Quantum Process Tomography: Measurement & Reconstruction

Hello. In our last lesson, we contrasted randomized benchmarking (RB) and quantum process tomography (QPT), establishing them as complementary tools. We saw that while RB offers a scalable benchmark of average gate performance, QPT provides a complete, high-resolution description of a quantum process, essential for detailed error diagnostics. You learned that QPT reconstructs a full mathematical description of the process, like the process matrix $\chi$, but we left the "how" of this reconstruction as an open question.

This lesson directly addresses that question. Our goal is to derive the measurement scheme for quantum process tomography and the linear inversion procedure for reconstructing the process matrix ($\chi$-matrix). We will move from the high-level concept of QPT to the precise mathematical and experimental steps required to carry it out. Given your background, we will develop the formalism from first principles.

The $\chi$-Matrix Representation of a Quantum Process

Any physical process or noise acting on a quantum state, described by its density matrix $\rho$, can be represented as a completely positive, trace-preserving (CPTP) linear map, $\mathcal{E}$. This map is often written in the operator-sum representation:

where the operators $E_k$ are the Kraus operators of the channel, satisfying ${\sum }_k{E}_k^{†}{E}_k=I$ for a trace-preserving map.

While this representation is fundamental, it's not unique. For characterizing a process, it is more convenient to work with a fixed representation. We can do this by expanding the Kraus operators $E_k$ in a fixed, complete basis of operators $\{{\tilde{A}}_m\}$ on the state space. For an $n$-qubit system of dimension $d=2^n$, this basis has $d^2$ elements. A standard choice is the set of normalized Pauli operators.

Let's express each Kraus operator $E_k$ in this basis:

where $c_{km}$ are complex coefficients. Substituting this into the operator-sum representation gives:

We can now define a new matrix, $\chi$, which encapsulates all the information about the process:

This $\chi$ matrix is called the process matrix. It allows us to write the quantum process in the following canonical form:

The $\chi$-matrix is a $d^2\times d^2$ Hermitian, positive semidefinite matrix that completely and uniquely characterizes the quantum process $\mathcal{E}$ with respect to the chosen operator basis $\{{\tilde{A}}_m\}$. Our goal in QPT is to experimentally determine all the elements of $\chi$.

The QPT Measurement Scheme

To determine the $d^4-1$ real parameters that define the $\chi$-matrix, we need to perform a series of experiments. The general procedure involves three main steps:

1. Prepare a tomographically complete set of input states: We must prepare $d^2$ linearly independent input states, whose density matrices $\{{\rho }_j\}$ form a basis for the space of $d\times d$ matrices.
2. Apply the process $\mathcal{E}$: For each prepared input state ${\rho }_j$, we apply the unknown quantum process $\mathcal{E}$ we wish to characterize.
3. Characterize the output states: For each resulting state $\mathcal{E}({\rho }_j)$, we perform full quantum state tomography (QST) to reconstruct its density matrix, which we'll call ${\rho }_j^{\prime }$.

The following resource provides a clear illustration of this experimental workflow.

With the set of known input states $\{{\rho }_j\}$ and the experimentally measured output states $\{{\rho }_j^{\prime }=\mathcal{E}({\rho }_j)\}$, we now have the data needed to solve for $\chi$.

Deriving the Linear Inversion Procedure

The core of QPT is turning the experimental data into the $\chi$-matrix. This is achieved by setting up and solving a system of linear equations.

Let's formalize the derivation. The following resource provides a concise mathematical walkthrough.

Let's break down the logic step-by-step, following the notation from the VeriQloud wiki.

1. Represent the measured outputs. After performing QST on each output state, we have the set of density matrices $\{\mathcal{E}({\rho }_j)\}$. We can express each of these in our basis of density matrices $\{{\rho }_k\}$ (which can be the same basis as the inputs):

   $$\mathcal{E}({\rho }_j)=\sum _k{\lambda }_{jk}{\rho }_k$$

   The coefficients ${\lambda }_{jk}$ are what we determine from our experimental data. For each input $j$, we get a set of coefficients ${\lambda }_{jk}$ describing the corresponding output.

2. Represent the theoretical outputs. We also have our theoretical model for the output state in terms of the unknown $\chi$ matrix:

   $$\mathcal{E}({\rho }_j)=\sum _{m,n}{\chi }_{mn}{\tilde{A}}_m{\rho }_j{\tilde{A}}_n^{†}$$

3. Connect theory and experiment. To relate these two expressions, we must expand the term ${\tilde{A}}_m{\rho }_j{\tilde{A}}_n^{†}$ in our density matrix basis $\{{\rho }_k\}$:

   $${\tilde{A}}_m{\rho }_j{\tilde{A}}_n^{†}=\sum _k{\beta }_{jk}^{mn}{\rho }_k$$

   The tensor ${\beta }_{jk}^{mn}$ is not experimental. It's a set of coefficients that can be pre-calculated once we've chosen our operator basis $\{{\tilde{A}}_m\}$ and state basis $\{{\rho }_j\}$. It simply describes how the basis operators transform the basis states.

