Editing 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]], 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 <math>N</math> dimensions, A set of <math>N^2</math> 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==
* <math>N</math>: Dimension of state space
* <math>\varepsilon</math>: Quantum process operator. This is the linear map which completely describes the dynamics of a quantum system, <math>\rho \xrightarrow[]{} \frac{\varepsilon(\rho)}{tr(\varepsilon(\rho))}</math>. The operator sum representation of <math>\varepsilon</math> is <math>\varepsilon(\rho) = \sum_{i}A_i\rho A^{\dagger}_i</math>
* <math>A_i</math>: Operators acting on the system alone, yet they completely describe the state changes of the system, including any possible unitary operation.
* <math>\tilde{A_i}</math>: Fixed set of operators used to describe <math>\varepsilon</math> which form a basis for the set of operators on the state space, so that <math>A_i = \sum_m a_{im}\tilde{A_m}</math>. This is done to related <math>A_i</math> to measurable operators.
* <math>a_{im}</math>: Set of complex numbers
* <math>\rho_j</math>: A set of linearly independent basis elements for the space of N x N matrices. A convienent choice is the set of projectors <math>|m\rangle\langle n|</math>
* <math>\chi</math>: Classical error correlation matrix which is positive Hermitian by definition. <math>\varepsilon</math> is completely described by this. <math>\chi_{mn} = \sum_i a_{im}a_{in}^{*}</math>
* <math>\lambda_{jk}</math>: Parameter which can be determined from <math>\varepsilon{(\rho_j)}</math>
* <math>\beta_{jk}^{mn}</math>: Complex numbers which can be determined by standard algorithms given the <math>\tilde{A_m}</math> operators and the <math>\rho_j</math> operators. This is A <math>N^4</math> x <math>N^4</math> matrix with columns indexed <math>mn</math> and rows indexed <math>ij</math>. <math>\tilde{A_m}\rho_j\tilde{A_n^{\dagger}} = \sum_k \beta^{mn}_{jk} \rho_k</math>.
* <math>\kappa</math>: the generalized inverse for the matrix <math>\beta</math>, satisfying the relation <math>\beta_{jk}^{mn} = \sum_{st, xy} \beta_{jk}^{st}\kappa^{xy}_{st}\beta^{mn}_{xy}</math>
* <math>U^{\dagger}</math>: Unitary matrix which diagonalizes <math>\chi</math>

==Properties==
* '''Figure of merit''': Density Matrix of the quantum process
* Multiple copies of the quantum state are required in this method.
* The process <math>\varepsilon</math> can be completely determined by <math>\chi</math> once the set of operators <math>\tilde{A_i}</math> has been fixed.
* <math>\chi</math> will contain <math>N^4 - N^2</math> parameters
* In the case of a “nonselective” quantum evolution, such as arises from uncontrolled interactions with an environment the <math>A_i</math> operators satisfy an additional completeness relation <math>\sum_i A_i^{\dagger} A_i = I</math>. This relation ensures that the trace factor <math>tr(\varepsilon(\rho))</math> is always equal to one, and thus the state change experienced by the system can be written <math>\rho \xrightarrow{} \varepsilon(\rho)</math>
* Quantum state tomography has to be performed multiple times in this experiment.
* This method is extremely resource intensive.
* The main drawback of this technique is that sometimes the computed solution of the density matrix will not be a valid density matrix. This issue occurs when not enough measurements are made. Due to this drawback, the maximum likelihood estimation technique [link here] is preferred over this linear inversion technique.
* Another drawback of this method is that an infinite number of measurement outcomes would be required to give the exact solution.
* Entanglement fidelity can be evaluated with the knowledge of <math>A_i</math> operators. This quantity can be used to measure how closely the dynamics of the quantum system under consideration approximates that of some ideal quantum system. This quantity can be determined robustly, because of its linear dependence on the experimental errors. Suppose the target quantum operation is a unitary quantum operation <math>U(\rho) = U\rho U^\dagger</math> and the actual quantum operation implemented experimentally is <math>\varepsilon</math>. Then the entanglement fidelity is <math>F_e(\rho, U, \varepsilon) = \sum_i {|tr(U^{\dagger}A_i \rho )|}^2 = \sum_{mn} \chi_{mn} tr(U^{\dagger}\tilde{A_m}\rho) tr(\rho \tilde{A_n^{\dagger}} U)</math>
* The minimum value of <math>F_e</math> over all possible states <math>\rho</math> is a single parameter which describes how well the experimental system implements the desired quantum logic gate
* Minimum fidelity of the gate operation is <math>F = min_{|\psi \rangle} \langle\psi| U^{\dagger} \varepsilon (|\psi\rangle\langle\psi|) U|\psi\rangle</math>
* Quantum channel capacity which is a measure of the amount of quantum information that can be sent reliably using a quantum communications channel which is described by a quantum operation <math>\varepsilon</math> can also be defined.
* This procedure can also be used to determine the form of the [[Lindblad operator]] used in Markovian master equations of a certain form.
* Quantum operations can also be used to describe measurements. For each measurement outcome, <math>i</math>, there is a associated quantum operation <math>\varepsilon_i</math>. The corresponding state change is given by <math>\rho \xrightarrow{} \varepsilon_i(\rho)/tr(\varepsilon_i(\rho))</math> where the probability of the measurement outcome occurring is <math>p_i = tr(\varepsilon_i(\rho))</math>. To determine the process the steps are the same, except the measurement has to be taken a large enough number of times that the probability pi can be reliably estimated. Next <math>\rho^{'}_{j}</math> is determined using tomography, thus obtaining <math>\varepsilon_i(\rho_j) = tr(\varepsilon_i(\rho_j))\rho^{'}_{j}</math> for each input <math>\rho_j</math>. The other steps remain to same to estimate <math>\varepsilon_i</math>.

