2021).
  * Cha  _et al._ [2021] P. Cha, P. Ginsparg, F. Wu, J. Carrasquilla, P. L. McMahon, and E.-A. Kim, Machine Learning: Science and Technology 3, 01LT01 (2021).
  * Cramer  _et al._ [2010] M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, Nature Communications 1, 149 (2010).
  * Baumgratz  _et al._ [2013] T. Baumgratz, D. Gross, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 111, 020401 (2013).
  * Beach  _et al._ [2019a] M. J. S. Beach, I. De Vlugt, A. Golubeva, P. Huembeli, B. Kulchytskyy, X. Luo, R. G. Melko, E. Merali, and G. Torlai, SciPost Phys. 7, 9 (2019a).
  * Note [1] The empirical distribution $q(\bm{x})$ is defined as follows: if a possible outcome $\bm{v}$ is contained $N_{\bm{v}}$ times in the set of $N_{\rm tot}$ existing measurements $\mathcal{D}$, then $q(\bm{v})=N_{\bm{v}}/N_{\rm tot}$. If $\bm{x}\notin\mathcal{D}$ has not been measured, $q(\bm{x})=0$.
  * Quc [2021] “Qucumber documentation,”  (2021).
  * IBM [2021] “Ibm quantum.”  (2021).
  * Aleksandrowicz  _et al._ [2019] G. Aleksandrowicz, T. Alexander, P. Barkoutsos, L. Bello, Y. Ben-Haim, D. Bucher, F. J. Cabrera-Hernández, J. Carballo-Franquis, A. Chen, J. Chow, A. Córcoles-Gonzales, A. J. Cross, A. Cross, J. Cruz-Benito, C. C., and authors, Zenodo (2019), 10.5281/zenodo.2562111.
  * [38] C. Hubig, F. Lachenmaier, N.-O. Linden, T. Reinhard, L. Stenzel, A. Swoboda, and M. Grundner, “The SyTen toolkit,” .
  * Hubig [2017] C. Hubig,  _Symmetry-Protected Tensor Networks_ , Ph.D. thesis, LMU München (2017).
  * Iadecola and Schecter [2020] T. Iadecola and M. Schecter, Phys. Rev. B 101, 024306 (2020).
  * Borla  _et al._ [2020] U. Borla, R. Verresen, F. Grusdt, and S. Moroz, Phys. Rev. Lett. 124, 120503 (2020).
  * Kebrič  _et al._ [2021] M. Kebrič, L. Barbiero, C. Reinmoser, U. Schollwöck, and F. Grusdt, Phys. Rev. Lett. 127, 167203 (2021).
  * Ferris and Vidal [2012] A. J. Ferris and G. Vidal, Phys. Rev. B 85, 165146 (2012).
  * Buser  _et al._ [2021] M. Buser, U. Schollwöck, and F. Grusdt, arXiv , 2110.14952 (2021), arXiv:2110.14952 [cond-mat.quant-gas] .
  * 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 authors, Science 374, 1242–1247 (2021).
  * Satzinger  _et al._ [2021] K. J. Satzinger, Y.-J. Liu, A. Smith, C. Knapp, M. Newman, C. Jones, Z. Chen, C. Quintana, X. Mi, A. Dunsworth, C. Gidney, I. Aleiner, F. Arute, K. Arya, J. Atalaya, and authors, Science 374, 1237–1241 (2021).
  * Huang  _et al._ [2020] H.-Y. Huang, R. Kueng, and J. Preskill, Nature Physics 16, 1050–1057 (2020).
  * Beach  _et al._ [2019b] M. J. S. Beach, I. De Vlugt, A. Golubeva, P. Huembeli, B. Kulchytskyy, X. Luo, R. G. Melko, E. Merali, and G. Torlai, SciPost Phys. 7, 9 (2019b).
  * Goodfellow  _et al._ [2016] I. Goodfellow, Y. Bengio, and A. Courville,  _Deep Learning_ (MIT Press, Cambridge, 2016).
  * Mehta  _et al._ [2019] P. Mehta, M. Bukov, C. Wang, A. G. Day, C. Richardson, C. K. Fisher, and D. J. Schwab, Physics Reports 810, 1–124 (2019).
  * Prosko  _et al._ [2017] C. Prosko, S.-P. Lee, and J. Maciejko, Phys. Rev. B 96, 205104 (2017).

