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Reconstructing quantum states with generative models
Juan Carrasquilla, Giacomo Torlai, Roger G. Melko, Leandro Aolita
TL;DR
Large quantum-state tomography faces exponential complexity, while existing efficient approaches have limited versatility or incur exponential scaling for mixed states. The paper reduces tomography to unsupervised learning of informationally complete measurement statistics with deep generative models. It reports scalable reconstruction of pure and mixed states across a broad range of structured systems, using accessible measurements and an approximate reconstruction certificate.
Problem
Large-scale quantum-device benchmarking requires density-matrix reconstruction despite the exponential complexity of generic quantum states and limitations of existing state-specific methods.
Method
The method parameterizes outcome probabilities from an informationally complete quantum measurement with neural-network generative models and uses single-particle measurements.
Results
The procedure reconstructs pure and mixed states described by a wide range of generative models and efficiently handles complex states including local-spin-model ground states in one and two dimensions.
Takeaways & Limitations
The approach provides a scalable machine-learning procedure for quantum-state reconstruction with easily available measurements and an approximate certificate of reconstruction.
Takeaways & Limitations
Existing efficient tomography methods have limited versatility, while MPS tomography becomes computationally intractable on higher-dimensional lattices.
Abstract
from arXiv · showhide
A major bottleneck in the quest for scalable many-body quantum technologies is the difficulty in benchmarking their preparations, which suffer from an exponential `curse of dimensionality' inherent to their quantum states. We present an experimentally friendly method for density matrix reconstruction based on deep neural-network generative models. The learning procedure comes with a built-in approximate certificate of the reconstruction and makes no assumptions on the state under scrutiny, making it both reliable and unconditional. It can efficiently handle a broad class of complex systems including prototypical states in quantum information, as well as ground states of local spin models common to condensed matter physics. The key insight is to reduce the state tomography task to an unsupervised learning problem of the statistics of an informationally complete set of quantum measurements. This constitutes a modern machine learning approach to the validation of large quantum devices, which may prove relevant as a neural-network ansatz over mixed states suitable for variational optimization.
INTRODUCTION
Quantum state tomography becomes impractical for large systems because generic states have exponentially complex descriptions, while existing efficient methods apply only to limited state classes. The paper introduces a generative-model approach that learns measurement statistics for arbitrary states using experimentally accessible single-particle measurements.
- Motivation: Large-system tomography is impractical because generic many-body states have exponentially complex descriptions.
- Limitations of existing methods: Existing efficient reconstruction methods exploit structural assumptions, but their applicability is limited across state classes and geometries.Examples include permutationally invariant tomography, compressed sensing, and tensor-network schemes.
- Limitations of existing methods: MPS tomography becomes computationally intractable on higher-dimensional lattices and is restricted in one dimension to states arising from short-time dynamics.
- Limitations of existing methods: RBM-based mixed-state reconstruction through purifications introduces exponential scaling in any spatial dimension.
- Proposed approach: The proposed ansatz combines tensor-network principles with neural-network generative models to parameterize outcome probabilities from a tomographically complete measurement on arbitrary states.The reconstruction represents the density matrix through the learned outcome distribution and single-qubit-factorable measurement tensors.
- Scope and validation: The method uses routinely available single-particle measurements, learns complex states in one and two dimensions, and approximately certifies reconstructions by sampling from the reconstructed distribution.
LEARNING STRATEGY
The method converts quantum state tomography into unsupervised learning of measurement-outcome statistics from an informationally complete POVM. Because these statistics determine the density matrix, a generative model can reconstruct the state from experimental samples.
- Informational completeness means measurement statistics contain all information about the state and specify it unambiguously.
- An informationally complete measurement lets the density matrix be unambiguously inferred from the probability distribution of outcomes.
- The procedure learns a model Pmodel(a) for experimental outcome statistics using expressive neural generative models.
- Possible generative models include variational autoencoders, generative adversarial networks, restricted Boltzmann machines, and recurrent neural networks.
- The demonstrations use local informationally complete POVMs, including tetrahedral measurements with four outcomes per qubit.
NUMERICAL EXPERIMENTS
Numerical experiments test generative-model reconstruction on noisy Bell and GHZ states and on ground states of local spin models. Reconstructed distributions achieve high fidelity, correlations agree with synthetic states, and RNN training scales favorably with system size.
- For Bell states under local depolarizing noise, the KL divergence decreases to values near zero for all noise probabilities p.
- Classical and quantum fidelities approach one with similar training behavior, supporting classical fidelity as a practical reconstruction-quality measure.
- RNN models are faster to train than RBMs and may support variational energy minimization for ground and thermal states.
- For locally depolarized GHZ states, FC(PRNN, PGHZ) quickly approaches unity, while noiseless states require significantly more training samples.
- The sample count required to reach FC(PRNN, PGHZ) = 0.99 scales approximately linearly with N for both tetrahedral and Pauli-6 POVMs.
- RNN reconstructions of Ising ground states attain classical fidelity FC(PRNN, PIsing) ≈0.998.
- Ground-state reconstructions reproduce one- and two-body correlations and total energies, including Heisenberg results with FC(PRNN, PHeisenberg) ≈0.98.
CONCLUSIONS AND OUTLOOK
The paper presents a scalable machine-learning procedure for reconstructing pure and mixed quantum states from accessible measurements, with certification based on measurement-statistics fidelity. Demonstrations cover quantum-information states and local-Hamiltonian ground states, connecting machine learning with tensor-network frameworks for validating quantum simulators.
