It has recently been realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties in realizing quantum computation is that decoherence tends to destroy the information in a superposition of states in a quantum computer, making long computations impossible. A further difficulty is that inaccuracies in quantum state transformations throughout the computation accumulate, rendering long computations unreliable. However, these obstacles may not be as formidable as originally believed. For any quantum computation with t gates, we show how to build a polynomial size quantum circuit that tolerates O(1 / log c t) amounts of inaccuracy and decoherence per gate, for some constant c; the previous bound was O(1 /t). We do this by showing that operations can be performed on quantum data encoded by quantum error-correcting codes without decoding this data. 1.

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Abstract. Consider a Boolean function χ: X → {0, 1} that partitions set X between its good and bad elements, where x is good if χ(x) = 1 and bad otherwise. Consider also a quantum algorithm A such that A|0 〉 = � x∈X αx|x 〉 is a quantum superposition of the elements of X, and let a denote the probability that a good element is produced if A|0 〉 is measured. If we repeat the process of running A, measuring the output, and using χ 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 / √ 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 χ(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 / √ 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 ∈ X such that χ(x) = 1. We obtain optimal quantum algorithms in a variety of settings. 1.

Current formal models for quantum computation deal only with unitary gates operating on “pure quantum states”. In these models it is difficult or impossible to deal formally with several central issues: measurements in the middle of the computation; decoherence and noise, using probabilistic subroutines, and more. It turns out, that the restriction to unitary gates and pure states is unnecessary. In this paper we generalize the formal model of quantum circuits to a model in which the state can be a general quantum state, namely a mixed state, or a “density matrix”, and the gates can be general quantum operations, not necessarily unitary. The new model is shown to be equivalent in computational power to the standard one, and the problems mentioned above essentially disappear. The main result in this paper is a solution for the subroutine problem. The general function that a quantum circuit outputs is a probabilistic function. However, the question of using probabilistic functions as subroutines was not previously dealt with, the reason being that in the language of pure states, this simply can not be done. We define a natural notion of using general subroutines, and show that using general subroutines does not strengthen the model. As an example of the advantages of analyzing quantum complexity using density matrices, we prove a simple lower bound on depth of circuits that compute probabilistic functions. Finally, we deal with the question of inaccurate quantum computation with mixed states. Using the so called “trace metric ” on density matrices, we show how to keep track of errors in the new model.

We provide a tight analysis of Grover’s algorithm for quantum database searching. We give a simple closed-form formula for the probability of success after any given number of iterations of the algorithm. This allows us to determine the number of iterations necessary to achieve almost certainty of finding the answer. Furthermore, we analyse the behaviour of the algorithm when the element to be found appears more than once in the table and we provide a new algorithm to find such an element even when the number of solutions is not known ahead of time. Finally, we provide a lower bound on the efficiency of any possible quantum database searching algorithm and we show that Grover’s algorithm comes within 2.62 % of being optimal.

Abstract. 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. 1

This paper proposes a novel evolutionary algorithm inspired by quantum computing, called a quantum-inspired evolutionary algorithm (QEA), which is based on the concept and principles of quantum computing, such as a quantum bit and superposition of states. Like other evolutionary algorithms, QEA is also characterized by the representation of the individual, the evaluation function, and the population dynamics. However, instead of binary, numeric, or symbolic representation, QEA uses a Q-bit, defined as the smallest unit of information, for the probabilistic representation and a Q-bit individual as a string of Q-bits. A Q-gate is introduced as a variation operator to drive the individuals toward better solutions. To demonstrate its effectiveness and applicability, experiments are carried out on the knapsack problem, which is a well-known combinatorial optimization problem. The results show that QEA performs well, even with a small population, without premature convergence as compared to the conventional genetic algorithm.

I improve the tight bound on quantum searching [4] to a matching bound, thus showing that for any probability of success Grover’s quantum searching algorithm is optimal. E.g. for near certain success we have to query the oracle π/4 √ N times, where N is the size of the search space. I also show that unfortunately quantum searching cannot be parallelized better than by assigning different parts of the search space to independent quantum computers. Earlier results left open the possibility of a more efficient parallelization. 1 Quantum searching Imagine we have N cases of which only one fulfills our conditions. E.g. we have a function which gives 1 only for one of N possible input values and gives 0 otherwise. Often an analysis of the algorithm for calculating the function will allow us to find the input value for which the output is 1. Here we consider the case where we don’t know better than to repeatedly calculate the function

A framework is presented for the design and analysis of quantum mechanical algorithms, the O ( N) step quantum search algorithm is an immediate consequence of this framework. It leads to several other search-type applications- an example is presented where the Walsh-Hadamard (W-H) transform of the quantum search algorithm is replaced by another transform tailored to the parameters of the problem. Also, it leads to quantum mechanical algorithms for problems not immediately connected with search- two such algorithms are presented for calculating the mean and median of statistical distributions. In order to classically estimate either the mean or median of a given distribution to a precision ε, needs Ω ε 2 – steps. The best known quantum mechanical algorithm for estimating the median takes steps, and that for estimating the mean takes O ε 1 –

Communication complexity has become a central complexity model. In that model, we count the amount of communication bits needed between two parties in order to solve certain computational problems. We show that for certain communication complexity problems quantum communication protocols are exponentially faster than classical ones. More explicitly, we give an example for a communication complexity relation (or promise problem) P such that: 1. The quantum communication complexity of P is O(log m). 2. The classical probabilistic communication complexity of P is \Omega\Gamma m 1=4 = log m). (where m is the length of the inputs). This gives an exponential gap between quantum communication complexity and classical probabilistic communication complexity. Only a quadratic gap was previously known. Our problem P is of geometrical nature, and is a finite precision variation of the following problem: Player I gets as input a unit vector x 2 R n and two orthogonal subspaces M 0 ...