Quantum State Tomography

We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary $n$-mode bosonic state $ρ$, the goal is to output a pure Gaussian state whose infidelity with $ρ$ is at most $\mathrm{opt} + ε$, where $\mathrm{opt}$ is the minimum infidelity achievable by any pure Gaussian state. We give efficient protocols achieving this in both the high and low fidelity regimes. When $\mathrm{opt}$ is below some universal constant, our protocol has runtime and copy complexity which is strongly polynomial in $n, 1/ε$ and $\log \log E$, where $E$ is the energy of the closest pure Gaussian state. For arbitrary $\mathrm{opt}$, our protocol uses $(n+1)^{\mathrm{poly}(1/ε)} \mathrm{poly}\left(1+\log\log(E)\right)$ copies and runtime. As a corollary, we obtain the first truly tolerant Gaussianity testing protocol for distinguishing whether $\mathrm{opt} > c + ε$ or $\mathrm{opt} < c - ε$, for any threshold $c\in(0,1)$. We also prove $\mathrm{poly}(n,1/ε)$ runtime is impossible, unless $\mathrm{NP}\subseteq\mathrm{BQP}$. Our protocols follow a shared paradigm: first, we iteratively use general Gaussian measurements combined with techniques from classical robust statistics to obtain a good warm start estimate, then we leverage non-Gaussian measurements to refine this warm start using convex and non-convex optimization methods. Interestingly, we prove that non-Gaussian measurements are necessary to match the strong agnostic guarantees we obtain, and in fact these guarantees are provably superior to what is possible for robustly estimating classical Gaussians.

Structured quantum-state learning not only depends on an expressive ansatz but also on an operational certificate that stays meaningful with finite measurements and imperfect implementation. We study pure one dimensional states learning by an inverse binary multiscale entanglement renormalization ansatz (MERA). In the learning procedure, the qubits removed during coarse graining are controlled coherently and measured together at the terminal register. We confirm that an ideal sequential and terminal measurement schedule delivers the same complete bit string distribution under matched causal operations, while normalized postselection can amplify perturbations inversely with prefix acceptance. A noise aware theorem introduces an individual calibrated total variation implementation budget to the finite shot certificate. The protocol is estimated on an open boundary transverse field Ising ground state. A frozen 8-qubit schedule using $560$ million simulated training measurements per run achieves fidelity above $0.99$ in all $60$ held-out runs, with a mean fidelity of $0.996886$. 1080 circuit-noise cells and 6480 confidence-coverage rows are covered by fixed-circuit robustness validation without a locked soundness violation. We then address architectural fairness at $n=16$ using three new studies. In a 120-run exact-gradient multistart diagnostic, MERA has higher fidelity in 58/60 paired restarts and lower long-range error in 60/60, although no run met the prespecified stationarity criterion. Finally, a causal cone-complete, parameter matched local circuit achieves $2.62\times$ greater aggregate gate exposure yet loses all 30 paired comparisons in fidelity, long-range error, energy, and entropy.

A fundamental question in physics is: When does classical behavior emerge from quantum systems? Bosonic Gaussian states provide a natural setting to explore this quantum-classical boundary, as they capture both the classical field behavior and the intrinsic quantum nature of light. Here, we address this problem from a learning-theoretic perspective by asking: When are bosonic Gaussian states classical to learn? That is, under what conditions (if any) can an n-mode bosonic Gaussian state be learned with as few samples, and with operations as simple, as are needed to learn a classical 2n-variate Gaussian distribution? We establish a smooth crossover in learnability governed by the state's thermal fluctuations: - Cold Gaussian states are non-classical to learn: When the covariance matrix satisfies $Σ\le(\frac12+O(\frac1n))I$, i.e. close to the vacuum covariance, tomography under single-copy (i.e., non-entangled) measurements fundamentally requires $Ω(n^3)$ copies, strictly exceeding the sample complexity $Θ(n^2)$ of learning classical Gaussian distributions. We show that this hardness persists even when few-copy entangled measurements are allowed. - Warm Gaussian states are classical to learn: When thermal fluctuations exceed the vacuum noise, parameterized by $Σ\ge(\frac12+ν)I$ for any parameter $ν>0$, we prove that single-copy tomography requires $N=Θ\left(n^2\min(n,1+ν^{-1})\right)$ copies. This bound is tight and is achieved by simple, non-adaptive, unentangled heterodyne measurements. Crucially, for $ν=Ω(1)$, the sample complexity drops to $Θ(n^2)$, matching the classical case. Our results tightly characterize a quantum-to-classical crossover in the learnability of bosonic Gaussian states, reveal a novel connection between fundamental physics and statistical learning theory, and have implications for real-world sensing experiments.

