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# Pauli transfer matrix direct reconstruction: channel characterization
without full process tomography

Simone Roncallo  0000-0003-3506-9027<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli Studi di Pavia, Via Agostino Bassi 6,
I-27100, Pavia, Italy  INFN Sezione di Pavia, Via Agostino Bassi 6, I-27100,
Pavia, Italy  Lorenzo Maccone  0000-0002-6729-5312<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli Studi di Pavia, Via Agostino Bassi 6,
I-27100, Pavia, Italy  INFN Sezione di Pavia, Via Agostino Bassi 6, I-27100,
Pavia, Italy  Chiara Macchiavello  0000-0002-2955-8759
<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli Studi di
Pavia, Via Agostino Bassi 6, I-27100, Pavia, Italy  INFN Sezione di Pavia, Via
Agostino Bassi 6, I-27100, Pavia, Italy

###### Abstract

We present a tomographic protocol for the characterization of multiqubit
quantum channels. We discuss a specific class of input states, for which the
set of Pauli measurements at the output of the channel directly relates to its
Pauli transfer matrix components. We compare our results to those of standard
quantum process tomography, showing an exponential reduction in the number of
different experimental configurations required by a single matrix element
extraction, while keeping the same number of shots. This paves the way for
more efficient experimental implementations, whenever a selective knowledge of
the Pauli transfer matrix is needed. We provide several examples and
simulations.

Quantum channels; Quantum operations; Quantum process tomography; Pauli
transfer matrix; Pauli-Liouville matrix;

##  I Introduction

Quantum channels, also known as quantum operations, describe the dynamics of
quantum systems that interact with their surrounding environment. Intuitively,
channels behave like quantum black boxes, mapping states into other states,
mathematically represented as linear completely-positive trace-preserving maps
[1]. They describe a wide variety of operations, including unitary
transformations, communication and teleportation protocols [2], or noisy
processes (e.g. models like the bit-flip, the depolarizing or the amplitude
damping channels [1], even in the presence of correlations within the system
[3, 4]).

Quantum process tomography (QPT) is the identification of an unknown quantum
channel, obtained by correlating a complete set of input states to a complete
set of measurements at their output. We call _experimental configurations_ ,
or configurations, each couple of choices of input state and output
measurement. For each configuration one must feed each state and perform the
same measurement for a certain number of times, called _shots_ , in order to
retrieve the necessary statistics. QPT can be achieved in two ways [5]:
directly, when the measurement outcome immediately enters the channel
reconstruction [6, 7, 8, 9, 10, 11](eventually after a post-processing
manipulation of the data), or indirectly, i.e. when it requires additional
techniques to analyse the output state (e.g. state tomography) [12, 13, 14].

QPT can be directly formulated in the vectorized representation (or Pauli-
Liouville representation), in which operators are mapped to column vectors and
channels to matrices, called _Pauli transfer matrices_ (PTM) [6, 7]. This
brings several advantages, e.g. the action of the channel, identified by the
PTM, becomes a matrix multiplication, simplifying its inversion and
manipulation in tasks like noise deconvolution [15, 16]. In this framework,
the purpose of QPT is the PTM reconstruction, which is achieved by combining
the outcome of different experimental configurations into each PTM entry. As
in the standard Kraus description [1], the number of experimental
configurations required by a full PTM tomography is $d^{4}$, where $d$ is the
dimension of the system (we do not consider overcomplete sets of states, whose
cost is even higher).

Figure 1: DPTM reconstruction of an unknown quantum channel $\Phi$. Two sets
of experimental configurations are considered. The first one provides the
characterization of the non-unital column of the channel PTM. The results in
each row are then subtracted to those of the second set, completing the
characterization of the remaining matrix elements. Independently from the
number of qubits, the reconstruction of each PTM entry requires at most 2
experimental configurations.

In this paper we focus on ancilla-free QPT for multiqubit quantum channels,
for which we propose an alternative approach that provides a direct
reconstruction of the PTM (DPTM). Indeed, we consider a particular set of
input states, for which each entry of the PTM is directly identified with few
elements of the set of local Pauli measurements at the output of the channel,
as summarized by Fig. 1. We then compare our results to generic and to
standard QPT (sQPT). We show that the DPTM reconstruction costs only $2$
experimental configurations for each PTM entry (decreasing to $1$ for unital
channels) independently of the dimension of the system, providing an
exponential speedup against the minimum number of configurations required by
sQPT for the same task. This exponential gain is lost if one needs to
reconstruct the whole PTM rather then few matrix elements. However, there are
many situations where few entries are sufficient to recover the required
channel characteristics, e.g. for multiparameter estimations in quantum
metrology [17] or in assessing the unitality of a quantum channel. In contrast
to other techniques like shadow tomography [18, 19], our approach does not
improve the statistics of the reconstruction, since the total number of shots
(or copies of the state) remains unchanged. Rather, it reduces the number of
configurations needed, whenever only some PTM matrix elements are required or
when one can introduce some prior knowledge of the channel in its
characterization (examples are discussed below). Another advantage of this
approach is that using less combinations of measurements can eventually reduce
the systematics due to hardware errors.

