Model-based optimal control is only as good as the model. To apply open-loop optimized pulses, one must know the Hamiltonian parameters, the noise and dissipation, and the distortions the control electronics impose on the pulses. System identification — learning this model from data — closes the gap between the simulated and the real device. It is the model-based counterpart of closed-loop optimization: instead of optimizing blindly on the device, learn a better model, then optimize on it.

What there is to learn

  • Hamiltonian parameters — frequencies, couplings, anharmonicities, cross-talk.
  • Noise and dissipation — decay and dephasing rates, Lindblad operators, noise spectra.
  • The control chain — transfer functions of cables and filters, amplifier non-linearities, timing offsets.

Learning Hamiltonians

  • Bayesian inference treats the unknown parameters as random variables and updates their distribution after each measurement. With sequential Monte Carlo and adaptive experiment design, it can run online, track parameters that change between experiments, and tolerate unknown noise [1].
  • Quantum-assisted learning. A trusted quantum simulator can learn the Hamiltonian of a larger, untrusted one [2]. A photonic quantum simulator has been used this way to learn the Hamiltonian of an NV-centre spin [3].
  • Learning the model itself. An automated, machine-learning agent can propose and test candidate Hamiltonians, not only their parameters, and identify which interactions are present [4].
  • Time traces. Ideas from linear systems theory reconstruct Hamiltonian parameters from time records of a few observables [5].
  • Many-body systems. Local Hamiltonians can be recovered from local measurements, with resources that scale linearly with system size [6]. Polynomially many local measurements on thermal states suffice to learn an interacting Hamiltonian accurately [7].

What can be learned is limited in principle. Controllable closed systems, for instance, can be identified only up to a unitary change of basis, and prior knowledge or particular measurements can help [8].

Learning noise and dissipation

Lindblad tomography reconstructs both the Hamiltonian and the Lindblad operators of a multi-qubit processor [9]. Learning the most accurate Markovian model from time-series data can be cast as a polynomial optimization and solved globally [10]; see QCPOP & polynomial optimization. Noise spectra are measured with tailored pulse sequences; see Filter functions & noise-tailored control.

Learning the control chain

Pulses arrive distorted. On superconducting processors, the step response of flux-control lines has been measured with the qubit itself (Cryoscope) and compensated by real-time digital filters [11]. A qubit can also act as a vector network analyzer, measuring a control line’s transfer function in the frequency domain and improving entangling gates [12]. Once known, distortions are built into the optimization; see Pulse parametrization & constraints and Control electronics & software stacks.

Machine learning and digital twins

Physics-inspired (“quantum-tailored”) machine-learning models can be trained on experimental data to characterize a superconducting qubit [13]. Optimizing controls on such a learned model yields pulses tailored to the actual device, including spurious modes and distortions [14]. Integrated toolsets cycle between optimal control, calibration and model refinement [15].

Verifying the result

Learned models and optimized operations must be validated. Characterization and benchmarking methods — randomized benchmarking, process and gate-set tomography — quantify how well the operations work [16]. Gate-set tomography, for example, can certify errors below rigorous fault-tolerance thresholds [17]. See Problem formulation & figures of merit.