Two-qubit state tomography.

To measure the density matrix of the output state of a two-qubit circuit, different combinations of I, X/2 and Y/2 operations are applied to the two qubits before measurements in the z-basis to determine the state probabilities \(P_{00}, P_{01}, P_{10}.\)

The density matrix rho is decomposed into an operator-sum representation \(\sum_{i, j} c_{ij} * \sigma_i \bigotimes \sigma_j\), where $i, j = 0, 1, 2, 3\( and \)\sigma_0 = I, \sigma_1 = \sigma_x, \sigma_2 = \sigma_y, \sigma_3 = \sigma_z$ are the single-qubit Identity and Pauli matrices.

Based on the measured probabilities probs and the transformations of the measurement operator by different basis rotations, one can build an overdetermined set of linear equations.

As an example, if the identity operation (I) is applied to both qubits, the measurement operators are \((I +/- \sigma_z) \bigotimes (I +/- \sigma_z)\). The state probabilities \(P_{00}, P_{01}, P_{10}\) thus obtained contribute to the following linear equations (setting \(c_{00} = 1\)):

\[ \begin{align} c_{03} + c_{30} + c_{33} &= 4*P_{00} - 1 \\ -c_{03} + c_{30} - c_{33} &= 4*P_{01} - 1 \\ c_{03} - c_{30} - c_{33} &= 4*P_{10} - 1 \end{align} \]

And if a Y/2 rotation is applied to the first qubit and a X/2 rotation is applied to the second qubit before measurement, the measurement operators are \((I -/+ \sigma_x) \bigotimes (I +/- \sigma_y)\). The probabilities obtained instead contribute to the following linear equations:

\[ \begin{align} c_{02} - c_{10} - c_{12} &= 4*P_{00} - 1 \\ -c_{02} - c_{10} + c_{12} &= 4*P_{01} - 1 \\ c_{02} + c_{10} + c_{12} &= 4*P_{10} - 1 \end{align} \]

Note that this set of equations has the same form as the first set under the transformation \(c_{03}\) <-> \(c_{02}, c_{30}\) <-> \(-c_{10}\) and \(c_{33}\) <-> \(-c_{12}\).

Since there are 9 possible combinations of rotations (each producing 3 independent probabilities) and a total of 15 unknown coefficients \(c_{ij}\), one can cast all the measurement results into a overdetermined set of linear equations, c) = probs. Here c is of length 15 and contains all the \(c_{ij}\)'s (except \(c_{00}\) which is set to 1), and mat is a 27 by 15 matrix having three non-zero elements in each row that are either 1 or -1.

The least-square solution to the above set of linear equations is then used to construct the density matrix rho.

See Vandersypen and Chuang, Rev. Mod. Phys. 76, 1037 for details and Steffen et al, Science 313, 1423 for a related experiment.

sampler The quantum engine or simulator to run the circuits.
first_qubit The first qubit under test.
second_qubit The second qubit under test.
circuit The circuit to execute on the qubits before tomography.
repetitions The number of measurements for each basis rotation.

A TomographyResult object that stores and plots the density matrix.