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The learnability of quantum states

2007
*
Proceedings of the Royal Society A
*

Traditional

doi:10.1098/rspa.2007.0113
fatcat:l5lmp5zdizap7ml5j5jvgtr7qe
*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. ...##
###
Learnability scaling of quantum states: Restricted Boltzmann machines

2019
*
Physical review B
*

Generative modeling with machine learning has provided a new perspective on

doi:10.1103/physrevb.100.195125
fatcat:2w7466wawze6rdeq5nrsfidof4
*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*. ...##
###
Stabiliser states are efficiently PAC-learnable
[article]

2018
*
arXiv
*
pre-print

Here, using results from

arXiv:1705.00345v2
fatcat:wz6p2xrubzfqvms42zsvpax3um
*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 ...##
###
Quantum Local Differential Privacy and Quantum Statistical Query Model
[article]

2022
*
arXiv
*
pre-print

In this work, we give a formal definition

arXiv:2203.03591v1
fatcat:kyzimg5ztzfdloms3klmmitqxm
*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 ...##
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Quantum versus Classical Learnability
[article]

2000
*
arXiv
*
pre-print

), there is a concept class which is polynomial-time

arXiv:quant-ph/0007036v1
fatcat:jkia4oqhyjcxhj3fvhtadiw2qy
*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. ...##
###
Learning DNF over the uniform distribution using a quantum example oracle

1995
*
Proceedings of the eighth annual conference on Computational learning theory - COLT '95
*

This

doi:10.1145/225298.225312
dblp:conf/colt/BshoutyJ95
fatcat:m6ojzpvyzndjha5zpbkgsnjyoq
*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. ...##
###
Learning DNF over the Uniform Distribution Using a Quantum Example Oracle

1998
*
SIAM journal on computing (Print)
*

This

doi:10.1137/s0097539795293123
fatcat:hcx2vqs3mraulkwfar5hbwk25m
*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. ...##
###
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]

2022
*
arXiv
*
pre-print

A well-known issue in benchmarking is that not everything about

arXiv:2206.06362v1
fatcat:pbutrcja45bstcudllwhkgv5aa
*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. ...##
###
Sample Complexity of Learning Parametric Quantum Circuits
[article]

2022
*
arXiv
*
pre-print

n^c gates and each gate acting on a constant number

arXiv:2107.09078v2
fatcat:w34x5ig4tfgwzos5rg7ru4bb24
*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. ...##
###
A single T-gate makes distribution learning hard
[article]

2022
*
arXiv
*
pre-print

In this work, we provide an extensive characterization

arXiv:2207.03140v1
fatcat:wvr7zbw7ondozfjv7pk2nbseva
*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), ...##
###
Quantum Learnability is Arbitrarily Distillable
[article]

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*). ...

##
###
On the Hardness of PAC-learning Stabilizer States with Noise

2022
*
Quantum
*

We consider

doi:10.22331/q-2022-02-02-640
fatcat:d66ihfibjfdcfmh2lt6y4ptkdq
*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. ...##
###
Pseudo-dimension of quantum circuits
[article]

2020
*
arXiv
*
pre-print

We characterize

arXiv:2002.01490v2
fatcat:fjyrdij74nfmjcth4wip5qtaka
*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. ...##
###
On the Hardness of PAC-learning Stabilizer States with Noise
[article]

2022
*
arXiv
*
pre-print

We consider

arXiv:2102.05174v3
fatcat:sl73w5s4sfgpvenwz2f7jrngfq
*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. ...
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