Machine learning quantum states in the NISQ era

I. INTRODUCTION

• As NISQ devices grow, conventional tomography breaks down, leaving validation of their behavior an open challenge.The paper contrasts this with Hamiltonian-driven discovery, which assumes a known microscopic model; data-driven reconstruction assumes no Hamiltonian knowledge but requires informationally complete measurements.Source: §I · p. 1

II. GENERATIVE MODELING

• Generative modeling learns a parameterized approximation of an unknown data distribution, with parameters optimized against the observed samples.The resulting model can generate configurations not encountered during training.Source: §I · p. 1, §II · p. 2

I. INTRODUCTION

• Single-site projective measurements can supply Born-rule data for unsupervised reconstruction of a quantum state when their type and number are sufficient.The paper identifies the scaling of data-driven reconstruction and its comparison with Hamiltonian-driven approaches as largely unanswered.Source: §I · p. 1
• A trained generative model can estimate observables inaccessible to the device, including Renyi entropies, and may mitigate preparation and measurement noise.Off-loading observable production to a parameterized model lets experimentalists focus on obtaining high-quality projective measurements.Source: §I · pp. 1–2
• The review develops RBM reconstruction from classical Ising distributions through quantum states and discusses experimental Rydberg-atom reconstruction.It also covers complex wavefunctions and density matrices; the experimental demonstration is described as a milestone despite noise challenges.Source: §I · p. 2

A. Artificial neural networks

• Neural networks represent information across interacting neurons, and the perceptron introduced supervised classification of input patterns.The account identifies neurons, graph-structured interactions, a teaching signal, and a learning rule as core ingredients.Source: A. Artificial neural networks · p. 2
• Single-layer perceptrons learn only linearly separable functions; backpropagation and the Boltzmann machine later helped revive neural-network research [12, 13, 14].The source describes the intervening decline as the first AI winter.Source: A. Artificial neural networks · pp. 2–3

1. The Hopfield model

• In the Hopfield model, asynchronous neuron updates monotonically lower energy until the network reaches a stable local minimum.With suitable learned interactions, stored memories become local minima; a sufficiently close corrupted memory can then be recovered.Source: §1 · p. 3

2. The Boltzmann machine

• At finite temperature, stochastic Boltzmann-machine updates equilibrate to a canonical distribution, whereas the zero-temperature limit recovers deterministic Hopfield dynamics.The stochastic dynamics minimize free energy rather than energy alone.Source: §2 · p. 3
• Boltzmann-machine learning adjusts interactions to make the model distribution mimic the data, increasing data-supported configurations and decreasing model-generated ones.The learning rule compares data and equilibrium averages in positive and negative phases.Source: §2 · p. 3
• Because a Boltzmann machine captures only pairwise correlations, hidden units and restricted interlayer connections yield the RBM’s tractable visible marginal.The figure contrasts a fully connected network with an RBM whose symmetric weights connect visible and hidden layers.Source: §2 · p. 4, Figure 1 · p. 4

B. Restricted Boltzmann machines

• With enough hidden neurons, an RBM can approximate any discrete binary distribution, although the worst-case hidden-unit count may grow exponentially with the visible layer.Its effective visible energy contains correlations among visible neurons at all orders.Source: B. Restricted Boltzmann machines · p. 4

1. Unsupervised learning

• RBM training minimizes KL divergence from the data distribution, with a gradient comprising data-driven and model-driven terms.The positive-phase term is an average over the data samples.Source: §1 · p. 4
• The negative phase is generally intractable and is approximated by Monte Carlo configurations drawn from a Markov chain on the model distribution.Source: §1 · p. 4
• Without within-layer connections, neurons in each layer are conditionally independent, allowing simultaneous layer-wise sampling in alternating block Gibbs updates.The restricted graph therefore aids the sampling stage used to estimate the negative phase.Source: §1 · pp. 4–5

2. Training by contrastive divergence

• Contrastive divergence uses k block Gibbs steps from a data sample to estimate the negative phase, avoiding the need to equilibrate the RBM at every training iteration.It is introduced to address the computational cost of equilibrium sampling and the high variance of the gradient.Source: §2 · p. 5
• Stochastic gradient descent updates parameters using a random data batch and learning rate; weight decay is a common regularizer against overfitting.Other algorithms can speed convergence and adapt the learning rate.Source: §2 · p. 5

