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.

Accelerogram processing using reliability bounds and optimal correction methods

This study addresses the problem of obtaining reliable velocities and displacements from accelerograms, a concern which often arises in earthquake engineering. A closed-form acceleration expression with random parameters is developed to test any strong-motion accelerogram processing method. Integration of this analytical time history yields the exact velocities, displacements and Fourier spectra. Noise and truncation can also be added. A two-step testing procedure is proposed and the original Volume II routine is used as an illustration. The main sources of error are identified and discussed. Although these errors may be reduced, it is impossible to extract the true time histories from an analog or digital accelerogram because of the uncertain noise level and missing data. Based on these uncertainties, a probabilistic approach is proposed as a new accelerogram processing method. A most probable record is presented as well as a reliability interval which reflects the level of error-uncertainty introduced by the recording and digitization process. The data is processed in the frequency domain, under assumptions governing either the initial value or the temporal mean of the time histories. This new processing approach is tested on synthetic records. It induces little error and the digitization noise is adequately bounded. Filtering is intended to be kept to a minimum and two optimal error-reduction methods are proposed. The "noise filters" reduce the noise level at each harmonic of the spectrum as a function of the signal-to-noise ratio. However, the correction at low frequencies is not sufficient to significantly reduce the drifts in the integrated time histories. The "spectral substitution method" uses optimization techniques to fit spectral models of near-field, far-field or structural motions to the amplitude spectrum of the measured data. The extremes of the spectrum of the recorded data where noise and error prevail are then partly altered, but not removed, and statistical criteria provide the choice of the appropriate cutoff frequencies. This correction method has been applied to existing strong-motion far-field, near-field and structural data with promising results. Since this correction method maintains the whole frequency range of the record, it should prove to be very useful in studying the long-period dynamics of local geology and structures.

Kinetic theory of normal quantum fluids

Close to equilibrium, a normal Bose or Fermi fluid can be described by an exact kinetic equation whose kernel is nonlocal in space and time. The general expression derived for the kernel is evaluated to second order in the interparticle potential. The result is a wavevector- and frequency-dependent generalization of the linear Uehling-Uhlenbeck kernel with the Born approximation cross section.

The theory is formulated in terms of second-quantized phase space operators whose equilibrium averages are the n-particle Wigner distribution functions. Convenient expressions for the commutators and anticommutators of the phase space operators are obtained. The two-particle equilibrium distribution function is analyzed in terms of momentum-dependent quantum generalizations of the classical pair distribution function h(k) and direct correlation function c(k). The kinetic equation is presented as the equation of motion of a two -particle correlation function, the phase space density-density anticommutator, and is derived by a formal closure of the quantum BBGKY hierarchy. An alternative derivation using a projection operator is also given. It is shown that the method used for approximating the kernel by a second order expansion preserves all the sum rules to the same order, and that the second-order kernel satisfies the appropriate positivity and symmetry conditions.

An investigation of detached shock waves

This investigation demonstrates an application of a flexible wall nozzle for testing in a supersonic wind tunnel. It is conservative to say that the versatility of this nozzle is such that it warrants the expenditure of time to carefully engineer a nozzle and incorporate it in the wind tunnel as a permanent part of the system. The gradients in the test section were kept within one percent of the calibrated Mach number, however, the gradients occurring over the bodies tested were only ± 0.2 percent in Mach number.

The conditions existing on a finite cone with a vertex angle of 75° were investigated by considering the pressure distribution on the cone and the shape of the shock wave. The pressure distribution on the surface of the 75° cone when based on upstream conditions does not show any discontinuities at the theoretical attachment Mach number.

Both the angle of the shock wave and the pressure distribution of the 75° cone are in very close agreement with the theoretical values given in the Kopal report, (Ref. 3).

The location of the intersection of the sonic line with the surface of the cone and with the shock wave are given for the cone. The blocking characteristics of the GALCIT supersonic wind tunnel were investigated with a series of 60° cones.

Radar methods for the exploration of glaciers

The problem of finding the depths of glaciers and the current methods are discussed briefly. Radar methods are suggested as a possible improvement for, or adjunct to, seismic and gravity survey methods. The feasibility of propagating electromagnetic waves in ice and the maximum range to be expected are then investigated theoretically with the aid of experimental data on the dielectric properties of ice. It is found that the maximum expected range is great enough to measure the depth of many glaciers at the lower radar frequencies if there is not too much liquid water present. Greater ranges can be attained by going to lower frequencies.

