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 $N$ dimensions, A set of 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^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

• 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} : Dimension of state space
• 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 \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 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 \varepsilon} is 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 \varepsilon(\rho) = \sum_{i}A_i\rho A^{\dagger}_i}
• 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_i} : Operators acting on the system alone, yet they completely describe the state changes of the system, including any possible unitary operation.
• 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 \tilde{A_i}} : Fixed set of operators used to describe Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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 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_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 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 |m\rangle\langle n|}
• 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 \chi} : Classical error correlation matrix which is positive Hermitian by definition. 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 \varepsilon} is completely described by this. 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 \chi_{mn} = \sum_i a_{im}a_{in}^{*}}
• 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 \lambda_{jk}} : Parameter which can be determined from 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 \varepsilon{(\rho_j)}}
• 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 \beta_{jk}^{mn}} : Complex numbers which can be determined by standard algorithms given the 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 \tilde{A_m}} operators and the 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} operators. This is A 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} 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 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 mn} and rows indexed 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 ij} . 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 \tilde{A_m}\rho_j\tilde{A_n^{\dagger}} = \sum_k \beta^{mn}_{jk} \rho_k} .
• 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 \kappa} : the generalized inverse for the matrix 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 \beta} , satisfying the relation 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 \beta_{jk}^{mn} = \sum_{st, xy} \beta_{jk}^{st}\kappa^{xy}_{st}\beta^{mn}_{xy}}
• 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 U^{\dagger}} : Unitary matrix which diagonalizes 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 \chi}

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*contributed by Rhea Parekh