Stochastic resonance for modulated gain in a single-mode laser

Stochastic resonance (SR) induced by the signal modulation is investigated, by introducing the signal-modulated gain into a single-mode laser system. Using the linear approximation method, we detailedly calculate the signal-to-noise ratio (SNR) of a gain-noise model of the single-mode laser, taking the cross-correlation between the quantum noise and pump noise into account. We find that, SR appears in the dependence of the SNR on the intensities of the quantum and the pump noises when the correlation coefficient between both the noises is negative; moreover, when the cross-correlation between the two noises is strongly negative, SR exhibits a resonance and a suppression versus the gain coefficient, meanwhile, the single-peaked SR and multi-peaked SR occur in the behaviors of the SNR as functions of the loss coefficient and the deterministic steady-state intensity. (c) 2005 Elsevier B.V. All rights reserved.

Analytical study of the layered XY model with VCE method

Motivated by experiments on liquid-crystal films, we study the development of specific heat anomaly of finite layer system. With the VCE method, we introduce the strong surface interaction into the layered XY model and get the results of the forth-order analytical expansion. The results show that when the strong surface interaction becomes strong enough, the order trend defeats the quantum noise and the specific heat peak moves abnormally to the high temperature with the number of layers decreasing.

Holevo-Ordering and the Continuous-Time Limit for Open Floquet Dynamics

Gough, John, (2004) 'Holevo-Ordering and the Continuous-Time Limit for Open Floquet Dynamics', Letters in Mathematical Physcis 67(3) pp.207-221 RAE2008

Comparative performance of multiview stereoscopic and mammographic display modalities for breast lesion detection.

PURPOSE: Mammography is known to be one of the most difficult radiographic exams to interpret. Mammography has important limitations, including the superposition of normal tissue that can obscure a mass, chance alignment of normal tissue to mimic a true lesion and the inability to derive volumetric information. It has been shown that stereomammography can overcome these deficiencies by showing that layers of normal tissue lay at different depths. If standard stereomammography (i.e., a single stereoscopic pair consisting of two projection images) can significantly improve lesion detection, how will multiview stereoscopy (MVS), where many projection images are used, compare to mammography? The aim of this study was to assess the relative performance of MVS compared to mammography for breast mass detection. METHODS: The MVS image sets consisted of the 25 raw projection images acquired over an arc of approximately 45 degrees using a Siemens prototype breast tomosynthesis system. The mammograms were acquired using a commercial Siemens FFDM system. The raw data were taken from both of these systems for 27 cases and realistic simulated mass lesions were added to duplicates of the 27 images at the same local contrast. The images with lesions (27 mammography and 27 MVS) and the images without lesions (27 mammography and 27 MVS) were then postprocessed to provide comparable and representative image appearance across the two modalities. All 108 image sets were shown to five full-time breast imaging radiologists in random order on a state-of-the-art stereoscopic display. The observers were asked to give a confidence rating for each image (0 for lesion definitely not present, 100 for lesion definitely present). The ratings were then compiled and processed using ROC and variance analysis. RESULTS: The mean AUC for the five observers was 0.614 +/- 0.055 for mammography and 0.778 +/- 0.052 for multiview stereoscopy. The difference of 0.164 +/- 0.065 was statistically significant with a p-value of 0.0148. CONCLUSIONS: The differences in the AUCs and the p-value suggest that multiview stereoscopy has a statistically significant advantage over mammography in the detection of simulated breast masses. This highlights the dominance of anatomical noise compared to quantum noise for breast mass detection. It also shows that significant lesion detection can be achieved with MVS without any of the artifacts associated with tomosynthesis.

Coherence Properties of Supercontinuum Spectra Generated in Photonic Crystal and Tapered Optical Fibers

Numerical simulations have been used in studies of the temporal and spectral features of supercontinuum generation in photonic crystal and tapered optical fibers. In particular, an ensemble average over multiple simulations performed with random quantum noise on the input pulse allows the coherence of the supercontinuum to be quantified in terms of the dependence of the degree of first-order coherence on the wavelength. The coherence is shown to depend strongly on the input pulse's duration and wavelength, and optimal conditions for the generation of coherent supercontinua are discussed. © 2002 Optical Society of America.