- The results demonstrate a scalable machine-learning procedure for reconstructing states and characterizing available quantum devices in the near-term era of approximate quantum computing.
- The procedure reconstructs pure and mixed states with structure captured by a broad range of generative models and easily available measurements.
- It includes an approximate certification scheme based on the classical fidelity of measurement statistics.
- The method was demonstrated on prototypical quantum-information states and ground states of local Hamiltonians relevant to condensed matter, cold atoms, and quantum simulators.
- The work combines state-of-the-art machine-learning algorithms with established tensor-network frameworks to help validate and characterize quantum simulators and commercially available quantum devices.
I. INFORMATIONALLY COMPLETE GENERALIZED MEASUREMENTS
The paper uses informationally complete generalized measurements to encode quantum-state information in outcome statistics. It discusses tetrahedral and Pauli POVMs, their physical implementation, and how overlap-matrix invertibility affects reconstruction and observable estimation.
- A POVM consists of non-negative operators that decompose the identity, and it is informationally complete when its elements span the full operator space.
- The tetrahedral POVM uses rank-1 projectors whose directions form a regular tetrahedron, making the measurement symmetric and its overlap matrix invertible.
- The tetrahedral POVM can be physically realized through Neumark dilation by coupling the system qubit to an ancilla and performing a projective measurement on both.
- The Pauli-6 POVM combines the eigenbases of σz, σx, and σy into one generalized measurement, implementable by randomly selecting a Pauli operator and measuring it.
- For a non-invertible overlap matrix, linear inversion and direct local-observable estimation are not possible, although classical-fidelity estimation remains feasible.
- Pauli-4 uses four outcomes and can be implemented similarly to Pauli-6 with classical post-processing that identifies three outcomes.
II. DIRECT STOCHASTIC ESTIMATION OF LOCAL OBSERVABLES
The method estimates expectation values of local observables directly from samples of a generative model, without explicitly constructing its density matrix. With suitable factorable POVMs and variance conditions, the sampling cost scales polynomially.
- Expectation values of observables acting on a constant number of qubits can be estimated directly from model samples without the reconstructed density matrix.
- The approach applies to local correlators and, under additional conditions, to non-local matrix-product-operator observables and fidelity with a constant-bond-dimension MPS.
- An informationally complete POVM expands an observable into outcome-dependent coefficients determined by a linear system involving the POVM overlap matrix.
- Because local observables have support on a constant number of qubits, the variance of the sampling estimator is independent of N.
- The estimator reaches arbitrary constant precision ε with runtime and sample count both scaling polynomially in N and ε^-1.
DATA AVAILABILITY
The paper provides numerical measurement data, generative-model implementations, and code for generating the manuscript’s datasets.
- Numerically generated measurements used for Fig. 4 are available with the paper’s implementation materials.
- The generative-model implementations and code for numerically generating the manuscript’s datasets are available online.
IV. GENERATIVE MODELS
The paper represents informationally complete quantum-measurement statistics with neural-network generative models. It uses RBMs whose visible units encode measurement outcomes and whose learned distributions can be sampled.
- RBM representation: RBMs model the measurement-outcome distribution using m-index visible units for an N-qubit system.Each qubit contributes m visible units, and its measurement outcome is encoded as a one-hot vector.
- RBM representation: A one-hot vector identifies one measurement outcome by containing a single 1 among m components.All other cells are zero, so each vector uniquely labels an outcome.
- Probability model: The model parameters include visible biases, hidden biases, and a four-index weight array coupling visible and hidden states.These parameters encode interactions between measurement-outcome states and hidden-unit states.
- Probability model: The RBM assigns probabilities through an energy function and its partition function over visible and hidden configurations.The joint probability is proportional to e^-E(v,h), while marginal modeling concerns the visible units.
- Training: RBMs are trained with block Gibbs sampling and contrastive divergence.Block Gibbs sampling updates groups of variables jointly conditional on the remaining variables.
B. Recurrent neural network models
The recurrent models learn joint measurement statistics autoregressively by processing an ordered sequence of qubit outcomes. GRU-based architectures address long-range dependencies, while tensor-network structure enables direct fidelity estimation for suitable target states.
- RNN construction: The autoregressive model represents a measurement string as a sequence of one-hot encoded qubit outcomes.The recurrent hidden state is updated as the sequence is processed.
- RNN construction: An RNN models each next measurement outcome conditional on all previously observed outcomes.The full distribution factorizes by the chain rule into sequential conditional probabilities.
- GRU architecture: GRUs adaptively capture dependencies at different scales through update and reset gates.The update gate interpolates the previous and candidate hidden states, while reset gates can suppress earlier sequence information.
- GRU architecture: The authors use a deep RNN with three stacked GRU units followed by a softmax layer, with hidden-state dimension 100.The GRU parameters are optimized by maximum likelihood estimation.
- Lattice ordering: For multidimensional lattices, an enumeration of sites defines a one-dimensional path along which the RNN processes the data.This strategy is illustrated for a 4 × 4 triangular lattice and used for the two-dimensional Heisenberg model.
- Fidelity estimation: Direct fidelity estimation is efficient for target states with efficient MPS representations, but its Monte Carlo variance must remain controlled.For GHZ states, the variance grows dramatically with N, making estimation unfeasible around N ≈8.
- Tensor-network sampling: MPS and MPO descriptions provide tractable sampling routes for suitable pure and mixed tensor-network states.Factorable POVMs and small MPO bond dimension allow sequential conditional probabilities to be computed and sampled.