# Scalable estimation of pure multi-qubit states

L. Pereira<EMAIL_ADDRESS>Instituto de Física Fundamental IFF-CSIC,
Calle Serrano 113b, Madrid 28006, Spain  L. Zambrano  Instituto Milenio de
Investigación en Óptica y Departamento de Física, Universidad de Concepción,
casilla 160-C, Concepción, Chile  A. Delgado  Instituto Milenio de
Investigación en Óptica y Departamento de Física, Universidad de Concepción,
casilla 160-C, Concepción, Chile

###### Abstract

We introduce an inductive $n$-qubit pure-state estimation method. This is
based on projective measurements on states of $2n$+1 separable bases or $2$
entangled bases plus the computational basis. Thus, the total number of
measurement bases scales as $O(n)$ and $O(1)$, respectively. Thereby, the
proposed method exhibits a very favorable scaling in the number of qubits when
compared to other estimation methods. Monte Carlo numerical experiments show
that the method can achieve a high estimation fidelity. For instance, an
average fidelity of $0.88$ on the Hilbert space of $10$ qubits is achieved
with 21 separable bases. The use of separable bases makes our estimation
method particularly well suited for applications in noisy intermediate-scale
quantum computers, where entangling gates are much less accurate than local
gates. We experimentally demonstrate the proposed method in one of IBM’s
quantum processors by estimating a 4-qubit Greenberger-Horne-Zeilinger states
with a fidelity close to $0.875$ via separable bases. Other 10-qubit separable
and entangled states achieve an estimation fidelity in the order of $0.85$ and
$0.7$, respectively.

###### pacs:

Valid PACS appear here

##  I Introduction

In recent years, great advances have been achieved in the development of
technologies based on quantum systems. These quantum technologies aim to
provide significant performance improvements compared to their classical
counterparts.

A contemporary example is quantum computing, where large numbers of two-
dimensional quantum systems (qubits) are controlled via canonical sets of
operations to solve certain computational tasks more efficiently than
classical computers [1, 2]. Several quantum systems, such as trapped ions [3,
4, 5, 6], superconducting circuits [7, 8, 9, 10], quantum dots [11] and
photonic platforms [12, 13, 14, 15], among others, have been proposed and
tested to encode qubits and implement quantum gates, in order to build quantum
simulators and quantum computers. Today, private companies provide cloud-based
access to their quantum hardware and software.

Currently, the above experimental platforms, among others, have evolved into
devices that have been described as noisy intermediate-scale quantum (NISQ)
technologies [16]. These are devices that operate with imperfect gates on a
number of qubits between 50 and up to a few hundred qubits. These devices are
beyond the simulation capabilities of current classical computing devices but
are still far from exhibiting a clear enough improvement in useful computing
power. In particular, noisy quantum gates and decoherence limit the depth of
the circuits and prevent the realization of complex algorithms [17, 18, 19].

Because of the properties of NISQ computers, the design of methods for the
characterization, certification, and benchmarking of these quantum devices has
become increasingly relevant and difficult [20]. For instance, the
characterization of states generated by this class of devices requires
measurements on a large set of bases. These are physically implemented by
applying sets of universal unitary transformations on the qubits followed by
measurements on the computational basis. Typically, an effort to reduce the
number of bases lead to a new set of bases that require the application of
entangling unitary transformations, such as, for instance, the controlled NOT
(CNOT) gate , whose implementation is associated to a large error.
Consequently, the estimation of states generated by NISQ devices could be
affected by a large build up of inaccuracy in the measurement procedure.

Several methods have been proposed to assess the performance of NISQ devices.
Randomized benchmarking [21, 22, 23] and direct fidelity estimation [24, 25]
are popular tools for this purpose. However, they do not give complete
information about the device in question. In contrast, quantum state
estimation [26] gives plenty of information about the device since it fully
determines the state of the system from a set of suitable chosen measurements.

Today, several protocols to perform quantum state estimation of
$d$-dimensional quantum systems are available. These attempt to estimate the
$d^{2}-1$ real parameters that characterize density operators. Standard
quantum tomography is based upon the measurement of the $d^{2}-1$ generalized
Gell-Mann matrices [27, 28, 29]. Quantum state estimation based on projections
onto the states of $d+1$ mutually unbiased bases has been suggested [30, 31,
32, 33, 34, 35, 36] to reduce the total number of measurement outcomes. This
number can be reduced to a minimum using a symmetric, informationally
complete, positive operator-valued measure [37, 38, 39, 40, 41, 42, 43], which
has exactly $d^{2}$ measurement outcomes.

Despite the progress made in the reduction of the total number of measurement
outcomes used by quantum state estimation methods, the exponential increase of
this number prevails in the case of multipartite systems. This is known as the
_curse of dimensionality_. For example, standard quantum tomography for a
$n$-qubit system requires the measurement of $3^{n}$ local Pauli settings. A
solution to overcome this problem is the use of a priori information about the
state to be determined. Compressed sensing [44, 45] allows a large reduction
in the total number of measurement as long as the rank of the state is known.
Another alternative is to consider states with special properties, such as,
for instance, matrix product states [46] or permutationally invariant states
[47], which also leads to a significant reduction in the total number of
measurements. The important case of pure states has also been studied. In this
case a set of five bases allows one to estimate with high accuracy all pure
quantum states [48, 49, 50]. Recently, a 3-bases-based tomographic scheme has
been introduced to estimate a single qudit [51]. The generalization of this
method to multiple qubits leads to entangled bases. Adaptivity has also been
introduced as a means of increasing the estimation accuracy of tomographic
methods [52, 53, 54, 55, 56] and even reaching fundamental accuracy limits
[57, 58].

Here, we present a method to estimate unknown pure quantum states in $n$-qubit
Hilbert spaces, as used by NISQ computers, by means of $mn+1$ local bases
(with $m$ an integer number), which correspond to the tensor product of $n$
single-qubit bases, or by the computational basis plus $m$ entangled bases,
that is, bases that cannot be decomposed as the tensor product of single-qubit
bases. Thereby, the number of bases scales linearly as $O(mn)$ and $O(m)$,
respectively, with the number $n$ of qubits. This is a great advantage over
standard methods, which are of exponential order. The $mn+1$ local bases also
provide an advantage in the estimation of states generated via NISQ devices,
such as current prototypes of quantum processors, because in this case
projective measurements can be carried out without the use of typically noisy
entangling gates. The computational basis together with $m=2$ entangled bases
also lead to a clear reduction with respect to the 5-bases tomographic method
[48, 49]. The method here proposed also improves over the 3-bases method,
since this requires to calculate the likelihood function of $2^{2^{n}-1}$
states to obtain an estimate. For a large number of qubits, this stage
exponentially increases the computational cost of the method. Furthermore, in
order to obtain distinguishable values of the likelihood, a large ensemble
size is required. Our estimation method does not require such a procedure.

