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

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 (CPTP) 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 [6–11](eventually after a post-processing manipulation of the data), or indirectly, i.e. when it requires additional techniques to analyze the output state (e.g. state tomography) [12–14].

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 d⁴, where d is the dimension of the system (we do not consider overcomplete sets of states, whose cost is even higher).

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 figure 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.

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

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

$$\begin{equation} \left\{\sigma_{\alpha_1} \otimes \sigma_{\alpha_2} \otimes {\ldots} \otimes \sigma_{\alpha_n} \ | \ \alpha_1, \alpha_2, {\ldots}, \alpha_n = 0,1,2,3 \right\} , \end{equation} \tag{ 1 }$$

with $\sigma_0 = \unicode{x1D7D9}_2$, $\sigma_1 = X$, $\sigma_2 = Y$ and $\sigma_3 = Z$. We write the Pauli basis in the following notation

$$\begin{equation} \left\{\mathcal{P}_k \ | \ k = 0,1,2,3,{\ldots},d^2-1 \right\} , \end{equation} \tag{ 2 }$$

with $d = 2^n$ and $\mathcal{P}_k$ given by the generic element of equation (1) in lexicographic order.

Consider a system with quantum state ρ. A quantum channel (or quantum operation) is a linear 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 $\{A_i\}_{0 \unicode{x2A7D} i \unicode{x2A7D} d^2-1}$, called Kraus operators, such that

$$\begin{equation} \Phi\left(\rho\right) = \sum_{i} A_i \rho A^{\dagger}_i . \end{equation} \tag{ 3 }$$

The CPTP condition implies that

$$\begin{equation} \sum_{i} A^\dagger_i A_i = \unicode{x1D7D9} , \end{equation} \tag{ 4 }$$

with $\unicode{x1D7D9}$ the identity operator.

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

$$\begin{equation} \Gamma_{ij} = \frac{1}{d}\mathrm{Tr}\left[\mathcal{P}_i\Phi\left(\mathcal{P}_j\right)\right] . \end{equation} \tag{ 5 }$$

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 $|{A}\rangle\rangle$, on which quantum channel acts through standard matrix multiplication

$$\begin{equation} \Phi\left(\rho\right) = \Gamma |{\rho}\rangle\rangle . \end{equation} \tag{ 6 }$$

This representation is equipped with the Hilbert–Schmidt inner product, so that $A_i = \langle \langle i | A\rangle \rangle$ and equation (5) reads

$$\begin{equation} \Gamma_{ij} = \langle\langle{i}|\Gamma |{j}\rangle\rangle , \end{equation} \tag{ 7 }$$

with $|{i}\rangle\rangle$ 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 equation (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 Φ be a quantum channel and Γ its $d^2 \times d^2$ PTM representation, whose definition is given in equation (5). The CPTP condition implies that

$$\begin{equation} \Gamma_{0j} = \delta_{0j} . \end{equation} \tag{ 8 }$$

with $0 \unicode{x2A7D} i,j \unicode{x2A7D} d^2-1$ and δ_ij denoting the Kronecker delta. If the channel is unital, i.e. $\Phi(\unicode{x1D7D9}) = \unicode{x1D7D9}$, then also

$$\begin{equation} \Gamma_{i0} = \delta_{i0} . \end{equation} \tag{ 9 }$$

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.

The goal of QPT is to reconstruct an unknown quantum channel Φ from the statistics of a collection of experimental configurations. Namely, by applying a set of measurement operators $\{E_i\}_{0\unicode{x2A7D} i \unicode{x2A7D} d^2-1}$ to different choices of input states $\{\rho_j\}_{0\unicode{x2A7D} j \unicode{x2A7D} d^2-1}$ [6, 7]. After having collected sufficient statistics (i.e. on a large number of shots), the outcome of each configuration reads

$$\begin{equation} p_{ij} = \mathrm{Tr}\left[E_i\Phi\left(\rho_j\right)\right] . \end{equation} \tag{ 10 }$$

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

$$\begin{equation} p = \alpha \Gamma \beta , \end{equation} \tag{ 11 }$$

with

$$\begin{equation} \alpha_{ij} = \frac{1}{d}\mathrm{Tr}\left[E_i \mathcal{P}_j\right] , \quad \beta_{ij} = \mathrm{Tr}\left[\mathcal{P}_i \rho_j\right] . \end{equation} \tag{ 12 }$$

We refer to α and β as reconstruction matrices, which, once inverted, provide the PTM in terms of the measurement data as

$$\begin{equation} \Gamma = \alpha^{-1} p \beta^{-1} , \end{equation} \tag{ 13 }$$

whenever α⁻¹ and β⁻¹ exist.

