There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Part I. Fundamental Concepts: 1. Introduction and overview 2. Introduction to quantum mechanics 3. Introduction to computer science Part II. Quantum Computation: 4. Quantum circuits 5. The quantum Fourier transform and its application 6. Quantum search algorithms 7. Quantum computers: physical realization Part III. Quantum Information: 8. Quantum noise and quantum operations 9. Distance measures for quantum information 10. Quantum error-correction 11. Entropy and information 12. Quantum information theory Appendices References Index.

Quantum Computation and Quantum Information

Spintronics, or spin electronics, involves the study of active control and manipulation of spin degrees of freedom in solid-state systems. This article reviews the current status of this subject, including both recent advances and well-established results. The primary focus is on the basic physical principles underlying the generation of carrier spin polarization, spin dynamics, and spin-polarized transport in semiconductors and metals. Spin transport differs from charge transport in that spin is a nonconserved quantity in solids due to spin-orbit and hyperfine coupling. The authors discuss in detail spin decoherence mechanisms in metals and semiconductors. Various theories of spin injection and spin-polarized transport are applied to hybrid structures relevant to spin-based devices and fundamental studies of materials properties. Experimental work is reviewed with the emphasis on projected applications, in which external electric and magnetic fields and illumination by light will be used to control spin and charge dynamics to create new functionalities not feasible or ineffective with conventional electronics.

/pdf/spintronics-fundamentals-and-applications-2dx38n6phu.pdf

Spintronics: Fundamentals and applications

Using the freedom of design that metamaterials provide, we show how electromagnetic fields can be redirected at will and propose a design strategy. The conserved fields-electric displacement field D, magnetic induction field B, and Poynting vector B-are all displaced in a consistent manner. A simple illustration is given of the cloaking of a proscribed volume of space to exclude completely all electromagnetic fields. Our work has relevance to exotic lens design and to the cloaking of objects from electromagnetic fields.

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Controlling Electromagnetic Fields

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

This report describes 3.091x: Introduction to Solid State Chemistry, one of the first 11 courses offered by MITx on edX, a platform for delivering massive open online courses (MOOCs). Offered in the Fall of 2012, 3.091x covers chemical principles explained by examination of the properties of materials, and follows the curriculum of a first-year course which has been offered at MIT for several decades. The report describes the course structure, in terms of the number of basic e-text, auto-graded problems, and video components. Following a methodology established for analysis of the first 17 HarvardX and MITx courses (Ho et al, HarvardX and MITx: The first year of open online courses), course registrants are described in terms of viewed, explored, and certified sub-populations, together with demographics. The diversity of student activity in the course, and the persistence of student interactions with the courseware, are addressed. The report concludes with descriptions of special features in 3.091x, including its use of web-browser based interactive chemistry simulations and randomized questions.

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3.091x Introduction to Solid-State Chemistry MITx on edX Course Report - 2013 Spring

Experimentally realizable quantum computers are rapidly approaching the threshold of quantum supremacy. Quantum Hamiltonian simulation promises to be one of the first practical applications for which such a device could demonstrate an advantage over all classical systems. However, these early devices will inevitably remain both noisy and small, precluding the use of quantum error correction. We use high-performance classical tools to construct, optimize, and simulate quantum circuits subject to realistic error models in order to empirically determine the "simulation capacity" of near-term simulation experiments implemented via quantum signal processing (QSP), describing the relationship between simulation time, system size, and resolution of QSP circuits which are optimally configured to balance algorithmic precision and external noise. From simulation capacity models, we estimate maximum tolerable error rate for meaningful simulation experiments on a near-term quantum computer. 
By exploiting symmetry inherent to the QSP circuit, we further demonstrate that its capacity for quantum simulation can be increased by at least two orders of magnitude if errors are systematic and unitary. We find that a device with $\epsilon^2=10^{-5}$ systematic amplitude errors could meaningfully simulate systems up to $n\approx16$ with an expected failure rate below $10\%$, whereas the largest system a device with a stochastic error rate of $p_\epsilon=10^{-5}$ could meaningfully simulate with the same rate of failure is between $n=3$ and $n=5$ (depending on the stochastic channel). Extrapolating from empirical results, we estimate that one would typically need a stochastic error rate below $p_\epsilon=10^{-8}$ to perform a meaningful $n=50$ simulation experiment with a failure rate below $10\%$, while the same experiment could tolerate systematic unitary errors with strength $\epsilon^2\approx10^{-6}$.