The present method estimates pure multi-qubit quantum states using an
inductive process. For an arbitrary $n$-qubit state $|\Psi\rangle$ we define
$n$ sets of $2^{n-j}$ reduced states of dimension $2^{j}$, with $j=1,2,...,n$
and with the reduced state of dimension $2^{n}$ equal to $|\Psi\rangle$ except
a global phase. We first show how to estimate the reduced states of dimension
$2$ and then show that the knowledge about reduced states of dimension
$2^{j-1}$ together with some measurement outcomes allows one to estimate the
reduced states of dimension $2^{j}$. This points to an inductive estimation
method: we first estimate the reduced states of dimension $2$, and in $n$
iterations we arrive to the reduced state of dimension $2^{n}$, completing the
estimation procedure. We show that the set of reduced states can be estimated
by projecting the state onto $mn+1$ local bases or $m$ entangled bases plus
the computational basis. Throughout the proposed estimation method, it is
necessary to solve several systems of linear equations that might have
vanishing determinants if the number of measurements is insufficient, leading
to the failure of the method. However, the presence of noise, such as ,finite
statistics, helps to mitigate this problem. We also show that a small increase
in the number of local or entangled bases solves this problem and improves the
overall fidelity of the estimation.

We study the present method through Monte Carlo numerical simulations. We
randomly generate, according to a Haar-uniform distribution, a set of unknown
pure states and calculate the average estimation fidelity as a function of the
number of qubits. Projective measurements on each basis are simulated
considering a fixed number of repetitions. We show that $2n+1$ separable bases
achieve an average estimation fidelity of 0.88 for $n=10$. We improve this
figure by considering larger number of bases, that is, $3n+1$ and $4n+1$,
which lead to an average estimation fidelity of 0.915 and 0.93 for $n=10$,
respectively. If the average estimation fidelity is calculated with respect to
the set of separable states, then for the above number of bases we obtain the
values $0.95,0.955$ and $0.96$, correspondingly. The use of entangled bases
leads to a less favorable picture, where $2,3$ and $4$ entangled bases plus
the computational basis lead to an average estimation fidelity of 0.2, 0.6 and
0.8 for $n=6$, respectively. These figures increase when estimating separable
states, where we achieve an average estimation fidelity of 0.75, 0.95 and 0.95
for $2,3$ and $4$ entangled bases plus the computational base, respectively.
The large gap between separable and entangled bases can be explained by
recalling that measurements are simulated with a fixed number of repetitions
or, equivalently, with a fixed ensemble size. Thereby, the total ensemble size
is much smaller in the case of the entangled bases, which decreases the
estimation fidelity.

We test the proposed method by means of experiments carried out using the
quantum processor _ibmq_manhattan_ provided by IBM. We consider the estimation
of fixed quantum states from 2 to 10 qubits using the local bases, and states
up to 4 qubits using the entangled bases. We first estimate two completely
separable states. One of these states is such that the matrices to be inverted
are ill-conditioned. Nevertheless, the estimation of these states via local
bases lead to very similar fidelities above 0.83 for $n=10$. The estimation
via entangled bases leads to fidelities in the interval $[0.1,0.3]$ for $n=4$.
We also consider the estimation of an $n/2$-fold tensor product of a Bell
state, where local bases for $n=10$ lead to a fidelity above $0.67$ and
entangled bases lead to a fidelity below 0.5 for $n=10$. Finally, we test the
estimation of a Greenberger-Horne-Zeilinger state for 2, 3 and 4 qubits. The
proposed method leads to en estimation fidelity of approximately 0.87 for
$n=4$ via local bases while in the case of entangled bases the fidelity is
approximately 0.81, 0.53 and 0.47 for 2, 3 and 4 entangled bases plus the
computational base, respectively. The large decrease in the estimation of
entangled states by means of entangled bases can be explained by the use of
low-accuracy entangling gates in the preparation of the states as well in the
projection to the states of the entangled bases.

Our simulations and experimental results indicate that the method proposed
here allows the efficient and scalable estimation of $n$-qubit pure states by
means of $mn+1$ separable bases in NISQ computers, while the $m$ entangled
bases together with the computational base can offer a large advantage in
quantum computers based on high accuracy entangling gates.

This article is organized as follows: in Section II we present the method and
its properties. In Section III we show the results of several Monte Carlo
simulations aimed at studying the overall behavior of the fidelity as a
function of the number of qubits. In Section IV we present the results of
implementing the proposed estimation method in IBM’s quantum processors. In
Section V we summarize and conclude.

##  II Method

Let us consider a $n$-qubit system described by the pure state

$\displaystyle|\Psi\rangle=\sum_{\alpha=0}^{2^{n}-1}c_{\alpha}e^{i\phi_{\alpha}}|\alpha\rangle_{n},$
(1)

where $c_{\alpha}\geq 0$ and
$|\alpha\rangle_{n}=|\alpha_{n-1}\rangle\otimes\cdots\otimes|\alpha_{0}\rangle$
is the $n$-qubit computational basis, with
$\alpha=\sum_{k=0}^{n-1}2^{k}\alpha_{k}$ the integer associated with the
$n$-bit binary string $\alpha_{n-1}\cdots\alpha_{0}$.

Our main aim is to estimate the values of the amplitudes $\\{c_{\alpha}\\}$
and the phases $\\{\phi_{\alpha}\\}$ with a total number of measurements that
does not scale exponentially with the number $n$ of qubits. For this purpose,
we employ an iterative algorithm based on estimating reduced states, which are
non-normalized $j$-qubit vectors defined by

$\displaystyle|\Psi_{\beta}^{j}\rangle=\sum_{\alpha=0}^{2^{j}-1}{}_{n}\langle
2^{n-j}\beta+\alpha|\Psi\rangle|\alpha\rangle_{j},$ (2)

with $1\leq j\leq n$ and $0\leq\beta\leq 2^{n-j}-1$. Thus, for a given state
$|\Psi\rangle$ we have $n$ sets of $2^{n-j}$ reduced states, one set for each
$j$ and one reduced state for each $\beta$.

