# Ultrafast quantum state tomography with feed-forward neural networks

Yong Wang<EMAIL_ADDRESS>College of Electronics and Information
Engineering, Tongji University, Shanghai 201804, China  Shuming Cheng
<EMAIL_ADDRESS>College of Electronics and Information
Engineering, Tongji University, Shanghai 201804, China  Shanghai Research
Institute for Artificial Intelligence Science and Technology, Tongji
University, Shanghai 201203, China  Institute for Advanced Study, Tongji
University, Shanghai, 200092, China  Li Li  College of Electronics and
Information Engineering, Tongji University, Shanghai 201804, China  Shanghai
Research Institute for Artificial Intelligence Science and Technology, Tongji
University, Shanghai 201203, China  Jie Chen  College of Electronics and
Information Engineering, Tongji University, Shanghai 201804, China  Shanghai
Research Institute for Artificial Intelligence Science and Technology, Tongji
University, Shanghai 201203, China

###### Abstract

Reconstructing the state of many-body quantum systems is of fundamental
importance in quantum information tasks, but extremely challenging due to the
curse of dimensionality. In this work, we present a quantum tomography
approach based on neural networks to achieve the ultrafast reconstruction of
multi-qubit states. Particularly, we propose a simple 3-layer feed-forward
network to process the experimental data generated from measuring each qubit
with a positive operator-valued measure, which is able to reduce the storage
cost and computational complexity. Moreover, the techniques of state
decomposition and $P$-order absolute projection are jointly introduced to
ensure the positivity of state matrices learned in the maximum likelihood
function and to improve the convergence speed and robustness of the above
network. Finally, it is tested on a large number of states with a wide range
of purity to show that we can faithfully tomography 11-qubit states on a
laptop within 2 minutes under noise. Our numerical results also demonstrate
that more state samples are required to achieve the given tomography fidelity
for the low-purity states, and the increased depolarizing noise induces a
linear decrease in the tomography fidelity.

##  1 Introduction

Quantum state tomography (QST) is a powerful tool to recover the state
information of the unknown many-body quantum systems from measurement
statistics, and hence plays an indispensable role in quantum information
processing tasks, with wide applications ranging from certifying fundamental
principles in quantum theory [1], benchmarking quantum devices [2], to
verifying quantum algorithms [3]. However, it is a challenging task due to the
curse of dimensionality in the sense that it admits an exponential growth of
measurement settings, memory cost, and computing resources as the number of
subsystems involved linearly increases [4].

Various approaches have been developed to accomplish the task of QST over the
past decades. Notable examples include the maximum likelihood estimation (MLE)
[5, 6, 7], linear regression [8, 9], and Bayesian tomography [10, 11]. The
state reconstruction speed could be improved by adopting compressed sensing
[12, 13, 14], permutationally invariant tomography [15, 16], or matrix-
product-state tomography [17, 18], however, they are only applicable for
limited classes of states and thus lack generality [19]. Furthermore, since
one fundamental physical constraint on the matrices reconstructed from these
methods is the positivity, some state-mapping techniques are needed. For
example, one is to parameterize the state matrix via the Cholesky
decomposition [5, 7, 10, 19, 13, 14, 20, 21] and another is the state
projection which maps negative eigenvalues of the learned matrix to those of
physical states [8, 9, 22, 11, 23, 24]. However, the problematic issues arise
that the decomposition method has limited convergence speed and reconstruction
accuracy while the projection needs extra computing resources [25]. Hence, it
still remains rather challenging for these methods to tomography multi-
dimensional quantum systems.

As the neural networks become more efficient and scalable to process the data
of high dimensionality, many neural-network QST (NN-QST) approaches have also
been proposed, such as feed-forward neural network (FNN) [26, 27, 28, 20, 21],
restricted Boltzmann machines [29, 30, 31], recurrent neural networks [32,
33], autoregressive models [34], generative adversarial networks (GANs) [19],
and transformers [35]. It has been demonstrated in [19] that half an hour is
enough to reconstruct the 6-qubit cat state by using conditional GANs.
However, it is still less explored how the neural net works for multi-qubit
states with a wide range of purity and whether the techniques of state
decomposition and/or projection can be applied to these network based QST to
improve their learning performances.

To tackle the above issues, here we propose a feed-forward neural network,
combining the state decomposition with state projection, to achieve the
ultrafast reconstruction of multi-qubit states. In particular, our network is
configured in a simple 3-layer structure where the input layer directly
processes the probability distributions (PDs) generated by measuring each
qubit, the hidden layer has fully-connected neurons in a linear number of
qubits, and the output yields a state matrix directly. Then, it is efficiently
trained by optimizing the maximum likelihood function and benchmarked by
quantum state fidelity between the learned state and the target. In this
training process, the transition matrix is decomposed into Hermitian matrices
which are then processed via a class of the projection methods called as
$P$-order absolute projection. Thus, we provide the first unified state-
mapping strategy which is able to not only speed-up its learning rate, but
also to improve its robustness towards state purity and noise. Finally, a
positive operator-valued measure (POVM) on each qubit in the product structure
is used to process the PDs, which reduces the memory cost to store and saves
the computational resource to process.

We further test our NN-QST on a large number of states, including trivial
product states, genuinely entangled W and GHZi states, all of these states
under depolarizing noise, and states randomly generated with exponential decay
of eigenvalues and fixed state purity as introduced in [23]. Numerical results
first confirm our NN-QST adopting the unified state-mapping strategy has
better convergence speed and state purity robustness than those using other
state-mapping methods. Next, we demonstrate that it has better performance in
the convergence time and robustness towards state purity and noise, in
comparison to various fast QST algorithms including the CG-APG algorithm [25]
and iterative MLE (iMLE) algorithm [36]. Surprisingly, it is found our
approach can fully tomography 11-qubit states on a laptop within 2 minutes,
while it takes nearly 3 hours for the CG-APG and iMLE algorithms to accomplish
the 10-qubit tomography and more than half an hour for conditional GANs to
tomography the 6-qubit cat state [19]. Moreover, we show more state samples
are needed for the net to achieve a given tomography fidelity for the low-
purity states. Finally, we investigate how the state purity affects the
tomography fidelity under noise and find it is robust to depolarizing noise.