Empirical determination of the simulation capacity of a near-term quantum computer

At the heart of both lossy compression and clustering is a trade-off between the fidelity and size of the learned representation. Our goal is to map out and study the Pareto frontier that quantifies this trade-off. We focus on the optimization of the Deterministic Information Bottleneck (DIB) objective over the space of hard clusterings. To this end, we introduce the primal DIB problem, which we show results in a much richer frontier than its previously studied Lagrangian relaxation when optimized over discrete search spaces. We present an algorithm for mapping out the Pareto frontier of the primal DIB trade-off that is also applicable to other two-objective clustering problems. We study general properties of the Pareto frontier, and we give both analytic and numerical evidence for logarithmic sparsity of the frontier in general. We provide evidence that our algorithm has polynomial scaling despite the super-exponential search space, and additionally, we propose a modification to the algorithm that can be used where sampling noise is expected to be significant. Finally, we use our algorithm to map the DIB frontier of three different tasks: compressing the English alphabet, extracting informative color classes from natural images, and compressing a group theory-inspired dataset, revealing interesting features of frontier, and demonstrating how the structure of the frontier can be used for model selection with a focus on points previously hidden by the cloak of the convex hull.

https://www.mdpi.com/1099-4300/24/6/771/pdf?version=1654155958

Pareto-Optimal Clustering with the Primal Deterministic Information Bottleneck

This report describes 8.MReV: Mechanics ReView, one of the first 11 courses offered by MITx on edX, a platform for delivering massive open online courses (MOOCs). Offered in the Summer of 2013, 8.MReV offers a second look at introductory Newtonian Mechanics, incorporating research pedagogy developed by the RELATE education group at MIT. This report describes the course structure, in terms of the number of basic e-text, auto-graded problems, and video components. Following a methodology established for analysis of the first 17 HarvardX and MITx courses (Ho et al, HarvardX and MITx: The first year of open online courses), course registrants are described in terms of viewed, explored, and certified sub-populations, together with demographics. The diversity of student activity in the course, and the persistence of student interactions with the courseware, are addressed. The report concludes with descriptions of special features in 8.MReV, including innovative pedagogy and pre-post testing.

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8.MReV Mechanics ReView MITx on edX Course Report - 2013 Spring

Quantum singular value transformation (QSVT) enables the application of polynomial functions to the singular values of near arbitrary linear operators embedded in unitary transforms, and has been used to unify, simplify, and improve most quantum algorithms. QSVT depends on precise results in representation theory, with the desired polynomial functions acting simultaneously within invariant two-dimensional subspaces of a larger Hilbert space. These two-dimensional transformations are largely determined by the related theory of quantum signal processing (QSP). While QSP appears to rely on properties specific to the compact Lie group SU(2), many other Lie groups appear naturally in physical systems relevant to quantum information. This work considers settings in which SU(1,1) describes system dynamics and finds that, surprisingly, despite the non-compactness of SU(1,1), one can recover a QSP-type ansatz, and show its ability to approximate near arbitrary polynomial transformations. We discuss various experimental uses of this construction, as well as prospects for expanded relevance of QSP-like ans\"atze to other Lie groups.

Isaac L. Chuang

Papers

3.091x Introduction to Solid-State Chemistry MITx on edX Course Report - 2013 Spring

Empirical determination of the simulation capacity of a near-term quantum computer

Pareto-Optimal Clustering with the Primal Deterministic Information Bottleneck

8.MReV Mechanics ReView MITx on edX Course Report - 2013 Spring

Quantum signal processing with continuous variables