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Learning Quantum Systems

Valentin Gebhart, Raffaele Santagati, Antonio Andrea Gentile, Erik M. Gauger, David Craig, Natalia Ares, Leonardo Banchi, Florian Marquardt, Luca Pezze', Cristian Bonato

arXiv:2207.00298v3quant-ph

TL;DR

Increasing quantum-system complexity makes efficient characterization, calibration and validation difficult because full descriptions scale exponentially. This Review synthesizes classical post-processing and adaptive approaches for learning states, dynamics, measurements and environmental interactions, highlighting Bayesian methods and neural networks. It surveys their demonstrated applications while noting exponential-scaling and interface limitations.

  • Problem

    Quantum states and dynamics become exponentially complex, creating challenges for efficiently characterizing, calibrating and validating quantum technologies.

  • Method

    The Review synthesizes classical post-processing and adaptive optimization methods for learning quantum states, dynamics, measurements and environmental interactions.

  • Results

    The reviewed approaches include scalable state-learning methods, specific-feature prediction, readout optimization, open-system reconstruction and sensing applications across quantum architectures.

  • Takeaways & Limitations

    Classical characterization and optimization remain useful for quantum technologies, including near-term noisy devices, despite full-system scaling challenges.

  • Takeaways & Limitations

    Quantum-process tomography retains exponential measurement and post-processing scaling, while quantum algorithms and quantum machine learning are outside the Review’s scope.

Abstract

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The future development of quantum technologies relies on creating and manipulating quantum systems of increasing complexity, with key applications in computation, simulation and sensing. This poses severe challenges in the efficient control, calibration and validation of quantum states and their dynamics. Although the full simulation of large-scale quantum systems may only be possible on a quantum computer, classical characterization and optimization methods still play an important role. Here, we review different approaches that use classical post-processing techniques, possibly combined with adaptive optimization, to learn quantum systems, their correlation properties, dynamics and interaction with the environment. We discuss theoretical proposals and successful implementations across different multiple-qubit architectures such as spin qubits, trapped ions, photonic and atomic systems, and superconducting circuits. This Review provides a brief background of key concepts recurring across many of these approaches with special emphasis on the Bayesian formalism and neural networks.

INTRODUCTION

Quantum-system learning is needed to characterize, control and validate increasingly complex systems whose full descriptions become intractable. The Review surveys classical post-processing and adaptive methods for learning states, dynamics and measurements, while delimiting its coverage and comparisons.

  • Motivation: Quantum states and dynamics grow exponentially in complexity, making full descriptions intractable and approximations challenging.
  • Motivation: Learning quantum systems supports gate calibration, state preparation, sensing optimization and studies of fundamental properties such as entanglement and nonlocality.
  • Scope: The Review covers classical post-processing and adaptive optimization for learning quantum states, dynamics, measurements and environmental interactions.
  • Scope: Covered methods span rigorous-guarantee approaches with worst-case bounds and heuristics that have been applied across diverse systems without general guarantees.
  • Limitations: The Review is not exhaustive, avoids prescribing which technique to use, and does not quantitatively compare rigorous-yet-pessimistic methods with heuristics.
  • Limitations: It excludes quantum-algorithm approaches, including quantum machine learning, while related reviews cover those topics and adjacent areas.

Quantum-state tomography

Quantum-state tomography infers an unknown state from measurements, but complete reconstruction requires extensive settings, repetitions and classical processing. Bayesian, compressed-sensing and adaptive approaches address parts of this burden under suitable assumptions.

  • Definition: Quantum-state tomography infers an unknown finite-dimensional quantum state from measurement outcomes, commonly for multiqubit systems.
  • Requirements: A tomographically complete measurement set must resolve the full space of possible states, often requiring many settings, repeated measurements and multiple state copies.
  • Bayesian inference: Bayesian inference updates prior beliefs using measurement likelihoods and normalization, with Born’s rule supplying quantum-system likelihoods.
  • Bayesian inference: Bayesian methods incorporate prior information, compute error bars and can select future measurement settings adaptively from current knowledge.
  • Scalable methods: Compressed sensing reduces reconstruction to a semidefinite programme for low-rank states and inherits rigorous performance and convergence guarantees.
  • Scaling: Full tomography scales exponentially with qubit number; estimating an arbitrary state to trace-distance error ϵ requires at least O(4^N/ϵ^2) copies.