In practice, the PTM is often reconstructed using a tomographic fitter instead of equation (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 equation (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

$$\begin{equation} \rho_0 = \frac{\unicode{x1D7D9}}{d} , \quad \rho_k = \frac{\unicode{x1D7D9} + \mathcal{P}_k}{d} \quad \text{for } k \neq 0 , \end{equation} \tag{ 14 }$$

which can be compactly written as ³

$$\begin{equation} \rho_k = \frac{1}{d}\left[ \left(1-\delta_{0k}\right)\unicode{x1D7D9} + \mathcal{P}_k \right]. \end{equation} \tag{ 15 }$$

These states are positive semidefinite and normalized [16]. Moreover, they are mixed, except for n = 1 and k ≠ 0. See the appendix for considerations on how they can be prepared. The channel Φ evolves ρ_k to

$$\begin{equation} \Phi\left(\rho_k\right) = \frac{1}{d}\left[ \left(1-\delta_{0k}\right)\Phi\left(\unicode{x1D7D9}\right) + \Phi\left(\mathcal{P}_k\right) \right]. \end{equation} \tag{ 16 }$$

Consider the ith element of the Pauli basis $\mathcal{P}_i$. As an observable, its expectation value against $\Phi(\rho_k)$ is

$$\begin{equation} \langle\mathcal{P}_i\rangle_{\Phi\left(\rho_j\right)} = \mathrm{Tr}\left[\mathcal{P}_i \Phi\left(\rho_j\right)\right] , \end{equation} \tag{ 17 }$$

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

$$\begin{equation} \langle\mathcal{P}_i\rangle_{\Phi\left(\rho_j\right)} = \left(1-\delta_{0j}\right) \Gamma_{i0} + \Gamma_{ij} . \end{equation} \tag{ 18 }$$

When j = 0 the second contribution vanishes, yielding

$$\begin{equation} \langle\mathcal{P}_i\rangle_{\Phi\left(\rho_0\right)} = \Gamma_{i0} . \end{equation} \tag{ 19 }$$

By subtracting this term from equation (18), we complete the PTM reconstruction as

$$\begin{equation} \Gamma_{ij} = \begin{cases} \langle\mathcal{P}_i\rangle_{\Phi\left(\rho_0\right)} \quad &\text{for } j = 0\\ \langle\mathcal{P}_i\rangle_{\Phi\left(\rho_j\right)}-\langle\mathcal{P}_i\rangle_{\Phi\left(\rho_0\right)} \quad &\text{for } j \neq 0 \end{cases} . \end{equation} \tag{ 20 }$$

More compactly, the DPTM equation reads

$$\begin{equation} \Gamma_{ij} = \langle\mathcal{P}_i\rangle_{\Phi\left(\rho_j\right)} - \left(1-\delta_{0j}\right)\langle\mathcal{P}_i\rangle_{\Phi\left(\rho_0\right)} . \end{equation} \tag{ 21 }$$

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. d² input states $\{\rho_j\}$ coupled to d² 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 ρ_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).

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.

	General	CPTP	Unitality	Pauli
Resources	d4	$d^2(d^2 - 1)$	$(d^2-1)^2$	$d^2 - 1$

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 equation (13) to (15). With $E_{i} = \mathcal{P}_i$, then $p_{ij} = \langle \mathcal{P}_i\rangle_{\Phi(\rho_j)}$ and the reconstruction matrices read

$$\begin{equation} \alpha_{ij} = \delta_{ij} , \quad \beta_{ij} = \left(1-\delta_{0j}\right)\delta_{i0} + \delta_{ij} , \end{equation} \tag{ 22 }$$

Their inversion yields

$$\begin{equation} \alpha^{-1}_{ij} = \delta_{ij} , \quad \beta_{ij}^{-1} = \delta_{ij} - \left(1-\delta_{0j}\right)\delta_{i0} . \end{equation} \tag{ 23 }$$

By direct substitution, this reduces equation (13) precisely to equation (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.

