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Mathematical aspects of quantum computing 2007
This book provides a comprehensive overview of the mathematical aspects of quantum computing. It will be useful for graduate students and researchers interested in quantum computing from different areas of physics, mathematics, informatics and computer science. The lecture notes in this volume are written in a self-contained style, and hence are accessible for graduate students and researchers with even less background in the topics.
eBook
Coding theory and quantum computing : an international conference on Coding Theory and Quantum Computing, May 20-24, 2003, University of Virginia
A conference, 'Coding Theory and Quantum Computing', was held in Charlottesville, VA, to provide an opportunity for computer scientists, mathematicians, and physicists to interact about subjects of common interest. This proceedings volume grew out of that meeting. It is divided into two parts: 'Coding Theory' and 'Quantum Computing'. In the first part, Harold Ward gives an introduction to coding theory. Other papers survey recent important work, such as coding theory applications of Grobner bases, methods of computing parameters of codes corresponding to algebraic curves, and problems in the theory of designs. The second part of the book covers a wide variety of directions in quantum information with an emphasis on understanding entanglement. The material presented is suitable for graduate students and researchers interested in coding theory and in quantum computing.
eBook
Topological Insulators and Topological Superconductors
This graduate-level textbook is the first pedagogical synthesis of the field of topological insulators and superconductors, one of the most exciting areas of research in condensed matter physics. Presenting the latest developments, while providing all the calculations necessary for a self-contained and complete description of the discipline, it is ideal for graduate students and researchers preparing to work in this area, and it will be an essential reference both within and outside the classroom. The book begins with simple concepts such as Berry phases, Dirac fermions, Hall conductance and its link to topology, and the Hofstadter problem of lattice electrons in a magnetic field. It moves on to explain topological phases of matter such as Chern insulators, two- and three-dimensional topological insulators, and Majorana p-wave wires. Additionally, the book covers zero modes on vortices in topological superconductors, time-reversal topological superconductors, and topological responses/field theory and topological indices. The book also analyzes recent topics in condensed matter theory and concludes by surveying active subfields of research such as insulators with point-group symmetries and the stability of topological semimetals. Problems at the end of each chapter offer opportunities to test knowledge and engage with frontier research issues. Topological Insulators and Topological Superconductors will provide graduate students and researchers with the physical understanding and mathematical tools needed to embark on research in this rapidly evolving field.
eBook
The golden ticket
The P-NP problem is the most important open problem in computer science, if not all of mathematics.The Golden Ticketprovides a nontechnical introduction to P-NP, its rich history, and its algorithmic implications for everything we do with computers and beyond. In this informative and entertaining book, Lance Fortnow traces how the problem arose during the Cold War on both sides of the Iron Curtain, and gives examples of the problem from a variety of disciplines, including economics, physics, and biology. He explores problems that capture the full difficulty of the P-NP dilemma, from discovering the shortest route through all the rides at Disney World to finding large groups of friends on Facebook. But difficulty also has its advantages. Hard problems allow us to safely conduct electronic commerce and maintain privacy in our online lives.
The Golden Ticketexplores what we truly can and cannot achieve computationally, describing the benefits and unexpected challenges of the P-NP problem.
eBook
Measuring Out-of-Time-Order Correlators on a Nuclear Magnetic Resonance Quantum Simulator
The idea of the out-of-time-order correlator (OTOC) has recently emerged in the study of both condensed matter systems and gravitational systems. It not only plays a key role in investigating the holographic duality between a strongly interacting quantum system and a gravitational system, it also diagnoses the chaotic behavior of many-body quantum systems and characterizes information scrambling. Based on OTOCs, three different concepts—quantum chaos, holographic duality, and information scrambling—are found to be intimately related to each other. Despite its theoretical importance, the experimental measurement of the OTOC is quite challenging, and thus far there is no experimental measurement of the OTOC for local operators. Here, we report the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator. We observe that the OTOC behaves differently in the integrable and nonintegrable cases. Based on the recent discovered relationship between OTOCs and the growth of entanglement entropy in the many-body system, we extract the entanglement entropy from the measured OTOCs, which clearly shows that the information entropy oscillates in time for integrable models and scrambles for nonintgrable models. With the measured OTOCs, we also obtain the experimental result of the butterfly velocity, which measures the speed of correlation propagation. Our experiment paves a way for experimentally studying quantum chaos, holographic duality, and information scrambling in many-body quantum systems with quantum simulators.