Every reduced state has some partial information about the full state. For
instance, if $j=1$ we have $2^{n-1}$ reduced states that are given by

$\displaystyle|\Psi_{\beta}^{1}\rangle=c_{2\beta}e^{i\phi_{2\beta}}|0\rangle+c_{2\beta+1}e^{i\phi_{2\beta+1}}|1\rangle.$
(3)

We can see that the amplitudes and the phases are the same entering in the
state $|\Psi\rangle$. Thereby, we can reduce the problem of estimating the
unknown state $|\Psi\rangle$ to the problem of estimating the set of reduced
states. The main obstacle of this approach is that measurements in quantum
mechanics do not contain information about global phases, and hence, if we try
to find every reduced state in an independent way using the results of some
measurements, we can only obtain the reduced states $|\Psi_{\beta}^{j}\rangle$
except a global phase, which in this case, would be a relative phase in
$|\Psi\rangle$. Thus, the only reduced state that fully characterizes the
system is $|\Psi_{0}^{n}\rangle$, which is the same as $|\Psi\rangle$.
Nevertheless, the remaining reduced states will also be useful in the task of
reconstructing $|\Psi\rangle$, because we can find the reduced states
$|\Psi_{\beta}^{j}\rangle$ (except a global phase) using the reduced states
$|\Psi_{\beta}^{j-1}\rangle$, as we will show.

For a large enough set of identical copies of the state $|\Psi\rangle$,
measurements in the computational base $\\{|\alpha\rangle\\}$ gives us a
histogram of observations such that we have approximations $p_{\alpha}$ to
$|\langle\alpha|\Psi\rangle|^{2}$. Then the amplitudes can be estimated as
$c_{\alpha}=\sqrt{p_{\alpha}}$. On the other hand, the phases can be
determined from projective measurements on the following states

$\displaystyle|P_{a\beta}^{j}\rangle=|\beta\rangle_{n-j}\otimes|+_{a}\rangle\otimes|-_{a}\rangle^{\otimes
j-1},$ (4)

where $|+_{a}\rangle=u_{a}|0\rangle+v_{a}e^{i\varphi_{a}}|1\rangle$ and
$|-_{a}\rangle=v_{a}|0\rangle-u_{a}e^{i\varphi_{a}}|1\rangle$ are orthonormal
single-qubit states, with $a=1,\dots,m$. Here, $m$ is the number of bases
$\\{|\pm_{a}\rangle\\}$ considered, which must be large enough to carry out
the algorithm. In addition, these bases have to be all different from the
computational one.

To start the algorithm, we set $j=1$ and the reduced states are given by Eq.
(3). The probabilities of projecting $|\Psi\rangle$ onto
$|P^{1}_{a\beta}\rangle$ are given by

$\displaystyle P^{1}_{a\beta}=$ $\displaystyle
u_{a}^{2}c_{2\beta}^{2}+v_{a}^{2}c_{2\beta+1}^{2}$
$\displaystyle+2u_{a}v_{a}c_{2\beta}c_{2\beta+1}[\cos(\varphi_{a})\cos(\phi_{2\beta+1}-\phi_{2\beta})$
$\displaystyle+\sin(\varphi_{a})\sin(\phi_{2\beta+1}-\phi_{2\beta})].$ (5)

These generate a set of equations, one equation for each value of $\beta$,
that are linear combinations of the cosine and sine of the relative phase
$\delta\phi_{\beta}=\phi_{2\beta+1}-\phi_{2\beta}$, with coefficients
depending on $|+_{a}\rangle$. Thereby, we can form a linear system of
equations $L_{\beta}\vec{a}_{\beta}=\vec{b}_{\beta}$ for trigonometric
functions of the relative phase, which is explicitly given by

$\displaystyle\begin{bmatrix}\cos\varphi_{1}&\sin\varphi_{1}\\\
\vdots&\vdots\\\
\cos\varphi_{m}&\sin\varphi_{m}\end{bmatrix}\begin{bmatrix}\cos\delta\phi_{\beta}\\\
\sin\delta\phi_{\beta}\end{bmatrix}=\begin{bmatrix}\tilde{P}^{1}_{1\beta}\\\
\vdots\\\ \tilde{P}^{1}_{m\beta}\end{bmatrix},$ (6)

where we have defined

$\displaystyle\tilde{P}_{a\beta}^{1}=\frac{1}{2u_{a}v_{a}c_{2\beta}c_{2\beta+1}}(P_{a\beta}^{1}-u_{a}^{2}c_{2\beta}^{2}-v_{a}^{2}c_{2\beta+1}^{2}).$
(7)

Since we know the projectors $|P^{1}_{a\beta}\rangle$ and the coefficients
$c_{j}$, this system can be solved for $m\geq 2$ inverting $L_{\beta}$ by the
Moore-Penrose pseudo-inverse $A^{+}=(A^{T}A)^{-1}A^{T}$. We can guarantee that
the pseudo-inversion is possible by suitably choosing the coefficients
$u_{a}$, $v_{a}$ and the phases $\varphi_{a}$ entering in the states
$|P_{a\beta}^{1}\rangle$. For example, taking the particular case of $m=2$ the
system has a solution

$\displaystyle\begin{bmatrix}\cos\delta\phi_{\beta}\\\
\sin\delta\phi_{\beta}\end{bmatrix}=\frac{1}{\det(L_{\beta})}\begin{bmatrix}\sin\varphi_{2}&-\sin\varphi_{1}\\\
-\cos\varphi_{2}&\cos\varphi_{1}\end{bmatrix}\begin{bmatrix}\tilde{P}^{1}_{1\beta}\\\
\tilde{P}^{1}_{2\beta}\end{bmatrix}$ (8)

as long as the determinant of $L_{\beta}$ does not vanish, that is,
$\det(L_{\beta})=\cos(\varphi_{1})\sin(\varphi_{2})-\sin(\varphi_{1})\cos(\varphi_{2})\neq
0$. Thereby, taking $\varphi_{1}=0$ and $\varphi_{2}=\pi/2$, or equivalently
$|\pm_{1}\rangle=u_{1}|0\rangle\pm v_{1}|1\rangle$,
$|\pm_{2}\rangle=u_{2}|0\rangle\pm iv_{2}|1\rangle$, the system of equations
can be always solved since $\det(L_{\beta})=1$, and the exponential of the
relative phase is given by
$e^{i\delta\phi_{\beta}}=\tilde{P}^{1}_{1\beta}+i\tilde{P}^{1}_{2\beta}$.
Therefore, the reduced states $|\Psi_{\beta}^{1}\rangle$ can be determined up
to a global phase,

