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:2608.02325

Learnable yet not simulable: a quantum resource theory of learning models

Xinbiao Wang, Yuxuan Du, Dacheng Tao

quant-phSubmitted Aug 3, 2026• Announced Aug 4, 2026• 0 citations
arXivHTMLPDF

Abstract

Quantum resource theory has sharpened our understanding of the intrinsic complexity of quantum systems, particularly their classical simulability. However, it remains unclear which quantum resource governs the classical learnability of quantum circuits, especially beyond the regime of efficient classical simulation. Here we close this knowledge gap by studying the expectation-value functions of families of tunable quantum circuits, with many applications in digital quantum simulation, quantum metrology, and quantum-system characterization. Specifically, we introduce a new resource measure, the dynamical stabilizer entropy (\DSE), which quantifies how broadly an expectation-value function is distributed across its frequency modes. By relating \DSE to operator stabilizer entropy, we establish a computational phase diagram that compares classical simulators with quantum-data-assisted classical surrogates. We first determine the \DSE-dependent learnability boundary of this diagram by deriving bounds on the sample complexity and runtime of classical surrogates, and by developing a \DSE-guided surrogate. We then complete the diagram by proving, under standard complexity-theoretic assumptions, the existence of circuit families that can be efficiently learned by this surrogate but cannot be efficiently emulated from their circuit descriptions alone. Numerical experiments on random and structured circuits with up to 80 qubits support the predicted \DSE-dependent computational landscape. These results establish a quantitative resource-theoretic framework for delineating the boundary between classical simulation and learning, motivate resource measures linking quantum resources to learnability, and guide the design of learning-based algorithms for scalable quantum systems beyond the reach of direct classical simulation.

Comments
61 Pages, 7 figures

Related papers

Comparing Classical Simulation and Sample-Based Learning of Quantum Systems

João Pedro d'El-Rey, Raúl O. Vallejos, Fernando de Melo

We investigate the relationship between two distinct classical approaches to quantum systems: direct simulation from a classical description and sample-based learning from measurement data. While both tasks ultimately aim to reproduce Born-rule statistics, complexity-theoretic results suggest that simulability and learnability need not coincide in general. Here we study this relationship empirically using a fixed deep energy-based generative model trained on measurement samples from controlled families of quantum states. We independently tune two quantum resources associated with classical simulation cost: entanglement, through the bond dimension of random matrix product states, and non-stabilizerness, through the number of T gates in Clifford-dominated circuits. Learning difficulty is characterized using two probes of neural-network complexity: the largest Hessian eigenvalue at convergence and Random Subspace Optimization. For both quantum resources, increasing simulation hardness systematically correlates with sharper loss landscapes and degraded reconstruction performance under constrained capacity. Our results indicate that, within the regimes studied here, classical learnability tracks known simulation complexity measures, suggesting that neural-network training dynamics can provide an empirical probe of quantum computational hardness.

quant-pharXiv:2605.28986•May 29, 2026•0 citations

The cost of simulating classically tractable quantum circuits and dynamics

Su Yeon Chang, Supanut Thanasilp, Zoë Holmes +1 more

Determining whether a quantum evolution can be efficiently simulated classically is central to understanding the boundary between classical and quantum computation. However, polynomial-time simulability is an asymptotic statement, and does not by itself determine whether the (quantum-inspired) classical simulation is actually practical. Indeed, different polynomial scalings can lead to vastly different computational costs, particularly when expensive preprocessing or quantum data acquisition is required. In this work, we ask whether classically simulable quantum dynamics are in practice more resource-efficient to simulate classically than to execute directly on quantum hardware. We analyze this question using three resource metrics, quantum sample, quantum time, and classical time complexity, for several widely studied classically simulable circuit families. Using representative hardware-level estimates, we identify regimes in which quantum simulation can be faster despite the existence of a polynomial-time classical algorithm, as well as regimes in which classical simulation remains more efficient. At the same time, the large quantum sampling cost needed to characterize unknown input states can make this polynomial-time classical simulation prohibitively expensive with current cloud-based hardware access prices. Ultimately, our work indicates that guarantees of classical simulability with polynomial resources alone are insufficient to determine the preferred implementation.

quant-pharXiv:2609.11847•Sep 11, 2026•0 citations

Optimal quantum reservoir learning in proximity to universality

Moein N. Ivaki, Matias Karjula, Tapio Ala-Nissilä

The study of the boundary between classically simulable and computationally complex quantum dynamics is fundamental to understanding which physical resources may enable enhanced information-processing capabilities. We investigate this within the framework of quantum reservoir computing by introducing a tunable N -qubit random circuit model, where a fraction p of Clifford gates are probabilistically substituted with nonstabilizing conditional- T ̂ gates. We establish a direct correspondence between the reservoir's performance on temporal processing tasks and its entanglement-spectrum statistics and long-range nonstabilizer resource content. To assess scalability, we study the scaling of the antiflatness of states in the large- N limit at a fixed circuit depth ratio d / N ∼ O ( 1 ) . This is taken as a witness to the concentration of measures, a known impediment to learning in thermalizing systems. We demonstrate that the learnability and scalability of the reservoir can be continuously controlled by the parameter p , allowing us to navigate from classically tractable to maximally expressive quantum dynamics. These architecture-agnostic results provide a general strategy for designing tunable and expressive quantum reservoirs, highlighting how certain nonclassical properties control average-case intrinsic learnability and functionality.

