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

were proposed in the early 1980’s [Benioff80] and shown to be at least as powerful as classical computers an important but not surprising result, since classical computers, at the deepest level, ultimately follow the laws of quantum mechanics. The description of quantum mechanical computers was formalized in the late 80’s and early 90’s [Deutsch85][BB92] [BV93] [Yao93] and they were shown to be more powerful than classical computers on various specialized problems. In early 1994, [Shor94] demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers. This is the problem of integer factorization, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) . ----------------------------------------------

A fast quantum mechanical algorithm for database search

Quantum mechanics can speed up a range of search applications over unsorted data. For example, imagine a phone directory containing $N$ names arranged in completely random order. To find someone's phone number with a probability of 50%, any classical algorithm (whether deterministic or probabilistic) will need to access the database a minimum of $0.5N$ times. Quantum mechanical systems can be in a superposition of states and simultaneously examine multiple names. By properly adjusting the phases of various operations, successful computations reinforce each other while others interfere randomly. As a result, the desired phone number can be obtained in only $O(\sqrt{N})$ accesses to the database.

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Quantum Mechanics Helps in Searching for a Needle in a Haystack

The science of quantum information has arisen over the last two decades centered on the manipulation of individual quanta of information, known as quantum bits or qubits. Quantum computers, quantum cryptography, and quantum teleportation are among the most celebrated ideas that have emerged from this new field. It was realized later on that using continuous-variable quantum information carriers, instead of qubits, constitutes an extremely powerful alternative approach to quantum information processing. This review focuses on continuous-variable quantum information processes that rely on any combination of Gaussian states, Gaussian operations, and Gaussian measurements. Interestingly, such a restriction to the Gaussian realm comes with various benefits, since on the theoretical side, simple analytical tools are available and, on the experimental side, optical components effecting Gaussian processes are readily available in the laboratory. Yet, Gaussian quantum information processing opens the way to a wide variety of tasks and applications, including quantum communication, quantum cryptography, quantum computation, quantum teleportation, and quantum state and channel discrimination. This review reports on the state of the art in this field, ranging from the basic theoretical tools and landmark experimental realizations to the most recent successful developments.

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Gaussian quantum information

日本物理学会誌及びJournal of the Physical Society of Japanの月刊について

Consider a Boolean function $\chi: X \to \{0,1\}$ that partitions set $X$ between its good and bad elements, where $x$ is good if $\chi(x)=1$ and bad otherwise. Consider also a quantum algorithm $\mathcal A$ such that $A |0\rangle= \sum_{x\in X} \alpha_x |x\rangle$ is a quantum superposition of the elements of $X$, and let $a$ denote the probability that a good element is produced if $A |0\rangle$ is measured. If we repeat the process of running $A$, measuring the output, and using $\chi$ to check the validity of the result, we shall expect to repeat $1/a$ times on the average before a solution is found. *Amplitude amplification* is a process that allows to find a good $x$ after an expected number of applications of $A$ and its inverse which is proportional to $1/\sqrt{a}$, assuming algorithm $A$ makes no measurements. This is a generalization of Grover's searching algorithm in which $A$ was restricted to producing an equal superposition of all members of $X$ and we had a promise that a single $x$ existed such that $\chi(x)=1$. Our algorithm works whether or not the value of $a$ is known ahead of time. In case the value of $a$ is known, we can find a good $x$ after a number of applications of $A$ and its inverse which is proportional to $1/\sqrt{a}$ even in the worst case. We show that this quadratic speedup can also be obtained for a large family of search problems for which good classical heuristics exist. Finally, as our main result, we combine ideas from Grover's and Shor's quantum algorithms to perform amplitude estimation, a process that allows to estimate the value of $a$. We apply amplitude estimation to the problem of *approximate counting*, in which we wish to estimate the number of $x\in X$ such that $\chi(x)=1$. We obtain optimal quantum algorithms in a variety of settings.

Quantum Amplitude Amplification and Estimation

Let T[0..N−1] be an unsorted table of N items, eachholding a value from an ordered set. For simplicity,assume that all values are distinct. The minimumsearchingproblem is to ﬁnd the index ysuch that T[y]is minimum. This clearly requires a linear number ofprobes on a classical probabilistic Turing machine.Here, we give a simple quantum algorithm whichsolves the problem using O(√N) probes. The mainsubroutine is the quantum exponential searching al-gorithm of [2], which itself is a generalization ofGrover’s recent quantum searching algorithm [3].Due to a general lower bound of [1], this is withina constant factor of the optimum.

A quantum algorithm for finding the minimum

We study some extensions of Grover's quantum searching algorithm. First, we generalize the Grover iteration in the light of a concept called amplitude amplification. Then, we show that the quadratic speedup obtained by the quantum searching algorithm over classical brute force can still be obtained for a large family of search problems for which good classical heuristics exist. Finally, as our main result, we combine ideas from Grover's and Shor's quantum algorithms to perform approximate counting, which can be seen as an amplitude estimation process.

Quantum Counting

This paper investigates various aspects of the nonlocal effects that can arise when entangled quantum information is shared between two parties. A natural framework for studying nonlocality is that of cooperative games with incomplete information, where two cooperating players may share entanglement. Here, nonlocality can be quantified in terms of the values of such games. We review some examples of non-locality and show that it can profoundly affect the soundness of two-prover interactive proof systems. We then establish limits on nonlocal behavior by upper-bounding the values of several of these games. These upper bounds can be regarded as generalizations of the so-called Tsirelson inequality. We also investigate the amount of entanglement required by optimal and nearly optimal quantum strategies.

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Peter F. Hoyer

Papers

Quantum Amplitude Amplification and Estimation

Quantum Amplitude Amplification and Estimation

A quantum algorithm for finding the minimum

Quantum Counting

Consequences and limits of nonlocal strategies