$\displaystyle|\tilde{\Psi}_{\beta}^{1}\rangle=c_{2\beta}|0\rangle+c_{2\beta+1}e^{i\delta\phi_{\beta}}|1\rangle,$
(9)

provided we measure the computational basis and at least two projectors
$|P_{a\beta}^{1}\rangle$ for every $\beta$ in $0,1,...,2^{n-1}-1$.
Nevertheless, when some of the computational basis probabilities $p_{\alpha}$
are null, we have to consider a particular rule to obtain the reduced state.
If any or both of the coefficients $c_{2\beta}=\sqrt{p_{2\beta+1}}$ or
$c_{2\beta+1}=\sqrt{p_{2\beta+1}}$ are null, the reduced state is determined
only by the computational basis, setting
$|\tilde{\Psi}_{\beta}^{1}\rangle=c_{2\beta}|0\rangle$,
$|\tilde{\Psi}_{\beta}^{1}\rangle=c_{2\beta+1}|1\rangle$ or
$|\tilde{\Psi}_{\beta}^{1}\rangle=\vec{0}$ in Eq. (9). This means that there
may be null reduced states, which will have an impact on the algorithm in
posterior iterations.

In the following iterations, that is, the case $j>1$, the reduced states can
be expressed as linear combinations of the previous ones,

$\displaystyle|\tilde{\Psi}_{\beta}^{j}\rangle=|0\rangle\otimes|\tilde{\Psi}_{2\beta}^{j-1}\rangle+e^{i\delta\phi_{\beta}^{j}}|1\rangle\otimes|\tilde{\Psi}_{2\beta+1}^{j-1}\rangle,$
(10)

with
$|\tilde{\Psi}_{\beta}^{j}\rangle=e^{-i\phi_{2^{j}\beta}}|\Psi_{\beta}^{j}\rangle$
the corresponding reduced states up to global phase and
$\delta\phi_{\beta}^{j}=\phi_{2^{j}(\beta+1/2)}-\phi_{2^{j}\beta}$ relative
phases. Thus, assuming that we know the reduced states of the previous
iteration (except for a global phase), we can determine the reduced state of
the current iteration (except for a global phase) simply by determining the
relative phase. Analogously to the first iteration, if any or both of the
previous reduced states $|\tilde{\Psi}_{2\beta}^{j-1}\rangle$ or
$|\tilde{\Psi}_{2\beta+1}^{j-1}\rangle$ are null, there is no relative phase
to determine and the next reduced state is simply obtained with the choice
$\delta\phi_{\beta}=0$. Otherwise, we have to determine the relative phase
from the projections $|P_{\beta}^{j}\rangle$. Considering $j>1$, the
probability of projecting the state $|\Psi\rangle$ onto the state
$|P^{j}_{a\beta}\rangle$ is given by

$\displaystyle P^{j}_{a\beta}=$ $\displaystyle u_{a}^{2}|\langle
W_{a}^{j}|\tilde{\Psi}_{2\beta}^{j-1}\rangle|^{2}+v_{a}^{2}|\langle
W_{a}^{j}|\tilde{\Psi}_{2\beta+1}^{j-1}\rangle|^{2}$
$\displaystyle+2u_{a}v_{a}\Re[e^{i(\delta\phi_{\beta}^{j}-\varphi_{a})}\langle\tilde{\Psi}_{2\beta}^{j-1}|W_{a}^{j}\rangle\langle
W_{a}^{j}|\tilde{\Psi}_{2\beta+1}^{j-1}\rangle],$ (11)

where $|W_{a}^{j}\rangle=|-_{a}\rangle^{\otimes j-1}$. Defining the quantities

$\displaystyle
X_{a\beta}^{j}=e^{-i\varphi_{a}}\langle\tilde{\Psi}_{2\beta}^{j-1}|W_{a}^{j}\rangle\langle
W_{a}^{j}|\tilde{\Psi}_{2\beta+1}^{j-1}\rangle$ (12)

and

$\displaystyle\tilde{P}_{a\beta}^{j}=\frac{1}{2u_{a}v_{a}}\big{(}P_{a\beta}^{j}-u_{a}^{2}|\langle
W_{a}^{j}|\tilde{\Psi}_{2\beta}^{j-1}\rangle|^{2}-v_{a}^{2}|\langle
W_{a}^{j}|\tilde{\Psi}_{2\beta+1}^{j-1}\rangle|^{2}\big{)},$ (13)

we obtain the following system of equations for the relative phases

$\displaystyle\begin{bmatrix}\Re X_{1\beta}^{j}&-\Im X_{1\beta}^{j}\\\
\vdots&\vdots\\\ \Re X_{m\beta}^{j}&-\Im
X_{m\beta}^{j}\end{bmatrix}\begin{bmatrix}\cos\delta\phi_{\beta}^{j}\\\
\sin\delta\phi_{\beta}^{j}\end{bmatrix}=\begin{bmatrix}\tilde{P}_{1\beta}^{j}\\\
\vdots\\\ \tilde{P}_{m\beta}^{j}\end{bmatrix},$ (14)

where $\Re$ and $\Im$ stand for real and imaginary part, respectively. Again,
the phases can be obtained by solving a linear system of equations
$L_{\beta}^{j}\vec{a}_{\beta}^{j}=\vec{b}_{\beta}^{j}$ given by Eq. (14)
through the Moore-Penrose pseudo-inverse. If for a fixed $j$, we are able to
pseudo-invert the matrices $L_{\beta}^{j}$, we can find all the
$|\tilde{\Psi}_{\beta}^{j}\rangle$. Then, we continue inductively with $j+1$
until reaching $j=n$. In that case, the protocol ends because the reduced
state $|\tilde{\Psi}_{0}^{n}\rangle$ is equal to the state $|\Psi\rangle$ up
to a global phase.