arXiv:2510.18623•Physical Review A•May 13, 2026•1 citation

Efficient learning for linear properties of bounded-gate quantum circuits

Yuxuan Du, Min-Hsiu Hsieh, Dacheng Tao

The vast and complicated many-qubit state space forbids us to comprehensively capture the dynamics of modern quantum computers via classical simulations or quantum tomography. Recent progress in quantum learning theory prompts a crucial question: can linear properties of a many-qubit circuit with d tunable RZ gates and G − d Clifford gates be efficiently learned from measurement data generated by varying classical inputs? In this work, we prove that the sample complexity scaling linearly in d is required to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. To address this challenge, we propose a kernel-based method leveraging classical shadows and truncated trigonometric expansions, enabling a controllable trade-off between prediction accuracy and computational overhead. Our results advance two crucial realms in quantum computation: the exploration of quantum algorithms with practical utilities and learning-based quantum system certification. We conduct numerical simulations to validate our proposals across diverse scenarios, encompassing quantum information processing protocols, Hamiltonian simulation, and variational quantum algorithms up to 60 qubits. It is currently unclear whether machine learning model could efficiently learn the dynamics of many-qubit bounded-gate quantum circuits. This work reveals the learnability of predicting linear properties of large- scale quantum circuits and proposes a learning algorithm that enables resource-efficient solutions for many practical quantum applications.

arXiv:2408.12199•Nature Communications•Apr 22, 2025•7 citations

Classical simulation of a quantum circuit with noisy magic inputs

Jiwon Heo, Sojeong Park, Changhun Oh

Magic states are essential for universal quantum computation and are widely viewed as a key source of quantum advantage, yet in realistic devices they are inevitably noisy. In this work, we characterize how noise on injected magic resources changes the classical simulability of quantum circuits and when it induces a transition from classically intractable behavior to efficient classical simulation. We adopt a resource-centric noise model in which only the injected magic components are noisy, while the baseline states, operations, and measurements belong to an efficiently simulable family. Within this setting, we develop an approximate classical sampling algorithm with controlled error and prove explicit noise-dependent conditions under which the algorithm runs in polynomial time. Our framework applies to both qubit circuits with Clifford baselines and fermionic circuits with matchgate baselines, covering representative noise channels such as dephasing and particle loss. We complement the analysis with numerical estimates of the simulation cost, providing concrete thresholds and runtime scaling across practically relevant parameter regimes.

quant-pharXiv:2601.10111•Jan 16, 2026•0 citations

Quantifying the magic of quantum channels

Xin Wang, Mark M. Wilde, Yuan Su

Abstract To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum ‘magic’ or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d , it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.

