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The learnability of quantum states

Scott Aaronson
2007 Proceedings of the Royal Society A  
Traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n.  ...  Besides possible implications for experimental physics, our learning theorem has two applications to quantum computing: first, a new simulation of quantum one-way communication protocols, and second, the  ...  Learning Quantum States We now turn to the problem of learning a quantum state. Let S be the set of two-outcome measurements on n qubits.  ... 
doi:10.1098/rspa.2007.0113 fatcat:l5lmp5zdizap7ml5j5jvgtr7qe

Learnability scaling of quantum states: Restricted Boltzmann machines

Dan Sehayek, Anna Golubeva, Michael S. Albergo, Bohdan Kulchytskyy, Giacomo Torlai, Roger G. Melko
2019 Physical review B  
Generative modeling with machine learning has provided a new perspective on the data-driven task of reconstructing quantum states from a set of qubit measurements.  ...  As increasingly large experimental quantum devices are built in laboratories, the question of how these machine learning techniques scale with the number of qubits is becoming crucial.  ...  We remark that a sample complexity linear in N is consistent with observations on the PAC-learnability of quantum states.  ... 
doi:10.1103/physrevb.100.195125 fatcat:2w7466wawze6rdeq5nrsfidof4

Stabiliser states are efficiently PAC-learnable [article]

Andrea Rocchetto
2018 arXiv   pre-print
Here, using results from the literature on the efficient classical simulation of quantum systems, we show that stabiliser states are efficiently PAC-learnable.  ...  In this model, quantum states have been shown to be Probably Approximately Correct (PAC)-learnable with sample complexity linear in the number of qubits.  ...  Acknowledgements I would like to thank Ronald de Wolf for helpful comments and careful reads of the manuscript and Scott Aaronson, Simon Benjamin, Fernando Brãndao, Toby Cubitt, Carlos González Guillén  ... 
arXiv:1705.00345v2 fatcat:wz6p2xrubzfqvms42zsvpax3um

Quantum Local Differential Privacy and Quantum Statistical Query Model [article]

Armando Angrisani, Elham Kashefi
2022 arXiv   pre-print
In this work, we give a formal definition of quantum local differential privacy and we extend the aforementioned result to quantum computation.  ...  The problem of private learning has been extensively studied in classical computer science.  ...  Grilo for discussions about quantum statistical query model and Mina Doosti for discussions about differential privacy  ... 
arXiv:2203.03591v1 fatcat:kyzimg5ztzfdloms3klmmitqxm

Quantum versus Classical Learnability [article]

Rocco A. Servedio, Steven J. Gortler
2000 arXiv   pre-print
), there is a concept class which is polynomial-time learnable in the quantum version but not in the classical version of the model.  ...  For each of the two learning models described above, we show that any concept class is information-theoretically learnable from polynomially many quantum examples if and only if it is information-theoretically  ...  Let |φ c t be the state of the quantum register at time t if the oracle responses are modified as stated above. Then ||φ c T − |φ c T | ≤ ǫ.  ... 
arXiv:quant-ph/0007036v1 fatcat:jkia4oqhyjcxhj3fvhtadiw2qy

Learning DNF over the uniform distribution using a quantum example oracle

Nader H. Bshouty, Jeffrey C. Jackson
1995 Proceedings of the eighth annual conference on Computational learning theory - COLT '95  
This quantum example oracle is a natural extension of the traditional PAC example oracle, and it immediately follows that all PAC-learnable function classes are learnable in the quantum model.  ...  We generalize the notion of PAC learning from an example oracle to a notion of efficient learning on a quantum computer using a quantum example oracle.  ...  The first author thanks Richard Cleve for an enlightening seminar on quantum computation.  ... 
doi:10.1145/225298.225312 dblp:conf/colt/BshoutyJ95 fatcat:m6ojzpvyzndjha5zpbkgsnjyoq

Learning DNF over the Uniform Distribution Using a Quantum Example Oracle

Nader H. Bshouty, Jeffrey C. Jackson
1998 SIAM journal on computing (Print)  
This quantum example oracle is a natural extension of the traditional PAC example oracle, and it immediately follows that all PAC-learnable function classes are learnable in the quantum model.  ...  We generalize the notion of PAC learning from an example oracle to a notion of efficient learning on a quantum computer using a quantum example oracle.  ...  The first author thanks Richard Cleve for an enlightening seminar on quantum computation.  ... 
doi:10.1137/s0097539795293123 fatcat:hcx2vqs3mraulkwfar5hbwk25m

Page 7889 of Mathematical Reviews Vol. , Issue 99k [page]

1999 Mathematical Reviews  
This quantum example oracle is a natural extension of the traditional PAC example oracle, and it immediately follows that all PAC-learnable function classes are learnable in the quan- tum model.  ...  Specif- Theory of computing 99k:68054 ically, we show that disjunctive normal form (DNF) is efficiently learnable with respect to the uniform distribution by a quantum algorithm using a quantum example  ... 