One might think that, analogously to the case $j=1$, the matrix
$L_{\beta}^{j}$ can be inverted for $m\geq 2$. Thereby, to completely estimate
the unknown state $|\Psi\rangle$ we would need a minimum of $2(2^{n}-1)$
different projections onto states $|P_{a\beta}^{j}\rangle$ in (4), because
there are $2^{n}-1$ different reduced states. However, now the matrix
$L_{\beta}^{j}$ to be pseudo-inverted does not only depends on the projectors
$|P^{j}_{a\beta}\rangle$, but it also depends on the reduced states of the
previous iteration, so that we cannot guarantee that the pseudo-inversion is
feasible. For example, taking $m=2$, the solution of the system of equations
(14) is

$\displaystyle\begin{bmatrix}\cos\delta\phi_{\beta}^{j}\\\
\sin\delta\phi_{\beta}^{j}\end{bmatrix}=\frac{1}{\det(L_{\beta}^{j})}\begin{bmatrix}-\Im
X_{2\beta}^{j}&\Im X_{1\beta}^{j}\\\ -\Re X_{2\beta}^{j}&\Re
X_{1\beta}^{j}\end{bmatrix}\begin{bmatrix}\tilde{P}_{1\beta}^{j}\\\
\tilde{P}_{2\beta}^{j}\end{bmatrix},$ (15)

with $\det(L_{\beta}^{j})=\Im(X_{1\beta}^{j}[X_{2\beta}^{j}]^{*})$ the
determinant of $L_{\beta}^{j}$. Clearly, this solution is only valid when
$\det(L_{\beta}^{j})\neq 0$. A non-invertible system of equations for a
$|\tilde{\Psi}_{\beta}^{j}\rangle$ means that all the equations are linearly
dependent and that it is sufficient to consider a single equation $\Re
X_{\beta}^{j}\cos(\delta\phi_{\beta}^{j})-\Im
X_{\beta}^{j}\sin(\delta\phi_{\beta}^{j})=\tilde{P}_{\beta}^{j}$, from which
the trigonometric functions of the relative phase cannot be solved. However,
using the identity
$\cos^{2}(\delta\phi_{\beta}^{j})+\sin^{2}(\delta\phi_{\beta}^{j})=1$, we can
obtain the trigonometric functions except for their corresponding quadrant,
that is, we have two possible estimates or ambiguities for the reduced states.
Thus, in the worst case, where the matrix $L_{\beta}^{j}$ cannot be pseudo-
inverted for any reduced state, the algorithm leads to a finite set of
$2^{n-1}$ possible estimates of $|\Psi\rangle$. This problem can be avoided by
increasing the number of $m$ of bases for given values of $j$ and $\beta$
until the systems can be solved. An alternative solution is to consider an
extra adaptive measurement to discriminate between the ambiguities.
Furthermore, since the probabilities are experimentally estimated by means of
a sample of finite size, with high probability the system of equations can be
solved due the finite sample noise, although not necessarily with high
accuracy since $L_{\beta}^{j}$ might have a bad condition number,
$\textrm{cond}(A)=||A||\times||A^{+}||$, with $||\cdot||$ a matrix norm.

In order to apply our method based on the inductive estimation of reduced
states we need to find an efficient implementation of projections onto states
$|P_{a\beta}^{j}\rangle$. For this purpose we consider the measurement of two
different sets of bases. The first set of $n\times m$ bases are given by

$\mathcal{L}_{ab}=\left\\{|\beta\rangle_{n-b}\otimes|\pm_{a}\rangle\otimes\cdots\otimes|\pm_{a}\rangle\right\\},$
(16)

with $0\leq\beta\leq 2^{n-b}-1$. These bases can be implement via local gates,
as shown in Fig. 1. These bases define a larger set of projections than
required by the method. However, these extra projections can also be used in
the algorithm, since
$|\beta\rangle_{n-j}\otimes|\pm_{a}\rangle\otimes\cdots\otimes|\pm_{a}\rangle$
form a system of $2^{j}$ equations at the $j$th-iteration. The second set of
$m$ bases are given by

$\mathcal{E}_{a}=\left\\{|\beta\rangle_{n-j}\otimes|+_{a}\rangle\otimes|-_{a}\rangle^{\otimes
j-1},|-_{a}\rangle^{\otimes n}\right\\},$ (17)

with $1\leq j\leq n$ and $0\leq\beta\leq 2^{n-j}-1$. Despite projections onto
members of the bases are local, these observables are entangled since many
multi-controlled $U$ gates are needed to implement them, as shown in Fig. 1.
In a quantum computer, these gates can be decomposed in terms of CNOT gates
[60].

(a) Local basis.

(b) Entangled basis.

Figure 1: Quantum circuits to implement 1(a) the local measurements
$\\{\mathcal{L}_{ab}\\}$ and 1(b) the entangled measurements
$\\{\mathcal{E}_{a}\\}$. The single qubit gate $U_{a}$ fulfill
$|+_{a}\rangle=U_{a}|0\rangle$ and $|-_{a}\rangle=U_{a}|1\rangle$.

As example, let us consider a 2-qubit system in the state

$\displaystyle|\Psi\rangle=c_{0}e^{i\phi_{0}}|0\rangle_{2}+c_{1}e^{i\phi_{1}}|1\rangle_{2}+c_{2}e^{i\phi_{2}}|2\rangle_{2}+c_{3}e^{i\phi_{3}}|3\rangle_{2}.$
(18)

Omitting the Kronecker product for compactness, the corresponding reduced
states are

$\displaystyle|\tilde{\Psi}^{1}_{0}\rangle$
$\displaystyle=c_{0}|0\rangle+c_{1}e^{i(\phi_{1}-\phi_{0})}|1\rangle,$ (19)
$\displaystyle|\tilde{\Psi}^{1}_{1}\rangle$
$\displaystyle=c_{2}|0\rangle+c_{3}e^{i(i\phi_{3}-\phi_{2})}|1\rangle,$ (20)
$\displaystyle|\tilde{\Psi}^{2}_{0}\rangle$
$\displaystyle=|0\rangle\big{(}c_{0}|0\rangle+c_{1}e^{i(\phi_{1}-\phi_{0})}|1\rangle\big{)}$
$\displaystyle\quad+e^{i(\phi_{2}-\phi_{0})}|1\rangle\big{(}c_{2}|0\rangle+c_{3}e^{i(i\phi_{3}-\phi_{2})}|1\rangle\big{)},$
(21)

and, the bases to be measured are

$\displaystyle\mathcal{L}_{a0}=$
$\displaystyle\\{\>|0\rangle|+_{a}\rangle,\>|0\rangle|-_{a}\rangle,\>|1\rangle|+_{a}\rangle,\>|1\rangle|-_{a}\rangle\>\\},$
(22) $\displaystyle\mathcal{L}_{a1}=$
$\displaystyle\\{\>|+_{a}\rangle|+_{a}\rangle,\>|+_{a}\rangle|-_{a}\rangle,\>|-_{a}\rangle|+_{a}\rangle,\>|-_{a}\rangle|-_{a}\rangle\>\\},$
(23)

or

$\displaystyle\mathcal{E}_{a}=\\{\>|0\rangle|+_{a}\rangle,\>|1\rangle|+_{a}\rangle,\>|+_{a}\rangle|-_{a}\rangle,\>|-_{a}\rangle|-_{a}\rangle\>\\}.$
(24)

(a) Randomly generated states, local bases

(b) Randomly generated separable states, local bases

(c) Randomly generated states, entangled bases

(d) Randomly generated separable states, entangled bases

Figure 2: Median fidelity (solid dots) between arbitrary states and its
estimates as a function of number of qubits. The colors label the number of
bases: For (a) and (b) we have $2n+1$ (blue), $3n+1$ (yellow) and $4n+1$
(purple) local bases. For (c) and (d) we have $2$ (blue), $3$ (yellow) and $4$
(purple) entangled bases plus the computational basis. Shaded areas represent
the corresponding interquartile range.