The reminder of this work is organized as follows. Sec. 2 introduces some
basic concepts related to QST, including the density matrix, MLE strategy,
state-mapping techniques, and quantum state fidelity. Sec. 3 details the
implementation of our NN-QST, unified state-mapping strategy, and the product-
structured POVM. Then, the corresponding numerical results are presented in
Sec. 4. Finally, the summary and outlook of the paper are discussed in Sec. 5.

##  2 Quantum state tomography

This section briefly introduces basic notations about QST and some commonly-
used techniques, such as state decomposition and projection, which are used to
guarantee the positivity of state matrices reconstructed from QST. Finally,
quantum state fidelity is introduced to evaluate the tomography performance.

###  2.1 Reconstructing the density matrix

The state of many-body quantum systems is fully characterized by the density
matrix $\rho$, which is a positive semi-definite (PSD) operator with unit
trace, i.e., ${\rm Tr}\left[{{\rho}}\right]=1$. Specifically, it can be
written in a form of

$\rho=\sum\limits_{i}{{p_{i}}|\psi_{i}\rangle\langle\psi_{i}|}.$ (1)

where the probability $p_{i}$ represents the occurrence of pure quantum state
$|{\psi_{i}}\rangle$, with $\sum\nolimits_{i}{{p_{i}}}=1$ and $p_{i}\geq 0$.
If $\rho=|\psi\rangle\langle\psi|$, then $\rho$ is pure. Otherwise, it is a
mixed state. Note that for the $N$-qubit system, generically, it requires
${d^{2}}-1$ real parameters to completely determine $\rho$, where $d={2^{N}}$
is the dimension of the Hilbert space. It is evident that as the number of
qubits $N$ increases, the number of parameters to describe $\rho$ will
increase exponentially.

To reconstruct the density matrix $\rho$ in Eq. (1), QST is decomposed of two
steps: The first is the measurement procedure which yields the outcome
statistics described by PDs from measuring on identically prepared copies of
the unknown quantum state. Each measurement $i$ is modeled as a positive-
operator-value measure (POVM) $\\{M^{i}_{k}\\}$ where the positive operator
$M^{i}_{k}$ satisfy $\sum_{k}M_{k}^{i}=\mathbb{I}$ for all $i$, and the
probability of each outcome is given by the Born rule $P^{i}_{k}={\rm
Tr}\left[{{M^{i}_{k}\rho}}\right]$ without statistical noise. And, the second
is the estimation procedure which estimate a physical $\hat{\rho}$ from those
measured PDs. Consequently, it can be formulated as the following optimization
problem:

$\displaystyle\hat{\rho}$ (2) $\displaystyle{\rm s.t.}$ $\displaystyle{\rm
Tr}\left[{{\hat{\rho}}}\right]=1\ {\rm and}\ \hat{\rho}\geq 0,$ $\displaystyle
P_{k}^{i}={\rm Tr}\left[{{{M^{i}_{k}}\hat{\rho}}}\right]={\rm
Tr}\left[{{{M^{i}_{k}}\rho}}\right]\ \forall\leavevmode\nobreak\ M^{i}_{k},$
$\displaystyle M^{i}_{k}\geq 0\ {\rm and}\
\sum\nolimits_{k}M^{i}_{k}=\mathbb{I}.$

However, it is challenging to directly solve the above problem due to the
curse of dimensionality that as the number of qubits linearly increases, it
requires an exponential growth of measurement settings to faithfully determine
$\rho$, memory cost to store $\boldsymbol{P}^{i}=(P_{1}^{i},\dots,P^{i}_{n})$,
and computational resources to process the PDs. In practice, the presence of
statistical noise leads us to obtain the frequency $f_{k}\propto{\rm
Tr}\left[{{M_{k}\rho}}\right]$, instead of the accurate probability of each
outcome, which makes the accurate tomography more challenging.

One efficient approach to QST is the maximum likelihood estimation [5, 6, 7].
In particular, MLE minimizes the negative log-likelihood function between the
observed frequency $f_{k}$ and the estimated probability $\hat{P}_{k}={\rm
Tr}\left[{{M_{k}\hat{\rho}}}\right]$:

$\displaystyle\rm{\mathop{minimize}\limits_{\hat{\rho}}}$
$\displaystyle-\sum\limits_{k}{{f_{k}}\log({\rm
Tr}\left[{{{M_{k}}\hat{\rho}}}\right])}$ (3) $\displaystyle{\rm s.t.}$
$\displaystyle{\rm Tr}\left[{{\hat{\rho}}}\right]=1\ {\rm and}\ \hat{\rho}\geq
0,$ $\displaystyle M_{k}\geq 0\ {\rm and}\ \sum\nolimits_{k}M_{k}=\mathbb{I}.$

where $\\{M_{k}\\}$ is one POVM, and $\hat{\rho}$ is the reconstructed
physical density matrix. Notably, it follows from the convexity of the
optimization function that there exists an unique solution to the MLE [37,
38]. Since no extra assumption is imposed by the MLE, it is also one of
commonly-used tomography methods [39], together with experimental verification
via the single-ion Zeeman qubit [40] and the polarization states of three
photons [41].

###  2.2 Mapping techniques to ensure the positivity of reconstructed states

In the optimization problems (2) and (3), the reconstructed state $\hat{\rho}$
should satisfy the fundamental physical constraint that it is a PSD operator
with trace being 1, however, this may not be always guaranteed. Thus, to solve
this problem, the techniques of state decomposition or state projection are
further employed to map the nonphysical states to physical ones.

####  2.2.1 Decomposition method

Note that an arbitrary density matrix $\hat{\rho}$ admits the Cholesky
decomposition,

$\hat{\rho}=\frac{T_{\hat{\rho}}^{\dagger}T_{\hat{\rho}}}{{\rm
Tr}\left[{{T_{\hat{\rho}}^{\dagger}T_{\hat{\rho}}}}\right]},$ (4)

where the transition matrix $T_{\hat{\rho}}$ is a complex lower triangular
matrix and $\dagger$ denotes the complex conjugate operation. Evidently,
$\hat{\rho}$ in the form (4) is automatically positive, and hence the QST
optimization (2) and (3) can be simplified to search over $T_{\hat{\rho}}$
without the positivity constraint, instead of $\hat{\rho}$.

