Adaptive quantum state tomography with neural networks

The study addresses the resource-intensive nature of quantum state tomography (QST), which estimates an unknown density matrix from repeated measurements on identically prepared states. QST scales poorly with system dimension due to the large number of parameters and measurements required. Adaptive schemes that choose subsequent measurements based on prior data have shown promise but often rely on Bayesian updates over particle approximations to the posterior, suffering from weight decay and expensive resampling. The research question is whether a neural-network-driven algorithm can replace the Bayesian update to enable fast, accurate, and adaptive QST without suffering from particle-weight decay, thereby reducing computational runtime while maintaining high reconstruction accuracy.

Prior adaptive QST methods include self-learning and adaptive Bayesian quantum tomography (ABQT), experimentally implemented in subsequent works. ABQT employs a particle filter approximation to the posterior with periodic resampling to mitigate weight decay, but resampling is computationally costly. As more data accumulate, the likelihood becomes sharply peaked, concentrating probability mass on few particles, exacerbating weight degeneracy and triggering frequent resampling. This motivates alternatives that retain adaptive benefits without expensive Bayesian posterior updates.

The proposed NAQT uses a custom recurrent neural network (RNN)-inspired architecture that operates on a particle bank B = {(ρ_i, w_i)} of candidate states with weights. At each measurement round: (1) the network ingests current particle states/weights and new measurement data to update particle weights and perturb particle states; (2) it outputs an updated estimate ρ̄ (a convex combination of valid particles, guaranteeing a physical density matrix) and a confidence score; and (3) it selects the next measurement by maximizing an information-gain heuristic I(Π, D) = H(P(Π|D)) – ⟨H(P(Π|ρ))⟩ over an accessible POVM set. The RNN cell is tailored to quantum tasks, using a differentiable implementation of POVM measurement in TensorFlow to simulate Born probabilities during training. The particle bank is updated every round by learned weight updates and learned perturbations of particles (in contrast to ABQT’s weight-only updates with occasional resampling). The learned update replaces the Bayes rule to avoid weight decay. Measurement adaptation follows the entropy-difference heuristic: choose Π that maximizes predicted information gain using the current bank. Training: The network is trained on simulated data for randomly sampled true states. For each training example, given a schedule of measurement batch sizes {M_t}, resampling times S_r, and adaptation times S_a, data D_t are simulated via Born’s rule for chosen POVMs. For each particle, the NN computes distances (e.g., L1/L2) between empirical probabilities and particle-induced probabilities, and outputs combined distances, perturbation sizes Δ (ε_i), and new weights w_i. A guess of the Born probabilities p_guess is computed from weighted particles; guesses over time are combined via softmax of learned guess scores. At resampling steps, particles are resampled according to weights and then perturbed: each particle is purified, random orthogonal perturbation directions are generated, and the best perturbation (minimizing distance to empirical probabilities) is selected, with step size ε_i learned by the NN. Measurement adaptation samples candidate POVM parameters, evaluates the mixed-entropy heuristic I, and selects the maximizing POVM for the next round. Loss is the distance between the reconstructed and true state across rounds, backpropagated end-to-end. Runtime scaling is achieved by scheduling resampling at exponentially increasing intervals (t1, 2t1, 4t1, …), making the number of resampling events logarithmic in total copies measured; since resampling is rate-limiting, overall runtime scales roughly logarithmically in measurements. The approach is agnostic to the number of qubits and POVM family, requiring only retraining with the specified measurement model.

- Accuracy: NAQT achieves reconstruction accuracy comparable to ABQT, evaluated via squared Bures distance d(ρ, ρ̂) = 2[1 − F(ρ, ρ̂)] with fidelity F(ρ, σ) = Tr(√ρ σ √ρ). - Runtime: For 100,000 copies measured, NAQT completes in ~2 s, while ABQT takes ~24,000 s (6.7 h) on the same hardware. For 10^5 copies, NAQT reconstructs in ≈5 s; ABQT extrapolates to multi-day runtimes. - Scaling fits (100 particles): NAQT runtime t ≈ 3.5 s + (1.3 s) log10(m + 300); ABQT runtime t ≈ (0.4 s) m, indicating logarithmic vs linear scaling with number of copies m. - Speedup: Orders-of-magnitude improvement, up to ~10^6× at 10^7 measurements (as noted by authors for same hardware). - Particle banks: NAQT used 100 particles; ABQT performance improves with particle count but empirically plateaus beyond ~30; reference ABQT results with 1000 particles were taken from prior work for comparison. - POVM robustness: NAQT shows nearly identical runtime behavior for product tetrahedron POVMs compared to basis POVMs, indicating robustness across measurement choices. - Practicality: The approach avoids weight decay and frequent resampling by learning update rules and using exponentially spaced resampling times, enabling on-the-fly adaptive QST.

NAQT replaces the Bayesian weight update with a learned neural update, avoiding weight degeneracy and reducing the need for expensive resampling. This enables a logarithmic runtime scaling due to planned, sparse resampling. The method retains adaptive measurement selection via an information-gain heuristic, yielding comparable accuracy to ABQT while providing dramatic runtime savings, making genuinely on-the-fly adaptive QST feasible. The architecture guarantees physicality of estimates and flexibly accommodates different POVMs without custom estimators or positivity constraints. The ability to retrain for different measurement models and system sizes, and to output functions of the state as needed, emphasizes the method’s generality. These findings suggest that machine learning can significantly improve practical prefactors in QST complexity, potentially extending tractable tomography to higher-dimensional systems.

The paper introduces NAQT, a meta-learning-based adaptive QST algorithm that integrates learned particle/weight updates with measurement adaptation. It matches ABQT’s reconstruction accuracy while reducing runtime by orders of magnitude and achieving logarithmic scaling in the number of copies measured. Contributions include: a custom RNN architecture with differentiable quantum simulations, a learned replacement for Bayesian updates that avoids weight decay, and a general framework easily retrainable for different POVMs and tasks. Future directions suggested by the authors include: extension to higher-dimensional (multi-qubit) systems, leveraging structure such as pure/low-rank states or specific channel classes, training to estimate functions of states (e.g., entanglement, fidelities), and incorporating realistic measurement noise during training for robustness.

- Demonstrations are on two-qubit systems; scalability to larger systems remains to be empirically validated although principles should carry over. - Exponential growth of complexity with qubit number remains inherent to QST; ML aims to improve prefactors rather than remove exponential scaling. - Practical NAQT experiments used 100 particles due to laptop memory constraints; performance with larger particle banks was not explored. - The method requires retraining for different POVM families or task specifications, incurring additional (though reported to be reasonable) training time. - ABQT runtime comparisons beyond 10^5 measurements involved extrapolation; and ABQT results with 1000 particles were taken from prior literature rather than recomputed on the same setup.