##  III Numerical Simulations

We are interested in how similar an unknown state and its estimate are. To
quantify this we use the fidelity defined by

$\displaystyle F=|\langle\Psi|\Psi_{est}\rangle|^{2},$ (25)

between an arbitrary state $|\Psi\rangle$ and its estimate
$|\Psi_{est}\rangle$. A vanishing fidelity indicates that $|\Psi\rangle$ and
$|\Psi_{est}\rangle$ can be perfectly distinguished. A unitary fidelity
indicates that $|\Psi\rangle$ and $|\Psi_{est}\rangle$ are equal. Thus, a good
pure-state estimation method will be characterized by high values of the
fidelity. We are interested in the behavior of the fidelity as a function of
the dimension, or equivalently, the number of qubits, and the impact on the
fidelity of shot noise or finite-statistics effects.

To study the performance of the proposed method we carried out numerical
experiments. For each number of qubits $n=2,3,\dots,10$ we randomly generate a
set $\\{|\Psi^{(i)}\rangle\\}$ of $100$ Haar-distributed pure states, which
play the role of the unknown state to be estimated. Afterward, we simulate
projective measurements on each $|\Psi^{(i)}\rangle$ using the entangled
bases, Eq. (17), and the local bases, Eq. (16). These measurements are
simulated considering $10^{13}$ repetitions for each base. Thereafter, using
the set of estimated probabilities obtained from the simulated measurements we
applied our estimation method to each state $|\Psi^{(i)}\rangle$, which leads
to a set $\\{|\Psi_{est}^{(i)}\rangle\\}$ of estimates. Finally, we calculate
the value of the fidelity between each unknown state and its estimate.
Thereby, we obtain a set of fidelity values for each number $n$ of qubits, the
statistics of which we study in the following figures.

Figure 2(a) shows the median of the fidelity as a function of the number $n$
of qubits for three different numbers $nm+1$ of local bases with $m=2,3$, and
$4$, from bottom to top. For each number of bases, the infidelity decreases as
the number of qubits increases. This is a general feature of quantum state
estimation methods. Since for each basis the number of repetitions is fixed,
the probabilities entering in the systems of linear equations, such as Eq.
(15), are estimated with a decreasing accuracy as $n$ increases, which leads
to lower fidelity values. Figure 2(a) also shows that the median value of the
fidelity can be increased, while keeping the number of repetitions fixed,
using a larger number of local bases. The proposed method requires the pseudo-
inverse for every matrix in Eq. (14). The quality of this inversion can be
improved by reducing the condition numbers of the linear equation systems,
which can be achieved by increasing the number of bases employed in the
estimation. An interesting feature of Fig. 2(a) is that the interquartile
range is very narrow for each basis, which indicates that the estimation
method generates very similar fidelity values for all the simulated states.

Figure 2(c) exhibits the median of the fidelity as a function of the number
$n$ of qubits for 2, 3 and 4 entangled bases plus the canonical basis. In this
case, the estimation method achieves a poor performance in comparison to the
use of local bases. The use of 2 entangled bases lead to two equations for
every reduced state, which generates an ill-conditioned equation system.
Thereby, a poor estimation of the phases of the probability amplitudes is
obtained, which propagates through the iterations of the algorithm leading to
a low fidelity. However, as Fig. 2(c) shows, the fidelity can be greatly
improved by increasing the number of entangled bases for the estimation. Also,
the simulation was carried out employing a fixed ensemble size of $2^{13}$ for
each base. This was done to allow a comparison between our simulations and the
results of experiments on the IBM’s quantum processors. The total ensemble
employed with the local and entangled bases scales as $mn\times 2^{13}$ and
$k\times 2^{13}$, respectively. Thereby, the total ensemble used with the
local bases is much larger than in the case of the entangled bases, which
leads to a better estimation via local bases. An increase of the ensemble size
used with the entangled bases could lead to an improvement of the fidelity.
Nevertheless, the entangled bases are capable of delivering for $n=6$ a
fidelity close to $0.9$.

Figures 2(b) and 2(d) also display the fidelity as a function of the number of
qubits for local and entangled bases, respectively. However, the simulation
has been carried out on the set of completely separable states. In both cases
we obtain an increase of the performance of the proposed estimation method,
achieving fidelities of around $0.95$ for $10$ qubits when employing local
bases. This result indicates that the performance of the proposed estimation
method on the set of entangled states should be closer to the depicted in
Figs. 2(a) and 2(c).

(a) $|\Phi_{1}^{n}\rangle$, local bases

(b) $|\Phi_{2}^{n}\rangle$, local bases

(c) $|\Phi_{3}^{n}\rangle$, local bases

(d) $|\Phi_{4}^{n}\rangle$, local bases

(e) $|\Phi_{1}^{n}\rangle$, entangled bases

(f) $|\Phi_{2}^{n}\rangle$, entangled bases

(g) $|\Phi_{3}^{n}\rangle$, entangled bases

(h) $|\Phi_{4}^{n}\rangle$, entangled bases

Figure 3: Fidelity achieved by projective measurements on $mn+1$ local bases
(upper row, $m=2$ (blue), $m=3$ (yellow), and $m=4$ (purple)) and $m$
entangled bases plus the computational base (lower row, $m=2$ (blue), $m=3$
(yellow), and $m=4$ (purple)), for states $|\Phi_{1}^{n}\rangle$,
$|\Phi_{2}^{n}\rangle$, $|\Phi_{3}^{n}\rangle$ and $|\Phi_{4}^{n}\rangle$,
columns from left to right, respectively, as a function of the number $n$ of
qubits.

##  IV Experimental Realization

We carried out an experimental demonstration of our estimation method using
the quantum processor _ibmq_manhattan_ developed by IBM. This NISQ device has
online access and can be programmed with Qiskit, an open-source development
framework for working with quantum computers in Python programming language.

