Quantum State Tomography

The latest version of this notebook is available on https://github.com/Qiskit/qiskit-tutorial.

Contributors

Christopher J. Wood$^{1}$ and Jay M. Gambetta$^{1}$

1. IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

Introduction

In this notebook we demonstrate how to design and run experiments to perform quantum state tomography using Qiskit, and demonstrate this using both simulators and the IBM Q Experience hardware.

The functions used in this notebook to implement state tomography may be imported from the qiskit.tools.qcvv.tomography module.

Experiment Design

The tomography experiments demonstrated in this notebook append a state preparation circuit with measurements of all qubits in each of the X, Y, and Z Pauli bases. For $n$-qubits this gives a total of $3^n$ measurement circuits which must be run, and the resulting counts for the $6^n$ measurement outcomes across all circuits give a tomographically overcomplete basis for reconstruction of the quantum state.

State Reconstruction Methods

State reconstruction may be done using a variety of methods. In this notebook we implement two simple cases:

• Linear Inversion
• Fast maximum likelihood (see J Smolin, JM Gambetta, G Smith, Phys. Rev. Lett. 108, 070502, open access: arXiv:1106.5458).

State tomography of an entangled Bell-state

We will demonstrate preparation, measurement, and tomographic reconstruction of a 2-qubit Bell state $\frac{1}{\sqrt2}\left(|00\rangle+|11\rangle\right)$.

State preparation

We begin with constructing a circuit to prepare the target state. This circuit requires a 2-qubit quantum register, and a 2-bit classical register to store measurement outcomes.

Visualization of the ideal state

Before we perform tomographic measurement, we will visualize the ideal output state for the preparation circuit. We may obtain the final state-vector by executing the circuit on Aer's "statevector_simulator" backend:

We may visualize the final state using the plot_state function. By default this creates a city plot, which is a 2D-bar plot of real and imaginary matrix elements of a density matrix $\rho$. Here, we instead ask it to plot the Pauli state vector.

We can compare the ideal entangled state to a non-entangled mixed state $\frac{1}{2}(|00\rangle\langle00| +|11\rangle\langle11|)$:

Building tomography measurement circuits

Next we must construct a family of circuits that implements a tomographically complete set of measurements of the qubit. The qiskit.tools.qcvv.tomography module contains functions to generate these measurement circuits for general n-qubit systems.

Specifying the measured qubits and measurement basis

A state tomography experiment is specified by a state_tomography_set object:

tomo_set = tomo.state_tomography_set(qubits, meas_basis)

where

• qubits is a list of the qubit indexes within a register (eg. [0, 1] for the first 2 qubits)
• meas_basis is the basis to measure each qubit in. The default value is meas_basis='Pauli'.

This contains information about which qubits are to be measured, and in what basis. The default option is to measure each single qubit in the X, Y, and Z bases. This results in $3^n$ measurement circuits that must be executed to gather count statistics for the tomographic reconstruction.

Adding measurement circuits to a Quantum Program

Once a state tomography set has been defined, the function to append the tomography circuits to a QuantumProgram is:

tomo_circuits = tomo.create_tomography_circuits(circuit, qr, cr, tomo_set)

where

• circuit is the preparation circuit 'name', to be appended with measurements.
• qr is the quantum register contained in circuit to be measured.
• cr is the classical register in circuit to store the measurement results.
• tomo_set is a state_tomography_set, the dict of tomography configurations.

The function output tomo_circuits is a list of quantum tomography circuits used to execute the tomography experiments. The measured qubits will be qr[j], with corresponding measurement outcomes stored in classical bits cr[j], where j corresponds to each qubit in the state tomography set.

Testing experiments on a simulator

Now that we have prepared the required circuits for state preparation and measurement, we should test them on a simulator before trying to run them on the real device.

We specify the device, and a number of experiment shots to perform to gather measurement statistics. The larger the number of shots, the more accurate our measurement probabilities will be compared to the true value.

Extracting tomography data from results

Next we extract the tomography data from the output results using:

data = tomo.tomography_data(results, 'name', state_tomo_set)

where

• results is the Result object returned from execution of state_tomo_set circuits.
• name is the name of the original circuit used to generate tomography data.
• state_tomo_set is the tomography set used for generation of tomography data.

This returns a dictionary that stores the measurement basis, and measured counts along with measurement basis configuration corresponding to each outcome.

Reconstructing state from count data

To reconstruct the maximum likelihood estimate of the measured quantum state, we use the following function:

tomo.fit_tomography_data(tomo_data, method, options)

where

• QP is the quantum program containing the measurement results
• circuits is the array of tomographic measurement circuits measured
• shots is the total number of shots for each measurement circuit
• total_qubits is the total number of qubits in the system (the length of shot outcome bitstrings)
• meas_qubits is an array of the measurement qubit indices

We can compare the reconstructed state to the target state vector. We use the Fidelity function, which for comparing a density matrix $\rho$ to a pure state $|\psi\rangle$ is given by $F = \sqrt{\langle \psi| \rho |\psi\rangle}$. This may be done by the function state_fidelity from the qiskit.tools.qi module.

Note that since our simulator is perfect, the output state should be exactly the Bell state, so we should obtain F = 1. Why is it not, in our case? Since we can never directly see the final state, we must obtain information about it via measurements. We would only obtain the true probabilities for the state in the limit of infinite measurement shots. Hence, we have statistical error in our reconstruction from imperfect information about the state itself. Try running with a different number of shots on the simulator and see how it affects the fidelity of the reconstruction.

Running on a real device

Now that we've checked that our simple tomography experiment worked, let's try it out on the IBM Quantum Experience! To do this, we must attach our API key, and it is good practice to set a limit on the number of credits to use:

You can monitor the status of your job, and ask for the result.

Note: a call to job.result() blocks until the job execution is done and result is available.

As before, we can extract the tomography data and reconstruct the measured density matrix for the 2-qubit Bell state prepared by our test circuit

Finally, we compare the fidelity of the experimentally measured state to the ideal state. Notice that the fidelity is lower than for the simulation. This is because in a real experiment, there is not only the statistical error in the reconstruction, but also experimental error in the device when implementing the gates in the preparation and measurement circuits, as well as error in the measurements themselves.