It immediately yields that the MLE with the state decomposition (4) becomes:

$\displaystyle\rm{\mathop{minimize}\limits_{T_{\hat{\rho}}}}$
$\displaystyle-\sum\limits_{k}{{f_{k}}\log(\mathrm{Tr}[{M_{k}}\frac{T_{\hat{\rho}}^{\dagger}T_{\hat{\rho}}}{{\rm
Tr}\left[{{T_{\hat{\rho}}^{\dagger}T_{\hat{\rho}}}}\right]}])}$ (5)
$\displaystyle{\rm s.t.}$ $\displaystyle M_{k}\geq 0\ {\rm and}\
\sum\nolimits_{k}M_{k}=\mathbb{I}.$

Here $T_{\hat{\rho}}$ could be a nonphysical state. It is remarked that the
difficulty in ensuring the positive $\hat{\rho}$ is essentially transferred to
optimize the objective function in the above problems, and there is evidence
to signal the slow convergence of these methods with decomposition [25].
Hence, another technique, called as state projection method, has been proposed
to speed up the estimation process [8, 9, 11, 22, 23, 24].

####  2.2.2 Projection method

Given a nonpositive Hermitian matrix $\hat{\rho}$ reconstructed from QST, the
projection method introduces a map $\mathcal{P}(\bullet)$ which maps its
eigenvalues to nonnegative ones with unit sum to produce a physical matrix
$\tilde{\rho}$. Indeed, following from the eigenvalue decomposition
$\hat{\rho}=Q\Lambda Q^{\dagger}$ where $\Lambda$ is the diagonal eigenvalue
matrix and $Q$ is the corresponding orthogonal matrix, we have

$\tilde{\rho}=\mathcal{P}(\hat{\rho})=\mathcal{P}(Q\Lambda
Q^{\dagger})=Q\mathcal{P}(\Lambda)Q^{\dagger}\equiv Q\Sigma Q^{\dagger}.$ (6)

Here $\Sigma=\mathcal{P}(\Lambda)$ is a positive diagonal matrix with trace
being 1, thus satisfying the physical constraints imposed by density matrices.
There are various ways to construct the map in Eq. (6), and one possible way
is subtracting a coefficient $c$ from all eigenvalues and then zeroing all
negative eigenvalues with

$\Sigma_{i}=\mathcal{P}(\Lambda_{i})=\max(\Lambda_{i},0)$ (7)

for each eigenvalue $\Lambda_{i}$. The known examples include the Frobenius
norm with alternative projection $\mathcal{F}[\cdot]$ [22] and simplex
projection $\mathcal{S}[\cdot]$ [23] to choose a proper coefficient $c$ or the
nuclear norm with projection to set $c$ to 0 [11]. For the non-Hermitian
matrix $\hat{\rho}$, using $(\hat{\rho}+\hat{\rho}^{\dagger})/2$ naturally
gives rise to a Hermitian matrix which then could be processed in the same
procedure.

Then, the MLE (3) can be solved with the projection method

$\displaystyle\rm{\mathop{minimize}\limits_{\hat{\rho}}}$
$\displaystyle-\sum\limits_{k}{{f_{k}}\log({\rm
Tr}\left[{{{M_{k}}\mathcal{P}(\hat{\rho})}}\right]})$ (8) $\displaystyle{\rm
s.t.}$ $\displaystyle M_{k}\geq 0\ {\rm and}\
\sum\nolimits_{k}M_{k}=\mathbb{I}.$

where $\mathcal{P}(\bullet)$ is any chosen projection method. Essentially, the
projection method pulls the estimate matrix $\hat{\rho}$ close to a physical
density matrix in the sense of matrix 2-norm [22, 23] and other operator norms
[11], to guarantee convergence speed and accuracy.

###  2.3 Quantum state fidelity

The QST performance can be evaluated via quantum state fidelity which measures
the distance between quantum states. Specifically, the quantum fidelity
$F_{q}$ for the state matrix $\hat{\rho}$ reconstructed from QST and the
target $\rho$ is defined as [42]

${F_{q}}({\hat{\rho}},{\rho})=\left(\mathrm{Tr}\left[\sqrt{\sqrt{{\hat{\rho}}}{\rho}\sqrt{{\hat{\rho}}}}\right]\right)^{2}.$
(9)

The state fidelity is symmetric, i.e.,
${F_{q}}({\hat{\rho}},{\rho})={F_{q}}({\rho},{\hat{\rho}})$. If the target
state is pure, then Eq. (9) simplifies to
${F_{q}}({\rho}=|\psi\rangle\langle\psi|,\hat{\rho})=\langle{\psi}|\hat{\rho}|{\psi}\rangle$.
Generally, computing Eq. (9) requires multiple square-root matrix operations,
which consume large computational resources. However, this computing cost can
be reduced, if the state information in QST is used.

Noting from the state projection method in Eq. (6) that
$\tilde{\rho}=\mathcal{P}(\hat{\rho})=Q\Sigma Q^{\dagger}$, we have

$\displaystyle F_{q}({\tilde{\rho}},{\rho})$ $\displaystyle=\left({\rm
Tr}\left[{{\sqrt{Q\sqrt{\Sigma}Q^{\dagger}\rho
Q\sqrt{\Sigma}Q^{\dagger}}}}\right]\right)^{2}$ (10) $\displaystyle=\left({\rm
Tr}\left[{{\sqrt{Q^{\prime}\Sigma^{\prime}{Q^{\prime}}^{\dagger}}}}\right]\right)^{2}$
$\displaystyle=\left({\rm
Tr}\left[{{Q^{\prime}\sqrt{\Sigma^{\prime}}{Q^{\prime}}^{\dagger}}}\right]\right)^{2}$
$\displaystyle=\left(\sum\limits_{i}\sqrt{\Sigma_{i}^{\prime}}\right)^{2}.$

The second equality follows from the eigenvalue decomposition
$Q\sqrt{\Sigma}Q^{\dagger}\rho
Q\sqrt{\Sigma}Q^{\dagger}=Q^{\prime}\Sigma^{\prime}{Q^{\prime}}^{\dagger}$
with the orthogonal matrix $Q^{\prime}$ and nonnegative diagonal matrix
$\Sigma^{\prime}$. Obviously, the fidelity (10) needs one single matrix-
decomposition operation, thus reducing much computational complexity.