Finally, we apply and simulate DPTM for two different scenarios. First, we
fully characterize a single-qubit amplitude damping channel, using only $4$
experimental configurations (with respect to the $8$ required by sQPT). Then,
we discuss the parameters extraction of a two-qubit correlated depolarizing
channel, for which DPTM requires $2$ configurations (with respect to the $15$
required by sQPT).

##  II Quantum channels and Pauli transfer matrix

We consider the Hilbert space of a $n$-qubits system. The basis for the set of
operators is

$\big{\\{}\sigma_{\alpha_{1}}\otimes\sigma_{\alpha_{2}}\otimes...\otimes\sigma_{\alpha_{n}}\
|\ \alpha_{1},\alpha_{2},...,\alpha_{n}=0,1,2,3\big{\\}}\ ,$ (1)

with $\sigma_{0}=\mathds{1}_{2}$, $\sigma_{1}=X$, $\sigma_{2}=Y$ and
$\sigma_{3}=Z$. We write the Pauli basis in the following notation

$\big{\\{}\mathcal{P}_{k}\ |\ k=0,1,2,3,...,d^{2}-1\big{\\}}\ ,$ (2)

with $d=2^{n}$ and $\mathcal{P}_{k}$ given by the generic element of Eq. 1 in
lexicographic order.

Consider a system with quantum state $\rho$. A _quantum channel_ (or quantum
operation) is a linear completely-positive trace-preserving (CPTP) map
$\rho\to\Phi(\rho)$ [1]. There are several ways to represent a quantum channel
[20], e.g. the _Kraus representation_ , in which $\Phi(\rho)$ is described by
a collection of operators $\\{A_{i}\\}_{0\leq i\leq d^{2}-1}$, called Kraus
operators, such that

$\Phi(\rho)=\sum_{i}A_{i}\rho A^{\dagger}_{i}\ .$ (3)

The CPTP condition implies that

$\sum_{i}A^{\dagger}_{i}A_{i}=\mathds{1}\ ,$ (4)

with $\mathds{1}$ the identity operator.

We consider the PTM representation [6, 7] (also known as Pauli-Liouville, or
superoperator, representation), which describes $\Phi$ as a $d^{2}\times
d^{2}$ matrix

$\Gamma_{ij}=\frac{1}{d}\Tr[\mathcal{P}_{i}\Phi(\mathcal{P}_{j})]\ .$ (5)

The PTM finds a natural application in the vectorized notation [6, 21], where
any operator $A$ is mapped to a $1\times d^{2}$ column vector
$\ket{A}\rangle$, on which quantum channel acts through standard matrix
multiplication

$\Phi(\rho)=\Gamma\ket{\rho}\rangle\ .$ (6)

This representation is equipped with the Hilbert-Schmidt inner product, so
that $A_{i}=\langle\langle i|A\rangle\rangle$ and Eq. 5 reads

$\Gamma_{ij}=\langle\bra{i}\Gamma\ket{j}\rangle\ ,$ (7)

with $\ket{i}\rangle$ denoting the vectorized Pauli basis operator
$\mathcal{P}_{i}$. A reshuffling transformation bijectively relates the PTM to
the Choi matrix, whose spectral decomposition can lead to the original Kraus
representation of the channel [20].

By straightforward application of Eq. 5, the following properties of the PTM
hold.

######  Proposition 1.

Consider the Hilbert space of a $n$-qubit system, with dimension $d=2^{n}$.
Let $\Phi$ be a quantum channel and $\Gamma$ its $d^{2}\times d^{2}$ PTM
representation, whose definition is given in Eq. 5. The CPTP condition implies
that

$\Gamma_{0j}=\delta_{0j}\ .$ (8)

with $0\leq i,j\leq d^{2}-1$ and $\delta_{ij}$ denoting the Kronecker delta.
If the channel is unital, i.e. $\Phi(\mathds{1})=\mathds{1}$, then also

$\Gamma_{i0}=\delta_{i0}\ .$ (9)

Further simplifications hold for Pauli channels, which are defined as those
CPTP maps whose Kraus operators belong to the Pauli basis only and whose PTM
is diagonal [22]. This specific class of channels includes the bit-flip, the
depolarizing or the dephasing noises [1], also in presence of correlations [3,
23]. See [15] for some examples of single-qubit noise models and PTM.

##  III QPT and Direct PTM reconstruction

The goal of QPT is to reconstruct an unknown quantum channel $\Phi$ from the
statistics of a collection of experimental configurations. Namely, by applying
a set of measurement operators $\\{E_{i}\\}_{0\leq i\leq d^{2}-1}$ to
different choices of input states $\\{\rho_{j}\\}_{0\leq j\leq d^{2}-1}$ [6,
7]. After having collected sufficient statistics (i.e. on a large number of
shots), the outcome of each configuration reads

$p_{ij}=\Tr[E_{i}\Phi(\rho_{j})]\ .$ (10)

Using the completeness relation, this can be written in terms of the channel
PTM as [6]

$p=\alpha\Gamma\beta\ ,$ (11)

with

$\alpha_{ij}=\frac{1}{d}\Tr[E_{i}\mathcal{P}_{j}]\
,\quad\beta_{ij}=\Tr[\mathcal{P}_{i}\rho_{j}]\ .$ (12)

We refer to $\alpha$ and $\beta$ as reconstruction matrices, which, once
inverted, provide the PTM in terms of the measurement data as

$\Gamma=\alpha^{-1}p\beta^{-1}\ ,$ (13)

whenever $\alpha^{-1}$ and $\beta^{-1}$ exist.