Numerical Simulations and Coherence Properties of Supercontinuum Generation in Photonic Crystal and Tapered Optical Fibers

Numerical simulations have been used to study broad-band supercontinuum generation in optical fibers with dispersion and nonlinearity characteristics typical and photonic crystal or tapered fibers structures. The simulations include optical shock and Raman nonlinearity terms, with quantum noise taken into account phenomenologically by including in the input field a noise seed of one photon per mode with random phase. For input pulses of 150-fs duration injected in the anomalous dispersion regime, the effect of modulational instability is shown to lead to severe temporal jitter in the output, and associated fluctuations in the spectral amplitude and phase across the generated supercontinuum. The spectral phase fluctuations are quantified by performing multiple simulations and calculating both the standard deviation of the phase and, more rigorously, the degree of first-order coherence as a function of wavelength across the spectrum. By performing simulations over a range of input pulse durations and wavelengths, we can identify the conditions under which coherent supercontinua with a well-defined spectral phase are generated.

Probing strongly correlated states of ultracold atoms in optical lattices

This thesis describes experiments which investigate ultracold atom ensembles in an optical lattice. Such quantum gases are powerful models for solid state physics. Several novel methods are demonstrated that probe the special properties of strongly correlated states in lattice potentials. Of these, quantum noise spectroscopy reveals spatial correlations in such states, which are hidden when using the usual methods of probing atomic gases. Another spectroscopic technique makes it possible to demonstrate the existence of a shell structure of regions with constant densities. Such coexisting phases separated by sharp boundaries had been theoretically predicted for the Mott insulating state. The tunneling processes in the optical lattice in the strongly correlated regime are probed by preparing the ensemble in an optical superlattice potential. This allows the time-resolved observation of the tunneling dynamics, and makes it possible to directly identify correlated tunneling processes.

Hybrid squeezing of solitonic resonant radiation in photonic crystal fibers

We report the existence of a kind of squeezing in photonic crystal fibers which is conceptually intermediate between four-wave-mixing-induced squeezing in which all the participant waves are monochromatic waves, and self-phase-modulation-induced squeezing for a single pulse in a coherent state. This hybrid squeezing occurs when an arbitrary short soliton emits quasimonochromatic resonant radiation near a zero-group-velocity-dispersion point of the fiber. Photons around the resonant frequency become strongly correlated due to the presence of the classical soliton, and a reduction of the quantum noise below the shot-noise level is predicted. © 2011 American Physical Society.

A comparison of denoising methods for X-ray fluoroscopic images

Fluoroscopic images exhibit severe signal-dependent quantum noise, due to the reduced X-ray dose involved in image formation, that is generally modelled as Poisson-distributed. However, image gray-level transformations, commonly applied by fluoroscopic device to enhance contrast, modify the noise statistics and the relationship between image noise variance and expected pixel intensity. Image denoising is essential to improve quality of fluoroscopic images and their clinical information content. Simple average filters are commonly employed in real-time processing, but they tend to blur edges and details. An extensive comparison of advanced denoising algorithms specifically designed for both signal-dependent noise (AAS, BM3Dc, HHM, TLS) and independent additive noise (AV, BM3D, K-SVD) was presented. Simulated test images degraded by various levels of Poisson quantum noise and real clinical fluoroscopic images were considered. Typical gray-level transformations (e.g. white compression) were also applied in order to evaluate their effect on the denoising algorithms. Performances of the algorithms were evaluated in terms of peak-signal-to-noise ratio (PSNR), signal-to-noise ratio (SNR), mean square error (MSE), structural similarity index (SSIM) and computational time. On average, the filters designed for signal-dependent noise provided better image restorations than those assuming additive white Gaussian noise (AWGN). Collaborative denoising strategy was found to be the most effective in denoising of both simulated and real data, also in the presence of image gray-level transformations. White compression, by inherently reducing the greater noise variance of brighter pixels, appeared to support denoising algorithms in performing more effectively. © 2012 Elsevier Ltd. All rights reserved.

Advanced Techniques for Image Quality Assessment of Modern X-ray Computed Tomography Systems

X-ray computed tomography (CT) imaging constitutes one of the most widely used diagnostic tools in radiology today with nearly 85 million CT examinations performed in the U.S in 2011. CT imparts a relatively high amount of radiation dose to the patient compared to other x-ray imaging modalities and as a result of this fact, coupled with its popularity, CT is currently the single largest source of medical radiation exposure to the U.S. population. For this reason, there is a critical need to optimize CT examinations such that the dose is minimized while the quality of the CT images is not degraded. This optimization can be difficult to achieve due to the relationship between dose and image quality. All things being held equal, reducing the dose degrades image quality and can impact the diagnostic value of the CT examination.

