Parsimony-optimal control-adapted frames and scalable frame-based characterization and control of multiqubit systems under multiaxis and non-Gaussian noise

The transfer filter-function formalism of Chalermpusitarak et al. (PRX Quantum 2, 030315 (2021)) reduces the dynamics of a controlled open quantum system to a finite set of control-adapted (CA) noise spectra by expanding every control matrix element over a finite frame tied to the available control capabilities. The frame size $N_{\#}$ measures the parsimony of the reduced model and sets the experimental cost of CA quantum noise spectroscopy and noise-optimized gate design. The paper left open the central optimization question — for fixed control capabilities $\mathscr{C}$ , which frame is maximally parsimonious while retaining sufficient predictive power at tolerance $\varepsilon$ — together with the construction of explicit, scalable frame-based characterization-and-control protocols for multiqubit systems under multiaxis and non-Gaussian noise, where higher-order spectra are required and the second-order weak-coupling truncation may fail. Follow-up work through 2026 has extended the framework to classical non-Gaussian dephasing noise on one and two qubits under digital control, with explicit sample-complexity bounds, but reports learning complexity growing exponentially with qubit number, treats only dephasing (no multiaxis noise) and instantaneous controls, and no work addresses the parsimony-optimality question. The surviving open core is the missing frame-selection theory — minimal frame size and optimal frame construction as a function of control capabilities and tolerance — and the scalable multiqubit protocols.

Background

Quantum devices are invariably coupled to uncontrolled environments, and the resulting noise is typically temporally correlated (non-Markovian) rather than memoryless. A standard tool for predicting and mitigating such noise is the filter-function formalism: for a qubit driven by open-loop control, the noise-induced degradation of an observable is expressed through overlap integrals between the noise power spectral density and filter functions determined by the applied control sequence — the mathematical origin of why dynamical-decoupling pulse sequences (sequences of inversion pulses that flip the system-environment coupling and average the noise away) suppress decoherence. Quantum noise spectroscopy (QNS) inverts this relationship: by running a family of control sequences with different filters and measuring the accumulated error, one reconstructs the noise spectrum, which can then inform noise-optimized gate design. Two limitations of the standard frequency-domain formalism motivated the source work: it is tied to stroboscopic, piecewise-constant control assumptions, and it offers no guarantee that information inferred in a QNS experiment suffices to predict the system under controls other than those used in the experiment.

Chalermpusitarak et al. replaced the Fourier basis with the signal-processing notion of a frame: a finite family of functions $\{\phi_n\}$ , $n=1,\dots,N_{\#}$ , that spans the relevant signal space without orthonormality, together with a dual frame $\{\tilde{\phi}_n\}$ such that any signal $x(t)$ is recovered as $x(t)\approx\sum_n\langle x,\tilde{\phi}_n\rangle\phi_n(t)$ . The key idea is to tie the frame to the experimentally available control capabilities $\mathscr{C}$ — the set of control waveforms the device can actually produce (for example, $M$ instantaneous pulses, or Gaussian pulses with widths in a given range). The control determines a set of control matrix elements $y^{(\alpha)}_{u,v}(t)\in L^2([0,T])$ , where $\alpha$ labels additive dephasing noise versus multiplicative control noise and $u,v$ label system operators. The finite-size-frame (FSF) condition holds to tolerance $\varepsilon\ge 0$ when every control matrix element generated by $\mathscr{C}$ is approximated by its finite frame expansion with $L^2$ error at most $\varepsilon$ . When the FSF condition holds, the open-system dynamics (computed to second order in a weak-coupling Dyson expansion, i.e., time-dependent perturbation theory in the system-environment coupling) is governed by finitely many control-adapted (CA) spectra — noise correlation functions indexed by frame labels, of which the ordinary power spectrum is the Fourier-frame special case. This is control-driven model reduction: instead of an infinite-dimensional environment model, prediction and gate design under any admissible control require only the CA spectra, and QNS becomes estimation of these spectra (CA-QNS). The authors demonstrated CA-QNS for non-stationary cyclostationary classical noise (noise whose correlation functions are periodic in time) using Walsh functions (the piecewise-constant functions taking values $\pm 1$ that form the natural frame for instantaneous digital control) as the digital frame, and single-qubit noise-optimized gate design in which the model-reduced optimal control was benchmarked against the full-knowledge optimal control and against uncontrolled natural decoherence, with no significant loss of predictive power. The frame size $N_{\#}$ is the parsimony measure of the reduced model: the number of CA spectra that must be estimated grows with $N_{\#}$ (roughly as its square per coupling type and perturbative order), so a smaller frame directly means fewer experimental settings and less data. The paper constructs frames structurally — Walsh/digital frames for instantaneous pulses, Gabor (windowed-Fourier) frames for smooth pulses — but establishes nothing about which frame is best, and its demonstrations are restricted to single-qubit, dephasing-dominated, second-order weak-coupling settings.

Problem Statement

For fixed control capabilities $\mathscr{C}$ and predictive tolerance $\varepsilon$ , characterize and construct the control-adapted frames of maximum parsimony: determine, or tightly bound, the minimal frame size $N_{\#}$ for which the finite-size-frame condition is satisfied to tolerance $\varepsilon$ simultaneously for all control matrix elements generated by $\mathscr{C}$ , together with the dependence of this optimum on $\mathscr{C}$ and $\varepsilon$ (including how it changes as the accessible control waveforms are enlarged or restricted). Construct, for multiqubit systems, explicit, scalable frame-based characterization-and-control protocols in the presence of multiaxis noise — noise coupling along several system axes, inducing relaxation as well as dephasing — and non-Gaussian noise, which requires higher-order control-adapted spectra beyond the second-order truncation of the weak-coupling Dyson expansion, with quantified estimation resources (number of experimental settings) and prediction error that remain controlled as the number of qubits grows, with the scaling of both quantities in the number of qubits made explicit. The two requested results serve one objective, resource-efficient control-aware model reduction of open-system dynamics: the parsimony theory identifies the representation in which the multiqubit protocols estimate and optimize — which is why the source calls the parsimony question of special significance in the multiqubit context — while the protocols are the operational test that a frame choice retains sufficient predictive power. Both results are required, and neither subsumes the other: a tight bound on $N_{\#}$ for given $\mathscr{C}$ and $\varepsilon$ resolves the parsimony deliverable on its own. The objective concerns general multiqubit control tasks under multiaxis and non-Gaussian noise; results restricted to one or two qubits under dephasing-only, digitally pulsed control do not satisfy it.

References

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