III. QUANTUM STATE RECONSTRUCTION

• Full quantum state tomography typically scales exponentially, restricting its practical use to relatively small systems.It reconstructs a density operator by maximum likelihood without assuming a phase structure or purity.Source: §III · p. 5
• MPS tomography can reconstruct states beyond full QST’s reach, but its one-dimensional, low-entanglement restriction limits its scope, including for some short-time dynamics.Permutationally invariant QST and compressed sensing are also identified as alternatives.Source: §III · p. 5
• The review presents unsupervised generative models as a framework for reconstructing many-body states, progressing from classical thermal states to pure wavefunctions and density operators.It focuses on RBMs, which are described as interpretable in statistical-physics terms and widely explored in classical and quantum reconstruction.Source: §III · p. 5

A. Classical limit

• In the classical limit, the target density operator reduces to a canonical Boltzmann distribution, making reconstruction unsupervised learning of Pβ(σ).The Ising model provides the example, with a two-dimensional system transitioning between low-temperature ferromagnetic order and a high-temperature disordered state.Source: A. Classical limit · p. 6
• Magnetization converges quickly in the classical Ising reconstruction, but specific heat needs more hidden units, especially near the critical point; deeper RBMs show no obvious improvement.Figure 2 compares reconstructed and dataset values as the number of hidden units increases.Source: A. Classical limit · p. 6, Figure 2 · p. 6

B. Positive wavefunctions

• For a real, positive wavefunction, the Born-rule measurement distribution uniquely characterizes the state, so the RBM can represent it through conventional probability learning.The review notes this assumption applies, for example, to ground states of stoquastic Hamiltonians.Source: B. Positive wavefunctions · p. 6
• With sufficiently many hidden units, an RBM can represent any positive quantum state to arbitrary accuracy.Source: B. Positive wavefunctions · p. 6
• Minimizing the KL divergence between measurement and RBM distributions makes the RBM wavefunction approximate the target positive state.Source: B. Positive wavefunctions · pp. 6–7

1. Measurement of physical observables

• Diagonal observables are estimated as thermal averages over the RBM distribution and checked against values from the training data.Block Gibbs sampling supplies the Monte Carlo approximation.Source: §1 · p. 7
• Off-diagonal observable expectations use a local estimate, which is efficiently Monte Carlo-estimable when the operator has polynomially many nonzero off-diagonal elements in N.Source: §1 · p. 7
• For α = 2, entanglement entropy is calculated by considering two identical replicas and computing the overlap between states with and without subregion A configurations swapped between replicas.Source: §1 · p. 7
• Figure 3 specifies magnetization for a 144-spin square lattice and entanglement entropy versus subsystem size for a 20-spin chain, using QMC and exact diagonalization datasets, respectively.Source: Figure 3 · p. 7
• The RBM can implement the swap calculation by replicating the trained neural network rather than the experimental NISQ hardware.The swap operator has also been measured for ultracold bosons in optical lattices using quantum interference.Source: §1 · p. 7

2. Reconstructing quantum spins on a lattice

• The reconstructed transverse-field Ising states yield longitudinal and transverse magnetizations and chain entanglement entropy through a swap operator between replicated neural networks.The example spans the model’s ferromagnetic-to-paramagnetic transition as the transverse field changes.Source: §2 · p. 8

C. Complex wavefunctions

• A single measurement basis cannot reveal sign or phase structure, so reconstructing states with such structure requires measurements in additional bases.These structures can occur in non-stoquastic frustrated spin systems and fermions.Source: C. Complex wavefunctions · p. 8
• The reviewed complex-state construction uses an additional RBM to capture phase alongside the positive RBM wavefunction’s amplitude.The review notes other options include a phase function parameterized by a feedforward network or complex-valued RBM weights and biases.Source: C. Complex wavefunctions · p. 8
• Configurations in the reference basis are sampled from pλ(σ), the squared-amplitude distribution, independently of the phase parameters.Source: C. Complex wavefunctions · p. 8

1. Learning the phase structure

• Phase learning uses measurements after local-basis rotations, whose number and type depend on the state’s phase structure, and minimizes total KL divergence across bases.The reported gradients use averages over measurement-data quasi-probabilities; standard contrastive-divergence training applies because the negative phase does not depend on phase parameters.Source: §1 · p. 8

D. Density operators

• A neural density operator must satisfy Hermiticity and positive semidefiniteness, requirements that are more stringent than those for an RBM wavefunction.Purification enforces positivity directly in the network representation.Source: D. Density operators · p. 9, §1 · p. 9

