Classical shadows constitute a protocol to estimate the expectation values of a collection of M observables acting on O(1) qubits of an unknown n-qubit state with a number of measurements that is independent of n and that grows only logarithmically with M. We propose a local variant of the quantum Wasserstein distance of order 1 of De Palma et al. [IEEE Trans. Inf. Theory 67, 6627–6643 (2021)] and prove that the classical shadow obtained measuring O(log n) copies of the state to be learned constitutes an accurate estimate with respect to the proposed distance. We apply the results to quantum generative adversarial networks, showing that quantum access to the state to be learned can be useful only when some prior information on such state is available.
REFERENCES
1.
G. M.
D’Ariano
, M. G.
Paris
, and M. F.
Sacchi
, “Quantum tomography
,” Adv. Imaging Electron Phys.
128
, 206
–309
(2003
).2.
R.
O’Donnell
and J.
Wright
, “Efficient quantum tomography
,” in Proceedings of the 48th Annual ACM Symposium on Theory of Computing, STOC’16
(Association for Computing Machinery
, New York, NY
, 2016
), pp. 899
–912
.3.
J.
Haah
, A. W.
Harrow
, Z.
Ji
, X.
Wu
, and N.
Yu
, “Sample-optimal tomography of quantum states
,” IEEE Trans. Inf. Theory
63
, 5628
–5641
(2017
).4.
A.
Anshu
and S.
Arunachalam
, “A survey on the complexity of learning quantum states
,” Nat. Rev. Phys.
6
, 59
–69
(2024
).5.
S.
Aaronson
, “Shadow tomography of quantum states
,” SIAM J. Comput.
49
, 368
–394
(2020
).6.
H.-Y.
Huang
, R.
Kueng
, and J.
Preskill
, “Predicting many properties of a quantum system from very few measurements
,” Nat. Phys.
16
, 1050
–1057
(2020
).7.
H.-Y.
Huang
, “Learning quantum states from their classical shadows
,” Nat. Rev. Phys.
4
, 81
(2022
).8.
H.-Y.
Huang
, R.
Kueng
, and J.
Preskill
, “Information-theoretic bounds on quantum advantage in machine learning
,” Phys. Rev. Lett.
126
, 190505
(2021
).9.
G.
De Palma
, M.
Marvian
, D.
Trevisan
, and S.
Lloyd
, “The quantum Wasserstein distance of order 1
,” IEEE Trans. Inf. Theory
67
, 6627
–6643
(2021
).10.
O.
Bratteli
and D. W.
Robinson
, Operator Algebras and Quantum Statistical Mechanics II: Equilibrium States Models in Quantum Statistical Mechanics
, Theoretical and Mathematical Physics
(Springer
, Berlin, Heidelberg
, 2013
).11.
P.
Naaijkens
, Quantum Spin Systems on Infinite Lattices: A Concise Introduction
, Lecture Notes in Physics
(Springer International Publishing
, 2017
).12.
C.
Rouzé
and D. S.
França
, “Learning quantum many-body systems from a few copies
,” Quantum
8
, 1319
(2024
).13.
E.
Onorati
, C.
Rouzé
, D. S.
França
, and J. D.
Watson
, “Efficient learning of ground and thermal states within phases of matter
,” Nat. Commun.
15
, 7755
(2024
).14.
G.
De Palma
and C.
Rouzé
, “Quantum concentration inequalities
,” Ann. Henri Poincare
23
, 3391
–3429
(2022
).15.
B. T.
Kiani
, G.
De Palma
, M.
Marvian
, Z.-W.
Liu
, and S.
Lloyd
, “Learning quantum data with the quantum earth mover’s distance
,” Quantum Sci. Technol.
7
, 045002
(2022
).16.
G.
De Palma
and D.
Trevisan
, “The Wasserstein distance of order 1 for quantum spin systems on infinite lattices
,” Ann. Henri Poincare
24
, 4237
–4282
(2023
).17.
L.
Arnaud
and N. J.
Cerf
, “Exploring pure quantum states with maximally mixed reductions
,” Phys. Rev. A
87
, 012319
(2013
).18.
G. D.
Palma
and D.
Pastorello
, “Quantum concentration inequalities and equivalence of the thermodynamical ensembles: An optimal mass transport approach
,” arXiv:2403.18617 [math-ph] (2024
).19.
20.
S. H.
Sack
, R. A.
Medina
, A. A.
Michailidis
, R.
Kueng
, and M.
Serbyn
, “Avoiding barren plateaus using classical shadows
,” PRX Quantum
3
, 020365
(2022
).21.
M.
Cerezo
, A.
Arrasmith
, R.
Babbush
, S. C.
Benjamin
, S.
Endo
, K.
Fujii
, J. R.
McClean
, K.
Mitarai
, X.
Yuan
, L.
Cincio
, and P. J.
Coles
, “Variational quantum algorithms
,” Nat. Rev. Phys.
3
, 625
–644
(2021
).22.
S.
Lloyd
and C.
Weedbrook
, “Quantum generative adversarial learning
,” Phys. Rev. Lett.
121
, 040502
(2018
).23.
J. R.
McClean
, S.
