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Matrix-product states for strongly correlated systems and quantum information processing [Elektronische Ressource] / vorgelegt von Hamed Saberi

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Matrix-product states for stronglycorrelated systems and quantuminformation processingHamed SaberiMu¨nchen 2008Matrix-product states for stronglycorrelated systems and quantuminformation processingHamed SaberiDissertationan der Fakult¨at fu¨r Physikder Ludwig–Maximilians–Universit¨atMu¨nchenvorgelegt vonHamed Saberiaus ArakMu¨nchen, den 12. Dezember 2008Erstgutachter: Prof. Dr. Jan von DelftZweitgutachter: Prof. Dr. Matthias ChristandlTag der m¨undlichen Pr¨ufung: 23. Januar 2009To my parentsviContentsAbstract xvI General Introduction and Basic Concpets 11 Introduction and motivation 32 DMRG and matrix-product states 72.1 Transition from NRG to DMRG . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Matrix-product states in the context of strongly correlated systems . . . . 122.3 DMRG leads to MPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 MPS in the context of quantum information processing 153.1 Definition and graphical representation of MPS . . . . . . . . . . . . . . . 153.2 Schmidt decomposition and singular value decomposition . . . . . . . . . . 173.3 Vidal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Prototypical low-dimensional MPS in quantum information theory . . . . . 213.5 MPS for sequential generation of entangled multiqubit states . . . . . . . . 23II Results 274 MPS comparison of NRG and the variational formulation of DMRG 294.1 Introduction . . . . . . . . . . . .

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Matrix-product states for strongly
correlated systems and quantum
information processing
Hamed Saberi
Mu¨nchen 2008Matrix-product states for strongly
correlated systems and quantum
information processing
Hamed Saberi
Dissertation
an der Fakult¨at fu¨r Physik
der Ludwig–Maximilians–Universit¨at
Mu¨nchen
vorgelegt von
Hamed Saberi
aus Arak
Mu¨nchen, den 12. Dezember 2008Erstgutachter: Prof. Dr. Jan von Delft
Zweitgutachter: Prof. Dr. Matthias Christandl
Tag der m¨undlichen Pr¨ufung: 23. Januar 2009To my parentsviContents
Abstract xv
I General Introduction and Basic Concpets 1
1 Introduction and motivation 3
2 DMRG and matrix-product states 7
2.1 Transition from NRG to DMRG . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Matrix-product states in the context of strongly correlated systems . . . . 12
2.3 DMRG leads to MPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 MPS in the context of quantum information processing 15
3.1 Definition and graphical representation of MPS . . . . . . . . . . . . . . . 15
3.2 Schmidt decomposition and singular value decomposition . . . . . . . . . . 17
3.3 Vidal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Prototypical low-dimensional MPS in quantum information theory . . . . . 21
3.5 MPS for sequential generation of entangled multiqubit states . . . . . . . . 23
II Results 27
4 MPS comparison of NRG and the variational formulation of DMRG 29
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Folded and unfolded representations of Wilson chain. . . . . . . . . . . . . 32
4.3 NRG treatment of folded Wilson chain . . . . . . . . . . . . . . . . . . . . 34
4.3.1 NRG matrix-product state arises by iteration . . . . . . . . . . . . 34
4.3.2 NRG truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3.3 Mutual information of opposite spins on site n . . . . . . . . . . . . 36
4.4 DMRG treatment of unfolded Wilson chain . . . . . . . . . . . . . . . . . . 40viii CONTENTS
4.4.1 Variational matrix-product state ansatz . . . . . . . . . . . . . . . . 40
4.4.2 VMPS truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4.3 Refolding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.5 Comparison of NRG and VMPS results . . . . . . . . . . . . . . . . . . . . 44
4.5.1 Ground state energies and overlaps . . . . . . . . . . . . . . . . . . 45
4.5.2 Comparison of eigenspectra and density of states . . . . . . . . . . 46
4.5.3 Comparison of energy eigenstates . . . . . . . . . . . . . . . . . . . 50
4.6 Cloning and variational improvement of NRG ground state . . . . . . . . 54
4.6.1 Mapping folded to unfolded states by cloning . . . . . . . . . . . . 56
4.6.2 Variational energy minimization of|Gi . . . . . . . . . . . . . . . . 57c
4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Constrained optimization of sequentially generated multiqubit states 63
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Restrictions on the number of ancilla levels . . . . . . . . . . . . . . . . . . 64
5.2.1 SVD for matrix approximation . . . . . . . . . . . . . . . . . . . . 65
5.2.2 SVD for MPS compression . . . . . . . . . . . . . . . . . . . . . . . 68
5.2.3 Variational optimization for ancilla dimension . . . . . . . . . . . . 69
5.3 Restrictions on the source-qubit interactions . . . . . . . . . . . . . . . . . 70
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6 Approximate sequential implementation of global operations 77
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Sequential decomposition of global unitaries within Frobenius metric . . . 78
6.2.1 Results for ancilla of dimension D = 2 . . . . . . . . . . . . . . . . 80
6.2.2 Results for ancilla of dimension D = 4 . . . . . . . . . . . . . . . . 81
6.3 Sequential decomposition of global isometries within Frobenius norm metric 82
6.3.1 The map 1→N . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3.2 The map M →N when 1<M <N . . . . . . . . . . . . . . . . . 84
6.4 Sequential decomposition of global unitaries within p-norm metric . . . . . 84
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
III Appendix 89
A Orthonormalization of B-matrices of unfolded Wilson chain 91
B Refolding 93Contents ix
C Calculation of the effective Hamiltonian within the MPS formalism 95
D Variational “cloning” of folded to unfolded states 103
E Matrix norm and singular value decomposition 107
IV Miscellaneous 109
Bibliography 111
Deutsche Zusammenfassung 119
Acknowledgements 121
Curriculum Vitae 125x Contents