Finally, quantum state fidelity $F_{q}$ can be upper bounded by the classical
fidelity $F_{c}$ between two probability distributions, i.e.,

$F_{q}({\hat{\rho}},{\rho})\leq{F_{c}(\hat{\rho},\rho|M_{k})}=\left(\sum\limits_{k}{\sqrt{\hat{P}_{k}P_{k}}}\right)^{2},$
(11)

where $\hat{P}_{k}={\rm Tr}\left[{{{M_{k}}{\hat{\rho}}}}\right]$ and
$P_{k}={\rm Tr}\left[{{{M_{k}}{\rho}}}\right]$. It is easy to calculate the
classical fidelity which provides useful bounds for benchmarking QST. Further,
$F_{c}$ can be approximated by sampling

$F_{c}(\hat{\rho},\rho|M_{k})=\left(\sum\limits_{k}{\hat{P}_{k}\sqrt{\frac{{P_{k}}}{{\hat{P}_{k}}}}}\right)^{2}\approx\left({\mathbb{E}_{{k}\sim{\hat{\boldsymbol{P}}}}}\left[{\sqrt{\frac{{P_{k}}}{{\hat{P}_{k}}}}}\right]\right)^{2}.$
(12)

The sampled value is generically smaller than the true value $F_{c}$.

##  3 Ultrafast neural-network quantum state tomography

As mentioned above, it becomes much less efficient for the traditional QST
approaches with the techniques of state decomposition and/or projection to
tomography the multi-qubit system with a large number of qubits. In this
section, we propose a QST approach which employs the neural network to speed
up the data-processing process and hence to achieve the ultrafast state
reconstruction. Surprisingly, it is found that we are able to achieve the full
tomography of 11-qubit states within 2 minutes, while it takes nearly 3 hours
to tomography the 10-qubit state for the CG-APG algorithm [25] and iMLE
algorithm [36].

The framework of our ultrafast NN-QST is shown in Fig. 1. First, the network
is built up as a simple 3-layer feed-forward neural network (FNN) which
processes the experimental data encoded by PDs and output a Hermitian matrix
$T_{\hat{\rho}}$. Then, the state decomposition method is used to ensure the
positive $\hat{\rho}$ and the projection method is further introduced to
improve the convergence speed of the training process. Moreover, the product-
structured POVM is employed to store and process the data, which can reduce
the memory cost and computational complexity. Finally, the reconstructed state
matrix is benchmarked via quantum state fidelity.

Figure 1: The framework of the ultrafast NN-QST. The mapping network is a
3-layer FNN which inputs the measured PD and outputs a transition matrix $T$.
Then, this transition matrix is efficiently processed by state decomposition
and state projection to yield a positive $\hat{\rho}$. Next, the network
parameters are optimized by minimizing the distance between original PDs and
measured PDs. Finally, it yields an estimate of the expected density matrix
after iterative optimizations and benchmarked via quantum state fidelity as
per Eq. (9).

###  3.1 The ultrafast feed-forward neural network

The FNN used in this work is easy to implement as it is only composed of the
input, hidden, and output layers. For the $N$-qubit system, the number of
neurons in each layer is chosen as:

  1. 1.

Input layer: the maximum is $N_{\rm in}=K^{N}$ equal to the dimension of the
PDs generated from measuring a POVM with $K$ elements on each qubit.

  2. 2.

Output layer: $N_{\rm out}=4^{N}$ which is the number of real parameters to
determine the density matrix $\rho$ for a $N$-qubit state.

  3. 3.

Hidden layer: $N_{\rm hidden}=\log N_{\rm out}=2N$ which is a linear number of
qubits.

The loss function for the network is chosen as the negative log-likelihood
function

$L(\hat{\rho})=-\sum\nolimits_{k}{{f_{k}}\log({\rm
Tr}\left[{{{M_{k}}\hat{\rho}}}\right])},$ (13)

which is commonly used in maximum likelihood tomography. Here $\hat{\rho}$
refers to the learned density matrix, $f_{k}$ is the measurement frequency,
and $M_{k}$ is one POVM element. And its performance in reconstructing the
target state $\rho$ is evaluated via quantum state fidelity
$F_{q}(\hat{\rho},\rho)$ in (9).

Noting that other neural nets can also be used to accomplish the QST task, we
propose the architecture of convolutional neural networks (CNNs) for QST in
Appendix A. We find that the CNN based QST requires preprocessing the PDs and
the convolution operations consume more computational resources, hence leading
to a slower convergence speed than that of the FNN used here, when the number
of qubits is greater than 8. A detailed experimental comparison between the
FNN and CNN is also given in the Appendix A.

###  3.2 The unified state-mapping strategy

We present a simple and efficient method which combines the state
decomposition with state projection to map the output matrix $T_{\hat{\rho}}$
to a physical state $\hat{\rho}$. Note first that if the matrix
$T_{\hat{\rho}}$ is Hermitian, then there is

$\displaystyle\hat{\rho}=\frac{T_{\hat{\rho}}^{\dagger}T_{\hat{\rho}}}{{\rm
Tr}\left[{{T_{\hat{\rho}}^{\dagger}T_{\hat{\rho}}}}\right]}$
$\displaystyle=\frac{Q\Lambda Q^{\dagger}Q\Lambda Q^{\dagger}}{{\rm
Tr}\left[{{Q\Lambda Q^{\dagger}Q\Lambda Q^{\dagger}}}\right]}$ (14)
$\displaystyle=\frac{Q\Lambda^{2}Q^{\dagger}}{{\rm
Tr}\left[{{\Lambda^{2}}}\right]}$ $\displaystyle=Q\Sigma Q^{\dagger}.$

Here $T_{\hat{\rho}}$ admits the eigenvalue decomposition $Q\Lambda
Q^{\dagger}$, and $\Sigma=\Lambda^{2}/{\rm Tr}\left[{{\Lambda^{2}}}\right]$ is
automatically a nonnegative diagonal matrix with unit trace. This can be
regarded as a unified strategy which first uses a simple projection method to
process the transition matrix $T_{\hat{\rho}}$ and simultaneously use the
state decomposition method to output a physical matrix $\hat{\rho}$ from the
processed $T_{\hat{\rho}}$.