To test our estimation method we prepare the following $n$-qubit states

$\displaystyle|\Phi_{1}^{n}\rangle=\frac{1}{\sqrt{2^{n}}}\left(|0\rangle-e^{i\pi/4}|1\rangle\right)^{\otimes
n},$ (26)
$\displaystyle|\Phi_{2}^{n}\rangle=\frac{1}{\sqrt{2^{n}}}\left(|0\rangle+e^{i\pi/4}|1\rangle\right)^{\otimes
n},$ (27)

and

$\displaystyle|\Phi_{3}^{n}\rangle=\Bigg{\\{}\begin{array}[]{cc}\frac{1}{2^{n/4}}\left(|00\rangle+|11\rangle\right)^{\otimes
n/2},&n\text{ even.}\\\
\frac{1}{2^{(n-1)/4}}\left(|00\rangle+|11\rangle\right)^{\otimes(n-1)/2}\otimes|0\rangle,&n\text{
odd.}\end{array}$ (30)

States $|\Phi_{1}^{n}\rangle$ and $|\Phi_{2}^{n}\rangle$ are strictly
separable and thus they can be prepared efficiently with the use of a single-
qubit gate acting on each qubit. The state $|\Phi_{3}^{n}\rangle$ has maximal
entanglement between pairs of qubits. For $n$ even (odd), this state can be
prepared with $n/2$ ($(n-1)/2$) CNOT gates and $n/2$ ($(n-1)/2$) single-qubit
gates. Thus, states $|\Phi_{1}^{n}\rangle$, $|\Phi_{2}^{n}\rangle$, and
$|\Phi_{3}^{n}\rangle$ can be generated by circuits with short depth. However,
the state $|\Phi_{3}^{n}\rangle$ is generated with a lower preparation
fidelity than states $|\Phi_{1}^{n}\rangle$ and $|\Phi_{2}^{n}\rangle$ due to
the fact that control-not gates are implemented with an error much larger than
the one achieved in the implementation of single-qubit gates.

We also perform the estimation of $n$-partite GHZ states,

$\displaystyle|\Phi_{4}^{n}\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle^{\otimes
n}+|1\rangle^{\otimes n}\right).$ (31)

These states are implemented by a long-depth circuit which contains $n-1$ CNOT
gates. Consequently, the fidelity achieved in its generation can be very low.
Thus, we focus our study in small number of qubits $n=2,3,4$ for both local
and entangled bases. Each basis was measured employing $10^{13}$ repetitions,
that is, the size of the ensemble of equally prepared copies of the unknown
quantum state. This is currently the maximal sample size that can be employed
in IBM’s quantum processors.

Figure 3 summarizes the results obtained by implementing our estimation method
on IBM’s quantum processor, where the fidelity between states
$|\Phi_{1}^{n}\rangle$, $|\Phi_{2}^{n}\rangle$, $|\Phi_{3}^{n}\rangle$,
$|\Phi_{4}^{n}\rangle$ and its corresponding estimates is displayed as a
function of the number $n$ of qubits for the cases of $mn+1$ local bases (with
$m=2,3$ and 4) and $m$ entangled bases plus the computational base. Shaded
areas correspond to the error in the values of the fidelity obtained by
bootstrapping method. The solid blue line is the maximal achievable value of
the fidelity of the state preparation stage $\mathcal{\mathcal{E}}$
considering a white noise model for the device based on the average error per
gate $r$ provided by IBM,

$\displaystyle\mathcal{E}_{noise}(|0\rangle^{\otimes
n}\rangle)=\left(1-\frac{2^{n}r}{2^{n}-1}\right)\mathcal{E}(|0\rangle^{\otimes
n})+\frac{2^{n}r}{2^{n}-1}I.$ (32)

The error model was applied on each gate necessary to prepare the states. We
use as average error per gate $r_{1}=5\times 10^{-4}$ for local gates and
$r_{2}=2\times 10^{-2}$ for CNOT gates.

According to Figs. 3(a) and 3(b), the estimation of local states
$|\Phi_{1}^{n}\rangle$ and $|\Phi_{2}^{n}\rangle$ through local bases leads
values that are comparable with the theoretical predictions shown in Figs.
2(a) and 2(b), which only consider noise due to finite statistics. For the
particular case of $n=10$ qubits, the theoretical predictions for the
estimation fidelity using local bases are within the intervals $[0.88,0.93]$
for randomly generated states, and $[0.95,0.96]$ for randomly generated
separable states, while the experiment leads to an estimation fidelity in the
interval $[0.82,0.89]$.

The estimation of entangled states $|\Phi_{3}^{n}\rangle$ and
$|\Phi_{4}^{n}\rangle$ by local bases, exhibited in Figs. 3(c) and 3(d),
respectively, also leads to good fidelities, albeit lower than in the case of
states $|\Phi_{1}^{n}\rangle$ and $|\Phi_{2}^{n}\rangle$. This is to be
expected since states $|\Phi_{3}^{n}\rangle$ and $|\Phi_{4}^{n}\rangle$
exhibit entanglement and thus are generated applying several CNOT gates, which
increase the preparation error. Nevertheless, for the state
$|\Phi_{3}^{n}\rangle$ with $n=10$ our method provides an estimation fidelity
close to $0.67$, where the maximum achievable value according to the noise
model is $0.9$ (blue solid line in Fig. 3(c)). A comparison between the
estimated fidelity for the bi-local state $|\Phi_{3}^{n}\rangle$ and GZH state
$|\Phi_{4}^{n}\rangle$ for $n=2,3$ and $4$ qubits indicates that these
entangled states are estimated with similar fidelities. This indicates that
the method delivers similar quality estimation for states with different type
of entanglement. The experimental realization of our estimation method shows
that the use of local bases allows us to estimate pure states of large numbers
of qubits. The fidelity of the estimation is meanly constrained by ensemble
size and number of local bases. An increase of any of these quantities leads
to an improvement in the quality of the estimation. In particular, states of
larger number of qubits can be reliably estimated by increasing the number of
local bases. As is shown in Figs. 3(a), 3(b), and 3(c) the use of $4n+1$ local
bases leads to higher values of the estimation fidelity than the cases of
$2n+1$ and $3n+1$ local bases.

