Localization and Scrambling of Quantum Information with Applications to Quantum Computation and Thermodynamics

As our demand for computational power grows, we encounter the question: What are the physical limits to computation? An answer is necessarily incomplete unless it can incorporate physics at the smallest scales, where we expect our near-term high-performance computing to occur. Microscopic physics -- namely, quantum mechanics -- behaves counterintuitively to our everyday experience, however. Quantum matter can occupy superpositions of states and build stronger correlations than are possible classically. This affects how quantum computers and quantum thermodynamic engines will behave. Though these properties may seem to overwhelmingly defeat our attempts to build a quantum computer at-first-glance, what is remarkable is that they can also be immensely helpful to computation. Quantum mechanics hinders and helps computation, and the nuanced details of how we perform computations are important. In this dissertation, we examine the transition between these two behaviors and connect it to a well-studied behavior in condensed matter physics, known as the many-body-localization transition. Our idea utilizes the fact that quantum many-body systems have an intrinsic fastest speed at which signals can travel. When this speed is maximal, we expect arbitrary universal quantum computation to be achievable, since strong quantum correlations, or entanglement, can be built quickly. When it is limited, however, the difficulty of the computation is classically simulatable. We demonstrate a similar transition in the amount of thermodynamic work that can be performed by a quantum system when entanglement is present. We first consider computations consisting of the evolution of a single particle or many noninteracting particles. When the number of such noninteracting particles is comparable to the total size of the system, we do not know of any way to simulate such computations classically. However, we find that we can still determine the fastest signal speed in such systems. We extend our result to interacting particles, which are universal for quantum computation, and observe a many-body-localization transition in a simple computational model using our algorithm. Finally, we apply ideas from quantum information to simulate the thermodynamic performance of a simple quantum system, showing that quantum effects can enable it to outperform its classical counterpart

The Computational Power of Non-interacting Particles

Shortened abstract: In this thesis, I study two restricted models of quantum computing related to free identical particles. Free fermions correspond to a set of two-qubit gates known as matchgates. Matchgates are classically simulable when acting on nearest neighbors on a path, but universal for quantum computing when acting on distant qubits or when SWAP gates are available. I generalize these results in two ways. First, I show that SWAP is only one in a large family of gates that uplift matchgates to quantum universality. In fact, I show that the set of all matchgates plus any nonmatchgate parity-preserving two-qubit gate is universal, and interpret this fact in terms of local invariants of two-qubit gates. Second, I investigate the power of matchgates in arbitrary connectivity graphs, showing they are universal on any connected graph other than a path or a cycle, and classically simulable on a cycle. I also prove the same dichotomy for the XY interaction. Free bosons give rise to a model known as BosonSampling. BosonSampling consists of (i) preparing a Fock state of n photons, (ii) interfering these photons in an m-mode linear interferometer, and (iii) measuring the output in the Fock basis. Sampling approximately from the resulting distribution should be classically hard, under reasonable complexity assumptions. Here I show that exact BosonSampling remains hard even if the linear-optical circuit has constant depth. I also report several experiments where three-photon interference was observed in integrated interferometers of various sizes, providing some of the first implementations of BosonSampling in this regime. The experiments also focus on the bosonic bunching behavior and on validation of BosonSampling devices. This thesis contains descriptions of the numerical analyses done on the experimental data, omitted from the corresponding publications.Comment: PhD Thesis, defended at Universidade Federal Fluminense on March 2014. Final version, 208 pages. New results in Chapter 5 correspond to arXiv:1106.1863, arXiv:1207.2126, and arXiv:1308.1463. New results in Chapter 6 correspond to arXiv:1212.2783, arXiv:1305.3188, arXiv:1311.1622 and arXiv:1412.678