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Robust quantum algorithms for oracle identification
, 2005
"... The oracle identification problem (OIP) was introduced by Ambainis et al. [4]. It is given as a set S of M oracles and a blackbox oracle f. Our task is to figure out which oracle in S is equal to the blackbox f by making queries to f. OIP includes several problems such as the Grover Search as specia ..."
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The oracle identification problem (OIP) was introduced by Ambainis et al. [4]. It is given as a set S of M oracles and a blackbox oracle f. Our task is to figure out which oracle in S is equal to the blackbox f by making queries to f. OIP includes several problems such as the Grover Search as special cases. In this paper, we provide a mostly optimal upper bound of query complexity for this problem: (i) For any oracle set S such that S  ≤ 2Nd (d < 1), we design an algorithm whose query complexity is O ( √ N log M / logN), matching the lower bound proved in [4]. (ii) Our algorithm also works for the range between 2Nd and 2N/log N (where the bound becomes O(N)), but the gap between the upper and lower bounds worsens gradually. (iii) Our algorithm is robust, namely, it exhibits the same performance (up to a constant factor) against the noisy oracles [2, 12, 21].
Certainty and Uncertainty in Quantum Information Processing
"... This survey, aimed at information processing researchers, highlights intriguing but lesser known results, corrects misconceptions, and suggests research areas. Themes include: certainty in quantum algorithms; the “fewer worlds ” theory of quantum mechanics; quantum learning; probability theory versu ..."
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This survey, aimed at information processing researchers, highlights intriguing but lesser known results, corrects misconceptions, and suggests research areas. Themes include: certainty in quantum algorithms; the “fewer worlds ” theory of quantum mechanics; quantum learning; probability theory versus quantum mechanics. This idiosyncratic survey delves into areas of quantum information processing of interest to researchers in fields like information retrieval, machine learning, and artificial intelligence. It overviews intriguing but lesser known results, corrects common misconceptions, and suggests research directions. Three types of applications of a quantum viewpoint on information processing are discussed: quantum algorithms and protocols; quantum proofs for classical
SingleQuery Learning from Abelian and nonAbelian Hamming Distance Oracles
, 2009
"... Abstract: We study the problem of identifying an nbit string using a single quantum query to an oracle that computes the Hamming distance between the query and hidden strings. The standard action of the oracle on a response register of dimension r is by powers of the cycle (1...r), all of which, of ..."
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Abstract: We study the problem of identifying an nbit string using a single quantum query to an oracle that computes the Hamming distance between the query and hidden strings. The standard action of the oracle on a response register of dimension r is by powers of the cycle (1...r), all of which, of course, commute. We introduce a new model for the action of an oracle—by general permutations in Sr—and explore how the success probability depends on r and on the map from Hamming distances to permutations. In particular, we prove that when r = 2, for even n the success probability is 1 with the right choice of the map, while for odd n the success probability cannot be 1 for any choice. Furthermore, for small odd n and r = 3, we demonstrate numerically that the image of the optimal map generates a nonabelian group of permutations. 1
Structure, randomness and complexity in quantum computation
"... This thesis explores the interplay between structure and randomness in quantum computation, with the goal being to characterise the types of structure that give quantum computers an advantage over classical computation. The thesis begins by giving a necessary and sufficient condition for one notion ..."
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This thesis explores the interplay between structure and randomness in quantum computation, with the goal being to characterise the types of structure that give quantum computers an advantage over classical computation. The thesis begins by giving a necessary and sufficient condition for one notion of a quantum walk to be defined on a directed graph, and goes on to derive conditions on the structure of graphs that allow a quantum advantage in a nonlocal graph colouring game. A lower bound on entanglementassisted quantum communication complexity based on informationtheoretic ideas is given, and applied to the communication complexity of random functions. New lower bounds on the probability of success of quantum state discrimination are derived, and are applied to the problem of distinguishing random quantum states. This result is used to show a quantum advantage in almost all instances of a boundederror singlequery oracle identification problem. Lower bounds, and almost optimal algorithms, are given for two models of quantum search of partially ordered sets. This leads to the development of an optimal quantum
Quantum Computation and Information Project, ERATO,
, 2005
"... The oracle identification problem (OIP) was introduced by Ambainis et. al. [4], which is given as a set S of M oracles and a hidden oracle f. Our task is to figure out which oracle in S is equal to the hidden f by doing queries to f. OIP includes several problems such as Grover Search as special cas ..."
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The oracle identification problem (OIP) was introduced by Ambainis et. al. [4], which is given as a set S of M oracles and a hidden oracle f. Our task is to figure out which oracle in S is equal to the hidden f by doing queries to f. OIP includes several problems such as Grover Search as special cases. In this paper, we design robust algorithms, i.e., those which are tolerant against noisy oracles, for OIP. Our results include: (i) For any oracle set S such that S  is polynomial in N, O ( √ N) queries are enough to identify the hidden oracle, which is obviously optimal since this OIP includes Grover Search as a special case. (ii) For the case that S  ≤ 2Nd (d < 1), we design an algorithm whose query complexity is O ( √ N log M / logN) and matches the lower bound proved in [4]. (iii) We can furthermore design a robust algorithm whose complexity changes smoothly between the complexity of (ii) and the complexity of recovering all information about the hidden oracle whose complexity is O(N) as showed by Buhrman et. al. in [11]. Thus our new algorithms are not only robust but also their query complexities are even better than the previous noiseless case [4]. 1
How Many Query Superpositions Are Needed to Learn?
"... Abstract. This paper introduces a framework for quantum exact learning via queries, the socalled quantum protocol. It is shown that usual protocols in the classical learning setting have quantum counterparts. A combinatorial notion, the general halving dimension, is also introduced. Given a quantum ..."
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Abstract. This paper introduces a framework for quantum exact learning via queries, the socalled quantum protocol. It is shown that usual protocols in the classical learning setting have quantum counterparts. A combinatorial notion, the general halving dimension, is also introduced. Given a quantum protocol and a target concept class, the general halving dimension provides lower and upper bounds on the number of queries that a quantum algorithm needs to learn. For usual protocols, the lower bound is also valid even if only involution oracle teachers are considered. Under some protocols, the quantum upper bound improves the classical one. The general halving dimension also approximates the query complexity of ordinary randomized learners. From these bounds we conclude that quantum devices can allow moderate improvements on the query complexity. However, any quantum polynomially query learnable concept class must be also polynomially learnable in the classical setting. 1