Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.25010

The bottleneck dimension of quantum operations

Pavel Sekatski

quant-phSubmitted Aug 25, 2026• Announced Aug 27, 2026• 0 citations
arXivHTMLPDF

Abstract

Programmable quantum devices nominally act on a Hilbert space whose dimension grows exponentially with the number of constituents, but the presence of noise makes it unlikely that they remain coherent across all of this immense Hilbert space. Then, what is the effective coherent quantum dimension that should be associated with such imperfect devices? To answer this question we here introduce an operational basis-independent framework which imposes a dimension bottleneck on the programmable transformations. Concretely we ask how strongly the quantum information they process can be compressed. Formalizing this idea we identify three inequivalent notions, termed $d$-compressibility, $d$-simulability and $d$-embeddability, which differ in the causal structure used to impose the bottleneck and form a strict hierarchy. The framework unifies several existing notions: joint measurability and simulability of quantum measurements, and the absolute dimensionality of state ensembles, are recovered as special cases. We illustrate the hierarchy with noisy qubit measurements in complementary bases, and we determine the white-noise thresholds at which the set of all noisy unitary channels in dimension $n$, a noisy universal quantum processor, becomes $d$-compressible, $d$-simulable and $d$-embeddable. The thresholds confirm the expectation -- maintaining coherence across the full Hilbert space becomes increasingly demanding as the nominal dimension $n$ increases.

Comments
10+4 pages

Related papers

Simulability of High-Dimensional Quantum Measurements

Marie Ioannou, Pavel Sekatski, Sébastien Designolle +3 more

We investigate the compression of quantum information with respect to a given set M of high-dimensional measurements. This leads to a notion of simulability, where we demand that the statistics obtained from M and an arbitrary quantum state ρ are recovered exactly by first compressing ρ into a lower-dimensional space, followed by some quantum measurements. A full quantum compression is possible, i.e., leaving only classical information, if and only if the set M is jointly measurable. Our notion of simulability can thus be seen as a quantification of measurement incompatibility in terms of dimension. After defining these concepts, we provide an illustrative example involving mutually unbiased bases, and develop a method based on semidefinite programming for constructing simulation models. In turn we analytically construct optimal simulation models for all projective measurements subjected to white noise or losses. Finally, we discuss how our approach connects with other concepts introduced in the context of quantum channels and quantum correlations.

arXiv:2202.12980•Physical Review Letters•Nov 4, 2022•19 citations

Efficient Noisy Quantum State and Process Tomography

Chenyang Li, Shengxin Zhuang, Yukun Zhang +4 more

Efficiently characterizing large quantum states and processes is a central yet notoriously challenging task in quantum information science, as conventional tomography methods typically require resources that grow exponentially with system size. Here, we introduce a structure-agnostic learning framework for noisy $n$-qubit quantum circuits under~i.i.d.~single-qubit noise. We first prove that quantum states with unital noise channels admit an efficient learnable representation in the logarithmic-depth regime. We then extend this framework to quantum process tomography under constant noise, deriving a unified protocol that applies to both unital and non-unital noisy channels and retains efficient guarantees for logarithmic-depth circuits. This process-learning formulation is input-agnostic and imposes no distributional assumptions on the input quantum states. We further study a more general regime with arbitrary noise strength. In this setting, low-weight Pauli propagation induces a terminal truncation whose threshold depends logarithmically on the inverse accuracy, leading to quasi-polynomial complexity and near-unit success probability in the average case. In contrast to the preceding two results, this arbitrary-noise guarantee does not impose any restriction on the circuit depth, and therefore covers arbitrary-depth circuits, including both the noiseless limit ($γ= 0$) and the strong-decoherence regime ($γ= Θ(1)$). Numerical simulations of two-dimensional Hamiltonian dynamics further demonstrate the accuracy and robustness of the approach, including for structured circuits beyond the random-circuit setting assumed in the theoretical analysis. These results provide a scalable and practically relevant route toward characterizing large-scale noisy quantum devices, addressing a key bottleneck in the development of quantum technologies.