##  Appendix A Quantum state representation in terms of a restricted Boltzmann
machine

Part of our active learning scheme is the training of a committee of
restricted Boltzmann machines (see Fig. 1). In this work, the implementation
of RBMs within the open-source software python package QuCumber [48] is used
for the training. The package is designed to learn quantum many body wave
functions from a set of projective measurements in different basis
configurations by representing the reconstructed state by RBMs. In this
section, we will present how to represent quantum states in terms of RBMs and
more details to the RBM training.

Firstly, a quantum state $|\Psi\rangle$ with only positive coefficients
$\Psi(\bm{x})=\langle\bm{x}|\Psi\rangle\geq 0$ will be considered, where
$\\{|\bm{x}^{b}\rangle\\}$ with
$\\{|\bm{x}^{b}\rangle\\}=|x_{1}^{b},\dots,x_{N}^{b}\rangle$ is a reference
basis $b$ for the Hilbert space of $N$ quantum degrees of freedom (i.e. for
two qubits it consists of the states $|00\rangle$, $|01\rangle$, $|10\rangle$
and $|11\rangle$). In this case, the representation of a quantum state in
terms of a RBM is straight forward: For an infinite number of measurements in
a reference basis, i.e. in the $b=z$ basis, the measurements adhere to Born’s
rule and the probability of finding a measurement result $\bm{x}$ is

$q(\bm{x})=|\Psi(\bm{x})|^{2}.$

QuCumber creates a RBM network with a probability distribution

$\displaystyle p_{\lambda}(\bm{v})=$
$\displaystyle\frac{1}{Z}\prod_{j=1}^{V}\mathrm{exp}\left(b_{j}v_{j}\right)$
(13)
$\displaystyle\quad\times\prod_{i=1}^{H}\left\\{1+\mathrm{exp}\left(\sum_{j}^{V}W_{ij}v_{j}+c_{i}\right)\right\\},$
(14)

where $\bm{v}$ and $\bm{h}$ are the visible and hidden nodes of the RBM and
$W_{ij}$ the weights between visible node $i$ and hidden node $j$ [49]. For a
finite data set $\mathcal{D}=(\bm{x}_{1},\bm{x}_{2},\dots)$ it trains the RBM
such that the Kullback-Leibler divergence (see Eq. (2)) is minimized. The
minimization is performed by gradient descent, which involves the calculation
of expectation values with respect to the distributions of the data and the
model. To calculate the expectation value over the model distribution one
usually uses Markov Chain Monte Carlo Gibbs sampling. Due to the restricted
nature of RBMs hidden and visible units are conditional independent and hence
the conditional probabilities factorize. Consequently, it is possible to
calculate the conditional distributions of all visible / hidden nodes in
parallel by taking $\bm{h}_{t+1}\propto p(\bm{h}|\bm{v}_{t})$ and
$\bm{v}_{t+1}\propto p(\bm{v}|\bm{h}_{t+1})$, where $t$ measures the number of
steps in the Monte Carlo chain. For large $t\to\infty$ it is guaranteed to
converge [50]. A slight modification which simplifies the training process is
called contrastive divergence. Hereby, only $k$ iterations of Gibbs sampling
are performed (contrastive divergence steps).

For positive wave functions the RBM representation can be defined as

$\psi_{\lambda}(\bm{x})=\sqrt{\frac{p_{\lambda}(\bm{x}}{Z_{\lambda}}}),$

with a normalization constant $Z_{\lambda}$ [48].

For more general wave functions with complex-valued coefficients like

$\displaystyle|\Psi\rangle=\sum_{x_{1},\dots
x_{N}}\Phi_{x_{1},\dots,x_{N}}\mathrm{e}^{i\varphi_{x_{1},\dots,x_{N}}}|x_{1},\dots,x_{N}\rangle$
(15)

