Entanglement in superconducting heterostructures, and quantum circuit simulation hardware

We begin this dissertation by studying noise correlations in superconducting heterostructures of various geometries. In recent years there has been a resurgence of interest in the nonlocal transport properties of superconducting heterostructures due to the possibility of their serving as a source of electronic entanglement in solid state quantum information processors. Devices designed for this purpose are called Cooper pair splitting devices. The utility of these devices as entanglement sources is known to have connections to the positivity of noise cross correlations in spatially separated leads. In Chapter 1 we outline the theoretical prerequisites for this work, outlining the scattering theory framework based on the Bogoliubov-de Gennes equations we adopt. Within this framework we apply a methodology first introduced by Demers and by Blonder, Tinkham and Klapwijk (BTK) in the early 1980s to find the scattering matrix for our superconducting structures. The current, local and nonlocal shot noise can all be expressed in terms of the underlying scattering processes. This framework allows us to investigate the behavior of the current and noise correlations in the structure as we change the geometry and other key system parameters such as the system size, superconducting phase difference and temperature. We also introduce the Andreev approximation, a commonly used approximation which simplifies the scattering theory for superconducting heterostructures. In Chapter 2, we study the local and nonlocal shot noise in a quasi-1D normal-superconducting-normal (NSN) geometry using material parameters relevant to high-T [subscript c] superconductivity. The scattering and shot noise distributions are studied in the short, intermediate and long system size limits, allowing us to examine the qualitative differences in these three parameter regimes. This allows us to, for example, identify the signatures of over-the-gap geometric resonances in the shot noise distributions that appear in the long system size limit. We also break the nonlocal shot noise distributions down further and study the individual contributions to the nonlocal shot due to particle-particle, hole-hole and particle-hole scattering processes. In Chapter 3, we extend our investigation of superconducting heterostructures to the more complicated NSNSN geometry. A novel feature introduced in the geometry is the presence of subgap quasibound states, which show up as resonances in the scattering matrix. We show that these quasibound states dramatically impact the nonlocal shot noise distributions in the system. At energies near the quasibound states the dominant transmission channel through the system is a process called particle-hole transmission, which results in sharp positive peaks in the nonlocal shot noise distribution of the system. The behavior of the nonlocal noise correlations as we change the size of the superconducting and normal regions is investigated and it is found that there is a "sweet spot'' with respect to the size of the superconducting regions that maximizes the positivity of the nonlocal noise distributions as well as a periodic-like behavior in the positivity of the noise distributions with respect to the normal region size. The results of the full scattering theory for the NSNSN geometry are compared to the results obtained using the Andreev approximation, where we find that the Andreev approximation breaks down at energies close to the quasibound state energies. In the second half of this dissertation we focus on work related to the development of a prototype special-purpose quantum circuit simulation device based on commercial off-the-shelf high-speed analog signal processing hardware. In Chapter 4 we introduce the embedding scheme used to represent quantum states and quantum gates in the frequency domain of a classical analog voltage signal. Experimental results are presented from an early two-qubit prototype device for the fidelity of the state generation and gate application circuits. In Chapter 5, a more in-depth investigation into the modeling of classical errors within our signal processing based simulation method is performed in terms of the effects this noise has on the results of the quantum computation being simulatied. It is shown, for example, that additive white gaussian noise (AWGN) in our system has the same effect as applying a depolarizing channel to the qubits in the simulation. We then perform a simulation of a simple quantum error correction (QEC) protocol using the device and show that, even in the presence of classical noise in the simulation hardware, an overall enhancement in the performance of gate operations as a result of applying QEC is observed