A recent push from the medical and scientific community towards using lower doses has spawned new dose reduction technologies such as automatic exposure control (i.e., tube current modulation) and iterative reconstruction algorithms. In theory, these technologies could allow for scanning at reduced doses while maintaining the image quality of the exam at an acceptable level. Therefore, there is a scientific need to establish the dose reduction potential of these new technologies in an objective and rigorous manner. Establishing these dose reduction potentials requires precise and clinically relevant metrics of CT image quality, as well as practical and efficient methodologies to measure such metrics on real CT systems. The currently established methodologies for assessing CT image quality are not appropriate to assess modern CT scanners that have implemented those aforementioned dose reduction technologies.

Thus the purpose of this doctoral project was to develop, assess, and implement new phantoms, image quality metrics, analysis techniques, and modeling tools that are appropriate for image quality assessment of modern clinical CT systems. The project developed image quality assessment methods in the context of three distinct paradigms, (a) uniform phantoms, (b) textured phantoms, and (c) clinical images.

The work in this dissertation used the “task-based” definition of image quality. That is, image quality was broadly defined as the effectiveness by which an image can be used for its intended task. Under this definition, any assessment of image quality requires three components: (1) A well defined imaging task (e.g., detection of subtle lesions), (2) an “observer” to perform the task (e.g., a radiologists or a detection algorithm), and (3) a way to measure the observer’s performance in completing the task at hand (e.g., detection sensitivity/specificity).

First, this task-based image quality paradigm was implemented using a novel multi-sized phantom platform (with uniform background) developed specifically to assess modern CT systems (Mercury Phantom, v3.0, Duke University). A comprehensive evaluation was performed on a state-of-the-art CT system (SOMATOM Definition Force, Siemens Healthcare) in terms of noise, resolution, and detectability as a function of patient size, dose, tube energy (i.e., kVp), automatic exposure control, and reconstruction algorithm (i.e., Filtered Back-Projection– FPB vs Advanced Modeled Iterative Reconstruction– ADMIRE). A mathematical observer model (i.e., computer detection algorithm) was implemented and used as the basis of image quality comparisons. It was found that image quality increased with increasing dose and decreasing phantom size. The CT system exhibited nonlinear noise and resolution properties, especially at very low-doses, large phantom sizes, and for low-contrast objects. Objective image quality metrics generally increased with increasing dose and ADMIRE strength, and with decreasing phantom size. The ADMIRE algorithm could offer comparable image quality at reduced doses or improved image quality at the same dose (increase in detectability index by up to 163% depending on iterative strength). The use of automatic exposure control resulted in more consistent image quality with changing phantom size.

Based on those results, the dose reduction potential of ADMIRE was further assessed specifically for the task of detecting small (<=6 mm) low-contrast (<=20 HU) lesions. A new low-contrast detectability phantom (with uniform background) was designed and fabricated using a multi-material 3D printer. The phantom was imaged at multiple dose levels and images were reconstructed with FBP and ADMIRE. Human perception experiments were performed to measure the detection accuracy from FBP and ADMIRE images. It was found that ADMIRE had equivalent performance to FBP at 56% less dose.

Using the same image data as the previous study, a number of different mathematical observer models were implemented to assess which models would result in image quality metrics that best correlated with human detection performance. The models included naïve simple metrics of image quality such as contrast-to-noise ratio (CNR) and more sophisticated observer models such as the non-prewhitening matched filter observer model family and the channelized Hotelling observer model family. It was found that non-prewhitening matched filter observers and the channelized Hotelling observers both correlated strongly with human performance. Conversely, CNR was found to not correlate strongly with human performance, especially when comparing different reconstruction algorithms.

The uniform background phantoms used in the previous studies provided a good first-order approximation of image quality. However, due to their simplicity and due to the complexity of iterative reconstruction algorithms, it is possible that such phantoms are not fully adequate to assess the clinical impact of iterative algorithms because patient images obviously do not have smooth uniform backgrounds. To test this hypothesis, two textured phantoms (classified as gross texture and fine texture) and a uniform phantom of similar size were built and imaged on a SOMATOM Flash scanner (Siemens Healthcare). Images were reconstructed using FBP and a Sinogram Affirmed Iterative Reconstruction (SAFIRE). Using an image subtraction technique, quantum noise was measured in all images of each phantom. It was found that in FBP, the noise was independent of the background (textured vs uniform). However, for SAFIRE, noise increased by up to 44% in the textured phantoms compared to the uniform phantom. As a result, the noise reduction from SAFIRE was found to be up to 66% in the uniform phantom but as low as 29% in the textured phantoms. Based on this result, it clear that further investigation was needed into to understand the impact that background texture has on image quality when iterative reconstruction algorithms are used.