quant-pharXiv:2603.01521•Quantum Science and Technology•Mar 3, 2026•0 citations

Absolute Dimensionality of Quantum Ensembles

Alexander Bernal, Gabriele Cobucci, Martin J. Renner +1 more

The dimension of a quantum state is traditionally seen as the number of superposed distinguishable states in a given basis. We propose an absolute, i.e., basis-independent, notion of dimensionality for ensembles of quantum states. It is based on whether a quantum ensemble can be simulated with states confined to arbitrary lower-dimensional subspaces and classical postprocessing. In order to determine the absolute dimension of quantum ensembles, we develop both analytical witness criteria and a semidefinite programming criterion based on the ensemble's information capacity. Furthermore, we construct explicit simulation models for arbitrary ensembles of pure quantum states subject to white noise, and in natural cases we prove their optimality. Also, efficient numerical methods are provided for simulating generic ensembles. Finally, we discuss the role of absolute dimensionality in high-dimensional quantum information processing.

arXiv:2409.01752•Physical Review Letters•Dec 13, 2024•3 citations

Inaccessible information in probabilistic models of quantum systems, non-contextuality inequalities and noise thresholds for contextuality

Iman Marvian

Classical probabilistic models of (noisy) quantum systems are not only relevant for understanding the non-classical features of quantum mechanics, but they are also useful for determining the possible advantage of using quantum resources for information processing tasks. A common feature of these models is the presence of inaccessible information, as captured by the concept of preparation contextuality: There are ensembles of quantum states described by the same density operator, and hence operationally indistinguishable, and yet in any probabilistic (ontological) model, they should be described by distinct probability distributions. In this work, we quantify the inaccessible information of a model in terms of the maximum distinguishability of probability distributions associated to any pair of ensembles with identical density operators, as quantified by the total variation distance of the distributions. We obtain a family of lower bounds on this maximum distinguishability in terms of experimentally measurable quantities. In the case of an ideal qubit this leads to a lower bound of, approximately, 0.07. These bounds can also be interpreted as a new class of robust preparation non-contextuality inequalities. Our non-contextuality inequalities are phrased in terms of generalizations of max-relative entropy and trace distance for general operational theories, which could be of independent interest. Under sufficiently strong noise any quantum system becomes preparation non-contextual, i.e., can be described by models with zero inaccessible information. Using our non-contextuality inequalities, we show that this can happen only if the noise channel has the average gate fidelity less than or equal to 1/D(1+1/2+...+1/D), where D is the dimension of the Hilbert space.

arXiv:2003.05984•Mar 12, 2020•5 citations

Diagnosing quantum reservoirs at scale based on expressivity and coverage

Laia Domingo, Oriol Balló-Gimbernat, Fernando Vilariño

Quantum reservoirs offer a hardware-friendly route to quantum machine learning, replacing trainable circuits with fixed random dynamics and a classical readout. Because the reservoir is not optimized, performance depends entirely on the choice of reservoir family, yet existing diagnostics demand resources that grow exponentially with system size. We introduce a scalable, hardware-agnostic framework built on two complementary quantities. The first is a task-independent order-statistics (ORS) expressivity score, which compares only the largest output probabilities of a reservoir ensemble against an analytical Haar baseline. It never reconstructs the full output distribution, is cost-independent of Hilbert-space dimension, and admits a closed-form depolarizing noise correction, making it directly usable on hardware. The second is the task-dependent effective rank $R_{\mathrm{eff}}$ of the feature matrix, which measures how much input-dependent information reaches the readout. We validate the ORS score against established complexity diagnostics and confirm it remains informative under simulated noise and on IBM quantum hardware. Across synthetic and real quantum extreme learning machine and quantum reservoir computing benchmarks, ORS captures the intrinsic expressivity hierarchy of reservoir families while $R_{\mathrm{eff}}$ determines when that expressivity becomes usable predictive information.

