О математической структуре моделей квантовых вычислений на основе минимизации гамильтониана / On the mathematical structure of quantum models of computation based on hamiltonian minimisation Биамонте Джейкоб Дэниел

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Биамонте Джейкоб Дэниел.
О математической структуре моделей квантовых вычислений на основе минимизации гамильтониана = On the mathematical structure of quantum models of computation based on hamiltonian minimisation : On the mathematical structure of quantum models of computation based on hamiltonian minimisation : диссертация ... доктора физико-математических наук : 01.01.03 / Jacob Daniel Biamonte; [Место защиты: ФГАОУ ВО «Московский физико-технический институт (национальный исследовательский университет)»]. - Москва, 2021. - 241 с. : ил.
Оглавление диссертациидоктор наук Биамонте Джейкоб Дэниел
Chapter 1. The Algebra of Programming Hamiltonian Ground
States
1.1 P- vs. NP problems and mathematical physics
1.2 Mathematical structures connecting Ising models and quantum
states
1.3 Computation and the Ising model
1.4 Low-energy subspace embedding
1.5 Two-body reductions
1.5.1 Karnaugh map codomain extension
Chapter 2. The Structure of Quantum vs Probabilistic
Computation
2.1 Defining Mechanics
2.1.1 Stochastic time evolution
2.1.2 Quantum time evolution
2.1.3 Stochastic Hamiltonians
2.1.4 Quantum Hamiltonians
2.1.5 Observables
2.2 Walks on Graphs: quantum vs stochastic
2.2.1 Normalized Laplacians
2.2.2 Stochastic walk
2.2.3 Quantum walks
2.2.4 Perron's theorem
2.3 Subadditivity of entropy of stochastic generators
2.4 Google page rank—a ground eigenvector problem
2.5 Kitaev's quantum phase estimation algorithm
2.6 Finding the ground state of a graph on a quantum processor
Chapter 3. Tensor Networks and Parameterised Quantum Circuits
3.1 Clifford gates
3.2 Tensor network building blocks
3.2.1 Reversible logic
3.2.2 Heisenberg picture
3.2.3 Stabilizer tensor theory
Chapter 4. Variational Quantum Search and Optimization
4.1 Random vs quantum complexity
4.2 Variational quantum computation framework
4.3 Variational quantum search
4.4 On Kitaev's ^-controlled U factorization
4.5 QAOA vs optimization by adiabatic and quantum annealing ... 141 4.5.1 Approximating adiabatic evolution
4.6 Computational phase transitions
4.6.1 Thermal states at the SAT phase transition
4.7 Low-depth quantum circuits
4.7.1 A combinatorial quantum circuit area law
4.8 Reachability deficits
Chapter 5. Variational Quantum Computation
5.1 Notions of quantum computational universality
5.2 Maximizing projection onto a circuit
5.3 Maximizing projection onto the history state
5.4 Discussion
Chapter 6. Gadget Hamiltonian Constructions in Ground State
Computation
6.1 Hamiltonian complexity
6.2 Local Hamiltonian is QMA-complete
6.2.1 Real Hamiltonian is QMA-complete
6.3 Exact ZZZ-gadget from Z, ZZ
6.4 Perturbation theory
6.5 Improved subdivision gadget
6.6 YY-gadget from XX, ZZ, X, Z
Chapter 7. Conclusion and Open Problems
List of symbols
List of abbreviations
Glossary of terms
Bibliography
List of Figures
List of Tables
Alphabetical Index