To further investigate this phenomenon with more realistic textures, two anthropomorphic textured phantoms were designed to mimic lung vasculature and fatty soft tissue texture. The phantoms (along with a corresponding uniform phantom) were fabricated with a multi-material 3D printer and imaged on the SOMATOM Flash scanner. Scans were repeated a total of 50 times in order to get ensemble statistics of the noise. A novel method of estimating the noise power spectrum (NPS) from irregularly shaped ROIs was developed. It was found that SAFIRE images had highly locally non-stationary noise patterns with pixels near edges having higher noise than pixels in more uniform regions. Compared to FBP, SAFIRE images had 60% less noise on average in uniform regions for edge pixels, noise was between 20% higher and 40% lower. The noise texture (i.e., NPS) was also highly dependent on the background texture for SAFIRE. Therefore, it was concluded that quantum noise properties in the uniform phantoms are not representative of those in patients for iterative reconstruction algorithms and texture should be considered when assessing image quality of iterative algorithms.

The move beyond just assessing noise properties in textured phantoms towards assessing detectability, a series of new phantoms were designed specifically to measure low-contrast detectability in the presence of background texture. The textures used were optimized to match the texture in the liver regions actual patient CT images using a genetic algorithm. The so called “Clustured Lumpy Background” texture synthesis framework was used to generate the modeled texture. Three textured phantoms and a corresponding uniform phantom were fabricated with a multi-material 3D printer and imaged on the SOMATOM Flash scanner. Images were reconstructed with FBP and SAFIRE and analyzed using a multi-slice channelized Hotelling observer to measure detectability and the dose reduction potential of SAFIRE based on the uniform and textured phantoms. It was found that at the same dose, the improvement in detectability from SAFIRE (compared to FBP) was higher when measured in a uniform phantom compared to textured phantoms.

The final trajectory of this project aimed at developing methods to mathematically model lesions, as a means to help assess image quality directly from patient images. The mathematical modeling framework is first presented. The models describe a lesion’s morphology in terms of size, shape, contrast, and edge profile as an analytical equation. The models can be voxelized and inserted into patient images to create so-called “hybrid” images. These hybrid images can then be used to assess detectability or estimability with the advantage that the ground truth of the lesion morphology and location is known exactly. Based on this framework, a series of liver lesions, lung nodules, and kidney stones were modeled based on images of real lesions. The lesion models were virtually inserted into patient images to create a database of hybrid images to go along with the original database of real lesion images. ROI images from each database were assessed by radiologists in a blinded fashion to determine the realism of the hybrid images. It was found that the radiologists could not readily distinguish between real and virtual lesion images (area under the ROC curve was 0.55). This study provided evidence that the proposed mathematical lesion modeling framework could produce reasonably realistic lesion images.

Based on that result, two studies were conducted which demonstrated the utility of the lesion models. The first study used the modeling framework as a measurement tool to determine how dose and reconstruction algorithm affected the quantitative analysis of liver lesions, lung nodules, and renal stones in terms of their size, shape, attenuation, edge profile, and texture features. The same database of real lesion images used in the previous study was used for this study. That database contained images of the same patient at 2 dose levels (50% and 100%) along with 3 reconstruction algorithms from a GE 750HD CT system (GE Healthcare). The algorithms in question were FBP, Adaptive Statistical Iterative Reconstruction (ASiR), and Model-Based Iterative Reconstruction (MBIR). A total of 23 quantitative features were extracted from the lesions under each condition. It was found that both dose and reconstruction algorithm had a statistically significant effect on the feature measurements. In particular, radiation dose affected five, three, and four of the 23 features (related to lesion size, conspicuity, and pixel-value distribution) for liver lesions, lung nodules, and renal stones, respectively. MBIR significantly affected 9, 11, and 15 of the 23 features (including size, attenuation, and texture features) for liver lesions, lung nodules, and renal stones, respectively. Lesion texture was not significantly affected by radiation dose.