Efficient quantum-state tomography

Efficient tomography exploits structure, symmetry and neural-network representations to reduce the burden of reconstructing large quantum states. These methods can reach highly entangled systems of about 100 qubits, while neural networks support several learning and control tasks.

  • Efficient quantum-state tomography: Heuristic ansätze improve scalability and enable tomography of highly entangled quantum states of up to about 100 qubits.
  • Symmetry: Permutation symmetry can reduce the number of tomographic measurements to scale quadratically with the number of qubits.
  • Tensor networks: Tensor networks represent quantum states through matrix product states; small bond dimension makes their description and tomography efficient, scaling linearly with qubit number.
  • Neural-network tomography: Restricted Boltzmann machines model quantum-state amplitudes and phases using parameters associated with visible and hidden layers.
  • Neural-network tomography: Generative adversarial networks reconstruct states and processes through a generator that produces approximations and a discriminator that distinguishes genuine from generated states.
  • Neural-network applications: Neural networks support measurement classification and regression, dynamics prediction, quantum-state and process tomography, and optimal quantum control.
  • Neural-network applications: Neural-network size and training must be chosen carefully because excessive expressivity or long schedules can compromise trainability and generalizability through overfitting.

Extracting specific features of a quantum state

When only selected properties of a quantum state are needed, tailored measurements and classical post-processing can avoid full quantum-state tomography. Classical shadows extend this strategy to predict many observables from independent measurements, with scaling caveats for some observables.

  • Extracting specific features of a quantum state: Tailored measurements and post-processing can avoid the intractability of full quantum-state tomography when only specific properties are required.Applications include entanglement entropies, Fisher information, spin-squeezing parameters and entanglement witnesses.
  • Extracting specific features of a quantum state: A linear number of measurements can PAC-learn future outcome probabilities when measurements are randomly drawn from a fixed distribution.
  • Extracting specific features of a quantum state: Shadow tomography predicts expectation values for an exponential number of arbitrary observables using measurements of a polynomial number of copies, but requires coherent parallel measurements.
  • Extracting specific features of a quantum state: Classical shadows remove the need for coherent measurements by using independent measurements of a polynomial number of copies to estimate exponentially many target observables.Their scaling depends on the observables’ norms and can become exponential for certain observables.
  • Extracting specific features of a quantum state: Classical shadows reconstruct estimates through random unitary measurements, inversion of the resulting channel, and statistical inference such as median-of-means.The intermediate shadow need not be a valid quantum state, but it yields estimates of target functions.
  • Extracting specific features of a quantum state: Classical shadows estimate nonlinear functions and, when combined with classical machine-learning models, provide provable advantages in tasks including ground-state prediction and topological-phase classification.

Optimizing qubit readout

Qubit readout can be optimized by processing time-resolved signals beyond simple thresholding. Bayesian filters and neural networks address relaxation and other readout imperfections, while averaged readout can also support learning.

  • Optimizing qubit readout: Single-shot readout processes time-resolved signals into histograms and thresholds to distinguish prepared ground and excited states.
  • Optimizing qubit readout: Excited-state relaxation creates asymmetric readout histograms that reduce fidelity under threshold classification.
  • Optimizing qubit readout: Clustering can discard relaxation signals and increase superconducting-qubit readout fidelity, although some implementations produce lower-quality outliers than repetitive thresholding.
  • Optimizing qubit readout: Nonlinear Bayesian filters improve threshold fidelity by using full time-resolved signals and modeling relaxation or stochastic turn-on times.
  • Optimizing qubit readout: Neural networks improve readout across platforms, including NV centers, quantum-dot spin qubits, multiplexed superconducting qubits and trapped-ion multiqubit classification.
  • Optimizing qubit readout: Single-shot readout is not required in every setting because averaged readout can be combined with Bayesian inference.

