matrix harmonic oscillator and different ways to encode quantum information in it. By calculating the mutual information between the system and a reference which purifies the encoded information, we identify a transition temperature,

{T}_{c}

, below which the encoded quantum information is protected from thermal noise for a memory time scaling as

{N}^{2}

. Conversely, for temperatures higher than

{T}_{c}

, the information is quickly destroyed by thermal noise. Second, we relax the requirement of gauge invariance and study a matrix harmonic oscillator model with only global symmetry. Finally, we further relax even the symmetry requirement and propose a model that consists of a large number

{N}^{2}

of qubits, with interactions derived from an approximate

SU(N)

symmetry. In both ungauged models, we find that the effects of gauging can be mimicked using an energy penalty to give a similar result for the memory time. The final qubit model also has the potential to be realized in the laboratory.

DOI: 10.1103/physrevd.108.086008

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In this proposal, the spins are required to lie spatially close to each other, and this might complicate the realization of a scalable architecture. \n \nIn the first part of the thesis we thus propose a scheme that allows the constraint on the positioning of the qubits to be relaxed. This is achieved by introducing a ferromagnetic coupler between the distant qubits, to which it is coupled via a dipolar interaction. Most importantly, our proposal is applicable to any type of spin qubits and in particular to the technologically very relevant silicon-based qubits and NV-centers in diamond to which previous coupling schemes do not apply. \n \nAs additional key element, a quantum computer needs a memory capable of reliably storing quantum information in the presence of thermal fluctuations. \n \nThis brings us to the second part of this thesis, where we consider self-correcting memories, for which the protection against thermal noise is built-in at the hardware level. 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Our analysis leads to an energy penalty for the creation of defects that grows linearly with the size L of the memory and thus to a lifetime increasing exponentially with L. \n \nIn the third part of this thesis, we study spin systems that support anyons, i.e., particles with fractional statistics. Similar to the toric code, such systems are topologically ordered: they are immune to local perturbations and quantum gates are implemented by non-local operations, namely the exchange of anyons, whose outcomes depend only on the topology of the exchange. Here again the fault-tolerance is achieved at the level of the hardware and physical systems supporting non-abelian anyons are thus promising platforms for quantum computation. \n \nWe focus on spin systems that exhibit some of these properties and specifically on variations of the honeycomb quantum spin model. We first investigate the exact solution of the honeycomb model in detail and derive an explicit formula for the projector onto the physical subspace. We use this result to study inhomogeneous open spin ladders, related to the honeycomb model, which can be tuned between topological and non-topological phases. We test the robustness of Majorana end states (MES) which emerge at the boundary between sections in different topological phases. Furthermore, we present a trijunction setup where MES can be braided. This is of interest since MES in these spin ladders potentially follow non-abelian braiding statistics. Finally, we study the ground states of the aforementioned ladders and show that they are free of vortices when the signs of the spin couplings are all positive or negative. To prove this, we use exact reflection-positivity-based methods as well as approximate methods. \n \nIn the last part of the thesis, we provide an extension of the Mermin-Wagner theorem to a system of lattice spins that are spin-coupled to itinerant and interacting charge carriers. We prove that neither (anti-) ferromagnetic nor helical long-range order is possible in one and two dimensions at any finite temperature (in the absence of spin-orbit). The fundamental question whether spontaneous ordering of the lattice spins occurs in these systems is of interest in the context of quantum computation; the polarization of nuclear spins coupled to a two-dimensional electron gas is a possible route towards the reduction of decoherence induced by the fluctuating Overhauser field in gate-defined quantum dots. \n \n

Jan 1, 20130 citations

In this paper, we explore the possibility of building a quantum memory that is robust to thermal noise using large

N

matrix quantum mechanics models. First, we investigate the gauged

SU(N)

{T}_{c}

, below which the encoded quantum information is protected from thermal noise for a memory time scaling as

{N}^{2}

. Conversely, for temperatures higher than

{T}_{c}

{N}^{2}

of qubits, with interactions derived from an approximate

SU(N)

In this paper, we explore the possibility of building a quantum memory that is robust to thermal noise using large

N

matrix quantum mechanics models. First, we investigate the gauged

SU(N)

T_c

, below which the encoded quantum information is protected from thermal noise for a memory time scaling as

N^2

. Conversely, for temperatures higher than

T_c

N^2

of qubits, with interactions derived from an approximate

SU(N)