Quantum communication underpins secure information processing and scalable quantum networks. In particular, remote state preparation (RSP) enables efficient quantum state transfer, but accurately estimating target states under complex noise remains challenging. Here, we propose a Transformer-based Quantum State Characterizer (TQSC) model for noisy RSP experiments. Our model reconstructs experimentally prepared pure and mixed photonic polarization states from noisy measurements in complex scattering environments, while its attention patterns provide physically grounded insights into correlations among the measured observables. The method achieves a mean estimator-target fidelity exceeding 99.999% under complex scattering and dynamic Gaussian noise, while its robustness and generalization are further examined using Qiskit-simulated Bloch-ball states.Furthermore, in a practical MNIST image transmission task with held-out states, the decoded bit error rate is reduced from 50.34% to zero after TQSC post-processing. The TQSC model enables accurate tomographic characterization under dynamic noise and provides physically grounded post-hoc insights, holding promise for intelligent quantum information processing applications.

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.

We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most $t$ samples. For sufficiently small $\varepsilon$, estimating an unknown state on $\mathbb{C}^d$ of rank at most $r$ to trace norm error $\varepsilon$ with constant success probability requires, and is achievable with, $Θ\left(\frac{dr}{\varepsilon^2}\mathop{\mathrm{max}}\left\{1,\frac{r}{\sqrt{t}}\right\}\right)$ samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most $t$ samples improve the complexity of algorithms making single-sample measurements by at most a factor $\sqrt{t}$. Further, measuring order $r^2$ samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on $t$ samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.

Quantum state tomography is a fundamental technique for estimating the state of a quantum system from measured data and plays a crucial role in evaluating the performance of quantum devices. However, standard estimation methods become computationally prohibitive as the system size increases due to the exponential growth of the density matrix, describing a quantum state, with the number of qubits. We propose a low-rank tensor-network framework for mixed-state quantum state tomography based on a block tensor train (Block-TT) factorization. Specifically, the density matrix is represented as the contraction of a Block-TT with its Hermitian transpose, yielding a TT analogue of the Burer-Monteiro factorization. This parameterization guarantees Hermiticity and positive semidefiniteness by construction while compressing the number of optimization variables from exponential to linear in the number of qubits. Building on this representation, we develop single-site and two-site density matrix renormalization group (DMRG) algorithms for estimating quantum states from compressed measurements. The resulting methods operate directly on the compressed parameterization, support adaptive rank refinement, and exploit efficient tensor-network contractions for expectation-value evaluation. The framework is applicable to a broad class of low-rank quantum states, including pure states, nearly pure states, and ground states that admit accurate tensor-network approximations. Numerical experiments demonstrate accurate state reconstruction from limited measurements together with substantial reductions in memory requirements and computational cost compared with conventional low-rank tomography methods.

Parameterised graph theory studies how the complexity of graph-theoretic problems depends on structural parameters of the input graph. This perspective has proved useful in analysing tensor-network simulation (Markov and Shi, 2008). Its implications for tensor-network representations and tomography are less well understood. In particular, which graph parameters determine whether a tensor-network state (TNS) admits a tractable matrix product state (MPS) or tree tensor network (TTN) representation, and which control the complexity of learning the state? We address these questions using parameterised graph theory. First, we show that cutwidth and tree-cutwidth bound the bond dimension overhead required to represent a TNS as an MPS or TTN. In the TTN case, tree-cutwidth also bounds the local dimension of the grouped subsystems. The proofs are based on entanglement rerouting, a tensor-network analogue of rerouting information in a classical network. Second, we derive graph-dependent upper bounds on the sample and computational complexity of realisable TNS tomography, with exponents that depend on cutwidth, tree-cutwidth, and a new graph parameter, learning complexity, which we bound in terms of degree and treewidth. We obtain these results by extending the disentangling MPS learner of (Cramer et al., 2010), as analysed further in (Bakshi et al., 2025; Lin et al., 2025), to TTNs and to tensor networks on arbitrary known graphs. Finally, we extend the framework beyond the realisable setting. For an arbitrary input state, our agnostic learner outputs a pure state whose fidelity is within additive error $ε$ of the optimum over tensor-network states on the given graph with a given bond dimension, with explicit graph-dependent bounds on sample and computational complexity.

Bosonic quantum systems provide a hardware-efficient platform for quantum information processing but remain challenging to characterise due to their large Hilbert space and the high measurement cost of state tomography. Existing approaches estimate the fidelity with respect to a single target state, making them unsuitable for applications in which physically equivalent states differ by phase space translations, rotations, or other transformations. Here, we introduce an adaptive reconstruction technique that estimates the fidelity with respect to a family of bosonic states while reconstructing the underlying Wigner function from a small number of measurements. The method combines a physics-informed parametric model with Bayesian inference, bootstrap, and active learning to iteratively select the most informative phase space sampling points. We implement the approach on a circuit quantum electrodynamics platform and benchmark it on Schrödinger cat states with amplitudes $α\in[1,3]$. The reconstruction yields reproducible fidelity estimates within a few minutes, remains robust to substantial displacements and rotations in phase space despite using a mismatched prior, and is sensitive to subtle state imperfections. We further compare the adaptive strategy with existing Wigner function sampling protocols experimentally, demonstrating the advantage of adaptive sampling for measurement-efficient fidelity estimation with respect to a family of cat states. Finally, we incorporate the reconstructed fidelity into the figure of merit used in a proof-of-principle closed-loop quantum optimal control experiment, demonstrating the applicability of the method to autonomous optimisation of bosonic quantum states.

