Quantum Speed Limits to Operator Growth

Supervisor’s Foreword
Abstract
Publications Related to This Thesis
Acknowledgements
Contents
Acronyms
1 Introduction
	References
2 Operator Growth in Krylov Space
	2.1 A Hopping Problem in Krylov Space
		2.1.1 Krylov Complexity
	2.2 Scrambling, Chaos and Complexity
		2.2.1 Universal Operator Growth Hypothesis
	2.3 Symmetries in Krylov Space
		2.3.1 Closure of the Complexity Algebra
		2.3.2 Complexity Growth Under the Simplicity Hypothesis
	2.4 Summary
	References
3 Dispersion Bound on Krylov Complexity
	3.1 The Dispersion Bound
	3.2 Equivalence Between Saturation and Simplicity Hypothesis
		3.2.1 The Dispersion Bound for Complexity Algebras
		3.2.2 Deviation from the Speed Limit
	3.3 Complexity Growth at the Speed Limit
		3.3.1 Two-Level System
	3.4 Complexity Growth in Generic Systems
		3.4.1 Random Matrix Ensemble
	3.5 Summary
	References
4 A Brief History of Quantum Speed Limits in Isolated Systems
	4.1 Mandelstam and Tamm
	4.2 Margolus and Levitin
	4.3 Tightness of the Unified QSL
	4.4 Geometric Interpretation of the MT QSL
	4.5 QSL and Quantum Metrology
	4.6 Summary
	References
5 QSLs on Operator Flows
	5.1 Liouville Space Formalism
	5.2 Hamiltonian Flows
		5.2.1 Wegner Flow and Dephasing
	5.3 Two QSLs for Operators
		5.3.1 Standard Trigonometric Inequalities
		5.3.2 Refined OQSLs
		5.3.3 Operators with Maximal Speed
		5.3.4 Hamiltonian Formulation of the OQSLs
	5.4 The OQSLs in the Schrödinger Picture
	5.5 A ML OQSL for Driven Hamiltonians
	5.6 Summary
	References
6 QSLs on Correlation Functions
	6.1 Correlation Functions in Quantum Dynamics
		6.1.1 The Theory of Quantum Linear Response
	6.2 QSLs for Autocorrelation Functions
		6.2.1 Examples
	6.3 Bounds on Dynamical Susceptibilities
		6.3.1 Applications
	6.4 A QSL Bound on the Fisher Information
	6.5 Summary
	References
7 A Geometric Operator Quantum Speed Limit
	7.1 Geometric OQSL
	7.2 Geodesic Operator Flows
		7.2.1 Two-Level Observable
		7.2.2 General Condition of Saturation
	7.3 Geodesic Toda Flow
	7.4 Geometric OQSL on Krylov Complexity
		7.4.1 Geometric OQSL for the Complexity Algebra
	7.5 Summary
	References
8 Conclusions
	8.1 Main Results
	8.2 Discussion
	References
Appendix A Krylov Complexity in the Super-Heisenberg Picture
A.1  Complexity Flow at the Speed Limit
A.2  A Differential Equation for Krylov Complexity
Appendix B Geodesic Solution to the Toda Flow