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

$$\begin{equation} \mathcal{P}_{0} = \unicode{x1D7D9}_2 , \quad \mathcal{P}_{1} = X , \quad \mathcal{P}_{2} = Y , \quad \mathcal{P}_{3} = Z . \end{equation} \tag{ 24 }$$

By substitution in equation (15), the set of input states reads

$$\begin{align} &\rho_0 = \frac{1}{2}\left( |{0}\rangle\!\langle{0}| + |{1}\rangle\!\langle{1}|\right) , \end{align} \tag{ 25 }$$

$$\begin{align} &\rho_1 = |{+}\rangle\!\langle{+}| , \end{align} \tag{ 26 }$$

$$\begin{align} &\rho_2 = |{i}\rangle\!\langle{i}| , \end{align} \tag{ 27 }$$

$$\begin{align} &\rho_3 = |{0}\rangle\!\langle{0}| . \end{align} \tag{ 28 }$$

with $\sqrt{2}|{\pm}\rangle = |{0}\rangle \pm |{1}\rangle$ and $\sqrt{2}|{\pm i}\rangle = |{0}\rangle \pm i|{1}\rangle$. In this case only ρ₀ is mixed, we discuss its preparation in the appendix. The remaining states are pure and can be prepared as

$$\begin{equation} |{\psi_1}\rangle = H|{0}\rangle , \quad |{\psi_2}\rangle = SH|{0}\rangle , \quad |{\psi_3}\rangle = |{0}\rangle , \end{equation} \tag{ 29 }$$

with H and S respectively the Hadamard and the phase gates [1]. Given a state ρ, 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

$$\begin{align} &\langle \mathcal{P}_1 \rangle = \mathrm{Tr}\left[ZH \rho H \right], \end{align} \tag{ 30 }$$

$$\begin{align} &\langle \mathcal{P}_2 \rangle = \mathrm{Tr}\left[ ZHS^{\dagger} \rho SH\right], \end{align} \tag{ 31 }$$

$$\begin{align} &\langle \mathcal{P}_3 \rangle = \mathrm{Tr}\left[ Z \rho \right], \end{align} \tag{ 32 }$$

where we used

$$\begin{equation} \mathcal{P}_1 = HZH , \quad \mathcal{P}_2 = SH Z HS^{\dagger} . \end{equation} \tag{ 33 }$$

Each change of basis translates the respective Pauli measurements to a simple count of the 0, 1 occurrences in the computational basis ⁴ .

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

$$\begin{align} &\Gamma_{i0} = M^D_{i0} , \nonumber\\ &\Gamma_{ij} = M^D_{ij} - M^D_{i0} \quad \text{for } j \neq 0 . \end{align} \tag{ 34 }$$

In terms of the reconstruction matrices α_D and β_D in equation (23), this gives $\alpha_D^{-1} = \unicode{x1D7D9}$ and

$$\begin{equation} \beta_D^{-1} = \begin{pmatrix} +1 & -1 & -1 & -1 \\ 0 & +1 & 0 & 0 \\ 0 & 0 & +1 & 0 \\ 0 & 0 & 0 & +1 \end{pmatrix} . \end{equation} \tag{ 35 }$$

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 sQPT ⁵ in which the set of Pauli measurements is usually coupled to the following set of states

$$\begin{equation} \begin{split} &\rho^Q_0 = |{1}\rangle\!\langle{1}| , \quad \rho^Q_1 = |{+}\rangle\!\langle{+}| , \\ &\rho^Q_2 = |{i}\rangle\!\langle{i}| \ , \quad \rho^Q_3 = |{0}\rangle\!\langle{0}| . \end{split} \end{equation} \tag{ 36 }$$

