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

Fault tolerant quantum computation by anyons

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed to be 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 that are generally thought to be hard on classical computers and that 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, for example, the number of digits of the integer to be factored.

/pdf/polynomial-time-algorithms-for-prime-factorization-and-rtdp7rr21z.pdf

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

This article provides an introduction to surface code quantum computing. We first estimate the size and speed of a surface code quantum computer. We then introduce the concept of the stabilizer, using two qubits, and extend this concept to stabilizers acting on a two-dimensional array of physical qubits, on which we implement the surface code. We next describe how logical qubits are formed in the surface code array and give numerical estimates of their fault tolerance. We outline how logical qubits are physically moved on the array, how qubit braid transformations are constructed, and how a braid between two logical qubits is equivalent to a controlled-not. We then describe the single-qubit Hadamard, Ŝ and T operators, completing the set of required gates for a universal quantum computer. We conclude by briefly discussing physical implementations of the surface code. We include a number of Appendices in which we provide supplementary information to the main text. © 2012 American Physical Society.

/pdf/surface-codes-towards-practical-large-scale-quantum-2hn4avaywa.pdf

Surface codes: Towards practical large-scale quantum computation

Simulating quantum mechanics is known to be a difficult computational problem, especially when dealing with large systems However, this difficulty may be overcome by using some controllable quantum system to study another less controllable or accessible quantum system, ie, quantum simulation Quantum simulation promises to have applications in the study of many problems in, eg, condensed-matter physics, high-energy physics, atomic physics, quantum chemistry and cosmology Quantum simulation could be implemented using quantum computers, but also with simpler, analog devices that would require less control, and therefore, would be easier to construct A number of quantum systems such as neutral atoms, ions, polar molecules, electrons in semiconductors, superconducting circuits, nuclear spins and photons have been proposed as quantum simulators This review outlines the main theoretical and experimental aspects of quantum simulation and emphasizes some of the challenges and promises of this fast-growing field

Quantum Simulation

I show that for any number of oracle lookups up to about $\ensuremath{\pi}/4\sqrt{N},$ Grover's quantum searching algorithm gives the maximal possible probability of finding the desired element. I explain why this is also true for quantum algorithms which use measurements during the computation. I also show that unfortunately quantum searching cannot be parallelized better than by assigning different parts of the search space to independent quantum computers.

https://arxiv.org/pdf/quant-ph/9711070v2.pdf

Grover’s quantum searching algorithm is optimal

We show that the time evolution of the wave function of a quantum–mechanical many–particle system can be simulated precisely and efficiently on a quantum computer. The time needed for such a simulation is comparable to the time of a conventional simulation of the corresponding classical system, a performance which can9t be expected from any classical simulation of a quantum system. We then show how quantities of interest, like the energy spectrum of a system, can be obtained. We also indicate that ultimately the simulation of quantum field theory might be possible on large quantum computers.

Simulating quantum systems on a quantum computer

We present fast and highly parallelized versions of Shor's algorithm. With a sizable quantum computer it would then be possible to factor numbers with millions of digits. The main algorithm presented here uses FFT-based fast integer multiplication. The quick reader can just read the introduction and the ``Results'' section.

Fast versions of Shor's quantum factoring algorithm

I make a rough estimate of the accuracy threshold for fault tolerant quantum computing with concatenated codes First I consider only gate errors and use the depolarizing channel error model I will follow PShor (quant-ph/9505011) for fault tolerant error correction (FTEC) and the fault tolerant implementation of elementary operations on states encoded by the 7-qubit code A simple computer simulation suggests a threshold for gate errors of the order \epsilon \approx 10^{-3} or better I also give a simple argument that the threshold for memory errors is about 10 times smaller, thus \epsilon \approx 10^{-4}

Threshold Estimate for Fault Tolerant Quantum Computation

This is a short introduction to quantum computers, quantum algorithms and quantum error correcting codes. Familiarity with the principles of quantum theory is assumed. Emphasis is put on a concise presentation of the principles avoiding lengthy discussions.

https://arxiv.org/pdf/quant-ph/9811006v1.pdf

Christof Zalka

Papers

Grover’s quantum searching algorithm is optimal

Simulating quantum systems on a quantum computer

Fast versions of Shor's quantum factoring algorithm

Threshold Estimate for Fault Tolerant Quantum Computation

An Introduction to Quantum Computers