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.

We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log(n/ε)). We also give an upper bound of O(n(log n) 2 log log n) on the circuit size of the exact QFT modulo 2 n, for which the best previous bound was O(n 2). As an application of the above depth bound, we show that Shor’s factoring algorithm may be based on quantum circuits with depth only O(log n) and polynomial-size, in combination with classical polynomial-time pre- and post-processing. In the language of computational complexity, this implies that factoring is in the complexity class ZPP BQNC, where BQNC is the class of problems computable with bounded-error probability by quantum circuits with polylogarithmic depth and polynomial size. Finally, we prove an Ω(log n) lower bound on the depth complexity of approximations of the

One of the most important quantum algorithms is Grover's search algorithm [G96]. Quantum searching can be used to speed up the search for solutions to NP-complete problems e.g. 3SAT. Even so, the best known quantum algorithms for 3SAT are considered inefficient.

If two parties, Alice and Bob, share some number, n, of partially entangled pairs of qubits, then it is possible for them to concentrate these pairs into some smaller number of maximally entangled states. We present a simplied version of the algorithm for such entanglement concentration, and we describe ecient networks for implementing these operations. 1 Introduction The state of a single pure quantum bit, or qubit, is described by a vector in a 2-dimensional Hilbert space spanned by basis vectors j0i and j1i. The state of n pure qubits (i.e. an n-qubit register) is described by a vector in a 2 n - dimensional Hilbert space which is the tensor product of the 2-dimensional spaces for the states of each of the n qubits. Consider a 2-qubit register in a state described by the vector j i = 1 p 2 j00i + 1 p 2 j11i. We call a pair of particles in this state an EPR pair, named after Einstein, Podolsky and Rosen, who discussed such particle pairs in their 1935 paper [EPR35]. It ca...

Feynman [1] in his talk during the First Conference on the Physics of Computation held at MIT in Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. In effect, they follow the same logical paradigm as (multiparticle) interferometers. We show how most known quantum algorithms, including quantum algorithms for factorizing and counting, may be cast in this manner, Quantum searching is described as inducing a desired relative phase between two eigenvectors to yield constructive interference on the sought elements and destructive interference on the remaining terms. 1981 observed that it appears to be impossible to simulate a general quantum evolution on a classical probabilistic computer in an efficient way. He pointed out that any classical simulation of quantum evolution appears to involve an exponential slowdown in time as compared to the natural evolution since the amount of information required to describe the evolving quantum state in classical terms generally grows exponentially in time. However, instead of viewing this as an obstacle, Feynman regarded it as an opportunity. If it requires so much computation to work out what will happen in a complicated multiparticle interference experiment, then, he argued, the very act of setting up such an experiment and measuring the outcome is tantamount to performing a complex computation. Indeed, all quantum multiparticle interferometers are quantum computers and some interesting computational problems can be based on estimating internal phase shifts in these interferometers. This approach leads to a unified picture of quantum algorithms and has been recently discussed in detail by Cleve. [2] Let us start with the textbook example of quantum interference, namely the double-slit experiment, which, in a more rnodern version, can be rephrased in terms of Mach-Zehnder interferometry

If two parties, Alice and Bob, share some number, n, of partially entangled pairs of qubits, then it is possible for them to concentrate these pairs into some smaller number of maximally entangled states. We present a simplified version of the algorithm for such entanglement concentration, and we describe efficient networks for implementing these operations. 1