Following then from the state projection (6), we are able to easily generalize
the above result to

$\displaystyle\hat{\rho}=\frac{\mathcal{P}(T_{\hat{\rho}}^{\dagger})\mathcal{P}(T_{\hat{\rho}})}{{\rm
Tr}\left[{{\mathcal{P}(T_{\hat{\rho}}^{\dagger})\mathcal{P}(T_{\hat{\rho}})}}\right]}$
$\displaystyle=\frac{Q\mathcal{P}(\Lambda)^{2}Q^{\dagger}}{{\rm
Tr}\left[{{\mathcal{P}(\Lambda)^{2}}}\right]}.$ (15)

Here we introduce a new class of projection methods: $P$-order absolute map
defined as

$\mathcal{A}(T_{\hat{\rho}})_{P}=Q|\Lambda|^{P/2}Q^{\dagger},$ (16)

where the tunable parameter $P$ can be used to adjust the weight of different
eigenvalues of $\Lambda$, and it immediately gives rise to $P$-order absolute
projection

$\hat{\rho}=\mathcal{A[\cdot]}_{P}=\frac{Q|\Lambda|^{P}Q^{\dagger}}{{\rm
Tr}\left[{{|\Lambda|^{P}}}\right]}.$ (17)

It is easy to find Eq. (14) is a special case of $\mathcal{A[\cdot]}_{P}$ with
$P=2$. Since more parameter degrees of freedom is assigned to the above
projection (17), it is suitable for dealing with more complex tomographic
conditions, which is numerically confirmed in the following sections.

Finally, it is remarked that there exists an alternate way to combine these
two methods in which the decomposition method is used in the early stage to
achieve a fast convergence speed and then the projection method is adopted in
the latter stage to guarantee a high learning accuracy [25]. However, our
numerical results in Sec. 4.4 show that it becomes inefficient to tomography
the large-qubit state and its implementation in neural networks is extremely
difficult due to switching. Besides, the issues about the switching conditions
between these two methods and the corresponding computational cost remain to
be examined carefully.

###  3.3 The product-structured POVM

Suppose a POVM $\left\\{M_{k}\right\\}_{k=1}^{K}$ is performed on each qubit.
Then, the product structure of single-qubit POVM is used to form a general
general POVM $\left\\{M_{\bf{k}}\right\\}$ for the $N$-qubit state where
${\bf{k}}=\left({k_{1}},\dots,{k_{N}}\right)$ with
${k_{i}}\in\left\\{1,\dots,K\right\\}$ and
${M_{\bf{k}}}={M_{k_{1}}}\otimes\cdots\otimes{M_{k_{N}}}$. The number of
measurement elements$\left\\{M_{\bf{k}}\right\\}$ is $K^{N}$, thus yielding a
$K^{N}$-dimensional PD.

For any $N$-qubit POVM in the above product structure, the tensor product and
trace operations for obtaining $P_{\bf{k}}={\rm
Tr}\left[{{M_{\bf{k}}\rho}}\right]$ can be converted into the product of
matrices. We take the 2-qubit case as an example. Given matrices
$M_{i}=\begin{bmatrix}a_{i}&b_{i}\\\ c_{i}&d_{i}\end{bmatrix}$ and
$\rho=\begin{bmatrix}A&B\\\ C&D\end{bmatrix}$, where $A$, $B$, $C$, $D$ are
block matrix elements, there is

$\displaystyle P_{(i,j)}$ (18) $\displaystyle=\mathrm{Tr}[(M_{i}\otimes
M_{j})\rho]$ $\displaystyle=\mathrm{Tr}[(M_{i}\otimes
M_{j})(\begin{bmatrix}1&0\\\ 0&0\end{bmatrix}\otimes A+\begin{bmatrix}0&1\\\
0&0\end{bmatrix}\otimes B+\cdots)]$
$\displaystyle=a_{i}\mathrm{Tr}[M_{j}A]+b_{i}\mathrm{Tr}[M_{j}B]+c_{i}\mathrm{Tr}[M_{j}C]+d_{i}\mathrm{Tr}[M_{j}D]$
$\displaystyle=\mathcal{V}_{M_{i}}^{T}\begin{bmatrix}\mathcal{V}_{A}&\mathcal{V}_{B}&\mathcal{V}_{C}&\mathcal{V}_{D}\end{bmatrix}^{T}\mathcal{V}_{M_{j}}.$

And $\mathcal{V}_{X}$ is the column vectorization of matrix $X$, such as
$\mathcal{V}_{M_{i}}^{T}=\begin{bmatrix}a&b&c&d\end{bmatrix}$. This form can
greatly reduce the computational and storage costs, which has been verified in
[25] the cost of computing the probabilities is reduced from $O(K^{N}4^{N})$
to $O(K^{N+1})$. Thus, we choose the product-structured POVM in the FNN.

##  4 Numerical experiments and results

We test our NN-QST on a large number of multi-qubit states with a wide range
of purity in this section. Specifically, the target states are chosen as:

  * •

the $N$-qubit states

$\rho=p|\psi\rangle\langle\psi|+\frac{1-p}{d}\mathbb{I}$ (19)

with $0\leq p\leq 1$ and $d=2^{N}$. And the pure state $|{\psi}\rangle$ ranges
from the product state

$|{\psi}\rangle_{P}={\left({\frac{{|{0}\rangle+|{1}\rangle}}{{\sqrt{2}}}}\right)^{\otimes
N}},$ (20)

to genuinely entangled GHZi and W states respectively, i.e.,

$\displaystyle|{\psi}\rangle_{G}$
$\displaystyle=\frac{{{{|{0}\rangle}^{\otimes N}}+i{{|{1}\rangle}^{\otimes
N}}}}{{\sqrt{2}}},$ (21) $\displaystyle|{\psi}\rangle_{W}$
$\displaystyle=\frac{1}{{\sqrt{N}}}(|{100\dots}\rangle+\dots+|{\dots
001}\rangle).$ (22)

  * •

the $N$-qubit states randomly generated with exponential decay of eigenvalues
and fixed state purity [23].