Boixo
, V. N.
Smelyanskiy
, R.
Babbush
, and H.
Neven
, “Barren plateaus in quantum neural network training landscapes
,” Nat. Commun.
9
, 4812
(2018
).24.
M.
Arjovsky
, S.
Chintala
, and L.
Bottou
, “Wasserstein generative adversarial networks
,” in Proceedings of the 34th International Conference on Machine Learning, Proceedings of Machine Learning Research (PMLR)
, edited by D.
Precup
and Y. W.
Teh
(PMLR
, 2017
), Vol. 70
, pp. 214
–223
.25.
M.
Cerezo
, A.
Sone
, T.
Volkoff
, L.
Cincio
, and P. J.
Coles
, “Cost function dependent barren plateaus in shallow parametrized quantum circuits
,” Nat. Commun.
12
, 1791
(2021
).26.
E. A.
Carlen
and J.
Maas
, “An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker–Planck equation is gradient flow for the entropy
,” Commun. Math. Phys.
331
, 887
–926
(2014
).27.
E. A.
Carlen
and J.
Maas
, “Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance
,” J. Funct. Anal.
273
, 1810
–1869
(2017
).28.
E. A.
Carlen
and J.
Maas
, “Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems
,” J. Stat. Phys.
178
, 319
–378
(2020
).29.
C.
Rouzé
and N.
Datta
, “Concentration of quantum states from quantum functional and transportation cost inequalities
,” J. Math. Phys.
60
, 012202
(2019
).30.
N.
Datta
and C.
Rouzé
, “Relating relative entropy, optimal transport and Fisher information: A quantum HWI inequality
,” Ann. Henri Poincare
21
, 2115
–2150
(2020
).31.
T.
Van Vu
and Y.
Hasegawa
, “Geometrical bounds of the irreversibility in Markovian systems
,” Phys. Rev. Lett.
126
, 010601
(2021
).32.
M.
Wirth
, “A dual formula for the noncommutative transport distance
,” J. Stat. Phys.
187
, 19
(2022
).33.
L.
Gao
, M.
Junge
, and N.
LaRacuente
, “Fisher information and logarithmic sobolev inequality for matrix-valued functions
,” Ann. Henri Poincare
21
, 3409
–3478
(2020
).34.
Y.
Chen
, T. T.
Georgiou
, L.
Ning
, and A.
Tannenbaum
, “Matricial Wasserstein-1 distance
,” IEEE Control Syst. Lett.
1
, 14
–19
(2017
).35.
E. K.
Ryu
, Y.
Chen
, W.
Li
, and S.
Osher
, “Vector and matrix optimal mass transport: Theory, algorithm, and applications
,” SIAM J. Sci. Comput.
40
, A3675
–A3698
(2018
).36.
Y.
Chen
, T. T.
Georgiou
, and A.
Tannenbaum
, “Matrix optimal mass transport: A quantum mechanical approach
,” IEEE Trans. Autom. Control
63
, 2612
–2619
(2018
).37.
Y.
Chen
, T. T.
Georgiou
, and A.
Tannenbaum
, “Wasserstein geometry of quantum states and optimal transport of matrix-valued measures
,” in Emerging Applications of Control and Systems Theory
(Springer
, 2018
), pp. 139
–150
.38.
J.
Agredo
, “A Wasserstein-type distance to measure deviation from equilibrium of quantum Markov semigroups
,” Open Syst. Inf. Dyn.
20
, 1350009
(2013
).39.
J.
Agredo
, “On exponential convergence of generic quantum Markov semigroups in a Wasserstein-type distance
,” Int. J. Pure Appl. Math.
107
, 909
–925
(2016
).40.
K.
Ikeda
, “Foundation of quantum optimal transport and applications
,” Quantum Inf. Process.
19
, 25
(2020
).41.
F.
Golse
, C.
Mouhot
, and T.
Paul
, “On the mean field and classical limits of quantum mechanics
,” Commun. Math. Phys.
343
, 165
–205
(2016
).42.
E.
Caglioti
, F.
Golse
, and T.
Paul
, “Towards optimal transport for quantum densities
,” Ann. Sc. Norm. Super.-Cl. Sci.
24
, 49
(2022
).43.
F.
Golse
, “The quantum N-body problem in the mean-field and semiclassical regime
,” Philos. Trans. R. Soc., A
376
, 20170229
(2018
).44.
F.
Golse
and T.
Paul
, “The Schrödinger equation in the mean-field and semiclassical regime
,” Arch. Ration. Mech. Anal.
223
, 57
–94
(2017
).45.
F.
Golse
and T.
Paul
, “Wave packets and the quadratic Monge–Kantorovich distance in quantum mechanics
,” C. R. Math.
356
, 177
–197
(2018
).46.
E.
Caglioti
, F.
Golse
, and T.
Paul
, “Quantum optimal transport is cheaper
,” J. Stat. Phys.
181
, 149
–162
(2020
).47.
S.
Friedland
, M.
Eckstein
, S.
Cole
, and K.
Życzkowski
, “Quantum Monge-Kantorovich problem and transport distance between density matrices
,” Phys. Rev. Lett.