In practice, the PTM is often reconstructed using a tomographic fitter instead
of Eq. 13, for example a least-squares minimization or a maximum likelihood
estimation [6, 24, 25]. We do not consider any of these methods in our
analysis, instead we compare our DPTM technique (which can equally benefit
from them) to the reconstruction provided by Eq. 13.

We now discuss an alternative procedure that provides a direct PTM
reconstruction (DPTM) from the experimental data.

Consider a $n$-qubit system, prepared in one of the following states

$\rho_{0}=\frac{\mathds{1}}{d},\quad\rho_{k}=\frac{\mathds{1}+\mathcal{P}_{k}}{d}\quad\text{for
}k\neq 0\ ,$ (14)

which can be compactly written as111Index summation is always made explicit.

$\rho_{k}=\frac{1}{d}\big{[}(1-\delta_{0k})\mathds{1}+\mathcal{P}_{k}\big{]}\
.$ (15)

These states are positive semidefinite and normalized [16]. Moreover, they are
mixed, except for $n=1$ and $k\neq 0$. See Appendix A for considerations on
how they can be prepared. The channel $\Phi$ evolves $\rho_{k}$ to

$\Phi(\rho_{k})=\frac{1}{d}\big{[}(1-\delta_{0k})\Phi(\mathds{1})+\Phi(\mathcal{P}_{k})\big{]}\
.$ (16)

Consider the $i$-th element of the Pauli basis $\mathcal{P}_{i}$. As an
observable, its expectation value against $\Phi(\rho_{k})$ is

$\langle\mathcal{P}_{i}\rangle_{\Phi(\rho_{j})}=\Tr[\mathcal{P}_{i}\Phi(\rho_{j})]\
,$ (17)

This mathematically represents each configuration outcome (combining input
states and measurements at the output of the channel), which, expanded with
Eq. 16, depends on the PTM as

$\langle\mathcal{P}_{i}\rangle_{\Phi(\rho_{j})}=(1-\delta_{0j})\Gamma_{i0}+\Gamma_{ij}\
.$ (18)

When $j=0$ the second contribution vanishes, yielding

$\langle\mathcal{P}_{i}\rangle_{\Phi(\rho_{0})}=\Gamma_{i0}\ .$ (19)

By subtracting this term from Eq. 18, we complete the PTM reconstruction as

$\Gamma_{ij}=\begin{cases}\langle\mathcal{P}_{i}\rangle_{\Phi(\rho_{0})}\quad&\text{for
}j=0\\\
\langle\mathcal{P}_{i}\rangle_{\Phi(\rho_{j})}-\langle\mathcal{P}_{i}\rangle_{\Phi(\rho_{0})}\quad&\text{for
}j\neq 0\end{cases}\ .$ (20)

More compactly, the DPTM equation reads

$\Gamma_{ij}=\langle\mathcal{P}_{i}\rangle_{\Phi(\rho_{j})}-(1-\delta_{0j})\langle\mathcal{P}_{i}\rangle_{\Phi(\rho_{0})}\
.$ (21)

This consistently satisfies all the properties of Proposition 1.

The equation of DPTM establishes a direct relation between $\Gamma_{ij}$ and
the corresponding configuration
$\langle\mathcal{P}_{i}\rangle_{\Phi(\rho_{j})}$. When the channel is
completely unknown, DPTM requires as many resources as QPT, i.e. $d^{2}$ input
states $\\{\rho_{j}\\}$ coupled to $d^{2}$ Pauli measurements
$\\{\mathcal{P}_{i}\\}$. However, if some prior knowledge on the channel is
already available (e.g. unitality or its, eventually incomplete, Kraus
representation), it is possible to drop those combinations of $\rho_{j}$ and
$\mathcal{P}_{i}$ already fixed by the initial information on $\Gamma_{ij}$.
In this case the number of configurations reduces and DPTM requires fewer
resources than QPT (see Table 1 for some examples).

| General | CPTP | Unitality | Pauli
---|---|---|---|---
Resources | $d^{4}$ | $d^{2}(d^{2}-1)$ | $(d^{2}-1)^{2}$ | $d^{2}-1$
Table 1: Example of the DPTM cost for different type of constraints, in terms
of the number of experimental configurations to collect all the data and
complete the reconstruction.

As discussed in the next section, similar reductions apply to any partial
extraction of the PTM, e.g. in the characterization of known theoretical
models, in the estimation of unknown quantum parameters, or in testing the
unitality of an unknown quantum channel. DPTM can efficiently solve this last
task, requiring at most $d^{2}-1$ configurations.