  * •

the $N$-qubit pure state $|{\psi}\rangle$ passing through the depolarizing
channel

$\rho=(1-\lambda)|\psi\rangle\langle\psi|+\frac{\lambda}{d}\mathbb{I}$ (23)

where the noise strength $0\leq\lambda\leq 1$, and the pure state
$|{\psi}\rangle$ from the product, GHZi, and W states. Note that although this
class is equivalent to Eq. (19), the target states used to benchmark the QST
methods are distinct in the sense that the former are pure states while the
latter chooses states after the depolarizing channel.

Numerical experiments are then performed to study the following problems:

  1. P1.

How different state decomposition and/or projection methods affect the
performance of the NN-QST.

  2. P2.

How the state purity affects the tomography fidelity of the NN-QST using
different state-mapping methods.

  3. P3.

How much advantage in the convergence time does the NN-QST provide, in
comparison to previous QST algorithms.

  4. P4.

How robust towards statistical and depolarizing noise are the NN-QST and other
QST algorithms.

###  4.1 Experimental setups

Key settings in the FNN are given below:

  * •

The Leaky rectified linear units activation function is used in the hidden
layer.

  * •

The Rprop optimizer is used and the learning rate is fixed at 0.001.

And the single-qubit POVM is chosen as
${\left\\{{{M_{a}}=\frac{1}{4}\left({\mathbb{I}}+{{\bf{s}}_{a}}\cdot{\boldsymbol{\sigma}}\right)}\right\\}_{a\in\left\\{{0,1,2,3}\right\\}}}$
with

$\displaystyle{{\bf{s}}_{0}}$
$\displaystyle=\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right),{{\bf{s}}_{1}}=\left(-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right),$
(24) $\displaystyle{{\bf{s}}_{2}}$
$\displaystyle=\left(-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right),{{\bf{s}}_{3}}=\left(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right)$

where
${\boldsymbol{\sigma}}=({{\bf{\sigma}}_{1}},{{\bf{\sigma}}_{2}},{{\bf{\sigma}}_{3}})$
are the Pauli operator vector. Evidently, we have $K=4$. Note that our method
can be easily applied to the case involving multiple POVMs on each qubit and
adaptive local measurements. Here, in order to save computational cost and to
easily compare with other QST algorithms, we are restricted to the above fixed
tetrahedral POVM.

All numerical experiments are run on the laptop with a single Intel(R)
Core(TM) i5 CPU @ 2.50GHz with 16GB RAM, and a single NVIDIA GeForce GTX 1070
GPU with 8GB RAM. The traditional MLE-based QST algorithms are based on the
Matlab platform and uses the CPU to accelerate operations, while the NN-QST is
based on the Pytorch platform and uses GPU to accelerate operations. The codes
are available at github:QST-NNGMs-FNN [43].

###  4.2 Fast convergence of the unified state-mapping methods

Figure 2: The fidelity of the NN-QST with ten different mapping methods as a
function of run time, on 10-qubit states. For each mapping method, experiments
are implemented with (a) 50 random mixed states with exponential decay of
eigenvalues initialized by uniformly distributed purity, and (b-d) 10 random
mixed states of product, W, and GHZi states initialized by uniformly
distributed $p$, respectively. The solid lines show the mean fidelity over
states and the black dashed line represents the fidelity of 1. The number of
iterations is fixed at 1000.

We focus on the following ten mapping methods:

  * •

The decomposition method $\rm Chol_{\Delta}$ with the complex lower triangular
matrix (4).

  * •

The $\mathcal{F}[\cdot]$ and $\mathcal{S}[\cdot]$ projection methods (7).

  * •

The simple unified method $\rm Chol_{H}$ (14).

  * •

The unified method based the $P$-order absolute projection
$\mathcal{A[\cdot]}_{P}$ (17), with $P=0.5,1,1.5,2,3,4$,

and test how the choice of state-mapping methods affects the convergence speed
of the NN-QST.

As displayed in Fig. 2, the numerical results show our unified methods $\rm
Chol_{H}$ and $P$-order absolute projection $\mathcal{A[\cdot]}_{P}$ with
$P=2$ admits the advantages in convergence speed and tomography accuracy, as
both can converge to the perfect fidelity of 1 within 100 seconds on all
tested states. Indeed, these two methods are mathematically identical, whereas
it is easier to calculate the state fidelity with $\rm Chol_{H}$, thus leading
to a faster convergence. Although the decomposition method $\rm Chol_{\Delta}$
has the fastest convergence speed in the early stage, it achieves a worse
state fidelity of less than 1 in the final estimate process. During the
training process within 100 seconds, the projection method
$\mathcal{F}[\cdot]$ only reconstructs the states with exponentially decaying
eigenvalues with fidelity $1$, while $\mathcal{S}[\cdot]$ fails for all
states.

It is interesting to point out that as $P$ increases, the performance of the
$\mathcal{A[\cdot]}_{P}$ projections on average tends to become better before
$P\leq 2$ and then worse after $P\geq 2$. Hence, the critical point is $P=2$.
However, it is also found that in Fig. 2(c) the $\mathcal{A[\cdot]}_{P}$ with
$P=1.5$ performs better on the W states and in Fig. 2(d) the unified
projection with $P=1$ and $1.5$ on the GHZi states. Hence, the flexible $P$ in
the unified state-mapping method (17) can have a better performance in
reconstructing different states.

###  4.3 State purity robustness of the unified state-mapping methods

The state purity is an important property of the state, encoding the
structural information of the state. Thus, we further analyze the tomography
accuracy of different state-mapping methods when the states with a wide range
of purity are considered.

Figure 3: The tomography fidelity of the NN-QST with ten mapping methods
versus state purity on 10-qubit states. The purity of the tested state $\rho$
is ${\rm Tr}\left[{{\rho^{2}}}\right]$, with the range being from 0 to 1 (the
purity of the pure state is 1). For each mapping method, experiments are
implemented with (a) 50 random mixed states with exponential decay of
eigenvalues initialized by uniformly distributed purity, and (b-d) 10 random
mixed states of product, W, and GHZi states initialized by uniformly
distributed $p$, respectively, of which the purity corresponds to $p$ is
$(1-1/2^{10})p^{2}+1/2^{10}$. The number of iterations is fixed at 1000.