129
, 110402
(2022
).48.
S.
Cole
, M.
Eckstein
, S.
Friedland
, and K.
Życzkowski
, “Quantum optimal transport
,” Math. Phys. Anal. Geom.
26
, 14
(2023
).49.
R.
Duvenhage
, “Optimal quantum channels
,” Phys. Rev. A
104
, 032604
(2021
).50.
R.
Bistroń
, M.
Eckstein
, and K.
Życzkowski
, “Monotonicity of a quantum 2-Wasserstein distance
,” J. Phys. A: Math. Theor.
56
, 095301
(2023
).51.
T.
Van Vu
and K.
Saito
, “Thermodynamic unification of optimal transport: Thermodynamic uncertainty relation, minimum dissipation, and thermodynamic speed limits
,” Phys. Rev. X
13
, 011013
(2023
).52.
R.
Duvenhage
, “Quadratic Wasserstein metrics for von Neumann algebras via transport plans
,” J. Oper. Theory
88
, 289
–308
(2022
).53.
R.
Duvenhage
, “Wasserstein distance between noncommutative dynamical systems
,” J. Mathemat. Anal. Applic.
527
(1
), 127353
(2023
).54.
R.
Duvenhage
, S.
Skosana
, and M.
Snyman
, “Extending quantum detailed balance through optimal transport
,” Transpor. Rev. Mathemat. Phys.
(published online) (2022
).55.
G.
De Palma
and D.
Trevisan
, “Quantum optimal transport with quantum channels
,” Ann. Henri Poincare
22
, 3199
–3234
(2021
).56.
R.
Duvenhage
and M.
Snyman
, “Balance between quantum Markov semigroups
,” Ann. Henri Poincare
19
, 1747
–1786
(2018
).57.
J.
Agredo
and F.
Fagnola
, “On quantum versions of the classical Wasserstein distance
,” Stochastics
89
, 910
–922
(2017
).58.
K.
Życzkowski
and W.
Slomczynski
, “The Monge distance between quantum states
,” J. Phys. A: Math. Gen.
31
, 9095
(1998
).59.
K.
Życzkowski
and W.
Slomczynski
, “The Monge metric on the sphere and geometry of quantum states
,” J. Phys. A: Math. Gen.
34
, 6689
(2001
).60.
I.
Bengtsson
and K.
Życzkowski
, Geometry of Quantum States: An Introduction to Quantum Entanglement
(Cambridge University Press
, 2017
).61.
L.
Li
, K.
Bu
, D.
Enshan Koh
, A.
Jaffe
, and S.
Lloyd
, “Wasserstein complexity of quantum circuits
,” arXiv:2208.06306v1 (2022
).62.
S.
Aaronson
, “The learnability of quantum states
,” Proc. R. Soc. A
463
, 3089
–3114
(2007
).63.
D. E.
Koh
and S.
Grewal
, “Classical shadows with noise
,” Quantum
6
, 776
(2022
).64.
S.
Becker
, N.
Datta
, L.
Lami
, and C.
Rouzé
, “Classical shadow tomography for continuous variables quantum systems
,” IEEE Transac. Infor. Theory
70
(5
), 3427
–3452
(2024
).65.
J.
Lukens
, K.
Law
, and R.
Bennink
, “A Bayesian analysis of classical shadows
,” npj Quantum Inf.
7
, 113
–967
(2021
).66.
T.
Zhang
, J.
Sun
, X.-X.
Fang
, X.-M.
Zhang
, X.
Yuan
, and H.
Lu
, “Experimental quantum state measurement with classical shadows
,” Phys. Rev. Lett.
127
, 200501
(2021
).67.
C.
Hadfield
, S.
Bravyi
, R.
Raymond
, and A.
Mezzacapo
, “Measurements of quantum Hamiltonians with locally-biased classical shadows
,” Commun. Math. Phys.
391
, 951
–967
(2022
).68.
A.
Zhao
, N. C.
Rubin
, and A.
Miyake
, “Fermionic partial tomography via classical shadows
,” Phys. Rev. Lett.
127
, 110504
(2021
).69.
S.
Shivam
, C. W.
von Keyserlingk
, and S. L.
Sondhi
, “On classical and hybrid shadows of quantum states
,” SciPost Phys.
14
, 094
(2023
).70.
H.-Y.
Hu
, S.
Choi
, and Y.-Z.
You
, “Classical shadow tomography with locally scrambled quantum dynamics
,” Phys. Rev. Res.
5
, 023027
(2023
).71.
K.
Bu
, D.
Enshan Koh
, R. J.
Garcia
, and A.
Jaffe
, “Classical shadows with pauli-invariant unitary ensembles
,” npj Quantum Inf.
10
, 6
(2024
).72.
C.
Bădescu
and R.
O’Donnell
, “Improved quantum data analysis
,” in Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021
(Association for Computing Machinery
, New York, NY
, 2021
), pp. 1398
–1411
.73.
S.
Chen
, J.
Cotler
, H.-Y.
Huang
, and J.
Li
, “Exponential separations between learning with and without quantum memory
,” in 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS
(IEEE
, 2021
), pp. 574
–585
.© 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