LEARNING QUANTUM DYNAMICS

Learning quantum dynamics supports fidelity assessment and sensing optimization. The review therefore considers both assumption-free approaches and model-based approaches that simplify characterization.

  • LEARNING QUANTUM DYNAMICS: Learning quantum dynamics helps establish channel fidelity, gate fidelity and optimal parameter encoding for communication, computing and sensing.
  • LEARNING QUANTUM DYNAMICS: The review first considers assumption-free methods for learning dynamics, followed by approaches that use specific models to simplify characterization.
  • LEARNING QUANTUM DYNAMICS: Quantum process tomography is presented as the full-reconstruction framework for unknown quantum dynamics.

Quantum process tomography

Quantum process tomography reconstructs an unknown quantum process from known inputs and measurements of outputs, but full reconstruction is more demanding than state tomography. Its costs, physicality constraints and gauge freedom motivate compressed-sensing, optimization and heuristic alternatives.

  • Quantum process tomography: Quantum process tomography reconstructs an unknown completely-positive trace-preserving process that maps input quantum states to output states.
  • Quantum process tomography: Complete process reconstruction requires known input states spanning all possible initial states and tomographically complete measurements of the outputs.
  • Quantum process tomography: Full quantum process tomography is more challenging than state tomography because it must satisfy completely-positive trace-preserving constraints.
  • Quantum process tomography: The measurement and classical post-processing cost of quantum process tomography scales exponentially with system size, while maximum-likelihood methods are additionally sensitive to preparation, gate and measurement errors.
  • Quantum process tomography: Compressed sensing reduces reconstruction cost when the process has latent structure such as sparsity, low Kraus rank or a simple interaction graph.
  • Quantum process tomography: Gate-set tomography reconstructs quantum objects only up to gauge freedom, so its learned model is not unique.
  • Quantum process tomography: Alternative heuristic approaches use tensor networks, GANs and neural networks to reduce resources or extend process characterization to specialized settings.

Reconstructing Hamiltonian quantum dynamics

Hamiltonian dynamics can be reconstructed using time-series, linear-systems, Bayesian, and adaptive model-selection methods, with efficiency depending on structured parametrizations and prior knowledge.

  • Reconstructing Hamiltonian quantum dynamics: Hamiltonian learning reconstructs dynamics by estimating a parameterized H(t) or H(x,τ) from measurement data and known control settings.Structured ansätze reduce the number of parameters relative to a general Hamiltonian.
  • Reconstructing Hamiltonian quantum dynamics: Achieving ϵ average precision typically requires quantum resources scaling as ϵ^-2, while neural-network training can be time-consuming.Gray-box approaches combine neural networks with physically interpretable quantum-mechanical structure.
  • Reconstructing Hamiltonian quantum dynamics: Time traces can be fitted or analyzed with Fourier methods, while eigenstate realization algorithms apply linear-systems theory to regularly sampled experimental data.The latter represents evolution through a linear differential equation after decomposing the Hamiltonian into Hermitian operators.
  • Reconstructing Hamiltonian quantum dynamics: Bayesian quantum Hamiltonian learning infers unknown parameters for a chosen parametrization by combining measurement outcomes with Bayes’ rule.The evolution time is typically the tunable control parameter.
  • Reconstructing Hamiltonian quantum dynamics: Quantum model learning agents address unknown parametrizations by generating candidate Hamiltonians, training their parameters, comparing performance, and discarding weaker models.Candidate terms can be combined through tree searches or genetic algorithms, with comparisons based on Bayes factors or modified Elo ratings.
  • Reconstructing Hamiltonian quantum dynamics: Polynomially many local observables can suffice to reconstruct local Hamiltonians because their parameter space scales polynomially with qubit number.This contrasts with worst-case characterization methods that typically scale exponentially.