The relation between QPT and DPTM can be understood by applying the set of
states of Eq. 15 to Eq. 13. With $E_{i}=\mathcal{P}_{i}$, then
$p_{ij}=\langle\mathcal{P}_{i}\rangle_{\Phi(\rho_{j})}$ and the reconstruction
matrices read

$\alpha_{ij}=\delta_{ij}\
,\quad\beta_{ij}=(1-\delta_{0j})\delta_{i0}+\delta_{ij}\ ,$ (22)

Their inversion yields

$\alpha^{-1}_{ij}=\delta_{ij}\
,\quad\beta_{ij}^{-1}=\delta_{ij}-(1-\delta_{0j})\delta_{i0}\ .$ (23)

By direct substitution, this reduces Eq. 13 precisely to Eq. 21: from a
mathematical point of view, DPTM represents a particular choice of input
states for QPT. Nevertheless, in the next section we show that the specific
choice of states of DPTM guarantees a faster reconstruction, in terms of the
number of experiments required for each PTM entry.

##  IV Comparison with standard QPT

In this section we explicitly compare DPTM with sQPT. We label with $D$ and
$Q$ the quantities and results related respectively to DPTM and sQPT.

For $n=1$, the basis of the set of operators contains four elements

$\mathcal{P}_{0}=\mathds{1}_{2}\ ,\quad\mathcal{P}_{1}=X\
,\quad\mathcal{P}_{2}=Y\ ,\quad\mathcal{P}_{3}=Z\ .$ (24)

By substitution in Eq. 15, the set of input states reads

$\displaystyle\rho_{0}=\frac{1}{2}\big{(}\ket{0}\\!\bra{0}+\ket{1}\\!\bra{1}\big{)}\
,$ (25) $\displaystyle\rho_{1}=\ket{+}\\!\bra{+}\ ,$ (26)
$\displaystyle\rho_{2}=\ket{i}\\!\bra{i}\ ,$ (27)
$\displaystyle\rho_{3}=\ket{0}\\!\bra{0}\ .$ (28)

with $\sqrt{2}\ket{\pm}=\ket{0}\pm\ket{1}$ and $\sqrt{2}\ket{\pm i}=\ket{0}\pm
i\ket{1}$. In this case only $\rho_{0}$ is mixed, we discuss its preparation
in Appendix A. The remaining states are pure and can be prepared as

$\ket{\psi_{1}}=H\ket{0}\ ,\quad\ket{\psi_{2}}=SH\ket{0}\
,\quad\ket{\psi_{3}}=\ket{0}\ ,$ (29)

with $H$ and $S$ respectively the Hadamard and the phase gates [1]. Given a
state $\rho$, each Pauli measurement $\langle\mathcal{P}_{i}\rangle$ can be
always obtained from a $Z$-measurement, i.e. by applying a unitary
transformation $\rho\to U\rho U^{\dagger}$, so that

$\displaystyle\langle\mathcal{P}_{1}\rangle=\Tr\left[ZH\rho H\right]\ ,$ (30)
$\displaystyle\langle\mathcal{P}_{2}\rangle=\Tr\left[ZHS^{\dagger}\rho
SH\right]\ ,$ (31)
$\displaystyle\langle\mathcal{P}_{3}\rangle=\Tr\left[Z\rho\right]\ ,$ (32)

where we used

$\mathcal{P}_{1}=HZH\ ,\quad\mathcal{P}_{2}=SHZHS^{\dagger}\ .$ (33)

Each change of basis translates the respective Pauli measurements to a simple
count of the $0$, $1$ occurrences in the computational basis.222For example,
if $\ket{\psi_{f}}=c_{0}\ket{0}+c_{1}\ket{1}$ is a pure final state,
eventually obtained by applying a unitary transformation to the evolved one,
then $\langle Z\rangle_{\rho_{f}}=|c_{0}|^{2}-|c_{1}|^{2}$.

The DPTM configurations
$M^{D}_{ij}:=\langle\mathcal{P}_{i}\rangle_{\Phi(\rho_{j})}$ follow by
coupling each input state to the set of Pauli measurements in all the possible
ways, so that

$\begin{split}&\Gamma_{i0}=M^{D}_{i0}\ ,\\\
&\Gamma_{ij}=M^{D}_{ij}-M^{D}_{i0}\quad\text{for }j\neq 0\ .\end{split}$ (34)

In terms of the reconstruction matrices $\alpha_{D}$ and $\beta_{D}$ in Eq.
23, this gives $\alpha_{D}^{-1}=\mathds{1}$ and

$\beta_{D}^{-1}=\begin{pmatrix}+1&-1&-1&-1\\\ 0&+1&0&0\\\ 0&0&+1&0\\\
0&0&0&+1\end{pmatrix}\ .$ (35)

When the channel is unital, the identification is completely direct and
$\Gamma_{ij}=M^{D}_{ij}$. For Pauli channels, it further simplifies to
$\Gamma_{kk}=M^{D}_{kk}$. In both cases, the uncertainty precisely comes from
the measurement, without the need for any error propagation.