4. Form the linear system. Now we substitute this back into our theoretical expression for $\mathcal{E}({\rho }_j)$:

   $$\mathcal{E}({\rho }_j)=\sum _{m,n}{\chi }_{mn}\left(\sum _k{\beta }_{jk}^{mn}{\rho }_k\right)=\sum _k\left(\sum _{m,n}{\beta }_{jk}^{mn}{\chi }_{mn}\right){\rho }_k$$

   By comparing this with our experimental expression, $\mathcal{E}({\rho }_j)={\sum }_k{\lambda }_{jk}{\rho }_k$, and leveraging the linear independence of the basis states ${\rho }_k$, we can equate the coefficients:

   $${\lambda }_{jk}=\sum _{m,n}{\beta }_{jk}^{mn}{\chi }_{mn}$$

This is the central result. It is a large system of linear equations. The left side, ${\lambda }_{jk}$, contains our measured data. The right side contains the product of a known, pre-calculated tensor ${\beta }_{jk}^{mn}$ and our unknown process matrix ${\chi }_{mn}$.

5. Solve for $\chi$. We can "flatten" the matrices $\lambda$ and $\chi$ into vectors and the tensor $\beta$ into a matrix $\mathbf{B}$, rewriting the system as: $$\vec{\lambda }=\mathbf{B}\vec{\chi }$$ The solution for the process matrix is then found by matrix inversion: $$\vec{\chi }={\mathbf{B}}^{-1}\vec{\lambda }$$ This final step is the linear inversion that gives us the $\chi$-matrix.

An Alternative Formulation: The Supermatrix

In many practical implementations and research papers, you'll encounter a slightly different but equivalent formalism based on a "supermatrix" or "superoperator."

This approach works by "vectorizing" the density matrices. The col operation stacks the columns of a density matrix into a single column vector. The action of the quantum process $\mathcal{E}$ can then be described by a simple matrix-vector multiplication:

Here, $M$ is the $d^2\times d^2$ supermatrix.

To find $M$, we prepare $d^2$ input states \{\rho_j\}\ and measure the corresponding output states $\{{\rho }_j^{\prime }\}$. We then construct two matrices, $R_{\text{in}}$ and $R_{\text{out}}$, whose columns are the vectorized density matrices:

From the definition of $M$, we have $R_{\text{out}}=M\cdot R_{\text{in}}$. Since the input states were chosen to be linearly independent, $R_{\text{in}}$ is invertible. We can therefore solve for the experimentally observed supermatrix $M_{\text{obs}}$ directly:

The supermatrix $M$ and the process matrix $\chi$ are simply different representations of the same underlying linear map, $\mathcal{E}$. They are related by a change of basis transformation. The $\chi$ matrix is the representation in the Pauli basis, which is often more physically intuitive, while the supermatrix $M$ is the representation in the computational basis ($|i⟩⟨j|$), which is often more direct to compute from experimental data.

Conclusion

In this lesson, we have demystified the inner workings of quantum process tomography. We moved beyond the high-level concept and derived the explicit mathematical procedure for reconstructing the process $\chi$-matrix.

Key Takeaways:

• A quantum process $\mathcal{E}$ can be fully described by a $d^2\times d^2$ positive semidefinite $\chi$-matrix in a chosen operator basis.
• The QPT measurement scheme involves preparing a basis of $d^2$ input states, applying the process, and performing state tomography on each of the $d^2$ output states.
• The raw data from QST gives the coefficients ${\lambda }_{jk}$ which describe the output states in a chosen basis.
• By relating the measured ${\lambda }_{jk}$ to the theoretical expression involving ${\chi }_{mn}$ via a pre-calculable tensor ${\beta }_{jk}^{mn}$, we form a large system of linear equations.
• Linear inversion of this system yields the $\chi$-matrix, thus completing the characterization of the quantum process.
• An alternative, equivalent formulation uses a supermatrix $M$ that acts on vectorized density matrices, which can be computed directly via $M=R_{\text{out}}{R}_{\text{in}}^{-1}$.

Preview of the next lesson:
Having established the theoretical and mathematical foundation, our next step is to put it into practice. In the next lesson, we will implement quantum process tomography for a single-qubit noisy gate and analyze the reconstructed process matrix. You will see how to generate the experimental data on a simulator and use the linear algebra we've discussed today to reconstruct and interpret a real $\chi$-matrix.