Information-computation gaps in quantum learning via low-degree likelihood

Sitan Chen, Weiyuan Gong, Jonas Haferkamp, Yihui Quek
Proceedings of Thirty Ninth Conference on Learning Theory, PMLR 336:1186-1278, 2026.

Abstract

In a variety of physically relevant settings for learning from quantum data, there is an established recipe for measuring polynomially many copies of that data such that the resulting measurement readouts contain enough information to reconstruct the underlying system. Yet designing protocols that can computationally efficiently extract that information remains largely an art, and there are important cases where we believe this to be impossible, that is, where there is an information-computation gap. While there is a large array of tools in the classical literature for giving evidence for average-case hardness of statistical inference problems, the corresponding tools in the quantum literature are far more limited. One such framework in the classical literature, the low-degree method, makes predictions about hardness of inference problems based on the failure of estimators given by low-degree polynomials. In this work, we extend this framework to the quantum setting and show a number of new information-computation gaps for quantum learning. We establish a general connection between state designs and low-degree hardness. We use this to obtain the first information-computation gaps for learning Gibbs states of random, sparse, non-local Hamiltonians. We also use it to prove hardness for learning random shallow quantum circuit states in a challenging model where states can be measured in round-based adaptively chosen bases. To our knowledge, the ability to model adaptivity within the low-degree framework was open even in classical settings. In addition, we also obtain a low-degree hardness result for quantum error mitigation against strategies with single-qubit measurements. We define a new quantum generalization of the planted biclique problem and identify the threshold at which this problem becomes computationally hard for protocols that perform local measurements. Interestingly, the complexity landscape for this problem shifts when going from local measurements to more entangled single-copy measurements. We show average-case hardness for the “standard” variant of Learning Stabilizers with Noise (Poremba et al., 2024) and for agnostically learning product states (Bakshi et al., 2024a).

Cite this Paper

BibTeX
@InProceedings{pmlr-v336-chen26d, title = {Information-computation gaps in quantum learning via low-degree likelihood}, author = {Chen, Sitan and Gong, Weiyuan and Haferkamp, Jonas and Quek, Yihui}, booktitle = {Proceedings of Thirty Ninth Conference on Learning Theory}, pages = {1186--1278}, year = {2026}, editor = {Hanneke, Steve and Lattimore, Tor}, volume = {336}, series = {Proceedings of Machine Learning Research}, month = {29 Jun--03 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v336/main/assets/chen26d/chen26d.pdf}, url = {https://proceedings.mlr.press/v336/chen26d.html}, abstract = {In a variety of physically relevant settings for learning from quantum data, there is an established recipe for measuring polynomially many copies of that data such that the resulting measurement readouts contain enough information to reconstruct the underlying system. Yet designing protocols that can computationally efficiently extract that information remains largely an art, and there are important cases where we believe this to be impossible, that is, where there is an information-computation gap. While there is a large array of tools in the classical literature for giving evidence for average-case hardness of statistical inference problems, the corresponding tools in the quantum literature are far more limited. One such framework in the classical literature, the low-degree method, makes predictions about hardness of inference problems based on the failure of estimators given by low-degree polynomials. In this work, we extend this framework to the quantum setting and show a number of new information-computation gaps for quantum learning. We establish a general connection between state designs and low-degree hardness. We use this to obtain the first information-computation gaps for learning Gibbs states of random, sparse, non-local Hamiltonians. We also use it to prove hardness for learning random shallow quantum circuit states in a challenging model where states can be measured in round-based adaptively chosen bases. To our knowledge, the ability to model adaptivity within the low-degree framework was open even in classical settings. In addition, we also obtain a low-degree hardness result for quantum error mitigation against strategies with single-qubit measurements. We define a new quantum generalization of the planted biclique problem and identify the threshold at which this problem becomes computationally hard for protocols that perform local measurements. Interestingly, the complexity landscape for this problem shifts when going from local measurements to more entangled single-copy measurements. We show average-case hardness for the “standard” variant of Learning Stabilizers with Noise (Poremba et al., 2024) and for agnostically learning product states (Bakshi et al., 2024a).} }
Endnote
%0 Conference Paper %T Information-computation gaps in quantum learning via low-degree likelihood %A Sitan Chen %A Weiyuan Gong %A Jonas Haferkamp %A Yihui Quek %B Proceedings of Thirty Ninth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2026 %E Steve Hanneke %E Tor Lattimore %F pmlr-v336-chen26d %I PMLR %P 1186--1278 %U https://proceedings.mlr.press/v336/chen26d.html %V 336 %X In a variety of physically relevant settings for learning from quantum data, there is an established recipe for measuring polynomially many copies of that data such that the resulting measurement readouts contain enough information to reconstruct the underlying system. Yet designing protocols that can computationally efficiently extract that information remains largely an art, and there are important cases where we believe this to be impossible, that is, where there is an information-computation gap. While there is a large array of tools in the classical literature for giving evidence for average-case hardness of statistical inference problems, the corresponding tools in the quantum literature are far more limited. One such framework in the classical literature, the low-degree method, makes predictions about hardness of inference problems based on the failure of estimators given by low-degree polynomials. In this work, we extend this framework to the quantum setting and show a number of new information-computation gaps for quantum learning. We establish a general connection between state designs and low-degree hardness. We use this to obtain the first information-computation gaps for learning Gibbs states of random, sparse, non-local Hamiltonians. We also use it to prove hardness for learning random shallow quantum circuit states in a challenging model where states can be measured in round-based adaptively chosen bases. To our knowledge, the ability to model adaptivity within the low-degree framework was open even in classical settings. In addition, we also obtain a low-degree hardness result for quantum error mitigation against strategies with single-qubit measurements. We define a new quantum generalization of the planted biclique problem and identify the threshold at which this problem becomes computationally hard for protocols that perform local measurements. Interestingly, the complexity landscape for this problem shifts when going from local measurements to more entangled single-copy measurements. We show average-case hardness for the “standard” variant of Learning Stabilizers with Noise (Poremba et al., 2024) and for agnostically learning product states (Bakshi et al., 2024a).
APA
Chen, S., Gong, W., Haferkamp, J. & Quek, Y.. (2026). Information-computation gaps in quantum learning via low-degree likelihood. Proceedings of Thirty Ninth Conference on Learning Theory, in Proceedings of Machine Learning Research 336:1186-1278 Available from https://proceedings.mlr.press/v336/chen26d.html.

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