the probability distribution underlying the outcomes of projective
measurements in the reference basis does not contain all possible information
about the unknown quantum state, because the information about the phase is
lost when only considering the underlying probability distribution of
measurements in one basis $q(\bm{x})=|\Psi(\bm{x})|^{2}$. In this case
QuCumber represents the quantum state as defined in Eq. (1) and trains two
RBMs with parameters $\lambda$ and $\mu$ separately. The RBM with parameters
$\lambda$ models the amplitude of the RBM wave function, the second RBM with
parameters $\mu$ the phase
$\theta_{\mu}(\bm{x})=\mathrm{log}\,p_{\mu}(\bm{x})$. Firstly $\lambda$ is
optimized such that
$|\Psi(\bm{x}^{b})|^{2}=|\psi_{\lambda,\mu}(\bm{x}^{b})|^{2}$ for measurements
$\bm{x}^{b}$ in the reference basis $b$, which is set to the $z$-basis by
default. Secondly, measurements in other basis configurations are considered
to determine the phase $\theta_{\mu}$ by training another RBM [11].

Figure 12: Measurement of a quantum system with two qubits ($q_{0}$ and
$q_{1}$) in the $xy$ basis configuration. Therefore $q_{0}$ is rotated from
the $z$ to the $x$ axis by applying the Hadamard gate, and $q_{1}$ to the $y$
axis by application of a combination of $S$ and Hadamard gates (see equations
(18) and (21)). Figure generated with Qiskit [37].

The information about the phase is extracted via the rotation of single qubits
within a quantum circuit into another basis. QuCumber uses by default the $z$
basis as reference basis and rotates to the $x$ and $y$ bases to extract phase
information. The rotation of a single qubit to the $x$ basis is achieved by
applying the Hadamard gate

$\displaystyle H=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rr}1&1\\\ 1&-1\\\
\end{array}\right),$ (18)

the rotation to $y$ by applying a combination of the Hadamard gate and the
S-adjoint gate,

$\displaystyle K=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rr}1&1\\\ i&-i\\\
\end{array}\right).$ (21)

The rotations are performed by using the pre-defined quantum operations of
Qiskit as shown for an exemplary rotation from $zz$ to $xy$ in figure 12.

The representation of a complex wave function in terms of the RBMs as
explained above is implemented with the QuCumber package by using the
ComplexWaveFunction method.

##  Appendix B Details of active learning QST

When using the AL procedure as described in section 2, within each cycle (step
3.)), $4$ RBMs are used if not stated otherwise, with $1000$ epochs, a
learning rate of $l=0.07$ and contrastive divergence steps $k=100$.

###  B.1 IBM quantum states

Figure 13: Reconstruction results for a pure RBM reconstruction of quantum
states prepared on an IBM device ($5$ qubits) without AL. The number of
samples is varied by keeping the number of configurations fixed ($6$
configurations). Figure 14: Reconstruction results for a pure RBM
reconstruction of real quantum states ($5$ qubits) without AL. The number of
configurations is varied by keeping the number of samples fixed ($2000$
samples).

For the generation of states on a classical quantum simulator we use the Aer
simulator, which is designed to mimic the execution of an actual device [36].
For real quantum states the devices ibmq bogota and ibmq quito were used,
which both consist of $5$ superconducting qubits.

In figures 13 and 14 the results for a pure RBM reconstruction of real quantum
states ($5$ qubits) without AL are shown. In 13 the number of samples is
varied by keeping the number of configurations fixed ($6$ configurations). In
14 the number of configurations is varied when fixing the number of samples
($2000$ samples). It can be seen that it is difficult and very time consuming
to find the most efficient set of measurements and configurations by such
scans. Furthermore, the number of samples and configurations with the best
reconstruction varies from state to state. This makes our active learning
scheme very appreciable since it chooses the number of samples and
configurations on its own and only the number of samples at the beginning and
per query have to be fixed by the user. As can be seen from section 3 equally
good results can be obtained for all states when setting the number of samples
per query to $1$ or $2$. Hence, the only free parameter is the number of
samples at the beginning of the AL.

###  B.2 DMRG states

Figure 15: Examplary learning curves for a LGT state with $19$ qubits and
$h/t=1$.

Figure 16: Correlator over distance (top) and density (bottom) for the KCS
state with $h/t=0$ and $7$ qubits for target state (black) and the
reconstructed states using AL (blue) and the baseline (green).

Figure 17: Correlator over distance (top) and density (bottom) for the KCS
state with $h/t=1$, $\mu/t=1$ and $7$ qubits for target state (black) and the
reconstructed states using AL (blue) and the baseline (green).