The Pauli measurements precisely matches the one of equations (30)–(32), with still $\alpha^{-1}_Q = \unicode{x1D7D9}$. However, the different choice of input states modifies β_Q , whose inverse now reads

$$\begin{equation} \beta^{-1}_Q = \frac{1}{2}\begin{pmatrix} +1 & -1 & -1 & -1 \\ 0 & +2 & 0 & 0 \\ 0 & 0 & +2 & 0 \\ +1 & -1 & -1 & +1 \end{pmatrix} . \end{equation} \tag{ 37 }$$

By substitution in equation (13), with $M^Q_{ij} : = \langle \mathcal{P}_i \rangle_{\Phi\left(\rho^Q_j\right)}$ given by equation (11), it follows

$$\begin{align} \Gamma_{i0} & = \frac{1}{2}\left(M^Q_{i0} + M^Q_{i3}\right) , \end{align} \tag{ 38 }$$

$$\begin{align} \Gamma_{i1} & = \frac{1}{2}\left(-M^Q_{i0} + 2M^Q_{i1} - M^Q_{i3}\right) , \end{align} \tag{ 39 }$$

$$\begin{align} \Gamma_{i2} & = \frac{1}{2}\left(-M^Q_{i0} + 2M^Q_{i2} - M^Q_{i3}\right) , \end{align} \tag{ 40 }$$

$$\begin{align} \Gamma_{i3} & = \frac{1}{2}\left(-M^Q_{i0} + M^Q_{i3}\right) . \end{align} \tag{ 41 }$$

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 eight configurations to sQPT, while only three 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 equation (15), while its reconstruction matrix shows high sparsity

$$\begin{equation} \beta_D^{-1} = \begin{pmatrix} +1 & -1 & \dots & -1 \\ 0 & \ddots & 0 & 0 \\ \vdots & 0 & \ddots & \vdots \\ 0 & 0 & \dots & +1 \end{pmatrix} . \end{equation} \tag{ 42 }$$

On the other hand, the n-qubit states for sQPT can be obtained by taking all the possible n-fold Kronecker products of ${\rho_i^Q}$. Equation (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 ⁶ . 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 Φ. Let Γ be its Pauli transfer matrix, reconstructed using DPTM and sQPT. Then

$$\begin{equation} \max_{i,j}||\Gamma_{ij}||_{D} \unicode{x2A7D} \min_{i,j} ||\Gamma_{ij}||_{Q} < \max_{i,j} ||\Gamma_{ij}||_{Q} , \end{equation} \tag{ 43 }$$

with the strict equality satisfied only for n =  Indeed

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

while

$$\begin{align} \min_{i,j} ||\Gamma_{ij}||_{Q} & = 2^n , \end{align} \tag{ 45 }$$

$$\begin{align} \max_{i,j} ||\Gamma_{ij}||_{Q} & = 3^n . \end{align} \tag{ 46 }$$

Proof. Equation (44) is a direct consequence of equation (20). On the other hand, the sQPT reconstruction for n-qubit yields

$$\begin{equation} \Gamma = M^Q\left(\beta^{-1}_Q\right)^{\otimes n} . \end{equation} \tag{ 47 }$$

From equations (37), (45) and (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 figure 2.

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

5.1. Amplitude damping characterization

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

$$\begin{equation} A_0 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-p} \end{pmatrix} \ A_1 = \begin{pmatrix} 0 & \sqrt{p} \\ 0 & 0 \ \end{pmatrix} , \end{equation} \tag{ 48 }$$

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

$$\begin{align} \Gamma = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \sqrt{1-p} & 0 & 0 \\ 0 & 0 & \sqrt{1-p} & 0 \\ \,p & 0 & 0 & 1 - p \end{pmatrix} . \end{align} \tag{ 49 }$$

The purpose of this section is to test both DPTM and sQPT in the characterization of the non-trivial components Γ₁₁, Γ₂₂, Γ₃₀ and Γ₃₃ (namely using equations (34) and (38)–(41)).

In figure 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.

