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

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

Entanglement purification protocols (EPPs) and quantum error-correcting codes (QECCs) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbitrary quantum state |\ensuremath{\xi}〉 can be transmitted at some rate Q through a noisy channel \ensuremath{\chi} without degradation. We prove that an EPP involving one-way classical communication and acting on mixed state M^(\ensuremath{\chi}) (obtained by sharing halves of Einstein-Podolsky-Rosen pairs through a channel \ensuremath{\chi}) yields a QECC on \ensuremath{\chi} with rate Q=D, and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts ${\mathit{D}}_{1}$(M) and ${\mathit{D}}_{2}$(M) that can be locally distilled from it by EPPs using one- and two-way classical communication, respectively, and give an exact expression for E(M) when M is Bell diagonal. While EPPs require classical communication, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way communication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way communication is available. We exhibit a family of codes based on universal hashing able to achieve an asymptotic Q (or D) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single-error-correcting quantum block code. We prove that iff a QECC results in high fidelity for the case of no error then the QECC can be recast into a form where the encoder is the matrix inverse of the decoder. \textcopyright{} 1996 The American Physical Society.

Mixed State Entanglement and Quantum Error Correction

We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Boolean values (x,y) to (x,x ⊕y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2 n )) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary n-bit unitary operations.

Elementary gates for quantum computation.

We present a scheme of quantum computation that consists entirely of one-qubit measurements on a particular class of entangled states, the cluster states. The measurements are used to imprint a quantum logic circuit on the state, thereby destroying its entanglement at the same time. Cluster states are thus one-way quantum computers and the measurements form the program.

http://www2.fisica.unlp.edu.ar/materias/qc/RB2.pdf

A one-way quantum computer.

We identify a 2-bit quantum gate that is sufficient to build any quantum logic network. The existence of such a 2-bit universal gate considerably simplifies the search for physical realizations of quantum computational networks. We propose an explicit construction of this gate, which is based on cavity QED techniques and may be realizable with current technology.

Realizable Universal Quantum Logic Gates.

Focusing of and imaging with atoms by means of a spherical Fresnel zone plate has been observed for the first time. An intense beam of metastable helium atoms with atomic de Broglie wavelength ${\ensuremath{\lambda}}_{\mathrm{dB}}$=0.5--2.5 \AA{} is passed through either a single or a double slip with dimensions in the 10-\ensuremath{\mu}m range. This transverse intensity distribution is imaged by a Fresnel zone plate, 210 \ensuremath{\mu}m in diameter and with an innermost zone diameter of 18.76 \ensuremath{\mu}m. For ${\ensuremath{\lambda}}_{\mathrm{dB}}$=1.96 \AA{} the focal length is 0.45 m. The imaging properties of the zone plate are presented and compared with numerical calculations.

Imaging and focusing of atoms by a fresnel zone plate.

We report the observation of the splitting of an atomic beam by the optical Stern-Gerlach effect. A beam of metastable He atoms interacts with a nearly or exactly resonant laser field with a well-defined intensity gradient perpendicular to the atomic beam. The coherent splitting of the atomic beam is produced in the case of exact resonance. For the nonresonant case, a deflection of the atomic beam is observed. We have studied the splitting and deflection of the atomic beam as a function of laser detuning from resonance. Our results are in good agreement with theoretical predictions.

Experimental demonstration of the optical Stern-Gerlach effect.

We report the development of an atom interferometer that uses optical standing waves as phase gratings and operates in the time domain. The observed signal is entirely caused by the wave nature of the atomic center-of-mass motion. The opportunities to measure recoil frequency and gravity acceleration are demonstrated.

Tycho Sleator

Papers

Elementary gates for quantum computation.

Realizable Universal Quantum Logic Gates.

Imaging and focusing of atoms by a fresnel zone plate.

Experimental demonstration of the optical Stern-Gerlach effect.

Time-Domain de Broglie Wave Interferometry