The results are given of two expeditions in two different years to the Seward Glacier in the Yukon Territory. Experiments were conducted on a small valley glacier whose depth was determined by seismic sounding. Many echoes were received but their identification was uncertain. Using the best echoes, a profile was obtained each year, but they were not in exact agreement with each other. It could not be definitely established that echoes had been received from bedrock. Agreement with seismic methods for a considerable number of glaciers would have to be obtained before radar methods could be relied upon. The presence of liquid water in the ice is believed to be one of the greatest obstacles. Besides increasing the attenuation and possibly reflecting energy, it makes it impossible to predict the velocity of propagation. The equipment used was far from adequate for such purposes, so many of the difficulties could be attributed to this. Partly because of this, and the fact that there are glaciers with very little liquid water present, radar methods are believed to be worthy of further research for the exploration of glaciers.

Oxygen relationships in intermittent sand filtration of wastewaters

A model for some of the many physical-chemical and biological processes in intermittent sand filtration of wastewaters is described and an expression for oxygen transfer is formulated.

The model assumes that aerobic bacterial activity within the sand or soil matrix is limited, mostly by oxygen deficiency, while the surface is ponded with wastewater. Atmospheric oxygen reenters into the soil after infiltration ends. Aerobic activity is resumed, but the extent of penetration of oxygen is limited and some depths may be always anaerobic. These assumptions lead to the conclusion that the percolate shows large variations with respect to the concentration of certain contaminants, with some portions showing little change in a specific contaminant. Analyses of soil moisture in field studies and of effluent from laboratory sand columns substantiated the model.

The oxygen content of the system at sufficiently long times after addition of wastes can be described by a quasi-steady-state diffusion equation including a term for an oxygen sink. Measurements of oxygen content during laboratory and field studies show that the oxygen profile changes only slightly up to two days after the quasi-steady state is attained.

Results of these hypotheses and experimental verification can be applied in the operation of existing facilities and in the interpretation of data from pilot plant-studies.

Formal languages, part theory, and change

A general definition of interpreted formal language is presented. The notion “is a part of" is formally developed and models of the resulting part theory are used as universes of discourse of the formal languages. It is shown that certain Boolean algebras are models of part theory.

With this development, the structure imposed upon the universe of discourse by a formal language is characterized by a group of automorphisms of the model of part theory. If the model of part theory is thought of as a static world, the automorphisms become the changes which take place in the world. Using this formalism, we discuss a notion of abstraction and the concept of definability. A Galois connection between the groups characterizing formal languages and a language-like closure over the groups is determined.

It is shown that a set theory can be developed within models of part theory such that certain strong formal languages can be said to determine their own set theory. This development is such that for a given formal language whose universe of discourse is a model of part theory, a set theory can be imbedded as a submodel of part theory so that the formal language has parts which are sets as its discursive entities.

Relativistic velocity - potential hydrodynamics and stellar stability

The equations of relativistic, perfect-fluid hydrodynamics are cast in Eulerian form using six scalar "velocity-potential" fields, each of which has an equation of evolution. These equations determine the motion of the fluid through the equation

U_ʋ=µ^-1 (ø,_ʋ + αβ,_ʋ + ƟS,_ʋ).

Einstein's equations and the velocity-potential hydrodynamical equations follow from a variational principle whose action is

I = (R + 16π p) (-g)^1/2 d⁴x,

where R is the scalar curvature of spacetime and p is the pressure of the fluid. These equations are also cast into Hamiltonian form, with Hamiltonian density –T₀⁰ (-g^oo)^-1/2.

The second variation of the action is used as the Lagrangian governing the evolution of small perturbations of differentially rotating stellar models. In Newtonian gravity this leads to linear dynamical stability criteria already known. In general relativity it leads to a new sufficient condition for the stability of such models against arbitrary perturbations.

By introducing three scalar fields defined by

ρ ᵴ = ∇λ + ∇x(xi + ∇xɣi)

(where ᵴ is the vector displacement of the perturbed fluid element, ρ is the mass-density, and i, is an arbitrary vector), the Newtonian stability criteria are greatly simplified for the purpose of practical applications. The relativistic stability criterion is not yet in a form that permits practical calculations, but ways to place it in such a form are discussed.