quant-pharXiv:2607.09445•Jul 13, 2026•0 citations

Operational framework for quantum measurement simulability

Leonardo Guerini, Jessica Bavaresco, Marcelo Terra Cunha +1 more

We introduce a framework for simulating quantum measurements based on classical processing of a set of accessible measurements. Well-known concepts such as joint measurability and projective simulability naturally emerge as particular cases of our framework, but our study also leads to novel results and questions. First, a generalisation of joint measurability is derived, which yields a hierarchy for the incompatibility of sets of measurements. A similar hierarchy is defined based on the number of outcomes necessary to perform a simulation of a given measurement. This general approach also allows us to identify connections between different kinds of simulability and, in particular, we characterise the qubit measurements that are projective-simulable in terms of joint measurability. Finally, we discuss how our framework can be interpreted in the context of resource theories.

arXiv:1705.06343•Journal of Mathematical Physics•Sep 1, 2017•65 citations
Programmable quantum devices nominally act on a Hilbert space whose dimension grows exponentially with the number of constituents, but the presence of noise makes it unlikely that they remain coherent across all of this immense Hilbert space. Then, what is the effective coherent quantum dimension that should be associated with such imperfect devices? To answer this question we here introduce an operational basis-independent framework which imposes a dimension bottleneck on the programmable transformations. Concretely we ask how strongly the quantum information they process can be compressed. Formalizing this idea we identify three inequivalent notions, termed ddd-compressibility, ddd-simulability and ddd-embeddability, which differ in the causal structure used to impose the bottleneck and form a strict hierarchy. The framework unifies several existing notions: joint measurability and simulability of quantum measurements, and the absolute dimensionality of state ensembles, are recovered as special cases. We illustrate the hierarchy with noisy qubit measurements in complementary bases, and we determine the white-noise thresholds at which the set of all noisy unitary channels in dimension nnn, a noisy universal quantum processor, becomes ddd-compressible, ddd-simulable and ddd-embeddable. The thresholds confirm the expectation -- maintaining coherence across the full Hilbert space becomes increasingly demanding as the nominal dimension nnn increases.
We investigate the compression of quantum information with respect to a given set M of high-dimensional measurements. This leads to a notion of simulability, where we demand that the statistics obtained from M and an arbitrary quantum state ρ are recovered exactly by first compressing ρ into a lower-dimensional space, followed by some quantum measurements. A full quantum compression is possible, i.e., leaving only classical information, if and only if the set M is jointly measurable. Our notion of simulability can thus be seen as a quantification of measurement incompatibility in terms of dimension. After defining these concepts, we provide an illustrative example involving mutually unbiased bases, and develop a method based on semidefinite programming for constructing simulation models. In turn we analytically construct optimal simulation models for all projective measurements subjected to white noise or losses. Finally, we discuss how our approach connects with other concepts introduced in the context of quantum channels and quantum correlations.
Efficiently characterizing large quantum states and processes is a central yet notoriously challenging task in quantum information science, as conventional tomography methods typically require resources that grow exponentially with system size. Here, we introduce a structure-agnostic learning framework for noisy nnn-qubit quantum circuits under~i.i.d.~single-qubit noise. We first prove that quantum states with unital noise channels admit an efficient learnable representation in the logarithmic-depth regime. We then extend this framework to quantum process tomography under constant noise, deriving a unified protocol that applies to both unital and non-unital noisy channels and retains efficient guarantees for logarithmic-depth circuits. This process-learning formulation is input-agnostic and imposes no distributional assumptions on the input quantum states. We further study a more general regime with arbitrary noise strength. In this setting, low-weight Pauli propagation induces a terminal truncation whose threshold depends logarithmically on the inverse accuracy, leading to quasi-polynomial complexity and near-unit success probability in the average case. In contrast to the preceding two results, this arbitrary-noise guarantee does not impose any restriction on the circuit depth, and therefore covers arbitrary-depth circuits, including both the noiseless limit (γ=0γ= 0γ=0) and the strong-decoherence regime (γ=Θ(1)γ= Θ(1)γ=Θ(1)). Numerical simulations of two-dimensional Hamiltonian dynamics further demonstrate the accuracy and robustness of the approach, including for structured circuits beyond the random-circuit setting assumed in the theoretical analysis. These results provide a scalable and practically relevant route toward characterizing large-scale noisy quantum devices, addressing a key bottleneck in the development of quantum technologies.