The NN-QST with ten state-mapping methods are experimented on 10-qubit states
generated with random purity. As shown in Fig. 3, our unified methods $\rm
Chol_{H}$ and $\mathcal{A[\cdot]}_{P}$ with $P=2$ have excellent robustness
against state purity in the sense that both can achieve the perfect fidelity
on states of various purities, while other methods only work well for special
states with narrow range of purity and are sensitive to the varying state
purity. Moreover, for the $\rm Chol_{\Delta}$ decomposition and
$\mathcal{F}[\cdot]$ projection, their quantum state fidelity is positively
correlated with purity, except for mixed states with exponentially decaying
eigenvalues, and the purity has a great effect on the final fidelity of the
projection $\mathcal{S}[\cdot]$. It is also noted that for the
$\mathcal{A[\cdot]}_{P}$ projection method, there is the negative correlation
between the fidelity and purity at $P\leq 2$ and positive correlation at
$P\geq 2$. Consequently, it becomes insensitive to the state purity when $P$
is around 2. This tendency further suggests that if the purity information is
known, then we use $\mathcal{A[\cdot]}_{P}$ with smaller $P$ for low-purity
states and larger $P$ for high-purity states. And, if purity is an unknown
parameter, then $\mathcal{A[\cdot]}_{2}$ is a good choice. This is the reason
that we choose $\rm Chol_{H}$ as the default state-mapping method for the NN-
QST.

###  4.4 Fast convergence of the NN-QST on a large number of qubits

Figure 4: The convergence time for the NN-QST ($\rm Chol_{H}$, red), CG-APG
(blue), and iMLE (orange) implementations to reach fidelity of 0.99 versus the
number of qubits. For each algorithm, experiments are implemented with (a) 50
random mixed states with exponential decay of eigenvalues initialized by
uniformly distributed purity, and (b-d) 50 random mixed states of product, W,
and GHZi states initialized by uniformly distributed $p$, respectively. A data
point is obtained by averaging the convergence time over 50 states and the
shaded area represents the standard deviation around the mean. The CG-APG and
iMLE algorithms are only performed up to 9-qubit experiments due to the huge
time cost, and the NN-QST performs up to 11-qubit experiments due to GPU
memory usage. The number of iterations is fixed at 1000.

We then test how fast our neural network and other QST algorithms can
tomography various states with fidelity $0.99$. In particular, it is shown in
Fig. 4 that the NN-QST outperforms over other two algorithms in the
convergence time for the states of above $7$ qubits. Even for the states with
a small number of qubits, our method can accomplish the tomography task within
1 minute, close to those of other two algorithms. Note that the CG-APG and
iMLE algorithms are only tested up to 9-qubit experiments due to the huge time
cost for a larger number of qubits, while the NN-QST still can. Surprisingly,
it is able to achieve the tomography fidelity $0.99$ for 11-qubit mixed states
within 100 seconds. Thus, the network-based QST is more scalable to deal with
the multi-qubit system. Moreover, it is found that the NN-QST admits a better
performance on all tested states, further demonstrating the tomography
stability on a wide range of states.

###  4.5 Noise robustness of the NN-QST

All above experiments are implemented without noise, however, it is
impractical to be free from noise induced by finite measurements and
experimental environment. In this subsection, we consider the tomography
performance of three QST algorithms under two types of noise: statistical
noise and depolarizing noise. The statistical noise arises from the difficulty
in obtaining ideal measurement probabilities from a finite number of
measurements, and the depolarizing noise has a certain probability to change
target states into the maximally mixed state.

####  4.5.1 Statistical noise robustness

Figure 5: Tomography experiments of 8-qubit random mixed states with
exponential decay of eigenvalues in various sample sizes. We consider three
QST algorithms: the NN-QST ($\rm Chol_{H}$, red), CG-APG (blue), and iMLE
(orange). (a) The average fidelity of three QST algorithms as a function of
sample sizes. (b) The average time to reach fidelity of 0.99 or complete the
iteration as a function of sample sizes. (c) The fidelity of three QST
algorithms versus state purity where $10^{4}$, $10^{5}$, and $10^{6}$ samples
are used ($X\\_n$ represents that the number of samples used by the $X$
algorithm is $10^{n}$). All experiments in each sample size are performed 50
times with the uniformly distributed purity and the shaded area represents the
standard deviation around the mean. The black dashed line represents the
fidelity of 1. The number of iterations is fixed at 500.

We examine the effects of different sample sizes on the tomography performance
of three QST algorithms on 8-qubit random mixed states. All measurement
probabilities are calculated with one product-structured POVM and a certain
number of samples from sampling a multinomial distribution of measurement
probabilities.

First, it is natural to find in Fig. 5(a) that more samples improves the
tomography fidelity of all QST algorithms.The NN-QST achieves a higher average
fidelity than the CG-APG and iMLE algorithms when the same number of samples
is consumed. And, given the same high fidelity, it can save one to two orders
of sample resources. Then, Fig. 5(b) shows that the time consumption of the
NN-QST is two orders of magnitude less than other two algorithms, under the
same samples condition. These results indicate that the NN-QST completes
fidelity convergence on more states with increasing samples and a slow
decrease in time, while other two algorithms almost remain unchanged in time,
indicating that the convergence is completed in few states. Finally, it is
demonstrated that the tomography fidelity is positively correlated with state
purity, as illustrated in Fig. 5(c). Given a fixed state purity, the NN-QST
achieves a higher fidelity than other algorithms under the same number of
samples. And the NN-QST achieves a fidelity of 0.99 on states of wider purity
as the number of samples increases, while other algorithms can not reach
fidelity of 0.99 on all states.

####  4.5.2 Depolarizing noise robustness

Figure 6: The fidelity of three QST algorithms on 8-qubit states as a function
of noise strength $\lambda$. Here, we consider three sample sizes of $10^{4}$,
$10^{5}$, and $10^{6}$ ($X\\_n$ represents that the number of samples used by
the $X$ algorithm is $10^{n}$). For each sample size, experiments are with
(a-c) 50 pure states with a uniformly distributed $\lambda$ of the product, W,
and GHZi states, respectively. The black dotted line is the ideal fidelity
$1-(1-1/2^{8})\lambda$, if the QST algorithm perfectly reconstructs the mixed
state disturbed by depolarizing noise. The number of iterations is fixed at
500.

