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Correlated readout error

This experiment computes the assignment matrix characterising the readout error for two coupled qubits.

Description

Current quantum computers are plagued by noise and errors that reduce fidelities. Errors typically fall into one of three categories1:

  • state preparation errors
  • gate errors
  • measurement errors

Although it is generally not possible to disentangle state preparation and measurement (SPAM) errors, several works2345 have shown that it's possible to detect and correct readout errors in superconducting devices through purely classical noise models. Such models describe a noisy \(n\)-qubit measurement by a matrix of transition probabilities, \(A\), of size \(2^{n} \times 2^{n}\). In this matrix, commonly referred to as the assignment matrix, each element \(A_{ij}\) represents the probability of observing a measurement outcome \(j\) given the expected outcome is \(i\); the conditional probability is given by

\[A_{ij} = P(j|i).\]

For a two-qubit system (\(n=2\)):

\[ A = \begin{pmatrix} A_{00} & A_{01} & A_{02} & A_{03} \\ A_{10} & A_{11} & A_{12} & A_{13} \\ A_{20} & A_{21} & A_{22} & A_{23} \\ A_{30} & A_{31} & A_{32} & A_{33} \end{pmatrix}, \]

where \(i, j = 0, 1, 2, 3\) correspond to the qubits being in the states \(|00\rangle, |01\rangle, |10\rangle, |11\rangle\), respectively.

Experiment steps

  1. A sequence of \(\pi\)-gates (\(R_x(\pi)\)) are applied to the chosen pair of qubits, preparing them in the following states: \(|00\rangle, |01\rangle, |10\rangle, |11\rangle\).

  2. The resonator transmission is measured for each prepared state.

Analysis steps

  1. The qubit states are determined from the \(I\)-\(Q\) data by employing a composite discriminator, which combines the Readout Discriminators for each qubit. The composite discriminator includes optional error mitigation along the sample dimension.

  2. The populations are plotted on a 2D colourmap of intended (prepared) state against measured state (left figure below) and the values for the assignment matrix are extracted from this.

image

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  1. Benjamin Nachman and Michael R. Geller. Categorizing readout error correlations on near term quantum computers. 2021. arXiv:2104.04607. ↩

  2. J. M. Chow, L. DiCarlo, J. M. Gambetta, A. Nunnenkamp, Lev S. Bishop, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. Detecting highly entangled states with a joint qubit readout. Phys. Rev. A, 81:062325, Jun 2010. doi:10.1103/PhysRevA.81.062325. ↩

  3. 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. ↩

  4. Sergey Bravyi, Sarah Sheldon, Abhinav Kandala, David C. Mckay, and Jay M. Gambetta. Mitigating measurement errors in multiqubit experiments. Phys. Rev. A, 103:042605, Apr 2021. doi:10.1103/PhysRevA.103.042605. ↩

  5. Paul D. Nation, Hwajung Kang, Neereja Sundaresan, and Jay M. Gambetta. Scalable mitigation of measurement errors on quantum computers. PRX Quantum, 2:040326, Nov 2021. doi:10.1103/PRXQuantum.2.040326. ↩