For comparison, we consider the standard protocol of sQPT333As an example, we
use the preparation and measurement basis from [1], which are the same applied
by default Qiskit Experiments library. in which the set of Pauli measurements
is usually coupled to the following set of states

$\begin{split}&\rho^{Q}_{0}=\ket{1}\\!\bra{1}\
,\quad\rho^{Q}_{1}=\ket{+}\\!\bra{+}\ ,\\\ &\rho^{Q}_{2}=\ket{i}\\!\bra{i}\ \
,\quad\rho^{Q}_{3}=\ket{0}\\!\bra{0}\ .\end{split}$ (36)

The Pauli measurements precisely matches the one of Eqs. 30, 31 and 32, with
still $\alpha^{-1}_{Q}=\mathds{1}$. However, the different choice of input
states modifies $\beta_{Q}$, whose inverse now reads

$\beta^{-1}_{Q}=\frac{1}{2}\begin{pmatrix}+1&-1&-1&-1\\\ 0&+2&0&0\\\
0&0&+2&0\\\ +1&-1&-1&+1\end{pmatrix}\ .$ (37)

By substitution in Eq. 13, with
$M^{Q}_{ij}:=\langle\mathcal{P}_{i}\rangle_{\Phi\left(\rho^{Q}_{j}\right)}$
given by Eq. 11, it follows

$\displaystyle\Gamma_{i0}$
$\displaystyle=\frac{1}{2}\left(M^{Q}_{i0}+M^{Q}_{i3}\right)\ ,$ (38)
$\displaystyle\Gamma_{i1}$
$\displaystyle=\frac{1}{2}\left(-M^{Q}_{i0}+2M^{Q}_{i1}-M^{Q}_{i3}\right)\ ,$
(39) $\displaystyle\Gamma_{i2}$
$\displaystyle=\frac{1}{2}\left(-M^{Q}_{i0}+2M^{Q}_{i2}-M^{Q}_{i3}\right)\ ,$
(40) $\displaystyle\Gamma_{i3}$
$\displaystyle=\frac{1}{2}\left(-M^{Q}_{i0}+M^{Q}_{i3}\right)\ .$ (41)

This implies that, for the same PTM entry, sQPT requires more experimental
configurations than DPTM, and also more resources in terms of post-processing
recombinations of the data into the desired outcome. For example, a full
characterization of a single-qubit Pauli channel (which has diagonal PTM)
costs 8 configurations to sQPT, while only 3 for DPTM.

We end this section by comparing the cost of sQPT and DPTM for the
reconstruction of a single PTM entry on $n$-qubit systems. In this framework,
DPTM scales always in the same way: its states are given by Eq. 15, while its
reconstruction matrix shows high sparsity

$\beta_{D}^{-1}=\begin{pmatrix}+1&-1&\dots&-1\\\ 0&\ddots&0&0\\\
\vdots&0&\ddots&\vdots\\\ 0&0&\dots&+1\end{pmatrix}\ .$ (42)

On the other hand, the $n$-qubit states for sQPT can be obtained by taking all
the possible $n$-fold Kronecker products of ${\rho_{i}^{Q}}$. Eq. 12 implies
that the reconstruction matrix reads $(\beta_{Q})^{\otimes n}$, with inverse
$(\beta_{Q}^{-1})^{\otimes n}$.

To compare the performance of the two methods we define $||\Gamma_{ij}||_{D}$
and $||\Gamma_{ij}||_{Q}$ as the number of experimental configurations
respectively required by DPTM and sQPT.444For example, in the single-qubit
case Eq. 39 implies that $||\Gamma_{i1}||_{Q}=3$. We give the following
theorem, which states that DPTM performs exponentially better than sQPT in
single-entry reconstructions, independently of the number of qubits.

######  Theorem 1.

Consider an $n$-qubit quantum system, and a quantum (unknown) channel $\Phi$.
Let $\Gamma$ be its Pauli transfer matrix, reconstructed using DPTM and sQPT.
Then

$\max_{i,j}||\Gamma_{ij}||_{D}\leq\min_{i,j}||\Gamma_{ij}||_{Q}<\max_{i,j}||\Gamma_{ij}||_{Q}\
,$ (43)

with the strict equality satisfied only for $n=1$. Indeed

$||\Gamma_{ij}||_{D}=\begin{cases}1\quad\text{for }j=0\\\ 2\quad\text{for
}j\neq 0\end{cases}\ ,$ (44)

while

$\displaystyle\min_{i,j}||\Gamma_{ij}||_{Q}$ $\displaystyle=2^{n}\ ,$ (45)
$\displaystyle\max_{i,j}||\Gamma_{ij}||_{Q}$ $\displaystyle=3^{n}\ .$ (46)

###### Proof.

Eq. 44 is a direct consequence of Eq. 20. On the other hand, the sQPT
reconstruction for $n$-qubit yields

$\Gamma=M^{Q}(\beta^{-1}_{Q})^{\otimes n}\ .$ (47)

From Eq. 37, Eq. 45 and Eq. 46 follow by respectively counting the minimum and
maximum number of non-zero entries in the $n$-fold Kronecker product
$\beta^{-1}_{Q}$. ∎

We summarize the content of this theorem in Fig. 2.