The use of entangled bases shows a different picture, where the physical
realization of the method and the particular class of states to be estimated
lead to a reduction of the fidelity when compared to the case of local bases.
Figs. 3(e) and 3(f) exhibit the estimation achieved for states
$|\Phi_{1}^{n}\rangle$ and $|\Phi_{2}^{n}\rangle$, respectively. A comparison
for $n=4$ with Figs. 3(a) and 3(b) shows an acute decrease of the fidelity
from approximately 0.995 to 0.2. A similar decrease can also be observed in
the case of states $|\Phi_{3}^{n}\rangle$ and $|\Phi_{4}^{n}\rangle$. The low
achieved fidelity finds its origin in the circuits employed to implement the
measurements on the entangled bases. For $n=2,3$, and $4$, 2 entangled bases
require the use of 2, 7 and 27 control-note gates, respectively. This number
becomes 3027 for $n=10$. Since the control-not gate has a high error rate in
NISQ-devices, the implemented bases differ significantly from the actual bases
$\mathcal{E}_{a}$ to be implemented. Besides, this also affects the generated
states. Fig. 3(f) shows that for $n=3$ the use of $2$ and $3$ entangled bases
plus the computational base lead to a fidelity value of $0.1$ much lower than
the case of using $4$ entangled bases plus the computational base. This
originates in the state $|\Phi_{2}^{3}\rangle$ to be reconstructed, which for
$2$ and $3$ entangled bases exhibits ill-conditioned equation systems. This is
not present in the case of $4$ entangled bases, where the fidelity is in the
order of $0.95$. Let us note that this effect does not appear in Fig. 3(b),
where the use of $mn+1$ local bases allow us to obtain well-conditioned
equation systems. Ultimately, the combinations of these factors leads to a low
estimation fidelity.

##  V Conclusions

Estimating states of $d$-dimensional quantum systems requires a minimal number
of measurement outcomes that scales quadratically with the dimension. In the
case of composite systems, such as quantum computers, the scaling becomes
exponential in the number of subsystems, which makes the estimation by generic
methods unfeasible except for few-component systems. NISQ computers are
characterized by low-accuracy entangling gates, which are required for
implementing measurements of arbitrary observables, and by a fixed number of
repetitions, which constrains the size of statistical samples and decreases
the estimation accuracy as the number of qubits increases. Therefore,
estimating $n$-qubit states of NISQ computers is difficult task.

We have proposed a method to estimate pure states of $n$-qubit systems that is
well suited for NISQ computers. The method is based on the reconstruction of
the so-called reduced states $|\tilde{\Psi}_{\beta}^{j}\rangle$, which for an
unknown state $|\Psi\rangle$ are $n$ sets of $2^{n-j}$ non-normalized states,
one set for each $j=1,2,...,n$. If we know the reduced states
$|\tilde{\Psi}_{\beta}^{j}\rangle$ for a fixed $j$, then it is easy to
reconstruct the reduced states with $|\tilde{\Psi}_{\beta}^{j+1}\rangle$ from
the results of well defined set of measurements. The method begins by
measuring a set of projectors that allow us to reconstructs all the reduced
states for $j=1$, and iterating, the rest of reduced states until reaching
$|\tilde{\Psi}_{\beta}^{n}\rangle$, the estimate of the unknown pure state.

The set of measurements employed by the proposed method corresponds to
projective measurements onto a set of bases. We have first shown that $mn+1$
bases, with $m$ an integer number equal or greater than 2, allow us to
estimate most pure states up to a null-measure set. Thereby, a total of
$(mn+1)2^{n}$ measurement outcomes is required. This number compares favorably
with other estimation methods. Mutually unbiased bases , SIC-POVM, and
compressed sensing require $(2^{n}+1)2^{n}$, $2^{2n}$, and in the order of
$2^{2n}n^{2}$ measurement outcomes, respectively, to reconstruct pure states.
The $mn+1$ bases are local, that is, they can be cast as the tensor product of
$n$ single-qubit bases. Projective measurements onto the states of these bases
are carried out on a NISQ computer by applying local gates onto each qubit
followed by a projection onto the computational base. Thus, no entangling
gates are necessary. In principle, $2n+1$ bases are enough. We have shown, by
means of numerical simulations, that the use of a larger number of local bases
increases the estimation accuracy for larger numbers of qubits. However, if
high-accuracy entangling gates are available, then the set of measurements
employed by the proposed method can be reduced to projective measurements on
$m$ entangled bases plus the computational base, with $m$ an integer number
equal or greater than 3. In this case, an increase in the number of entangled
bases also allows us to increase the estimation accuracy.

We tested the proposed estimation method in the IBM’s quantum processing
units. The estimation of local pure states via local bases up to 10 qubits
provides fidelities that agree with the numerical simulations and are above
90%. The estimation of entangled states via local bases was also tested. In
this case, the estimation of tensor product of two-qubit Bell states up to
$n=10$ led to fidelities above 70%. In this case, however, the preparation of
the state is affected by large errors due to the use of entangling gates.
Finally, we tested the estimation of a GHZ state for $n=2,3$ and 4 qubits
obtaining fidelities above 86%. Entangled bases were also tested.
Nevertheless, due to the massive use of entangling gates in the preparation
and measurement stages very low fidelities were achieved, as expected. The
estimation via $k$ entangled bases plus the computational base, where $k$ is
high-enough but does not scale with the number $n$ of qubits, can lead to a
large fidelity in future fault-tolerating quantum architectures [59], where a
large numbers of qubits and high-accuracy entangling gates are required. Also,
it might be possible to increase the estimation fidelity by using alternative
implementations of the multi-control NOT gates that employ ancillary qubits.
This allows a reduction in the depth of the circuit required to implement
measurements on the proposed entangled bases [60].

The main characteristic of our protocol is its scalability, which is a
consequence of the a priori information about the states to be estimated. This
is an approach common among many estimation methods such as, for instance,
compressed sensing and matrix product states. The purity assumption is
reasonable in systems that are able to prepare high purity states or in
systems whose purity can be certified, for instance, through randomized
benchmarking [61]. However, this is not necessarily true on NISQ devices.
Decoherence and gate errors can reduce the quality of preparing a pure state,
so it really becomes a mixed state. In this case our protocol could not be
applied. Nevertheless, it has been shown that experimental results can be
improved by reducing the error of the raw data employing error-mitigation
techniques [49, 17, 62, 63]. Combining scalable error-mitigation methods with
our estimation method could extend its applicability to high-noise systems.

###### Acknowledgements.

This work was supported by ANID – Millennium Science Initiative Program –
ICN17-012\. AD was supported by FONDECYT Grant 1180558. LP was supported by
ANID-PFCHA/DOCTORADO-BECAS-CHILE/2019-72200275. LZ was supported by ANID-
PFCHA/DOCTORADO-NACIONAL/2018-21181021. We thank the IBM Quantum Team for
making multiple devices available to the CSIC-IBM Quantum Hub via the IBM
Quantum Experience.

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