The second study demonstrating the utility of the lesion modeling framework focused on assessing detectability of very low-contrast liver lesions in abdominal imaging. Specifically, detectability was assessed as a function of dose and reconstruction algorithm. As part of a parallel clinical trial, images from 21 patients were collected at 6 dose levels per patient on a SOMATOM Flash scanner. Subtle liver lesion models (contrast = -15 HU) were inserted into the raw projection data from the patient scans. The projections were then reconstructed with FBP and SAFIRE (strength 5). Also, lesion-less images were reconstructed. Noise, contrast, CNR, and detectability index of an observer model (non-prewhitening matched filter) were assessed. It was found that SAFIRE reduced noise by 52%, reduced contrast by 12%, increased CNR by 87%. and increased detectability index by 65% compared to FBP. Further, a 2AFC human perception experiment was performed to assess the dose reduction potential of SAFIRE, which was found to be 22% compared to the standard of care dose.

In conclusion, this dissertation provides to the scientific community a series of new methodologies, phantoms, analysis techniques, and modeling tools that can be used to rigorously assess image quality from modern CT systems. Specifically, methods to properly evaluate iterative reconstruction have been developed and are expected to aid in the safe clinical implementation of dose reduction technologies.

Prospective Estimation of Radiation Dose and Image Quality for Optimized CT Performance

X-ray computed tomography (CT) is a non-invasive medical imaging technique that generates cross-sectional images by acquiring attenuation-based projection measurements at multiple angles. Since its first introduction in the 1970s, substantial technical improvements have led to the expanding use of CT in clinical examinations. CT has become an indispensable imaging modality for the diagnosis of a wide array of diseases in both pediatric and adult populations [1, 2]. Currently, approximately 272 million CT examinations are performed annually worldwide, with nearly 85 million of these in the United States alone [3]. Although this trend has decelerated in recent years, CT usage is still expected to increase mainly due to advanced technologies such as multi-energy [4], photon counting [5], and cone-beam CT [6].

Despite the significant clinical benefits, concerns have been raised regarding the population-based radiation dose associated with CT examinations [7]. From 1980 to 2006, the effective dose from medical diagnostic procedures rose six-fold, with CT contributing to almost half of the total dose from medical exposure [8]. For each patient, the risk associated with a single CT examination is likely to be minimal. However, the relatively large population-based radiation level has led to enormous efforts among the community to manage and optimize the CT dose.

As promoted by the international campaigns Image Gently and Image Wisely, exposure to CT radiation should be appropriate and safe [9, 10]. It is thus a responsibility to optimize the amount of radiation dose for CT examinations. The key for dose optimization is to determine the minimum amount of radiation dose that achieves the targeted image quality [11]. Based on such principle, dose optimization would significantly benefit from effective metrics to characterize radiation dose and image quality for a CT exam. Moreover, if accurate predictions of the radiation dose and image quality were possible before the initiation of the exam, it would be feasible to personalize it by adjusting the scanning parameters to achieve a desired level of image quality. The purpose of this thesis is to design and validate models to quantify patient-specific radiation dose prospectively and task-based image quality. The dual aim of the study is to implement the theoretical models into clinical practice by developing an organ-based dose monitoring system and an image-based noise addition software for protocol optimization.

More specifically, Chapter 3 aims to develop an organ dose-prediction method for CT examinations of the body under constant tube current condition. The study effectively modeled the anatomical diversity and complexity using a large number of patient models with representative age, size, and gender distribution. The dependence of organ dose coefficients on patient size and scanner models was further evaluated. Distinct from prior work, these studies use the largest number of patient models to date with representative age, weight percentile, and body mass index (BMI) range.

With effective quantification of organ dose under constant tube current condition, Chapter 4 aims to extend the organ dose prediction system to tube current modulated (TCM) CT examinations. The prediction, applied to chest and abdominopelvic exams, was achieved by combining a convolution-based estimation technique that quantifies the radiation field, a TCM scheme that emulates modulation profiles from major CT vendors, and a library of computational phantoms with representative sizes, ages, and genders. The prospective quantification model is validated by comparing the predicted organ dose with the dose estimated based on Monte Carlo simulations with TCM function explicitly modeled.

Chapter 5 aims to implement the organ dose-estimation framework in clinical practice to develop an organ dose-monitoring program based on a commercial software (Dose Watch, GE Healthcare, Waukesha, WI). In the first phase of the study we focused on body CT examinations, and so the patient’s major body landmark information was extracted from the patient scout image in order to match clinical patients against a computational phantom in the library. The organ dose coefficients were estimated based on CT protocol and patient size as reported in Chapter 3. The exam CTDIvol, DLP, and TCM profiles were extracted and used to quantify the radiation field using the convolution technique proposed in Chapter 4.