The learnability of Pauli noise [article]

Senrui Chen, Yunchao Liu, Matthew Otten, Alireza Seif, Bill Fefferman, Liang Jiang
2022 arXiv   pre-print
A well-known issue in benchmarking is that not everything about quantum noise is learnable due to the existence of gauge freedom, leaving open the question of what information about noise is learnable  ...  Here we give a precise characterization of the learnability of Pauli noise channels attached to Clifford gates, showing that learnable information corresponds to the cycle space of the pattern transfer  ...  The boundary of learnability of quantum noise -a precise understanding of what information is learnable and what is not, still remains an open question.  ... 
arXiv:2206.06362v1 fatcat:pbutrcja45bstcudllwhkgv5aa

Sample Complexity of Learning Parametric Quantum Circuits [article]

Haoyuan Cai, Qi Ye, Dong-Ling Deng
2022 arXiv   pre-print
n^c gates and each gate acting on a constant number of qubits, the sample complexity is bounded by Õ(n^c+1).  ...  Here, we prove that physical quantum circuits are PAC (probably approximately correct) learnable on a quantum computer via empirical risk minimization: to learn a parametric quantum circuit with at most  ...  In this work, we write x ∈ X as an abbreviation of the n-qubit quantum state |ψ(x) , and similarly for y ∈ Y.  ... 
arXiv:2107.09078v2 fatcat:w34x5ig4tfgwzos5rg7ru4bb24

A single T-gate makes distribution learning hard [article]

Marcel Hinsche, Marios Ioannou, Alexander Nietner, Jonas Haferkamp, Yihui Quek, Dominik Hangleiter, Jean-Pierre Seifert, Jens Eisert, Ryan Sweke
2022 arXiv   pre-print
In this work, we provide an extensive characterization of the learnability of the output distributions of local quantum circuits.  ...  Our first result yields insight into the relationship between the efficient learnability and the efficient simulatability of these distributions.  ...  This work has been funded by the Cluster of Excellence MATH+ (EF1-11), the BMWK (PlanQK), the BMBF (Hybrid, QPIC-1), the DFG (CRC183, EI 519 20-1), the QuantERA (HQCC), the Munich Quantum Valley (K8),  ... 
arXiv:2207.03140v1 fatcat:wvr7zbw7ondozfjv7pk2nbseva

Quantum Learnability is Arbitrarily Distillable [article]

Joe H. Jenne, David R. M. Arvidsson-Shukur
2021 arXiv   pre-print
Quantum learning (in metrology and machine learning) involves estimating unknown parameters from measurements of quantum states.  ...  The quantum Fisher information matrix can bound the average amount of information learnt about the unknown parameters per experimental trial.  ...  We give the following theorem Theorem 2 (Arbitrary distillation of quantum learnability).  ... 
arXiv:2104.09520v1 fatcat:ucnauchrfrcdln5oxicuce5wna

On the Hardness of PAC-learning Stabilizer States with Noise

Aravind Gollakota, Daniel Liang
2022 Quantum  
We consider the problem of learning stabilizer states with noise in the Probably Approximately Correct (PAC) framework of Aaronson (2007) for learning quantum states.  ...  Our results position the problem of learning stabilizer states as a natural quantum analogue of the classical problem of learning parities: easy in the noiseless setting, but seemingly intractable even  ...  DL was supported by the Simons It from Qubit Collaboration and Scott Aaronson's Vannevar Bush Faculty Fellowship from the US Department of Defense.  ... 
doi:10.22331/q-2022-02-02-640 fatcat:d66ihfibjfdcfmh2lt6y4ptkdq

Pseudo-dimension of quantum circuits [article]

Matthias C. Caro, Ishaun Datta
2020 arXiv   pre-print
We characterize the expressive power of quantum circuits with the pseudo-dimension, a measure of complexity for probabilistic concept classes.  ...  Using these bounds, we exhibit a class of circuit output states out of which at least one has exponential state complexity, and moreover demonstrate that quantum circuits of known polynomial size and depth  ...  We also give two applications of these bounds, one in quantum state complexity, the other in learnability of quantum circuits.  ... 
arXiv:2002.01490v2 fatcat:fjyrdij74nfmjcth4wip5qtaka

On the Hardness of PAC-learning Stabilizer States with Noise [article]

Aravind Gollakota, Daniel Liang
2022 arXiv   pre-print
We consider the problem of learning stabilizer states with noise in the Probably Approximately Correct (PAC) framework of Aaronson (2007) for learning quantum states.  ...  Our results position the problem of learning stabilizer states as a natural quantum analogue of the classical problem of learning parities: easy in the noiseless setting, but seemingly intractable even  ...  DL was supported by the Simons It from Qubit Collaboration and Scott Aaronson's Vannevar Bush Faculty Fellowship from the US Department of Defense.  ... 
arXiv:2102.05174v3 fatcat:sl73w5s4sfgpvenwz2f7jrngfq
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