A unified approach to sparse signal processing

A unified view of the area of sparse signal processing is presented in tutorial form by bringing together various fields in which the property of sparsity has been successfully exploited. For each of these fields, various algorithms and techniques, which have been developed to leverage sparsity, are described succinctly. The common potential benefits of significant reduction in sampling rate and processing manipulations through sparse signal processing are revealed. The key application domains of sparse signal processing are sampling, coding, spectral estimation, array processing, compo-nent analysis, and multipath channel estimation. In terms of the sampling process and reconstruction algorithms, linkages are made with random sampling, compressed sensing and rate of innovation. The redundancy introduced by channel coding i

Joint Source-Channel Coding with Real Number BCH and Reed-Solomon Codes: Their Properties and Performance in the Presence of Additive Noise

This thesis investigates the joint source-channel coding properties of real number BCH and Reed-Solomon codes in the presence of additive noise. From previous results, it was known that additive noise can cause the error correction ability of a real number code to degrade. This degradation results in decoding failures. Knowing this, there are two main objectives of this research. The first objective is to determine under what conditions a given real number code is reliable. More specifically, for a given real number BCH or Reed-Solomon code, I sought to determine the highest additive noise level for which the real number code could still be accurately decoded within a specified probability of failure. Using these results, the second objective is to determine whether a real number code can obtain better joint source-channel performance than a comparable finite field code. During the investigation process, I formalized the source coding properties that had been mentioned in previous research. The frrst objective was met by deriving an upper bound to the probability of a decoding failure as a function of the signal to noise ratio, the transmission error magnitudes and the code parameters. These bounds assume that a full search decoding method is implemented. Siflce the full search method is impractical and the traditional decoding method performed poorly in the presence of additive noise, an alternate decoding algorithm was developed. This algorithm attempts to combine the directness of the traditional BCH decoding algorithm with the robustness of the full search decoder. The second objective was met with mixed success since deriving an accurate average channel coding performance for multiple error correcting codes proved elusive. However, simulated results for a four error correcting code is examined.Electrical Engineerin

Practical Quantum Chemistry on Near Term Quantum Computers

Solutions to the time-independent Schrödinger equation for molecular systems allow chemical properties to be studied without the direct need for the material. However, the dimension of this problem grows exponentially with the size of the quantum system under consideration making conventional treatment intractable. Quantum computers can efficiently represent and evolve quantum states. Their use offers a possible way to perform simulations on molecules previously impossible to model. However, given the constraints of current quantum computers even studying small systems is limited by the number of qubits, circuit depth and runtime of a chosen quantum algorithm. The work in this thesis is to explore and provide new tools to make chemical simulation more practical on near-term devices. First, the unitary partitioning measurement reduction strategy is explored. This reduces the runtime of the variational quantum eigensolver algorithm (VQE). We then apply this reduction technique to the contextual subspace method, which approximates a problem by introducing artificial symmetries based on the solution of noncontextual version of the problem that reduces the number of qubits required for simulation. We provide a modification to the original algorithm that makes an exponentially scaling part of the technique quadratic. Finally, we develop the projection-based embedding (PBE) technique to allow chemical systems to be studied using state-of-the-art classical methods in conjuncture with quantum computing protocols in a multiscale hierarchy. This allows molecular problems much larger than conventionally studied on quantum hardware to be approached

Full configuration interaction quantum Monte Carlo for coupled electron--boson systems and infinite spaces

We extend the scope of full configuration interaction quantum Monte Carlo (FCIQMC) to be applied to coupled fermion-boson hamiltonians, alleviating the a priori truncation in boson occupation which is necessary for many other wave function based approaches to be tractable. Detailing the required algorithmic changes for efficient excitation generation, we apply FCIQMC in two contrasting settings. The first is a sign-problem-free Hubbard--Holstein model of local electron-phonon interactions, where we show that with care to control for population bias via importance sampling and/or reweighting, the method can achieve unbiased energies extrapolated to the thermodynamic limit, without suffering additional computational overheads from relaxing boson occupation constraints. Secondly, we apply the method as a `solver' within a quantum embedding scheme which maps electronic systems to local electron-boson auxiliary models, with the bosons representing coupling to long-range plasmonic-like fluctuations. We are able to sample these general electron-boson hamiltonians with ease despite a formal sign problem, including a faithful reconstruction of converged reduced density matrices of the system