Parallel Quantum Computing Emulation

Quantum computers provide a fundamentally new computing paradigm that promises to revolutionize our ability to solve broad classes of problems. Surprisingly, the basic mathematical structures of gate-based quantum computing, such as unitary operations on a finite-dimensional Hilbert space, are not unique to quantum systems but may be found in certain classical systems as well. Previously, it has been shown that one can represent an arbitrary multi-qubit quantum state in terms of classical analog signals using nested quadrature amplitude modulated signals. Furthermore, using digitally controlled analog electronics one may manipulate these signals to perform quantum gate operations and thereby execute quantum algorithms. The computational capacity of a single signal is, however, limited by the required bandwidth, which scales exponentially with the number of qubits when represented using frequency-based encoding. To overcome this limitation, we introduce a method to extend this approach to multiple parallel signals. Doing so allows a larger quantum state to be emulated with the same gate time required for processing frequency-encoded signals. In the proposed representation, each doubling of the number of signals corresponds to an additional qubit in the spatial domain. Single quit gate operations are similarly extended so as to operate on qubits represented using either frequency-based or spatial encoding schemes. Furthermore, we describe a method to perform gate operations between pairs of qubits represented using frequency or spatial encoding or between frequency-based and spatially encoded qubits. Finally, we describe how this approach may be extended to represent qubits in the time domain as well.Comment: 9 pages, 4 figures, 2018 IEEE International Conference on Rebooting Computing (ICRC

Two-Qubit Gate Set Tomography with Fewer Circuits

Gate set tomography (GST) is a self-consistent and highly accurate method for the tomographic reconstruction of a quantum information processor's quantum logic operations, including gates, state preparations, and measurements. However, GST's experimental cost grows exponentially with qubit number. For characterizing even just two qubits, a standard GST experiment may have tens of thousands of circuits, making it prohibitively expensive for platforms. We show that, because GST experiments are massively overcomplete, many circuits can be discarded. This dramatically reduces GST's experimental cost while still maintaining GST's Heisenberg-like scaling in accuracy. We show how to exploit the structure of GST circuits to determine which ones are superfluous. We confirm the efficacy of the resulting experiment designs both through numerical simulations and via the Fisher information for said designs. We also explore the impact of these techniques on the prospects of three-qubit GST.Comment: 46 pages, 13 figures. V2: Minor edits to acknowledgment

Tomography of entangling two-qubit logic operations in exchange-coupled donor electron spin qubits

Scalable quantum processors require high-fidelity universal quantum logic operations in a manufacturable physical platform. Donors in silicon provide atomic size, excellent quantum coherence and compatibility with standard semiconductor processing, but no entanglement between donor-bound electron spins has been demonstrated to date. Here we present the experimental demonstration and tomography of universal 1- and 2-qubit gates in a system of two weakly exchange-coupled electrons, bound to single phosphorus donors introduced in silicon by ion implantation. We surprisingly observe that the exchange interaction has no effect on the qubit coherence. We quantify the fidelity of the quantum operations using gate set tomography (GST), and we use the universal gate set to create entangled Bell states of the electrons spins, with fidelity ~ 93%, and concurrence 0.91 +/- 0.08. These results form the necessary basis for scaling up donor-based quantum computers

Consistency of high-fidelity two-qubit operations in silicon

The consistency of entangling operations between qubits is essential for the performance of multi-qubit systems, and is a crucial factor in achieving fault-tolerant quantum processors. Solid-state platforms are particularly exposed to inconsistency due to the materials-induced variability of performance between qubits and the instability of gate fidelities over time. Here we quantify this consistency for spin qubits, tying it to its physical origins, while demonstrating sustained and repeatable operation of two-qubit gates with fidelities above 99% in the technologically important silicon metal-oxide-semiconductor (SiMOS) quantum dot platform. We undertake a detailed study of the stability of these operations by analysing errors and fidelities in multiple devices through numerous trials and extended periods of operation. Adopting three different characterisation methods, we measure entangling gate fidelities ranging from 96.8% to 99.8%. Our analysis tools also identify physical causes of qubit degradation and offer ways to maintain performance within tolerance. Furthermore, we investigate the impact of qubit design, feedback systems, and robust gates on implementing scalable, high-fidelity control strategies. These results highlight both the capabilities and challenges for the scaling up of spin-based qubits into full-scale quantum processors