The dimension of a quantum state is traditionally seen as the number of superposed distinguishable states in a given basis. We propose an absolute, i.e., basis-independent, notion of dimensionality for ensembles of quantum states. It is based on whether a quantum ensemble can be simulated with states confined to arbitrary lower-dimensional subspaces and classical postprocessing. In order to determine the absolute dimension of quantum ensembles, we develop both analytical witness criteria and a semidefinite programming criterion based on the ensemble's information capacity. Furthermore, we construct explicit simulation models for arbitrary ensembles of pure quantum states subject to white noise, and in natural cases we prove their optimality. Also, efficient numerical methods are provided for simulating generic ensembles. Finally, we discuss the role of absolute dimensionality in high-dimensional quantum information processing.
Classical probabilistic models of (noisy) quantum systems are not only relevant for understanding the non-classical features of quantum mechanics, but they are also useful for determining the possible advantage of using quantum resources for information processing tasks. A common feature of these models is the presence of inaccessible information, as captured by the concept of preparation contextuality: There are ensembles of quantum states described by the same density operator, and hence operationally indistinguishable, and yet in any probabilistic (ontological) model, they should be described by distinct probability distributions. In this work, we quantify the inaccessible information of a model in terms of the maximum distinguishability of probability distributions associated to any pair of ensembles with identical density operators, as quantified by the total variation distance of the distributions. We obtain a family of lower bounds on this maximum distinguishability in terms of experimentally measurable quantities. In the case of an ideal qubit this leads to a lower bound of, approximately, 0.07. These bounds can also be interpreted as a new class of robust preparation non-contextuality inequalities. Our non-contextuality inequalities are phrased in terms of generalizations of max-relative entropy and trace distance for general operational theories, which could be of independent interest. Under sufficiently strong noise any quantum system becomes preparation non-contextual, i.e., can be described by models with zero inaccessible information. Using our non-contextuality inequalities, we show that this can happen only if the noise channel has the average gate fidelity less than or equal to 1/D(1+1/2+...+1/D), where D is the dimension of the Hilbert space.
Quantum reservoirs offer a hardware-friendly route to quantum machine learning, replacing trainable circuits with fixed random dynamics and a classical readout. Because the reservoir is not optimized, performance depends entirely on the choice of reservoir family, yet existing diagnostics demand resources that grow exponentially with system size. We introduce a scalable, hardware-agnostic framework built on two complementary quantities. The first is a task-independent order-statistics (ORS) expressivity score, which compares only the largest output probabilities of a reservoir ensemble against an analytical Haar baseline. It never reconstructs the full output distribution, is cost-independent of Hilbert-space dimension, and admits a closed-form depolarizing noise correction, making it directly usable on hardware. The second is the task-dependent effective rank ReffR_{\mathrm{eff}}Reff​ of the feature matrix, which measures how much input-dependent information reaches the readout. We validate the ORS score against established complexity diagnostics and confirm it remains informative under simulated noise and on IBM quantum hardware. Across synthetic and real quantum extreme learning machine and quantum reservoir computing benchmarks, ORS captures the intrinsic expressivity hierarchy of reservoir families while ReffR_{\mathrm{eff}}Reff​ determines when that expressivity becomes usable predictive information.
We introduce a framework for simulating quantum measurements based on classical processing of a set of accessible measurements. Well-known concepts such as joint measurability and projective simulability naturally emerge as particular cases of our framework, but our study also leads to novel results and questions. First, a generalisation of joint measurability is derived, which yields a hierarchy for the incompatibility of sets of measurements. A similar hierarchy is defined based on the number of outcomes necessary to perform a simulation of a given measurement. This general approach also allows us to identify connections between different kinds of simulability and, in particular, we characterise the qubit measurements that are projective-simulable in terms of joint measurability. Finally, we discuss how our framework can be interpreted in the context of resource theories.