We introduce a theoretical framework for resource-efficient characterization and control of non-Markovian open quantum systems, which naturally allows for the integration of given, experimentally motivated, control capabilities and constraints. This is achieved by developing a transfer filter-function formalism based on the general notion of a frame and by appropriately tying the choice of frame to the available control. While recovering the standard frequency-based filter-function formalism as a special instance, this control-adapted generalization affords intrinsic flexibility and allows us to overcome important limitations of existing approaches. In particular, we show how to implement quantum noise spectroscopy in the presence of non-stationary noise sources, and how to effectively achieve control-driven model reduction for noise-tailored optimized quantum gate design.
We propose and analyze quantum state estimation (tomography) using continuous quantum measurements with resource limitations, allowing the global state of many qubits to be constructed from only measuring a few. We give a proof-of-principle investigation demonstrating successful tomographic reconstruction of an arbitrary initial quantum state for three different situations: single qubit, remote qubit, and two interacting qubits. The tomographic reconstruction utilizes only a continuous weak probe of a single qubit observable, a fixed coupling Hamiltonian, together with single-qubit controls. In the single qubit case, a combination of the continuous measurement of an observable and a Rabi oscillation is sufficient to find all three unknown qubit state components. For two interacting qubits, where only one observable of the first qubit is measured, the control Hamiltonian can be implemented to transfer all quantum information to the measured observable, via the qubit-qubit interaction and Rabi oscillation controls applied locally on each qubit. We discuss different sets of controls by analyzing the unitary dynamics and the Fisher information matrix of the estimation in the limit of weak measurement, and simulate tomographic results numerically. As a result, we obtained reconstructed state fidelities in excess of 0.98 with a few thousand measurement runs. P (R|ρ 0) = Tr[M † R M R ρ 0 ], (4) which is the same as the denominator of Eq. (2). In the case that the initial state ρ 0 is unknown, a Bayesian probability density function of a possible initial state can be obtained as: P (ρ |R) ∝ P (R|ρ)P (ρ) given a prior probability function of the unknown state P (ρ). This Bayesian probability function for an unknown initial state is the main quantity used in the state estimation process. We note that Eq. (2) is equivalent to the stochastic master equation [23, 24] in the limit of dt → 0. Decoherence, inefficiencies, or extra dephasing effects can be added to the above model, which only results in degrading the quality of the state estimation.
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