With effective methods to predict and monitor organ dose, Chapters 6 aims to develop and validate improved measurement techniques for image quality assessment. Chapter 6 outlines the method that was developed to assess and predict quantum noise in clinical body CT images. Compared with previous phantom-based studies, this study accurately assessed the quantum noise in clinical images and further validated the correspondence between phantom-based measurements and the expected clinical image quality as a function of patient size and scanner attributes.

Chapter 7 aims to develop a practical strategy to generate hybrid CT images and assess the impact of dose reduction on diagnostic confidence for the diagnosis of acute pancreatitis. The general strategy is (1) to simulate synthetic CT images at multiple reduced-dose levels from clinical datasets using an image-based noise addition technique; (2) to develop quantitative and observer-based methods to validate the realism of simulated low-dose images; (3) to perform multi-reader observer studies on the low-dose image series to assess the impact of dose reduction on the diagnostic confidence for multiple diagnostic tasks; and (4) to determine the dose operating point for clinical CT examinations based on the minimum diagnostic performance to achieve protocol optimization.

Chapter 8 concludes the thesis with a summary of accomplished work and a discussion about future research.

Design of a speed meter interferometer proof-of-principle experiment

The second generation of large scale interferometric gravitational wave (GW) detectors will be limited by quantum noise over a wide frequency range in their detection band. Further sensitivity improvements for future upgrades or new detectors beyond the second generation motivate the development of measurement schemes to mitigate the impact of quantum noise in these instruments. Two strands of development are being pursued to reach this goal, focusing both on modifications of the well-established Michelson detector configuration and development of different detector topologies. In this paper, we present the design of the world's first Sagnac speed meter (SSM) interferometer, which is currently being constructed at the University of Glasgow. With this proof-of-principle experiment we aim to demonstrate the theoretically predicted lower quantum noise in a Sagnac interferometer compared to an equivalent Michelson interferometer, to qualify SSM for further research towards an implementation in a future generation large scale GW detector, such as the planned Einstein telescope observatory.

Tomographic readout of an opto-mechanical interferometer

The quantum state of light changes its nature when being reflected off a mechanical oscillator due to the latter's susceptibility to radiation pressure. As a result, a coherent state can transform into a squeezed state and can get entangled with the motion of the oscillator. Full information of the state of light can only be gathered by a tomographic measurement. Here we demonstrate a tomographic interferometer readout by measuring arbitrary quadratures of the light field exiting a Michelson-Sagnac interferometer that contains a thermally excited high-quality silicon nitride membrane. A readout noise of 1.9 x 10(-16) mHz(-1/2) around the membrane's fundamental oscillation mode at 133 kHz has been achieved, going below the peak value of the standard quantum limit by a factor of 8.2 (9 dB). The readout noise was entirely dominated by shot noise in a rather broad frequency range around the mechanical resonance.

Quantum error correction with biased noise

Quantum computing offers powerful new techniques for speeding up the calculation of many classically intractable problems. Quantum algorithms can allow for the efficient simulation of physical systems, with applications to basic research, chemical modeling, and drug discovery; other algorithms have important implications for cryptography and internet security.

At the same time, building a quantum computer is a daunting task, requiring the coherent manipulation of systems with many quantum degrees of freedom while preventing environmental noise from interacting too strongly with the system. Fortunately, we know that, under reasonable assumptions, we can use the techniques of quantum error correction and fault tolerance to achieve an arbitrary reduction in the noise level.

In this thesis, we look at how additional information about the structure of noise, or "noise bias," can improve or alter the performance of techniques in quantum error correction and fault tolerance. In Chapter 2, we explore the possibility of designing certain quantum gates to be extremely robust with respect to errors in their operation. This naturally leads to structured noise where certain gates can be implemented in a protected manner, allowing the user to focus their protection on the noisier unprotected operations.

In Chapter 3, we examine how to tailor error-correcting codes and fault-tolerant quantum circuits in the presence of dephasing biased noise, where dephasing errors are far more common than bit-flip errors. By using an appropriately asymmetric code, we demonstrate the ability to improve the amount of error reduction and decrease the physical resources required for error correction.

In Chapter 4, we analyze a variety of protocols for distilling magic states, which enable universal quantum computation, in the presence of faulty Clifford operations. Here again there is a hierarchy of noise levels, with a fixed error rate for faulty gates, and a second rate for errors in the distilled states which decreases as the states are distilled to better quality. The interplay of of these different rates sets limits on the achievable distillation and how quickly states converge to that limit.