For the DMRG states, we use

$\displaystyle R_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rr}i&-i\\\ 1&1\\\
\end{array}\right)$ (24)

and

$\displaystyle R_{y}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rr}1&-i\\\
-i&1\\\ \end{array}\right).$ (27)

For lattice gauge model states we consider states with $\mu=10^{-7}\approx 0$,
since the convergence is much faster than for $\mu=0$.

In Fig. 15 the learning curve for a lattice gauge model state with $19$ qubits
and $h/t=1$ is shown. When using AL divergence, the $xxxxxxx$ configuration is
chosen to be the reference basis. One can see that only this choice of
reference basis leads to relatively good results for the density differences
(around $15\,\%$, see upper part for epoch $0$) and $17\,\%$ correlator
difference (bottom). The RBM training then reduces the density difference
further to
$|\bm{n}-\bm{n}_{\mathrm{target}}/|\bm{n}_{\mathrm{target}}|=9.59\pm 0.02\,\%$
and $|\bm{c}-\bm{c}_{\mathrm{target}}/|\bm{c}_{\mathrm{target}}|=12.4\pm
0.2\,\%$.

In Figs. 16 and 17 the local correlators over distance $d$ (top) and densities
over the system size are shown for KCS states with $7$ qubits and $h/t=0$,
$\mu/t=0$ and $h/t=1$, $\mu/t=1$ respectively. Similarly to the results for
$19$ qubits in Figs. 8 and 9 one can see that the agreement with the target
state is improved when using the AL reconstruction scheme. Also here, features
like the curvature of the correlator can be reproduced for AL, but not for the
baseline. For the parameters $h/t=1$ and $\mu/t=0$ the density increases to a
maximum in the middle of the chain. Even though this behaviour can be
reproduced by AL and baseline, the results for the baseline are of a around
half of the magnitude of the target state. In contrast, the AL reconstruction
yields densities with magnitudes much closer to the target state.

Moreover, when plotting the correlators for $h/t=0$ ($h/t=1$) with logarithmic
scales for $x$ and $y$ axes ($y$ axis) as shown for a system with $19$ qubits
in Fig. 18 (Fig. 19) one can observe the expected power-law (exponential)
decay for the target state. This power-law (exponential) decay is
reconstructed when using AL, but not for the baseline.

Figure 18: Correlator over distance for the KCS model state with $h/t=0$,
$\mu/t=0$ and $19$ qubits for target state (black) and the reconstructed
states using AL (blue) and the baseline (green) as in Fig. 8 but with
logarithmic scaling for $x$ and $y$ axes. Figure 19: Correlator over distance
for the KCS model state with $h/t=1$, $\mu/t=1$ and $19$ qubits for target
state (black) and the reconstructed states using AL (blue) and the baseline
(green) as in Fig. 9 but with logarithmic scaling for the $y$ axis.

##  Appendix C Kinetically constrained spin model and $\mathbb{Z}_{2}$ lattice
gauge theory

In this section we provide more background information on the kinetically
constrained quantum spin model, see Eq. (9), considered in the main text.

###  C.1 Mapping to a $\mathbb{Z}_{2}$ lattice gauge theory

The model in Eq. (9) can be mapped to a one-dimensional $\mathbb{Z}_{2}$
lattice gauge theory model, with U(1) matter [41, 42]. To this end, domain
walls in the spin model are mapped to hardcore bosons, which are coupled to
$\mathbb{Z}_{2}$ gauge fields defined on the links between the lattice sites
$i,j$. As explained below, one obtains the following equivalent Hamiltonian,

$\displaystyle\hat{\mathcal{H}}_{\mathbb{Z}_{2}}=$
$\displaystyle-t\sum_{\langle
i,j\rangle}\left(\hat{a}^{\dagger}_{i}\hat{\tau}^{z}_{\langle
i,j\rangle}\hat{a}_{j}+\text{h.c.}\right)$ $\displaystyle-h\sum_{\langle
i,j\rangle}\hat{\tau}^{x}_{\langle i,j\rangle}+\mu\sum_{\langle
i,j\rangle}\hat{n}_{j}.$ (28)

Here $\hat{a}^{\dagger}_{j}$ is the hardcore boson creation operator, defined
on site $j$, and $\hat{n}_{j}=\hat{a}^{\dagger}_{j}\hat{a}_{j}$ is the local
number operator. The $\mathbb{Z}_{2}$ gauge and electric fields are
represented with the Pauli matrices, defined on the links between neighboring
lattice sites, as $\hat{\tau}^{z}_{\langle i,i+1\rangle}$ and
$\hat{\tau}^{x}_{\langle i,i+1\rangle}$ respectively. This model is appealing
since it exhibits confinement of dynamical particles which is induced by any
nonzero $\mathbb{Z}_{2}$ electrical field term $h\neq 0$ [41, 42].