Figure 7: The time to reach a fidelity close to the ideal fidelity or complete
the iteration as a function of noise strength $\lambda$. We consider three
sample sizes of $10^{4}$, $10^{5}$, and $10^{6}$ ($X\\_n$ represents that the
number of samples used by the $X$ algorithm is $10^{n}$). For each sample
size, experiments are with (a-c) 50 pure states with a uniformly distributed
$\lambda$ of the product, W, and GHZi states, respectively. The number of
iterations is fixed at 500.

The tomography performances of three algorithms on 8-qubit pure states under
different depolarizing noise strength are then investigated. The measured
probability is obtained by measuring the pure states after the depolarizing
channel, with the sample sizes of $10^{4}$, $10^{5}$, and $10^{6}$. The
presence of depolarizing noise reduces the tomography fidelity for all
algorithms, as depicted in Fig. 6 where the black dashed line describes the
ideal fidelity with respect to the noise strength $\lambda$. For the product
and W states, the NN-QST can have a tomography fidelity, which is closer to
the ideal than other algorithms with a $\lambda$. And all of them performs
well on the GHZi state. It is noted that the increased number of samples makes
the tomography fidelity of each algorithm closer to the ideal fidelity for
each $\lambda$.

Furthermore, the NN-QST takes one or two orders of magnitude less time than
other algorithms to achieve a fidelity close to the ideal fidelity (plus or
minus 0.005) or finish the iteration under various $\lambda$, as shown in Fig.
7. Again, for the product and W states, the NN-QST achieves the near-ideal
fidelity faster than other algorithms. On the GHZi state, the three algorithms
are not close to the ideal fidelity except for complete noise strength
($\lambda$=1), and the time consumption of the NN-QST is lower to maintain the
same degree of closeness. Thus, the effect of depolarizing noise on the NN-QST
is weaker than on other QST algorithms.

##  5 Conclusion and Outlook

We have presented the NN-QST which utilizes the unified state-mapping
technique (17), to achieve the ultrafast reconstruction of multi-qubit states.
In comparison to various state-mapping strategies and fast QST methods,
numerous numerical results shows that our method has a fast convergence speed
to achieve a high tomography fidelity and a great robustness to the state
purity and noise. It is demonstrated that we are able to accomplish the
tomography task of 11-qubit states within 2 minutes on a laptop. Moreover, it
has been found that with respect to the NN-QST, reconstructing low-purity
states needs more samples to reach a given fidelity and the depolarizing noise
linearly reduces its tomography fidelity.

There are many interesting questions left for the future work. For example,
more than one-single POVM and/or adaptive measurements could be used to
improve the reconstruction fidelity for a more wide range of multi-qubit
states. And generative models could further be introduced to enhance the
learnability of the neural network and then to improve the NN-QST. Finally, it
would be interesting to test our method on real experimental setups.

## Acknowledgments

We thank Dr. Changkang Hu for useful discussions. This research is supported
by the National Natural Science Foundation of China (61903027, 72171172); the
National Key R&D Program of China (2018YFE0105000, 2018YFB1305304); the
Shanghai Municipal Science and Technology Major Project (2021SHZDZX0100); the
Shanghai Municipal Commission of Science and Technology (1951113210,
19511132101); the Shanghai Municipal Science and Technology Fundamental
Project (21JC1405400); and Shanghai Gaofeng & Gaoyuan Project for University
Academic Program Development (22-8).

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##  Appendix A NN-QST Based on CNN

Figure 8: Schematic diagram of the CNN structure for the NN-QST in the 6-qubit
experiments. The network consists of the convolution, max-pooling, and fully
connected layers. The input to the network comes from the probability
distributions obtained by measuring each qubit using one POVM with a product
structure, and the output is mapped into a density matrix.

Figure 9: The convergence time for the FNN (red) and CNN (orange)
implementations to reach fidelity of 0.99 versus the number of qubits. For
each network, experiments are implemented with (a) 50 random mixed states with
exponential decay of eigenvalues initialized by uniformly distributed purity,
and (b-d) 50 random mixed states of product, W, and GHZi states initialized by
uniformly distributed $p$, respectively. A data point is the average
convergence time over 50 states and the shaded area represents the standard
deviation around the mean. The number of iterations is fixed at 1000.

To explain the reason for choosing the FNN in Sec. 3.1, here, we replace the
FNN with a CNN with nearly the same amount of parameters to perform the NN-
QST, and other experimental setups such as the negative log-likelihood loss
function, Rprop optimizer with 0.001 learning rate, simple unified state-
mapping strategy $\rm Chol_{H}$, and one product-structured POVM, etc. are
consistent with Sec. 4.4.

The CNN structure for the 6-qubit experiments used here is shown in Fig. 8,
the CNN structure presented below is extended to $N$ qubits for full
measurements. The number of full measurements is $K^{N}$, where $K$ is the
number of elements of a single-qubit POVM, which is 4 here. To satisfy the
convolution operation, we convert the input probability distributions to
matrix form. The convolution layer has a kernel of size 3$\times$3, stride
length of 1, padding length of 1, 10 feature maps, and a rectified linear unit
(ReLU) activation function. The max-pooling layer has a kernel of size
$2^{N-3}\times 2^{N-3}$. The fully connected layer has an input layer of 640
neurons, a hidden layer of 4$N$ neurons, and an output of size $K^{N}$. The
input and output of the CNN are consistent with the FNN in Sec. 3.1.

To illustrate the performance difference between the FNN and CNN, we compare
the convergence time of the two mapping networks to achieve fidelity of 0.99
on various qubits. It is shown in Fig. 9 that the FNN outperforms over the CNN
for the most states of above 8 qubits. Even for the states with a small number
of qubits, the FNN can accomplish the tomography task within 1 minute, close
to those of the CNN. Thus, the FNN is more suitable to deal with the large-
qubit system.

Considering the tomographic requirements of large qubits and the loss of
scalability by using the CNN to transform one-dimensional input data, we
finally chose a 3-layer FNN as the mapping network. And, the architecture of
the CNN guarantees that a smaller number of parameters can be maintained under
the same tomographic performance, which will be exploited in future research.