We study the deformation of the thermofield-double (TFD) under evolution by a double-traced operator by computing its entanglement entropy. After saturation, the entanglement change leads to the temperature change. In Jackiw-Teitelboim gravity, the new temperature can be computed independently from two-point function by considering the Schwarzian modes. We will also derive the entanglement entropy from the Casimir associated with the SL(2,R) symmetry. From AdS/CFT correspondence, where TFD is dual to a two-sided black hole, such deformations correspond to the *coherent* shrinking or expansion of the black hole.

Next, we compute the entanglement entropy after coupling a system to the bath perturbatively as a function of κ, the system-bath coupling. At very early times where the entanglement entropy is a logarithmic function of time, the leading contribution is due to the terms of order 2s in the coupling where s is the number of replicas. In the middle time, the entanglement goes linear as a function of time. Assuming saturation at a later time, we will study the effect of an external perturbation to the entropy at an early time where it is related to the OTOCs. A major simplification appears when the system saturates the chaos bound.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Kitaev, Alexei}, } @phdthesis{10.7907/Z9DJ5CND, author = {Mozgunov, Evgeny}, title = {Slow Drive of Many-Body Localized Systems}, school = {California Institute of Technology}, year = {2017}, doi = {10.7907/Z9DJ5CND}, url = {https://resolver.caltech.edu/CaltechTHESIS:06052017-155729766}, abstract = {We investigate a many-body localized 1d spin chain with a Hamiltonian consisting of classical disordered Ising and a small transversal field. An existing perturbative diagonalization by Imbrie is simplified and reinterpreted in order to prove the anticipated form of the Lieb-Robinson bound and the area law in an eigenstate. We also show how to approximately reduce Imbrie’s unitary to a finite depth circuit. The concept of resonances in Imbrie’s work can be given a physical meaning as an avoided crossing of levels as functions of a magnetic field. For a slow drive of this field, we discuss the proofs of validity for an efficient classical simulation of such disordered systems, both isolated and in contact with the environment. Our results are applicable to Floquet systems and describe an unexpected mechanism of heating up over long times. We also revisit noisy quantum adiabatic annealers like the D-wave machine and find a nontrivial physics that can possibly be observed in them.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Kitaev, Alexei}, } @phdthesis{10.7907/6HJB-MC69, author = {Fefferman, William Jason}, title = {The Power of Quantum Fourier Sampling}, school = {California Institute of Technology}, year = {2014}, doi = {10.7907/6HJB-MC69}, url = {https://resolver.caltech.edu/CaltechTHESIS:05302014-131308138}, abstract = {How powerful are Quantum Computers? Despite the prevailing belief that Quantum Computers are more powerful than their classical counterparts, this remains a conjecture backed by little formal evidence. Shor’s famous factoring algorithm [Shor97] gives an example of a problem that can be solved efficiently on a quantum computer with no known efficient classical algorithm. Factoring, however, is unlikely to be NP-Hard, meaning that few unexpected formal consequences would arise, should such a classical algorithm be discovered. Could it then be the case that any quantum algorithm can be simulated efficiently classically? Likewise, could it be the case that Quantum Computers can quickly solve problems much harder than factoring? If so, where does this power come from, and what classical computational resources do we need to solve the hardest problems for which there exist efficient quantum algorithms?

We make progress toward understanding these questions through studying the relationship between classical nondeterminism and quantum computing. In particular, is there a problem that can be solved efficiently on a Quantum Computer that cannot be efficiently solved using nondeterminism? In this thesis we address this problem from the perspective of sampling problems. Namely, we give evidence that approximately sampling the Quantum Fourier Transform of an efficiently computable function, while easy quantumly, is hard for any classical machine in the Polynomial Time Hierarchy. In particular, we prove the existence of a class of distributions that can be sampled efficiently by a Quantum Computer, that likely cannot be approximately sampled in randomized polynomial time with an oracle for the Polynomial Time Hierarchy.

Our work complements and generalizes the evidence given in Aaronson and Arkhipov’s work [AA2013] where a different distribution with the same computational properties was given. Our result is more general than theirs, but requires a more powerful quantum sampler.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, } @mastersthesis{10.7907/3Q00-DD64, author = {Fefferman, William Jason}, title = {On Quantum Computing and Pseudorandomness}, school = {California Institute of Technology}, year = {2011}, doi = {10.7907/3Q00-DD64}, url = {https://resolver.caltech.edu/CaltechTHESIS:01312011-155543067}, abstract = {

The relationship between efficient verification and quantum computing is one of the most important and least well-understood questions in the theory of computation. In particular, is there a problem that can be solved efficiently on a quantum computer that cannot be verified? In this thesis we give evidence that BQP ⊄ PH, relating the classes of languages decidable with a quantum computer to a generalization of NP. In so doing we connect a question in pseudorandomness, first studied [BSW03] to the problem of finding an oracle relative to which BQP ⊄ PH. The primary technical challenge is to construct a unitary matrix, realized by an efficient quantum circuit and whose rows are supported on nearly disjoint subsets. Using this matrix and assuming the validity of the aforementioned question in pseudorandomness, we show an instantiation of the Nisan-Wigderson pseudorandom generator that can be broken with quantum computers, but not with the relevant mode of classical computation.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, month = {July}, advisor = {Kitaev, Alexei and Umans, Christopher M.}, }