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5.2. 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, 4, 23, 26, 27]. We start from the class of correlated Pauli channels [3, 4, 23], whose Kraus representation reads

$$\begin{equation} \Phi\left(\rho\right) = \sum_{\alpha_1, \alpha_2 = 0}^{3} p_{\alpha_1 \alpha_2} A_{\alpha_1 \alpha_2} \rho A_{\alpha_1 \alpha_2} , \end{equation} \tag{ 50 }$$

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

$$\begin{align} &p_{\alpha_{j}|\alpha_{i}} = \left(1-\mu\right)p_{\alpha_j} + \mu\delta_{\alpha_i \alpha_j} \, \end{align} \tag{ 51 }$$

$$\begin{align} &\vec{p} = \left[1-p,p_x,p_y,p_z\right]^{T} . \end{align} \tag{ 52 }$$

and $p = p_x + p_y + p_z$ ⁷ . A correlated depolarizing channel is obtained by choosing $\vec{p} = [1-3p/4,$ $p/4,p/4,p/4]^T$.

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

$$\begin{align} \Gamma_{44} & = 1 - p , \end{align} \tag{ 53 }$$

$$\begin{align} \Gamma_{66} & = \left(1-p\right)\left(\mu p - p + 1\right) . \end{align} \tag{ 54 }$$

Their inverses allow the parameter extraction as $p \leftarrow \Gamma_{44}$ and $\mu \leftarrow \Gamma_{44}$, Γ₆₆. 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 two-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

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

along with

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

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

$$\begin{equation} \Gamma_{44} = M^D_{44} , \quad \Gamma_{66} = M^D_{66} , \end{equation} \tag{ 57 }$$

with observables in the Pauli basis and input states given by equation (15). The implementation of these measurements is reported in figure 4.

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By choosing p = 0.25 and µ = 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

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

and

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

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).

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 d⁴ 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) two 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.

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.

The data that support the findings of this study are available upon reasonable request from the authors.

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

For single-qubit channels, ρ₀ in equation (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

$$\begin{equation} |{\Psi_0}\rangle = \frac{1}{\sqrt{2}}\sum_{k = 0}^{1}|{k}\rangle\otimes|{k}\rangle_a , \end{equation} \tag{ A1 }$$

and tracing out the ancillary qubit

$$\begin{equation} \rho_0 = \mathrm{Tr}_{a}\left[|{\Psi_0}\rangle\!\langle{\Psi_0}|\right] , \end{equation} \tag{ A2 }$$

or by classical random generation of its pure components

$$\begin{equation} |{\psi_0}\rangle \in \left\{|{0}\rangle, |{1}\rangle \right\} \quad \text{with } p\left(|{0}\rangle\right) = p\left(|{1}\rangle\right) = \frac{1}{2} . \end{equation} \tag{ A3 }$$

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

$$\begin{equation} \mathcal{E}_0\left(\psi\right) = \frac{1}{2}\psi + \frac{1}{2}X\psi X , \end{equation} \tag{ A4 }$$

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 equation (15)). Consider $\{\psi_k\}$ a set of d², 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

$$\begin{equation} \rho_k = \Lambda \psi_k , \end{equation} \tag{ A5 }$$

with Λ the PTM of $\mathcal{E}$. This problem is a QPT task, in which each combination of input-output states is already known. In this case, equation (13) reads

$$\begin{equation} \Lambda = \pi B^{-1} , \end{equation} \tag{ A6 }$$

where $E_i = \mathcal{P}_i$ and

$$\begin{align} B_{ij} & = \mathrm{Tr}\left[\mathcal{P}_i \psi_j\right] , \end{align} \tag{ A7 }$$

$$\begin{align} \pi_{ij} & = \mathrm{Tr}\left[\mathcal{P}_i\mathcal{E}\left(\psi_j\right)\right] = \mathrm{Tr}\left[\mathcal{P}_i \rho_j\right] . \end{align} \tag{ A8 }$$

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

When the states involved are those of DPTM and sQPT, the channel $\mathcal{E}$ provides a map between these two tomographic reconstructions, with π and B respectively given by β_D and $(\beta_Q)^{\otimes n}$. We summarize this protocol in figure A1.

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Returning to the original example, we consider n = 1. The channel that prepares the DPTM states $\{\rho_k\}$ (see equations (25)–(28)) from those of sQPT $\{\rho_k^Q\}$ (see equation (36)) reads

$$\begin{equation} \Lambda = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \end{pmatrix} , \end{equation} \tag{ A9 }$$

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