Continuous-variable quantum-state tomography usually reconstructs a truncated Fock-basis density matrix. We study a compact alternative: finite mixtures of displaced, squeezed Gaussian kets whose ρ = BB† construction is positive semidefinite by construction, fitted by per-sample homodyne likelihood with closed-form gradients, and composed with physical loss and noise channels. On the public propagating-light GKP dataset of Konno et al. (Science 2024), a rank-4 model with 92 real parameters matches the empirical full-rank maximum-likelihood frontier (255 parameters) at confidence-interval resolution on held-out likelihood. On a synthetic thermal-noise target that we prove lies outside the model family — no detection efficiency and no finite rank reproduces it exactly — the channel-composed model fitted blind exceeds a full-rank MLE run under the pre-declared 900-second baseline budget, a verdict that holds across all five pre-declared seed and noise configurations. We do not claim a universally superior method: comparisons on real data reuse observations across splits, the strongest baseline is test-selected, and the blind result covers one target class. The contribution is a compact, physically constrained model family together with a fully falsification-first research record — negative results, superseded scorings, and pre-declared protocols are all preserved in the accompanying repository.
Continuous-variable quantum-state tomography usually reconstructs a truncated Fock-basis density matrix. We study a compact alternative: finite mixtures of displaced, squeezed Gaussian kets whose ρ = BB† construction is positive semidefinite by construction, fitted by per-sample homodyne likelihood with closed-form gradients, and composed with physical loss and noise channels. On the public propagating-light GKP dataset of Konno et al. (Science 2024), a rank-4 model with 92 real parameters matches the empirical full-rank maximum-likelihood frontier (255 parameters) at confidence-interval resolution on held-out likelihood. On a synthetic thermal-noise target that we prove lies outside the model family — no detection efficiency and no finite rank reproduces it exactly — the channel-composed model fitted blind exceeds a full-rank MLE run under the pre-declared 900-second baseline budget, a verdict that holds across all five pre-declared seed and noise configurations. We do not claim a universally superior method: comparisons on real data reuse observations across splits, the strongest baseline is test-selected, and the blind result covers one target class. The contribution is a compact, physically constrained model family together with a fully falsification-first research record — negative results, superseded scorings, and pre-declared protocols are all preserved in the accompanying repository.
Neural networks can identify an optical quantum state from an image of its phase-space function, but a real measurement gives a finite record of N detection events. Heterodyne detection samples the Husimi Q function directly, whereas homodyne detection gives the Wigner function only after a reconstruction from several phases. We ask which scheme better identifies one of seven families of states, including Gottesman-Kitaev-Preskill (GKP) states, under loss, dephasing and Gaussian noise. We simulate finite records on identical states for two sets of families, one with equal mean photon number and one built from the code of Ahmed et al. For convolutional networks, heterodyne detection outperforms homodyne detection with filtered back-projection by 14.1 percentage points at N=100 events and 20% dephasing in the first set (adjusted p=0.003 over 24 cells across training runs), and it depends on the number of phases and on the reconstruction. At N=103 and 104 no difference is significant, but the intervals are wide. At N=100, a homodyne network without reconstruction does not differ significantly from heterodyne detection in the first set and is nominally worse in the second. Near thermal states we show to leading order that heterodyne detection damps phase-space structure of order n exponentially, whereas homodyne detection shares unit weight among n+1 angular components, a local result that does not explain the finite-record differences. Experimental records of a GKP state, photon-subtracted squeezed states and one mode of an Einstein-Podolsky-Rosen source are classified like simulated ones, but these low-power tests probe our counting-noise model, not the seven-class accuracy. A likelihood-ratio observer shows no such gap at N=100, so our evidence concerns a reconstruction-based pipeline, not homodyne detection in general.
Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confirm this intuition. Firstly, we show that a low-rank density matrix can be estimated using fewer copies of the state, i.e. the sample complexity of tomography decreases with the rank. Secondly, we show that unknown low-rank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. These techniques use simple Pauli measurements, and their output can be certified without making any assumptions about the unknown state. In this paper, we present a new theoretical analysis of compressed tomography, based on the restricted isometry property for low-rank matrices. Using these tools, we obtain near-optimal error bounds for the realistic situation where the data contain noise due to finite statistics, and the density matrix is full-rank with decaying eigenvalues. We also obtain upper bounds on the sample complexity of compressed tomography, and almost-matching lower bounds on the sample complexity of any procedure using adaptive sequences of Pauli measurements. Using numerical simulations, we compare the performance of two compressed sensing estimators—the matrix Dantzig selector and the matrix Lasso—with standard maximum-likelihood estimation (MLE). We find that, given comparable experimental resources, the compressed sensing estimators consistently produce higher fidelity state reconstructions than MLE. In addition, the use of an incomplete set of measurements leads to faster classical processing with no loss of accuracy. Finally, we show how to certify the accuracy of a low-rank estimate using direct fidelity estimation, and describe a method for compressed quantum process tomography that works for processes with small Kraus rank and requires only Pauli eigenstate preparations and Pauli measurements.