Reconstructing open quantum system dynamics

Open-system dynamics require models beyond isolated Hamiltonian evolution: Markovian processes use Lindblad generators, whereas non-Markovian processes retain history dependence and require richer formalisms.

  • Reconstructing open quantum system dynamics: For a memoryless environment, the quantum state evolves according to the Gorini–Kossakowski–Sudarshan–Lindblad master equation.The Hamiltonian describes coherent evolution, while Lindblad operators describe dissipative processes.
  • Reconstructing open quantum system dynamics: Lindblad tomography reconstructs H and the Lindblad operators from measurement data using maximum-likelihood estimation or time-trace and linear-system methods.Local Markovian dynamics can also be recovered from local measurements when suitable steady states are available.
  • Reconstructing open quantum system dynamics: Learning Markovian dynamics is harder than Hamiltonian learning because it involves a d^2×d^2-dimensional Liouvillian for a d-dimensional system.Heuristics and prior knowledge about expected noise processes can enable more efficient parameterizations.
  • Reconstructing open quantum system dynamics: Non-Markovian evolution depends on earlier states as well as the current state, and can be represented using time-nonlocal superoperators or process tensors.Process tensors provide access to multitime correlations but scale exponentially with the number of time steps.
  • Reconstructing open quantum system dynamics: Neural networks can reconstruct non-Markovian dynamics, although these approaches generally provide reduced physical insight.Some interpretability can be retained through effective time-dependent descriptions.

LEARNING QUANTUM MEASUREMENTS

Quantum detection tomography reconstructs realistic, non-projective measurements as POVMs from probe-state data, while optimization and self-characterizing methods address calibration challenges.

  • LEARNING QUANTUM MEASUREMENTS: Quantum detection tomography reconstructs detector operation beyond efficiency, linearity, dark counts, and spectral or temporal response, which can introduce systematic errors.The method targets the detector itself without requiring prior information about its operation.
  • LEARNING QUANTUM MEASUREMENTS: Noisy detectors are modeled by positive-operator valued measure operators satisfying positivity and normalization constraints.These constraints ensure nonnegative detection probabilities that sum to one.
  • LEARNING QUANTUM MEASUREMENTS: Standard QDT inverts the measurement relation using sampled probabilities and probe states spanning the relevant Hilbert subspace.For noisy projective measurements, a stochastic mapping can model the probability of observing each detector result.
  • LEARNING QUANTUM MEASUREMENTS: The detector mapping can be optimized by minimizing distinguishability between observed and modeled probabilities using Kullback–Leibler divergence, fidelity, gradient descent, or maximum likelihood.The resulting POVM elements are constructed from the optimized mapping.
  • LEARNING QUANTUM MEASUREMENTS: QDT has characterized optical photocounting, homodyne detection, qubit readout, and competing physical models of superconducting single-photon detectors.Self-characterizing approaches avoid precisely calibrated probe states, reducing possible systematic errors and self-reference.

APPLICATIONS IN QUANTUM SENSING AND CONTROL

Quantum sensing and imaging use learned estimators, Bayesian posteriors, entanglement, and deep learning to characterize parameters and improve measurement performance, with multiparameter estimation facing compatibility limits.