==Protocol Description==
'''Input''': <math>p_j, j = 1, ..., N^2</math>

'''Output''': Density matrix of the quantum process operator, <math>\varepsilon</math>

* Select a fixed set of operators <math>\tilde{A_i}</math> such that <math>A_i = \sum_m a_{im} \tilde{A_m}</math>
* The operator sum representation is <math>\varepsilon(\rho) = \sum_{i}A_i\rho A^{\dagger}_i</math>
* Hence, <math>\varepsilon(\rho) = \sum_{mn}\tilde{A_m}\rho \tilde{A^{\dagger}_n}\chi_{mn}</math>
* For <math>\rho_j = 1, 2, ..., N^2</math>:
** The process <math>\varepsilon</math> is performed on <math>\rho_j</math>
** The output state <math>\varepsilon(\rho_j)</math> is measured using quantum state tomography
**  <math>\varepsilon(\rho_j)</math> is expressed as a linear combination of basis states,  <math>\varepsilon(\rho_j) = \sum_k \lambda_{jk} \rho_k</math>
** Since <math>\tilde{A_m}\rho_j\tilde{A_n^{\dagger}} = \sum_k \beta^{mn}_{jk} \rho_k</math>, then <math>\sum_k \sum_{mn} \chi_{mn} \beta^{mn}_{jk} \rho_k = \sum_k \lambda_{jk} \rho_k</math>
** Since <math>\rho_k</math> is independent, <math>\lambda_{jk} = \sum_{mn}  \beta^{mn}_{jk} \chi_{mn}</math>
* From there <math>\chi_{mn}</math> is defined as, <math>\chi_{mn} = \sum_{jk} \kappa^{mn}_{jk}</math>
* Let the <math>U^{\dagger}</math> diagonalize <math>\chi</math>, <math>\chi_{mn} =\sum_{xy} U_{mn}d_x\delta_{xy}U^{*}_{ny}</math>
* From there, <math>A_i = \sqrt{d_i} \sum_j U_{ij} \tilde{A_j}</math>
* Thus <math>\varepsilon(\rho)</math> can be determined using <math>\varepsilon(\rho) = \sum_{i}A_i\rho A^{\dagger}_i</math>

==Further Information==

* [https://arxiv.org/abs/quant-ph/0402166 Maximum likelihood estimation] (also known as MLE or MaxLik) is a popular technique that overcomes the problem that the naive matrix inversion procedure in QPT, when performed on real (i.e., inherently noisy) experimental data, typically leads to an unphysical process matrix.
* The standard approach of process tomography is grossly inaccurate in the case where the states and measurement operators used to interrogate the system are generated by gates that have some systematic error, a situation all but unavoidable in any practical setting. These errors in tomography cannot be fully corrected through oversampling or by performing a larger set of experiments. Hence [[gate set tomography]] was introduced.

==Related Papers==
* Chuang et al arXiv:quant-ph/9610001v1: Prescription for experimental determination of the dynamics of a quantum black box 
* J. F. Poyatos, J. I. Cirac, and P. Zoller, PhysRevLett.78.390: Complete Characterization of a Quantum Process: The Two-Bit Quantum Gate
* M.W. Mitchell, et al., Phys. Rev. Lett. 91, 120402 (2003) 
* J. L. O’Brien et al  arXiv:quant-ph/0402166v2: Quantum process tomography of a controlled-not gate
* S. T. Merkel, J. M. Gambetta, J. A. Smolin, S. Poletto, A. D. Corcoles, B. R.
* Johnson, C. A. Ryan, and M. Steffen, Phys. Rev. A 87, 062119 (2013).

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>