I wanted to ask how do you implement a circuit that finds the non-diagonal values of the density matrix of a quantum state on IBM Q?


Basically you add measurements in different bases by applying gates before the (Z-basis) measurement.

See here the standard implementation:


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Density matrix of single qubit state can be estimated based on this formula

\begin{equation} \rho = \frac{\text{tr}(\rho)I+\text{tr}(X\rho)X+\text{tr}(Y\rho)Y+\text{tr}(Z\rho)Z}{2}, \end{equation}

where $X$, $Y$, $Z$ are Pauli matrices.

Obviously $\mathrm{tr}(\rho) = 1$.

Terms $\mathrm{tr}(X\rho)$, $\mathrm{tr}(Y\rho)$ and $\mathrm{tr}(Z\rho)$ can be estimated by measuring a quantum state in different bases:

  • In $z$ basis you simply measure the state
  • In $x$ basis Hadamard gate has to be applied before measurement
  • In $y$ basis $S^\dagger$ gate followed by Hadamard gate have to be applied before measurement

Value of $\mathrm{tr}(A\rho)$, where $A \in \{X,Y,Z\}$, is given by

$$ \mathrm{tr}(A\rho) = \frac{1}{m}\sum_{i=1}^{m}\lambda_{i}, $$

where $m$ is number of measurements (i.e. shots on IBM Q) and $\lambda_{i}$ is eigenvalue respective to measured state. Since Pauli matrices are Hermitian and unitary, their eigenvalues are -1 and +1. Moreover, eigenvalue -1 is assigned to such eigenstate that after measurement it is mapped to state $|1\rangle$, eigenvalue +1 is mapped to state $|0\rangle$.

Note: You can check this if you calculate eigenvectors and eigenvalues of Pauli matrices and then apply above mentioned gates. Eigenstates should be mapped to $|0\rangle$ and $|1\rangle$.

For practical purposes formula for calculation $\mathrm{tr}(A\rho)$ can be rewritten in terms of measured states probability distribution followingly

$$ \mathrm{tr}(A\rho) = P(|0\rangle) - P(|1\rangle), $$

because +1 is equivalent to measuring $|0\rangle$, hence $\frac{1}{m}\sum_{\lambda = 1}\lambda_i = \frac{\#(\lambda = 1)}{m} = P(\lambda = 1) = P(|0\rangle)$. Similarly for eigenvalue -1.

Note: probabilities are expressed as decimal number, not percentage!

To sum up how to do quantum tomography on IBM Q:

  1. Prepare a qubit in some state
  2. Measure it in $x$, $y$ and $z$ bases
  3. Use probabilities of measuring $|0\rangle$ and $|1\rangle$ for estimation of $\mathrm{tr}(A\rho)$
  4. Calculate density matrix $\rho$ (the first formula in this answer)
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