Reconstruction of the quantum density operator ρ from finite projective measurement statistics is a foundational requirement for quantum characterization, verification, and benchmarking (QCVB) in modern quantum processors. However, standard linear inversion systematically fails non-negativity (λmin < 0) in the presence of statistical shot noise, yielding unphysical indefinite operators that violate the Born rule and predict negative probabilities. Conversely, standard iterative Maximum Likelihood Estimation (MLE) via Hradil's RρR map or Semidefinite Programming (SDP) incurs high computational overhead and exhibits sensitivity to rank deficiencies, creating severe latency bottlenecks in automated quantum processing unit (QPU) calibration sweeps. Here, we present IdemQST, a deterministic quantum state tomography framework governed by idempotent projection operator theory (Π2 = Π). In an informationally complete orthogonal Pauli basis, sequential projection across measurement hyperplanes collapses in exactly one pass to the unconstrained linear inversion estimate ρlin, and subsequent Euclidean simplex spectral projection ΠΔ yields the exact global Frobenius-closest physical state on the quantum state cone Sd, establishing mathematical equivalence to Projected Linear Inversion (PLI; Smolin, Gambetta, and Smith, 2012). Rather than asserting novel algorithmic optimality for the simplex step, IdemQST re-articulates this foundation within metric idempotent operator algebra, integrates zero-heap static workspace caching (IdemKV), provides empirical dynamic drift cancellation via Dynamic Permutation Networks (DPN), and extends to incomplete or shadow tomography (M < 4N - 1) via cyclic Projections Onto Convex Sets (POCS). Across simulated projective Born measurements on one-, two-, and three-qubit systems under realistic Poisson shot noise (Nshots ∈ [102, 2 × 104]), IdemQST unconditionally eliminates negative eigenvalues (λmin ≥ 0), conserves unit trace to machine precision (10-15), and achieves state fidelities F > 0.994 under pure prior projection or large shot budgets. Crucially, across N=30 repeated timing benchmarks, IdemQST executes the core simplex spectral projection in 96.7 ± 44.3 μs for two-qubit Bell states (39.3× speedup over Hradil MLE and 41.1× over Diluted MLE) and 0.145 ± 0.035 ms (145.0 μs) for three-qubit GHZ states (d=8), achieving a 111.0× speedup over iterative Hradil MLE (16.08 ± 2.17 ms) and 113.6× over accelerated Diluted MLE (16.46 ± 1.58 ms; Šeháček et al., 2007). Furthermore, we incorporate State Preparation and Measurement (SPAM) readout mitigation via simplex-constrained assignment inversion, providing a deterministic, hardware-ready platform for real-time automated QPU calibration.
Bayesian quantum tomography requires efficient inference while preserving a posterior fixed by the prior and Born likelihood. Learned transport provides fast amortized samples, but reward tuning can reshape the generated distribution rather than improve exploration of this fixed target. We introduce GRPO-QPS, a target-preserving framework in which GRPO learns proposal behavior and an exact Metropolis correction preserves the posterior after training. Across the evaluated reconstruction benchmarks, GRPO-QPS improves over BuresTomFlow and Flow-GRPO on thermal, cat, Dicke, and cluster families, and it closely matches an exact two-qubit reference posterior. Tuned conventional MCMC is slightly stronger on several original continuous benchmarks where the available fixed proposals already match the posterior geometry well. To test whether this reflects a fundamental limitation of learned exploration, we evaluate a more challenging multimodal thermal posterior. At six qubits and 800 shots, the learned proposal achieves a minimum effective sample size of 102 per 1,000 likelihood calls, compared with 28 for prior independence, 27 for a tuned fixed mixture, and 20 for Haario adaptive Metropolis. A record-conditioned policy also transfers to unseen 3,000-shot records, matching or exceeding the strongest conventional baseline in all nine held-out seed-record comparisons. These results show that GRPO-QPS combines target-preserving Bayesian inference with broad gains over learned transport baselines and a sampling advantage when efficient exploration requires proposal geometry beyond the evaluated conventional kernels.
Certifying reliable sources for quantum information processing is still one of the major challenges towards practical applications. Continuous variable quantum information provides a very promising platform where important resources such as scalable entanglement come "for free" [1] . In that context, a crucial quantum feature to look for and investigate is the so-called non-Gaussianity, which, beyond its conceptual relevance, is a necessary resource for any quantum computational advantage as well as for effective error correction, entanglement distillation and more. Non-Gaussian states have been produced and probed using full tomography techniques [2] . However, performing a full tomography of multimode continuous variable systems is a hard problem. Indeed, systems aimed at overcoming classical approach are by definition too complex to be fully characterized, thus alternative approaches are required.
The exploration of tomography of bosonic Gaussian states is presumably as old as quantum optics, but only recently, their precise and rigorous study have been moving into the focus of attention, motivated by technological developments. In this work, we present an efficient and experimentally feasible Gaussian state tomography algorithm with provable recovery trace-distance guarantees, whose sample complexity depends only on the number of modes, and - remarkably - is independent of the state's photon number or energy, up to doubly logarithmic factors. Our algorithm yields a doubly-exponential improvement over existing methods, and it employs operations that are readily accessible in experimental settings: the preparation of an auxiliary squeezed vacuum, passive Gaussian unitaries, and homodyne detection. At its core lies an adaptive strategy that systematically reduces the total squeezing of the system, enabling efficient tomography. Quite surprisingly, this proves that estimating a Gaussian state in trace distance is generally more efficient than directly estimating its covariance matrix. Our algorithm is particularly well-suited for applications in quantum metrology and sensing, where highly squeezed - and hence high-energy - states are commonly employed. As a further contribution, we establish improved sample complexity bounds for standard heterodyne tomography, equipping this widely used protocol with rigorous trace-norm guarantees.