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Cross resonance quantum process tomography

The notebook implements QPT for a cross resonance gate.

Description

Quantum process tomography (QPT) is a procedure that reconstructs an unknown quantum process from measurement data, serving as a fundamental tool for diagnosing and characterizing quantum gates and circuits1. Essentially, it is possible to describe the change of \(\rho\) in a discrete-time step \(\Delta t\) as a quantum process* \(\mathcal{E}\): \(\rho(t_0) \rightarrow \rho(t_0 + \Delta t) =\mathcal{E}(\rho)\). The procedure to infer \(\mathcal{E}\) is called QPT 2.

In QPT, the goal is to characterise and reconstruct the quantum channel that describes the evolution of a quantum system under the influence of a particular circuit. The quantum channel is a mathematical representation of the transformation applied to a quantum state. One possible representation is achieved through the 'superoperator' \(\chi\) known as the quantum process matrix3, being a linear operator within the space of density matrices. \(\chi\) captures the action of the quantum channel on a density matrix, which represents the state of a quantum system, and its reconstruction via a finite number of experiments is the main goal of QPT. The Chi matrix \(\xi\) is defined as:

\[\rho_{ab} = \sum_{cd} \chi_{ab,cd} \rho_{cd}\]

Along these lines, this notebook implements QPT for a cross resonance gate. Refer to the Cross Resonance Amplitude Sweep experiment for further details.

Experimental steps

  1. Defining the custom_gates needed to perform cross-resonance QPT. We employ custom_r90s + custom\ _r180s. This circuit represents the quantum process circuit we aim to reconstruct.

  2. Executing the QuantumProcessTomography experiment, which consists of:

    1. Tomography Circuit: creating a quantum circuit (pt_circuit) for performing tomography on a specified quantum instruction (self.instr = zx_instr). We employ the class ProcessTomography from Qiskit 4, which consists of an experiment to reconstruct the quantum channel from measurement data, returning the target quantum channel for process tomography.

    2. Base Circuit: creating a base quantum circuit (base_circuit) that includes the definitions of gates from custom_gates instructions.

    3. Tomography Circuits: executing the transpiled tomography circuits on a quantum backend.

    4. Retrieve and Format Data: Measuring the resonator transmission and collecting the \(I\) and \(Q\) signals for each qubit and circuit.

Analysis steps

  1. Initializing a ProcessTomography object with the target quantum circuit (target_circ). The tomography is performed using the state vector simulator backend (aer_simulator_statevector) with a specified number of shots (statevector_shots) 4.

  2. Setting a random seed and configuring bootstrap** options for the process tomography analysis.

  3. Running ProcessTomography, results are stored in pt_experiment_data.

  4. Obtaning the \(\Xi\) matrix for the target circuit.

  5. Applying error mitigation using a composite_discriminator.

  6. Updating the experiment data with the sampled mitigated counts.

  7. Re-running ProcessTomography with the updated experiment data containing the sampled mitigated counts.

  8. Storing measured \(\chi\) matrix and fidelity.

* A general quantum process \(\mathcal{E}\) can be described by a set of Kraus operators, \(E_j\), such that \(E(\rho) = Pj E_j\rho E_j^{†}\).

** Bootstrapping is a resampling technique. This method involves the generation of multiple datasets by drawing samples with replacement from the original data. It is used to estimate the distribution of a statistic, providing valuable insights into the variability and uncertainty intrinsic to the data.

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  1. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010. doi:10.1017/CBO9780511976667. ↩

  2. Yanzhu Chen, Maziar Farahzad, Shinjae Yoo, and Tzu-Chieh Wei. Detector tomography on ibm quantum computers and mitigation of an imperfect measurement. Phys. Rev. A, 100:052315, Nov 2019. doi:10.1103/PhysRevA.100.052315. ↩

  3. Joel Yuen-Zhou, Jacob J Krich, Ivan Kassal, Allan S Johnson, and Alán Aspuru-Guzik. The process matrix and how to determine it: quantum process tomography. In Ultrafast Spectroscopy, 2053-2563, pages 1–1 to 1–9. IOP Publishing, 2014. doi:10.1088/978-0-750-31062-8ch1. ↩

  4. Qiskit Experiments. ProcessTomography. Accessed: 2024-06-27. URL: https://qiskit-extensions.github.io/qiskit-experiments/stubs/qiskit_experiments.library.tomography.ProcessTomography.html. ↩↩