Figure 2: Number of experimental configurations to reconstruct a single PTM
entry, plotted with respect to the number of qubits $n$ ($log_{2}$ for both
axes). As shown in Theorem 1, DPTM provides an exponential gain in terms of
resources required by a single $\Gamma_{ij}$ reconstruction, improving the
performance of sQPT $\forall n>1$. For the latter, the shadowed area
represents the possible configurations cost of a single $\Gamma_{ij}$
reconstruction (the exact value depends on the choice of $i$ and $j$). The
results are all compared with the number of resources needed for a full
process tomography (i.e. the reconstruction of the whole PTM), which requires
$d^{4}$ configurations. Figure 3: Reconstruction of the non-trivial components
of a single-qubit amplitude damping channel PTM. Simulated with Qiskit Aer,
with transition probability $p=0.25$. The results of DPTM and sQPT are
represented respectively by the left and right bars, for each non-trivial
$\Gamma_{ij}$. The dashed lines represent the theoretical values obtained from
Eq. 49. This characterization costs $4$ experimental configurations to DPTM,
and $8$ for sQPT. The number of shots for each experiment is 512. The error
bars represent the standard error and are obtained by propagating the
uncertainty from Eq. 34 and Eqs. 38, 39, 40 and 41).

##  V Applications

In this section we provide some examples and simulations, by comparing DPTM
and sQPT for a few channels.

###  V.1 Amplitude damping characterization

We consider a single-qubit amplitude damping channel with Kraus operators

$A_{0}=\begin{pmatrix}1&0\\\ 0&\sqrt{1-p}\end{pmatrix}\
A_{1}=\begin{pmatrix}0&\sqrt{p}\\\ 0&0\ \end{pmatrix}\ ,$ (48)

where $p$ describes the transition probability of the state $\ket{1}$ to
$\ket{0}$, e.g. when the system emits a photon [1]. The channel theoretical
PTM reads

$\Gamma=\begin{pmatrix}1&0&0&0\\\ 0&\sqrt{1-p}&0&0\\\ 0&0&\sqrt{1-p}&0\\\
p&0&0&1-p\end{pmatrix}\ .$ (49)

The purpose of this section is to test both DPTM and sQPT in the
characterization of the non-trivial components $\Gamma_{11}$, $\Gamma_{22}$,
$\Gamma_{30}$ and $\Gamma_{33}$ (namely using Eqs. 34, 38, 39, 40 and 41).

In Fig. 3 we plot the results of a simulation performed with Qiskit Aer for
$p=0.25$. All the results are compatible with the theoretical prediction, but
DPTM reduces the cost in experimental configurations: only $4$ against the $8$
required by sQPT. We notice differences in terms of statistical uncertainties,
which can be understood as follows: although sQPT combines more data into the
same entries, potentially worsening the error propagation, DPTM uses mixed
states, which can affect the statistics of the outcome, then leading to
similar variances, in this specific example.

###  V.2 Two-qubit correlated depolarizing channel

We consider a two-qubit correlated depolarizing channel, where the amount of
correlations is measured by a parameter $\mu\in[0,1]$ [3, 23, 4, 26, 27]. We
start from the class of correlated Pauli channels [3, 23, 4], whose Kraus
representation reads

$\Phi(\rho)=\sum_{\alpha_{1},\alpha_{2}=0}^{3}p_{\alpha_{1}\alpha_{2}}A_{\alpha_{1}\alpha_{2}}\rho
A_{\alpha_{1}\alpha_{2}}\ ,$ (50)

where
$A_{\alpha_{1}\alpha_{2}}=\sigma_{\alpha_{1}}\otimes\sigma_{\alpha_{2}}$. The
transition probabilities are given by the Markov chain
$p_{\alpha_{1}\alpha_{2}}=p_{\alpha_{1}}p_{\alpha_{2}|\alpha_{1}}$ [4, 28],
with

$\displaystyle
p_{\alpha_{j}|\alpha_{i}}=(1-\mu)p_{\alpha_{j}}+\mu\delta_{\alpha_{i}\alpha_{j}}\,$
(51) $\displaystyle\vec{p}=[1-p,p_{x},p_{y},p_{z}]^{T}\ .$ (52)

and $p=p_{x}+p_{y}+p_{z}$.555The case $\mu=0$ represents a memoryless channel,
that is when the qubits are completely uncorrelated, while $\mu=1$ describes a
full-memory channel, i.e. when the qubits exhibit complete correlation. A
correlated depolarizing channel is obtained by choosing
$\vec{p}=[1-3p/4,p/4,p/4,p/4]^{T}$.

(a)

(b)

Figure 4: Two-qubit circuits for the DPTM reconstruction of $\Gamma_{44}$ and
$\Gamma_{66}$. (a) Circuit for
$\Gamma_{44}=\langle\mathcal{P}_{4}\rangle_{\Phi(\rho_{4})}$, with
$\mathcal{P}_{4}=X\otimes I$ and
$\rho_{4}=[(\mathds{1}_{2}+X)\otimes\mathds{1}_{2}]/4=\ket{+}\\!\bra{+}\otimes\mathds{1}_{2}/2$.
The completely mixed state is prepared in the second qubit using the channel
$\mathcal{E}_{1}(\psi)=\psi/2+X\psi X/2$, with $\psi=\ket{0}\\!\bra{0}$. (b)
Circuit for $\Gamma_{66}=\langle\mathcal{P}_{6}\rangle_{\Phi(\rho_{6})}$, with
$\mathcal{P}_{6}=X\otimes Y$ and $\rho_{6}=(\mathds{1}+X\otimes Y)/4$. The
input state is prepared from $\psi=\ket{0}\\!\bra{0}$ using the channel
$\mathcal{E}_{2}(\psi)=U\psi U^{\dagger}/2+VU\psi U^{\dagger}V^{\dagger}/2$,
with $U=\text{CNOT}(S\otimes\mathds{1})(H\otimes\mathds{1})$ and
$V=(\mathds{1}\otimes Z)(\mathds{1}\otimes X)$.

