Self-consistent quantum process tomography

Quantum process tomography is a necessary tool for verifying quantum gates and diagnosing faults in architectures and gate design. We show that the standard approach of process tomography is grossly inaccurate in the case where the states and measurement operators used to interrogate the system are generated by gates that have some systematic error, a situation all but unavoidable in any practical setting. These errors in tomography cannot be fully corrected through oversampling or by performing a larger set of experiments. We present an alternative method for tomography to reconstruct an entire library of gates in a self-consistent manner. The essential ingredient is to define a likelihood function that assumes nothing about the gates used for preparation and measurement. In order to make the resulting optimization tractable, we linearize about the target, a reasonable approximation when benchmarking a quantum computer as opposed to probing a black-box function.

Figure 1

Reconstructing a perfect identity gate with imperfect tomography due to depolarizing errors on the SPAM gates as well as Gaussian random noise on the measurement outcomes. We plot the diamond norm distance between the bare estimate Eq. (8) and the identity versus the noise power. In (a) we vary the strength of the depolarizing error for a fixed gate set, with four elements corresponding to mapping the ground state to the four corners of a tetrahedron. The experimental noise power is obtained from [1]. In (b) we fix the depolarizing strength at ${\mathcal{E}}_{\mathrm{dep}}={10}^{-3}$ and vary the set of gates used for SPAM over four sets of gates of differing order.

Figure 2

The ratio of the error in reconstructing the identity to the average gate error plotted versus the average gate error for a single qubit using a tetrahedral gate set and no stochastic measurement noise. The estimated operation is the “closest” physical map to the bare reconstruction in a flat metric. Seven different error models are compared, four coherent errors (different random unitary maps applied to each gate, a single random unitary map applied to each gate, a detuning error applied i.i.d. to each gate sampled from a Gaussian distribution, and an over-rotation error applied i.i.d. to each gate sampled from a Gaussian distribution) as well three incoherent processes (amplitude damping, dephasing, and depolarizing errors).

Figure 3

Simulated error in reconstructing the identity versus average SPAM error in terms of the gate fidelity (left) and the diamond norm distance (right). We look at four gate sets and three types of SPAM error.

Figure 4

A comparison of standard QPT and the self-consistent method presented in this manuscript for individual random unitary errors for the set of gates mapping to the six cardinal directions. We plot both the diamond norm and fidelity error between the estimates and the actual gates (averaged over the set of gates) plotted versus the equivalent distance of the actual set from the ideal gates. We show the magnitude of the tomography error and the ratio of the error with respect to the actual gate error for both cases.

Figure 5

A comparison of least-squares estimator (negative log-likelihood function) for standard QPT and the linearized likelihood estimation method with SPAM errors is generated by random unitary maps of each gate. On the $x$ axis is the estimator of the ideal gate set in the absence of errors with respect to the measurement outcomes and on the $y$ axis the ratio of the estimated gate set's log likelihood to the ideal.

Figure 6

The $\mathcal{R}$ matrices for the ideal gates and the reconstruction from QPT and self-consistent tomography for each of the four measured gates on sample B.