Full Quantum Process Tomography with Linear inversion

Quantum process tomography is a method used to characterize a physical process in an open quantum system. This method is similar to Quantum state tomography [link], but the goal here is to characterize a quantum gate instead of a state. The method described here performs quantum process tomography using linear inversion.

Tags: Certification protocol, Tomography, Quantum process density matrix reconstruction, Linear inversion

Assumptions

Noise is assumed to be independent and identically distributed.
The initial states and final measurements are known.
Here the case of a “nonselective” quantum evolution, such as arises from uncontrolled interactions with an environment.

Outline

Quantum process tomography is used to characterise an unknown quantum dynamical process. The general way to describe state change or a process experienced by a quantum system is by using quantum operations. This quantum process operation is a linear map which completely describes the dynamics of the quantum system.

This linear map can be described by an operator sum representation, where the map is broken down into different operators acting on the system. These different operators completely describe the state changes of the system including any possible unitary operations. It is convenient to consider an equivalent description of the process using a fixed set of these operators, so as to relate them to measurable parameters.

This method consists of the following steps:

For a system with the state space having Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} dimensions, A set of $N^{2}$ linearly independent basis elements are selected and the corresponding quantum states are prepared.
The quantum process is performed on each of these states and the output state is measured using quantum state tomography.
These output states are expressed as a linear combination of the basis states. A mathematical relation is formed which is used to determine the different operators which describe the quantum process.

Hardware Requirements

Measurement device.
Quantum computational resources to perform the operation.

Notation

$N$ : Dimension of state space
$\varepsilon$ : Quantum process operator. This is the linear map which completely describes the dynamics of a quantum system, $\rho {\xrightarrow[{}]{}}{\frac {\varepsilon (\rho )}{tr(\varepsilon (\rho ))}}$ . The operator sum representation of $\varepsilon$ is $\varepsilon (\rho )=\sum _{i}A_{i}\rho A_{i}^{\dagger }$
$A_{i}$ : Operators acting on the system alone, yet they completely describe the state changes of the system, including any possible unitary operation.
${\tilde {A_{i}}}$ : Fixed set of operators used to describe $\varepsilon$ which form a basis for the set of operators on the state space, so that $A_{i}=\sum _{m}a_{im}{\tilde {A_{m}}}$ . This is done to related $A_{i}$ to measurable operators.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{im}} : Set of complex numbers
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_j} : A set of linearly independent basis elements for the space of N x N matrices. A convienent choice is the set of projectors $|m\rangle \langle n|$
$\chi$ : Classical error correlation matrix which is positive Hermitian by definition. $\varepsilon$ is completely described by this. $\chi _{mn}=\sum _{i}a_{im}a_{in}^{*}$
$\lambda _{jk}$ : Parameter which can be determined from $\varepsilon {(\rho _{j})}$
$\beta _{jk}^{mn}$ : Complex numbers which can be determined by standard algorithms given the ${\tilde {A_{m}}}$ operators and the $\rho _{j}$ operators. This is A $N^{4}$ x Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N^4} matrix with columns indexed $mn$ and rows indexed $ij$ . ${\tilde {A_{m}}}\rho _{j}{\tilde {A_{n}^{\dagger }}}=\sum _{k}\beta _{jk}^{mn}\rho _{k}$ .
$\kappa$ : the generalized inverse for the matrix $\beta$ , satisfying the relation $\beta _{jk}^{mn}=\sum _{st,xy}\beta _{jk}^{st}\kappa _{st}^{xy}\beta _{xy}^{mn}$
$U^{\dagger }$ : Unitary matrix which diagonalizes $\chi$