In this section we compare DPTM and sQPT, for the case in which we only
require the extraction of the parameters $p$ and $\mu$ from a set of
tomographic configurations. To this extent, we compute the theoretical PTM,
which yields the following relations

$\displaystyle\Gamma_{44}$ $\displaystyle=1-p\ ,$ (53)
$\displaystyle\Gamma_{66}$ $\displaystyle=(1-p)(\mu p-p+1)\ .$ (54)

Their inverses allow the parameter extraction as $p\leftarrow\Gamma_{44}$ and
$\mu\leftarrow\Gamma_{44}$, $\Gamma_{66}$. On one hand, these components are
provided by sQPT from $15$ experimental configurations $\\{M_{ij}^{Q}\\}$,
where the observables are those forming the Pauli basis and the states come
from all the possible $2$-fold Kronecker products of the single-qubit set
$\\{\rho_{i}^{Q}\\}$, with the reconstruction matrix
$(\beta_{Q}^{-1})^{\otimes 2}$. This gives the results as

$\begin{split}\Gamma_{44}=&-\frac{1}{4}M^{Q}_{40}-\frac{1}{4}M^{Q}_{43}+\frac{1}{2}M^{Q}_{44}+\\\
&+\frac{1}{2}M^{Q}_{47}-\frac{1}{4}M^{Q}_{4(12)}-\frac{1}{4}M^{Q}_{4(15)}\
,\end{split}$ (55)

along with

$\begin{split}\Gamma_{66}=\frac{1}{4}M^{Q}_{60}-\frac{1}{2}M^{Q}_{62}+\frac{1}{4}M^{Q}_{63}-\frac{1}{2}M^{Q}_{64}+M^{Q}_{66}+\\\
-\frac{1}{2}M^{Q}_{67}+\frac{1}{4}M^{Q}_{6(12)}-\frac{1}{2}M^{Q}_{6(14)}+\frac{1}{4}M^{Q}_{6(15)}\
.\end{split}$ (56)

On the other hand, DPTM provides the same components but using only $2$
configurations

$\Gamma_{44}=M^{D}_{44}\ ,\quad\Gamma_{66}=M^{D}_{66}\ ,$ (57)

with observables in the Pauli basis and input states given by Eq. 15. The
implementation of these measurements is reported in Fig. 4.

By choosing $p=0.25$ and $\mu=0.75$ so that $\Gamma_{44}=0.750$ and
$\Gamma_{66}\simeq 0.703$, we simulate DPTM and sQPT. With $2048$ shots and
using Qiskit Aer, we obtain

$\text{DPTM}\rightarrow\begin{cases}\Gamma_{44}=&0.749\pm 0.015\\\
\Gamma_{66}=&0.710\pm 0.016\end{cases}\ ,$ (58)

and

$\text{sQPT}\rightarrow\begin{cases}\Gamma_{44}=&0.756\pm 0.015\\\
\Gamma_{66}=&0.705\pm 0.016\end{cases}\ .$ (59)

We notice that both methods are in agreement with the theoretical prediction.
In this regard, DPTM performs better than sQPT: requiring fewer experimental
configurations under the same number of shots and with compatible errors.
Importantly, reducing the number of required settings will typically reduce
the systematics due to hardware errors (which are not considered in this
simulation).

##  VI Conclusions

In this work we applied QPT to the reconstruction of a multiqubit quantum
channel PTM. In general, QPT performs a set of measurements on different
experimental configurations (i.e. by changing the input state and/or the
observable at the output of the channel), combining them into each PTM entry
at the post-processing stage.

We presented an alternative technique that provides a direct reconstruction of
the PTM from the measurement outcomes. In principle, our approach differs from
sQPT only in the choice of the input states. However, this choice simplifies
both the experimental implementation and the post-processing reconstruction:
DPTM exponentially reduces the number of different configurations that
combines into a single PTM entry, while keeping the same number of shots of
sQPT.

Though both techniques require $d^{4}$ configurations for a full tomography of
the channel, DPTM truly shines when only a subset of the PTM has to be
reconstructed (e.g. in the extraction of the channel parameters under a given
theoretical model, or for biased characterizations of unknown channels, for
example of the Pauli type). While not improving the statistics of the results,
DPTM requires (at most) $2$ experimental configurations for each PTM entry,
independently of the dimension of the system: this allows for more efficient
(and scalable) experimental implementations of tomographic protocols, with
fewer computational circuits or setups of the optical table.