Journal Article
Encoding a qubit in a trapped-ion mechanical oscillator
The stable operation of quantum computers will rely on error correction, in which single quantum bits of information are stored redundantly in the Hilbert space of a larger system. Such encoded qubits are commonly based on arrays of many physical qubits, but can also be realized using a single higher-dimensional quantum system, such as a harmonic oscillator
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–
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. In such a system, a powerful encoding has been devised based on periodically spaced superpositions of position eigenstates
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. Various proposals have been made for realizing approximations to such states, but these have thus far remained out of reach
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. Here we demonstrate such an encoded qubit using a superposition of displaced squeezed states of the harmonic motion of a single trapped
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Ca
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ion, controlling and measuring the mechanical oscillator through coupling to an ancillary internal-state qubit
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. We prepare and reconstruct logical states with an average squared fidelity of 87.3 ± 0.7 per cent. Also, we demonstrate a universal logical single-qubit gate set, which we analyse using process tomography. For Pauli gates we reach process fidelities of about 97 per cent, whereas for continuous rotations we use gate teleportation and achieve fidelities of approximately 89 per cent. This control method opens a route for exploring continuous variable error correction as well as hybrid quantum information schemes using both discrete and continuous variables
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. The code states also have direct applications in quantum sensing, allowing simultaneous measurement of small displacements in both position and momentum
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A single logical qubit is encoded, manipulated and read out using a superposition of displaced squeezed states of the harmonic motion of a trapped calcium ion.
Journal Article
Learning the quantum algorithm for state overlap
Short-depth algorithms are crucial for reducing computational error on near-term quantum computers, for which decoherence and gate infidelity remain important issues. Here we present a machine-learning approach for discovering such algorithms. We apply our method to a ubiquitous primitive: computing the overlap Tr ( ) between two quantum states and . The standard algorithm for this task, known as the Swap Test, is used in many applications such as quantum support vector machines, and, when specialized to = , quantifies the Renyi entanglement. Here, we find algorithms that have shorter depths than the Swap Test, including one that has a constant depth (independent of problem size). Furthermore, we apply our approach to the hardware-specific connectivity and gate sets used by Rigetti's and IBM's quantum computers and demonstrate that the shorter algorithms that we derive significantly reduce the error-compared to the Swap Test-on these computers.
Journal Article
Universal control of a six-qubit quantum processor in silicon
Future quantum computers capable of solving relevant problems will require a large number of qubits that can be operated reliably
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. However, the requirements of having a large qubit count and operating with high fidelity are typically conflicting. Spins in semiconductor quantum dots show long-term promise
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but demonstrations so far use between one and four qubits and typically optimize the fidelity of either single- or two-qubit operations, or initialization and readout
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. Here, we increase the number of qubits and simultaneously achieve respectable fidelities for universal operation, state preparation and measurement. We design, fabricate and operate a six-qubit processor with a focus on careful Hamiltonian engineering, on a high level of abstraction to program the quantum circuits, and on efficient background calibration, all of which are essential to achieve high fidelities on this extended system. State preparation combines initialization by measurement and real-time feedback with quantum-non-demolition measurements. These advances will enable testing of increasingly meaningful quantum protocols and constitute a major stepping stone towards large-scale quantum computers.
The universal control of six qubits in a
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Si/SiGe quantum dot array is demonstrated, achieving Rabi oscillations for each qubit with visibilities of 93.5–98.0%, implying high readout and initialization fidelities.
Journal Article