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arXiv:2602.22057

Quantum tomography for non-iid sources

Leonardo Zambrano

quant-phSubmitted Feb 25, 2026• Announced Feb 26, 2026• 0 citations• Revised Jul 15, 2026 (v2)
v1 · Feb 25, 2026v2 · Jul 15, 2026
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Abstract

Quantum state and process tomography are typically analyzed under the assumption that devices emit independent and identically distributed (i.i.d.) states or channels. In realistic experiments, however, noise, drift, feedback, or adversarial behavior violate this assumption. We show that projected least-squares tomography remains statistically optimal even under fully adaptive state and channel preparation. Specifically, we prove that the sample complexity for reconstructing the time-averaged state or channel matches the optimal i.i.d. scaling for non-adaptive, single-copy measurements. For rank-$r$ states, the sample complexity is $\mathcal{O}(d r^2/ε^2)$ to achieve accuracy $ε$ in trace distance, while for process tomography it is $\mathcal{O}(d^6/ε^2)$ to achieve accuracy $ε$ in diamond distance. Thus, dropping the i.i.d. assumption does not increase the fundamental sample complexity of quantum tomography, but only changes the interpretation of the reconstructed object.

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Quantum state and process tomography are typically analyzed under the assumption that devices emit independent and identically distributed (i.i.d.) states or channels. In realistic experiments, however, noise, drift, feedback, or adversarial behavior violate this assumption. We show that projected least-squares tomography remains statistically optimal even under fully adaptive state and channel preparation. Specifically, we prove that the sample complexity for reconstructing the time-averaged state or channel matches the optimal i.i.d. scaling for non-adaptive, single-copy measurements. For rank-rrr states, the sample complexity is O(dr2/ε2)\mathcal{O}(d r^2/ε^2)O(dr2/ε2) to achieve accuracy εεε in trace distance, while for process tomography it is O(d6/ε2)\mathcal{O}(d^6/ε^2)O(d6/ε2) to achieve accuracy εεε in diamond distance. Thus, dropping the i.i.d. assumption does not increase the fundamental sample complexity of quantum tomography, but only changes the interpretation of the reconstructed object.
Quantum process tomography, the task of estimating an unknown quantum channel, is a central problem in quantum information theory. A long-standing open question is how many uses of an unknown channel are required to learn it in diamond distance, the standard metric for distinguishing quantum processes. While quantum state tomography is well understood, for general channels the problem remained open beyond the unitary case. Here we establish the query complexity of channel tomography with optimal dependence on the dimension parameters, at any fixed constant accuracy. We design an algorithm showing that any channel with input/output dimensions din,doutd_{\mathrm{in}},d_{\mathrm{out}}din​,dout​ and Kraus rank at most kkk can be learned to accuracy ε\varepsilonε using O(dindoutk/ε2)O(d_{\mathrm{in}}d_{\mathrm{out}}k/\varepsilon^{2})O(din​dout​k/ε2) channel uses. Conversely, we prove that Ω(dindoutk)Ω(d_{\mathrm{in}}d_{\mathrm{out}}k)Ω(din​dout​k) uses are necessary at constant accuracy and that, for non-minimal Kraus rank, a separate Ω(1/ε2)Ω(1/\varepsilon^{2})Ω(1/ε2) contribution is unavoidable. Since channels subsume states, unitaries, isometries, and measurements as special cases, our protocol provides a unified framework for these tomography tasks, yielding new guarantees for isometry and measurement tomography while recovering known optimal scalings for state and unitary tomography. Our algorithm follows the natural strategy of performing optimal tomography on the Choi state. The main technical contribution is to show that this suffices to control the induced diamond-distance error, avoiding the dimension loss incurred by a naive conversion from Choi-state trace distance to channel diamond distance. The protocol uses the channel non-adaptively to prepare Choi-state copies, purifies them in parallel, and performs optimal pure-state tomography on the resulting purifications. Hence, we reduce channel tomography to pure-state tomography.
Quantum process tomography (QPT) protocols are often designed by lifting quantum state tomography (QST) protocols through the Choi-Jamiolkowski isomorphism, at the cost of an extra step enforcing the trace-preserving (TP) constraint. Existing approaches fall into two classes. The folklore partial normalization, which sandwiches the density estimate between inverse square roots of its input marginal, has a closed form but is used as a heuristic, its only analyses being asymptotic and losing a factor polynomial in the dimension. Projection-based methods are principled but have no closed forms and scale poorly with dimension. We resolve this dichotomy by showing that the partial normalization is the exact projection onto the Choi states of quantum channels in fidelity (equivalently in Bures or purified distance). Consequently, any QST algorithm with guarantees in fidelity lifts to a QPT algorithm that inherits its sample complexity, through a single closed-form congruence that preserves the Choi rank and is computed in Kraus form without any eigendecomposition of Choi dimension. We lift two single-copy protocols and prove that the non-adaptive one (Fidelity Projected Least Squares) recovers the Choi rank and reaches infidelity ε\varepsilonε from a number of channel uses proportional to 1/ε1/\varepsilon1/ε. Numerically, for channels of low Choi rank, it improves on the state-of-the-art projected least squares in accuracy, and by orders of magnitude in the time and memory of the TP-regularization step and of the returned estimate.