## Acknowledgments

This work received support from MIUR Dipartimenti di Eccellenza 2018-2022,
Project No. F11I18000680001, from EU H2020 QuantERA ERA-NET Cofund in Quantum
Technologies, Quantum Information and Communication with High-dimensional
Encoding (QuICHE), Grant Agreement 731473 and 101017733, from the U.S.
Department of Energy, Office of Science, National Quantum Information Science
Research Centers, Superconducting Quantum Materials and Systems Center (SQMS),
Contract No. DE-AC02-07CH11359. L.M. acknowledges support from the PNRR MUR
Project PE0000023-NQSTI. C.M. acknowledges support from the National Research
Centre for HPC, Big Data and Quantum Computing, PNRR MUR Project
CN0000013-ICSC.

*

##  APPENDIX A MIXED STATE PREPARATION

In this section we address the problem of generating the set of DPTM input
states of Eq. 15. Indeed, this discussion applies to any set of input,
possibly mixed, states.

For single-qubit channels, $\rho_{0}$ in Eq. 25 is the only mixed state of the
set. Its implementation can be achieved either through an ancilla-assisted
approach: namely by preparing the system in the maximally entangled state

$\ket{\Psi_{0}}=\frac{1}{\sqrt{2}}\sum_{k=0}^{1}\ket{k}\otimes\ket{k}_{a}\ ,$
(60)

and tracing out the ancillary qubit

$\rho_{0}=\Tr_{a}\big{[}\ket{\Psi_{0}}\\!\bra{\Psi_{0}}\big{]}\ ,$ (61)

or by classical random generation of its pure components

$\ket{\psi_{0}}\in\\{\ket{0},\ket{1}\\}\quad\text{with
}p(\ket{0})=p(\ket{1})=\frac{1}{2}\ .$ (62)

In principle, this last approach is equivalent to preparing the initial state
$\psi=\ket{0}\\!\bra{0}$ and then applying a quantum channel

$\mathcal{E}_{0}(\psi)=\frac{1}{2}\psi+\frac{1}{2}X\psi X\ ,$ (63)

so that $\rho_{0}=\mathcal{E}_{0}(\psi)$.

Starting from this last consideration, we now discuss the multiqubit
generation of a set of possibly mixed states $\\{\rho_{k}\\}$ (e.g. the DPTM
input states of Eq. 15). Consider $\\{\psi_{k}\\}$ a set of $d^{2}$, linearly
independent, _pure_ states. The requirement of purity allows to start from
states that can be easily and procedurally generated. Let $\mathcal{E}$ be the
channel that maps each initial pure state to an element of the mixed set, i.e.
$\rho_{k}=\mathcal{E}(\psi_{k})\ \forall k$. Indeed, we can split the
preparation of a set of mixed states to that of a set of pure states and the
simulation of a quantum channel.

To determine $\mathcal{E}$ from $\\{\rho_{k}\\}$ and $\\{\psi_{k}\\}$, we move
again to the vectorized notation, which yields

$\rho_{k}=\Lambda\psi_{k}\ ,$ (64)

with $\Lambda$ the PTM of $\mathcal{E}$. This problem is a quantum process
tomography task, in which each combination of input-output states is already
known. In this case, Eq. 13 reads

$\Lambda=\pi B^{-1}\ ,$ (65)

where $E_{i}=\mathcal{P}_{i}$ and

$\displaystyle B_{ij}$ $\displaystyle=\Tr[\mathcal{P}_{i}\psi_{j}]\ ,$ (66)
$\displaystyle\pi_{ij}$
$\displaystyle=\Tr[\mathcal{P}_{i}\mathcal{E}(\psi_{j})]=\Tr[\mathcal{P}_{i}\rho_{j}]\
.$ (67)

Since both $\\{\rho_{k}\\}$ and $\\{\psi_{k}\\}$ are chosen, we can
theoretically compute $\Lambda$, and then procedurally simulate it using
circuits methods like PTM.to_instruction() from the Qiskit package.

Figure 5: Conversion between sQPT and DPTM, in the identification of an
unknown quantum channel $\Phi$. The circuit corresponds to the configuration
$\langle\mathcal{P}_{i}\rangle_{\Phi(\rho_{j})}$, with the system initially
prepared in the sQPT multiqubit input states $\\{\rho_{j}^{nQ}\\}$ (obtained
by taking all the possible $n$-fold Kronecker products of ${\rho_{i}^{Q}}$).
The conversion is implemented through the channel $\mathcal{E}$, with PTM
$\Lambda$ given by Eq. 65.

When the states involved are those of DPTM and sQPT, the channel $\mathcal{E}$
provides a map between these two tomographic reconstructions, with $\pi$ and
$B$ respectively given by $\beta_{D}$ and $(\beta_{Q})^{\otimes n}$. We
summarize this protocol in Fig. 5.

Returning to the original example, we consider $n=1$. The channel that
prepares the DPTM states $\\{\rho_{k}\\}$ (see Eqs. 25, 26, 27 and 28) from
those of sQPT $\\{\rho_{k}^{Q}\\}$ (see Eq. 36) reads

$\Lambda=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&\frac{1}{2}\end{pmatrix}\ ,$ (68)

which is almost trivial, since for $n=1$ the states $\\{\rho_{k}\\}$ and
$\\{\rho^{Q}_{k}\\}$ differ only for $k=0$.

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