Quantum state tomography is sample-starved, and the states one prepares live on a narrow, learnable manifold. A $k{=}0$ prior-only control shows that on concentrated families a prior estimate is already near-optimal, so ``high fidelity at few measurements'' can be family memorization rather than tomography; genuine measurement-efficiency needs a model that conditions on the measurements and demonstrably uses them. On a shared matrix-product-state (MPS) core parameterization we study two routes. ApproachA learns a generative prior over MPS cores with measurement-guided posterior inference (gold-standard-validated, but whose few-measurement accuracy the control shows is largely the prior). ApproachB, our main proposal, is a \emph{fixed-protocol amortized} MPS estimator trained once with a gauge-invariant fidelity loss; we deliberately do not rest it on a permutation-invariant set encoder (a plain MLP matches it). The decisive lever is the measurement design: motivated by the fact that local reduced density matrices determine a $χ$-MPS, conditioning on an \emph{informative local} Pauli set rather than random strings turns a modest, memorization-prone estimator into a high-fidelity one ($\approx\!0.95$, up to $+0.59$ over prior-only, decisively passing a shuffled-measurement control). A dropout ensemble, conformally recalibrated, gives $\approx\!90\%$-coverage intervals -- including for observables never measured, where a shot-based interval does not exist. Quality holds as the system grows (fidelity $0.90$ at $n{=}10$, gain \emph{growing} in $n$; $0.88$ at bond dimension $χ{=}4$), the parameterization is polynomial (native contraction to $20$ qubits), and we close the loop on IBM hardware ($5$ states at $0.97$ from hardware-measured Paulis).

Machine-learning approaches to quantum state tomography can achieve high reconstruction fidelity, but the physical structure used by the trained model often remains implicit. Here we ask whether a sparsified Kolmogorov-Arnold Network (KAN) can be used not only as a regressor, but also as an inspectable reconstruction rule whose internal organization can be checked against known Pauli structure. We study a controlled three-qubit GHZ-family benchmark in which all 63 non-identity Pauli expectation values are used to reconstruct three GHZ-subspace variables: the population imbalance $z$, the real off-diagonal component $c$, and the imaginary off-diagonal component $s$. Under finite-shot sampling and depolarizing noise, external ablation identifies the extended 12-channel GHZ-relevant Pauli set from the 63 measurements, with exact top-12 recovery across the tested shot counts and depolarizing-noise strengths. These support patterns remain stable across multi-seed random-initialization and noise-level analyses, and collapse under random-label controls. The dominant pruned input-hidden-output pathways organize Z-type population observables and X/Y off-diagonal observables in a pattern consistent with the analytic GHZ Pauli grouping, and sparse formula recovery recovers the canonical signed Pauli relations. The contribution of the KAN is therefore pathway-level structural interpretability within a neural reconstruction model, rather than superior sparse regression. Together with negative controls, these probes provide a consistency chain for auditing learned reconstruction rules against known physical structure.

We extend quantum state tomography with minimal cumulative disturbance, first investigated in [arXiv:2406.18370], to arbitrary finite-dimensional pure states. A learner sequentially receives fresh copies of an unknown pure state, chooses a rank-one projector for each copy using the previous outcomes, and performs the corresponding two-outcome projective measurement. The goal is to learn the state while keeping the chosen projectors close to the unknown state in order to minimize disturbance. The qubit solution relies on the special geometry of the Bloch sphere and does not extend directly to qudits, where pure states form a curved manifold. We show that this obstruction can be overcome by working locally on the pure-state manifold. The algorithm proceeds in epochs. In each epoch, it fixes a current estimate, measures pairs of nearby rank-one projectors obtained by moving in opposite tangent directions, and takes differences of the corresponding outcomes. This gives an exact linear observation of the tangent component of the error. The resulting local linear models are combined with a robust variance-adaptive estimator and a hot-start regularization that transfers precision across epochs. For every unknown pure state in dimension $d$, after $T$ measured copies, our protocol achieves cumulative regret $\mathcal{O}(d^3\log^2 T)$, and at each intermediate time $t\leq T$ its current estimate has online infidelity $\mathcal{O}(d^3\log(T)/t)$. Hence, pure-state tomography with essentially no cumulative disturbance is not a peculiarity of qubits but a geometric phenomenon that persists for qudits.