1. Latent space purification

• Latent-space purification represents a mixed state as a pure RBM wavefunction on an enlarged system, then traces out auxiliary variables to obtain a neural density operator.Embedding the auxiliary units in the network's latent space makes the trace analytic; parameters V encode mixing and vanish for pure states.Source: §1 · p. 9
• The NDO reconstruction cost has analytically calculable gradients, evaluated on training data after the appropriate unitary rotations; contrastive divergence approximates its partition-function term.The negative phase uses finite-step block Gibbs sampling of the diagonal distribution.Source: §1 · p. 9
• Because latent-space purification yields a positive density operator, this NDO ansatz is also suitable for simulating open-system quantum dynamics.The review notes numerical studies of this application.Source: §1 · p. 9
• Explicitly rotating the NDO back to the reference basis for gradient evaluation is feasible only when the measurement-basis unitary acts on sufficiently few degrees of freedom.An alternative represents the outcome distribution of an informationally complete POVM instead of parametrizing the quantum state directly.Source: §1 · p. 9

1. Noise mitigation

• Known measurement-error rates can be incorporated into an RBM to learn the de-noised distribution corresponding to projective measurements before the noise process.The experiment's noisy outcomes are modeled as a POVM generated from actual projective outcomes.Source: §1 · pp. 9–10
• Figure 4 tracks average Rydberg population, average transverse field, and bond-(3, 4) Renyi mutual information across detuning or bond position.The reconstruction includes the experimental wavefunction of a Rydberg-atom simulator [66].Source: Figure 4 · p. 10
• The noise-regularized RBM adds a noise layer for recorded outcomes τ and minimizes negative log-likelihood on noisy data, using a Bayesian posterior in the positive phase.The conditional distribution p(τ | σ) is set to match the experimental error rates.Source: §1 · p. 10
• The trained wavefunction approximates the state before measurement errors, and a related autoencoder strategy was applied to neural-network reconstruction of experimental photonic states [67].Source: §1 · p. 10

2. Application to a Rydberg-atom quantum simulator

• The simulator couples trapped 87Rb atoms' ground and Rydberg states with a uniform laser drive and van der Waals interactions, enabling a transition into broken-symmetry phases as detuning increases.The system is described as a programmable simulator of Ising-like quantum spins.Source: §2 · p. 10
• Under perfect adiabatic evolution, the state can be gauged positive, making occupation-number-basis measurements sufficient for reconstruction; real experimental states may be mixed through decoherence.When purity remains sufficiently high, a pure-state RBM can capture properties of local subsystems.Source: §2 · pp. 10–11
• The reconstructed average Rydberg population, average transverse field, and Renyi mutual information show overall good agreement with open-system dynamics simulations.The population is compared with experimental data and full Lindbladian master-equation simulations; transverse field and mutual information are also estimated.Source: §2 · p. 11
• For N = 8 atoms entering the Z2 phase, about 3,000 measurements are collected at successive detunings during the sweep and independently used to train noise-regularized RBMs.Direct RBM sampling gives access to observables diagonal in the measurement basis, while the model can also sample observables unavailable to the apparatus.Source: §2 · p. 11

• RBM theory for quantum state reconstruction is becoming better understood, and open-source software is helping make related methods accessible to experimentalists.The review says fundamental questions remain before these techniques are fully integrated with NISQ hardware.Source: §IV · p. 11
• The most well-studied cases are real-positive wavefunctions, mathematically equivalent to probability distributions, although RBMs can be used for complex wavefunctions as well.The review describes real-positive states as including ground states of stoquastic Hamiltonians.Source: §IV · p. 11
• For pure real-positive states, RBMs produce accurate, scalable estimators for observables absent from the original data, including basis-independent Renyi entanglement entropy via model replication.The review notes that this entropy is experimentally resource-intensive but relatively simple to implement in the trained model.Source: §IV · p. 11
• Complex-wavefunction reconstruction requires a phase convention, such as additional hidden layers or complex weights, and measurements in more than one basis.The review says experimental NISQ wavefunctions, including cold-atom implementations of the fermionic Hubbard model, may conceivably be reconstructed using this strategy.Source: §IV · p. 11
• The number of measurement bases needed for informational completeness remains poorly understood, ranging from one to exponential growth in qubit number for pure states.The review also identifies open questions about how wavefunction structure affects measurement needs and how those needs relate to RBM parameter scaling.Source: §IV · p. 11
• Density-matrix learning suggests an approximate reformulation of quantum state tomography, while informationally complete POVMs may improve scaling at the cost of more complicated measurements.The review further suggests that mixed-state reconstruction could inform an unsupervised-learning reformulation of process tomography.Source: §IV · pp. 11–12
• The review anticipates machine learning developing alongside quantum hardware for state preparation, measurement, verification, error correction, and other tasks.It presents machine learning of quantum many-body states as potentially important in the NISQ era and beyond.Source: §IV · p. 12