arXiv:1903.04483•New Journal of Physics•Sep 18, 2019•127 citations
Quantum resource theory has sharpened our understanding of the intrinsic complexity of quantum systems, particularly their classical simulability. However, it remains unclear which quantum resource governs the classical learnability of quantum circuits, especially beyond the regime of efficient classical simulation. Here we close this knowledge gap by studying the expectation-value functions of families of tunable quantum circuits, with many applications in digital quantum simulation, quantum metrology, and quantum-system characterization. Specifically, we introduce a new resource measure, the dynamical stabilizer entropy (\DSE), which quantifies how broadly an expectation-value function is distributed across its frequency modes. By relating \DSE to operator stabilizer entropy, we establish a computational phase diagram that compares classical simulators with quantum-data-assisted classical surrogates. We first determine the \DSE-dependent learnability boundary of this diagram by deriving bounds on the sample complexity and runtime of classical surrogates, and by developing a \DSE-guided surrogate. We then complete the diagram by proving, under standard complexity-theoretic assumptions, the existence of circuit families that can be efficiently learned by this surrogate but cannot be efficiently emulated from their circuit descriptions alone. Numerical experiments on random and structured circuits with up to 80 qubits support the predicted \DSE-dependent computational landscape. These results establish a quantitative resource-theoretic framework for delineating the boundary between classical simulation and learning, motivate resource measures linking quantum resources to learnability, and guide the design of learning-based algorithms for scalable quantum systems beyond the reach of direct classical simulation.
We investigate the relationship between two distinct classical approaches to quantum systems: direct simulation from a classical description and sample-based learning from measurement data. While both tasks ultimately aim to reproduce Born-rule statistics, complexity-theoretic results suggest that simulability and learnability need not coincide in general. Here we study this relationship empirically using a fixed deep energy-based generative model trained on measurement samples from controlled families of quantum states. We independently tune two quantum resources associated with classical simulation cost: entanglement, through the bond dimension of random matrix product states, and non-stabilizerness, through the number of T gates in Clifford-dominated circuits. Learning difficulty is characterized using two probes of neural-network complexity: the largest Hessian eigenvalue at convergence and Random Subspace Optimization. For both quantum resources, increasing simulation hardness systematically correlates with sharper loss landscapes and degraded reconstruction performance under constrained capacity. Our results indicate that, within the regimes studied here, classical learnability tracks known simulation complexity measures, suggesting that neural-network training dynamics can provide an empirical probe of quantum computational hardness.
Determining whether a quantum evolution can be efficiently simulated classically is central to understanding the boundary between classical and quantum computation. However, polynomial-time simulability is an asymptotic statement, and does not by itself determine whether the (quantum-inspired) classical simulation is actually practical. Indeed, different polynomial scalings can lead to vastly different computational costs, particularly when expensive preprocessing or quantum data acquisition is required. In this work, we ask whether classically simulable quantum dynamics are in practice more resource-efficient to simulate classically than to execute directly on quantum hardware. We analyze this question using three resource metrics, quantum sample, quantum time, and classical time complexity, for several widely studied classically simulable circuit families. Using representative hardware-level estimates, we identify regimes in which quantum simulation can be faster despite the existence of a polynomial-time classical algorithm, as well as regimes in which classical simulation remains more efficient. At the same time, the large quantum sampling cost needed to characterize unknown input states can make this polynomial-time classical simulation prohibitively expensive with current cloud-based hardware access prices. Ultimately, our work indicates that guarantees of classical simulability with polynomial resources alone are insufficient to determine the preferred implementation.
The study of the boundary between classically simulable and computationally complex quantum dynamics is fundamental to understanding which physical resources may enable enhanced information-processing capabilities. We investigate this within the framework of quantum reservoir computing by introducing a tunable N -qubit random circuit model, where a fraction p of Clifford gates are probabilistically substituted with nonstabilizing conditional- T ̂ gates. We establish a direct correspondence between the reservoir's performance on temporal processing tasks and its entanglement-spectrum statistics and long-range nonstabilizer resource content. To assess scalability, we study the scaling of the antiflatness of states in the large- N limit at a fixed circuit depth ratio d / N ∼ O ( 1 ) . This is taken as a witness to the concentration of measures, a known impediment to learning in thermalizing systems. We demonstrate that the learnability and scalability of the reservoir can be continuously controlled by the parameter p , allowing us to navigate from classically tractable to maximally expressive quantum dynamics. These architecture-agnostic results provide a general strategy for designing tunable and expressive quantum reservoirs, highlighting how certain nonclassical properties control average-case intrinsic learnability and functionality.
The vast and complicated many-qubit state space forbids us to comprehensively capture the dynamics of modern quantum computers via classical simulations or quantum tomography. Recent progress in quantum learning theory prompts a crucial question: can linear properties of a many-qubit circuit with d tunable RZ gates and G − d Clifford gates be efficiently learned from measurement data generated by varying classical inputs? In this work, we prove that the sample complexity scaling linearly in d is required to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. To address this challenge, we propose a kernel-based method leveraging classical shadows and truncated trigonometric expansions, enabling a controllable trade-off between prediction accuracy and computational overhead. Our results advance two crucial realms in quantum computation: the exploration of quantum algorithms with practical utilities and learning-based quantum system certification. We conduct numerical simulations to validate our proposals across diverse scenarios, encompassing quantum information processing protocols, Hamiltonian simulation, and variational quantum algorithms up to 60 qubits. It is currently unclear whether machine learning model could efficiently learn the dynamics of many-qubit bounded-gate quantum circuits. This work reveals the learnability of predicting linear properties of large- scale quantum circuits and proposes a learning algorithm that enables resource-efficient solutions for many practical quantum applications.
Magic states are essential for universal quantum computation and are widely viewed as a key source of quantum advantage, yet in realistic devices they are inevitably noisy. In this work, we characterize how noise on injected magic resources changes the classical simulability of quantum circuits and when it induces a transition from classically intractable behavior to efficient classical simulation. We adopt a resource-centric noise model in which only the injected magic components are noisy, while the baseline states, operations, and measurements belong to an efficiently simulable family. Within this setting, we develop an approximate classical sampling algorithm with controlled error and prove explicit noise-dependent conditions under which the algorithm runs in polynomial time. Our framework applies to both qubit circuits with Clifford baselines and fermionic circuits with matchgate baselines, covering representative noise channels such as dephasing and particle loss. We complement the analysis with numerical estimates of the simulation cost, providing concrete thresholds and runtime scaling across practically relevant parameter regimes.
Abstract To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum ‘magic’ or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d , it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.