  • APPLICATIONS IN QUANTUM SENSING AND CONTROL: Quantum sensing estimates a quantity θ from a probe state, parameter-dependent evolution, and POVM using measurement probabilities P(µ|θ)=Tr[EµΛθ[ρ]].This frames quantum sensing as a special case of quantum process tomography.
  • APPLICATIONS IN QUANTUM SENSING AND CONTROL: For continuous parameters, the quantum Cramér–Rao relation connects estimation uncertainty to quantum Fisher information, with asymptotic saturation by maximum likelihood in the frequentist setting.In the Bayesian setting, it gives the asymptotic posterior variance under an optimal POVM.
  • APPLICATIONS IN QUANTUM SENSING AND CONTROL: Neural networks have been trained to construct estimators or Bayesian posterior distributions from limited calibration data, and deep learning can improve quantum imaging under noise.Reported imaging aims include increasing fidelity, discovering deeper data structure, and overcoming shot and background noise.
  • APPLICATIONS IN QUANTUM SENSING AND CONTROL: Separable N-qubit states under collective spin rotation achieve at best FQ=N, whereas FQ>N requires entanglement and FQ=N^2 requires genuine multipartite entanglement.Quantum Fisher information can witness multipartite entanglement across several many-body phenomena.
  • APPLICATIONS IN QUANTUM SENSING AND CONTROL: For joint estimation of multiple parameters, non-commuting optimal measurements can prevent saturation of the quantum Cramér–Rao bound, requiring the Holevo–Cramér–Rao bound instead.The bound applies to the covariance matrix through the inverse quantum Fisher information matrix.

Adaptive methods for quantum sensing

Adaptive Bayesian protocols use measurement data to update parameter knowledge and select subsequent settings, while reinforcement learning agents learn feedback-control strategies. These approaches have been applied to quantum sensing, decoherence suppression, calibration, and control.

  • Adaptive Bayesian inference: Bayesian adaptive protocols update parameter distributions after each measurement and use them to compute optimal settings for the next measurement.Settings may be selected through heuristics or by optimizing variance, Fisher information, or Kullback–Leibler divergence.
  • Adaptive Bayesian inference: Adaptive techniques have improved optical phase-sensing sensitivity and have been used with NV-centre quantum sensors for nanoscale magnetic mapping and magnetic resonance.For online sensing, simplified near-optimal processing can outperform computationally intensive optimal methods when time is constrained.
  • Learning and controlling dynamics: Quantum Hamiltonian learning has suppressed decoherence in a single spin qubit by compensating in real time for nuclear-spin-bath fluctuations and classical noise.When the environment is quantum, measurement back-action can itself perturb the state and become part of the control process.
  • Learning and controlling dynamics: Closed adaptive loops alternate between learning a dynamical model from experimental data and designing improved control strategies based on that model.This framework integrates calibration, characterization, and control rather than treating them as separate processes.
  • Reinforcement learning: Model-free reinforcement learning uses an agent to learn feedback-control strategies without prior knowledge of the environment’s dynamics.Policy-gradient methods adjust a parameterized stochastic policy to maximize average cumulative reward, such as final-state fidelity.

OUTLOOK

The Review identifies limited comparative evidence and interpretability as open challenges, while highlighting integrated learning loops, physics-informed neural networks, and quantum processors as future directions. Joint measurements and coherent interfaces could expand quantum-system learning, but interfacing remains an obstacle.

  • OUTLOOK: Few detailed comparisons and limited rigorous complexity analyses make it difficult to prescribe which learning method suits a task under given assumptions.The Review calls for provable analyses of heuristic methods and numerical comparisons across techniques.
  • OUTLOOK: Quantum-state, dynamics, and measurement reconstruction should be approached jointly through self-calibrating tomography, GST, and closed integrated learning loops.In closed loops, Hamiltonian-learning information guides control-gate and pulse design, which further refines system characterization.
  • OUTLOOK: Neural ordinary differential equations incorporate known physical laws, differential equations, or Lagrangian functions into neural-network descriptions to improve physical interpretability.This addresses the open problem of interpreting black-box neural-network approaches.
  • OUTLOOK: Joint quantum measurements can outperform single measurements in distinguishing quantum states, even when the states are uncorrelated.The Review presents this as an advantage demonstrated by quantum-computing hardware, including current noisy intermediate-scale quantum devices.
  • OUTLOOK: Learning quantum states with a quantum processor requires a quantum-coherent interface between the system under study and the processor.Further research is needed on architectures that preserve coherence when interfacing sensors, devices, and systems with quantum computers.
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