The generator of the local $\mathbb{Z}_{2}$ gauge symmetry of Eq. (28) can be
written as [51]

$\hat{\mathcal{G}}_{j}=\hat{\tau}^{x}_{\langle
i-1,i\rangle}(-1)^{\hat{n}_{i}}\hat{\tau}^{x}_{\langle
i,i+1\rangle},\qquad[\hat{\mathcal{H}}_{\mathbb{Z}_{2}},\hat{\mathcal{G}}_{j}]=0.$
(29)

This leads to the $\mathbb{Z}_{2}$ Gauss law, requiring all states to be $+1$
eigenstates of $\hat{\mathcal{G}}_{j}$ (we assume no background charges):

$\hat{\mathcal{G}}_{j}|\psi\rangle=+|\psi\rangle.$ (30)

From here it is relatively straightforward to obtain back the constrained spin
model Eq. (9) from the main text. One notices that the charge configuration is
entirely determined by the $\mathbb{Z}_{2}$ electric fields due to the
constraints imposed by the $\mathbb{Z}_{2}$ Gauss law. This allows to
formulate the Hamiltonian entirely in terms of the gauge field, by identifying
the presence of a particle on a lattice site as anti-alignment of the
$\mathbb{Z}_{2}$ electric field defined on the links connecting that site.
This also means that the particles, or domain walls, are connected with the
$\mathbb{Z}_{2}$ electric fields of the same orientation, which we interpret
as strings and anti-strings which connect the $\mathbb{Z}_{2}$ charges.

Using the above interpretation it is straightforward to see that the first
term in model (9) corresponds to the kinetic term, where the number of domain
walls is conserved. This ensures two things in the lattice gauge
interpretation: the $\mathbb{Z}_{2}$ electric string remains attached to the
hopping particle, and the total number of particles is conserved. The second
term $\propto h$ in Eq. (9) induces energy cost to strings and as a result
confines the particle pairs into dimers. Finally, the Ising term is needed to
control the number of domain walls and thus the filling of the chain in the 1D
LGT interpretation.

###  C.2 The gauge-invariant equal-time Green’s function

The correlation function in Eq. (12) can be mapped to the $\mathbb{Z}_{2}$
invariant equal-time Green’s function defined as

${\bm{c}(|i-j|)=\left\langle\hat{a}^{\dagger}_{i}\prod_{i\leq
l<j}\hat{\tau}^{z}_{\langle l,l+1\rangle}\hat{a}_{j}\right\rangle}.$ (31)

This is once again done by taking into account the constraint imposed by the
Gauss law in Eq. (30). Both terms $\propto\left(1\pm
4\hat{S}^{x}_{l}\hat{S}^{x}_{l+1}\right),l\in\\{i,j\\}$ act as projectors to
the states where there is a site with a particle to be annihilated and an
empty site, where a particle can be created. The actual annihilation at site
$j$ and creation of the particle at site $i$ comes from the application of
$\prod_{l}\left(2\hat{S}^{z}_{l}\right)$ between the domain walls. Combining
both terms, first applying the projectors and then performing the annihilation
and creation of the particle, gives us the $\mathbb{Z}_{2}$ Green’s function
expressed entirely in terms of gauge fields. Note that the order is slightly
changed in the main text Eq.(12) by considering the spin anticommutation
property.

Such correlation function probes the confinement of particle pairs in the LGT
interpretation. It decays as a power law in the deconfined phase $h=0$ and
decays exponentially in the confined phase, which is the case for any non-zero
value of the field [41, 42]. On the other hand, the pair-pair correlation
function decays as a power-law function also in the confined phase, which
means that the effective dimers behave as Luttinger liquid [41].