Assumption-free reconstruction of quantum states from measurements is essential for benchmarking and certifying quantum devices, but it remains difficult due to the extensive measurement statistics and experimental resources it demands. An approach to alleviating these demands is provided by adaptive measurement strategies, which can yield up to a quadratic improvement in reconstruction accuracy for pure states by dynamically optimizing measurement settings during data acquisition. A key open question is whether these asymptotic advantages remain in realistic experiments, where readout is inevitably noisy. In this work, we analyze the impact of readout noise on adaptive quantum state tomography with readout-error mitigation, focusing on the challenging regime of reconstructing pure states using mixed-state estimators. Using analytical arguments based on Fisher information optimization and extensive numerical simulations using Bayesian inference, we show that any nonzero readout noise eliminates the asymptotic quadratic scaling advantage of adaptive strategies. We numerically investigate the behavior for finite measurement statistics for single- and two-qubit systems with exact readout-error mitigation and find a gradual transition from ideal to sub-optimal scaling. We furthermore investigate realistic scenarios where detector tomography is performed with a limited number of state copies for calibration, showing that insufficient detector characterization leads to estimator bias and limited reconstruction accuracy. Although our result imposes an upper bound on the reconstruction accuracy that can be achieved with adaptive strategies, we nevertheless observe numerically a constant-factor gain in reconstruction accuracy, which becomes larger as the readout noise decreases. This indicates potential practical benefits in using adaptive measurement strategies in well-calibrated experiments.
Quantum process tomography, the task of learning an unknown quantum channel from black-box access, is a central problem in quantum information. In this setting, protocols with quantum memory can coherently store and jointly process quantum information obtained from multiple channel uses, whereas protocols without quantum memory must measure after each use and retain only a classical transcript of the measurement outcomes. A fundamental open question is whether quantum memory provides a query-complexity advantage even when protocols without quantum memory may adapt their experiments based on all previous outcomes with unbounded classical computational power. In this work, we show that it does. We determine the optimal query complexity of quantum process tomography without quantum memory up to a constant factor to be Θ(din3dout3/ε2)Θ(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)Θ(din3​dout3​/ε2), where dind_{\mathrm{in}}din​ and doutd_{\mathrm{out}}dout​ are the channel input and output dimensions, respectively, and ε\varepsilonε is the target diamond-norm accuracy. More precisely, we prove that any incoherent protocol for this task, including adaptive protocols, requires Ω(din3dout3/ε2)Ω(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)Ω(din3​dout3​/ε2) queries, even when each channel use may be assisted by arbitrary fresh ancilla, and we present a non-adaptive, ancilla-free incoherent protocol achieving the matching upper bound O(din3dout3/ε2)O(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)O(din3​dout3​/ε2). Our results thereby generalize the optimal sample-complexity bounds for single-copy state tomography, recovered as the special case din=1d_{\mathrm{in}}=1din​=1. By contrast, coherent protocols with quantum memory achieve query complexity Θ(din2dout2/ε2)Θ(d_{\mathrm{in}}^2 d_{\mathrm{out}}^2/\varepsilon^2)Θ(din2​dout2​/ε2). Hence, our results establish a rigorous learning separation between quantum process tomography with and without quantum memory.
We present a two-step protocol for quantum measurement tomography that is light on classical co-processing cost and still achieves optimal sample complexity. Given measurement data from a known probe state ensemble, we first apply least-squares estimation to produce an unconstrained approximation of the POVM, and then project this estimate onto the set of valid quantum measurements. For a POVM with L outcomes acting on a d -dimensional system, we show that the protocol requires O ( ( d 3 + d 2 L ) / ϵ 2 ) samples to achieve error ϵ in worst-case distance, and O ( d 2 L / ϵ 2 ) samples in average-case distance. We further establish two matching sample complexity lower bounds of Ω ( ( d 3 + d 2 L ) / ϵ 2 ) and Ω ( d 2 L / ϵ 2 ) for any non-adaptive, single-copy POVM tomography protocol. Hence, our projected least squares POVM tomography is sample-optimal in both the dimension and the number of outcomes for both distances. Our method admits an analytic form when using global or local 2-designs as probe ensembles and enables rigorous non-asymptotic error guarantees. Finally, we also complement our findings with empirical performance studies carried out on a noisy superconducting quantum computer with flux-tunable transmon qubits.
We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most ttt samples. For sufficiently small ε\varepsilonε, estimating an unknown state on Cd\mathbb{C}^dCd of rank at most rrr to trace norm error ε\varepsilonε with constant success probability requires, and is achievable with, Θ(drε2max{1,rt})Θ\left(\frac{dr}{\varepsilon^2}\mathop{\mathrm{max}}\left\{1,\frac{r}{\sqrt{t}}\right\}\right)Θ(ε2dr​max{1,t​r​}) 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 ttt samples improve the complexity of algorithms making single-sample measurements by at most a factor t\sqrt{t}t​. Further, measuring order r2r^2r2 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 ttt 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.