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Quantum Computation and Lattice Problems
 Proc. 43rd Symposium on Foundations of Computer Science
, 2002
"... We present the first explicit connection between quantum computation and lattice problems. Namely, we show a solution to the uniqueSVP under the assumption that there exists... ..."
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Cited by 56 (4 self)
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We present the first explicit connection between quantum computation and lattice problems. Namely, we show a solution to the uniqueSVP under the assumption that there exists...
Quantum Lower Bound for Recursive Fourier Sampling
 Quantum Information and Computation
, 2003
"... We revisit the oftneglected 'recursive Fourier sampling' (RFS) prob lem, introduced by Bernstein and Vazirani to prove an oracle separation between B]] and BQ] . We show that the known quantum algorithm for RF q is essentially optimal, despite its seemingly wasteful need to un compute informa ..."
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Cited by 8 (3 self)
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We revisit the oftneglected 'recursive Fourier sampling' (RFS) prob lem, introduced by Bernstein and Vazirani to prove an oracle separation between B]] and BQ] . We show that the known quantum algorithm for RF q is essentially optimal, despite its seemingly wasteful need to un compute information. This implies that, to place BQ] outside of ]H [log] relative to an oracle, one needs to go outside the RFS framework. Our proof argues that, given any variant of RF q, either the adversary method of Ambainis yields a good quantum lower bound, or else there is an efficient classical algorithm. This technique may be of independent interest.
Quantum hidden subgroup algorithms on free groups, (in preparation
"... Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In thi ..."
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Cited by 6 (2 self)
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Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor’s quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an ”alphabetic building blocks approach, ” we use these primitives to form an ”algorithmic toolkit ” for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover’s algorithm is most surprisingly “almost ” a QHS algorithm, and how this result suggests the possibility of an even more complete ”algorithmic tookit ” beyond the QHS algorithms. Contents
Classical and quantum polynomial reconstruction via Legendre symbol evaluation
 Journal of Complexity
, 2004
"... We consider the problem of recovering a hidden monic polynomial f(X) of degree d ≥ 1 over a finite field Fp of p elements given a black box which, for any x ∈ Fp, evaluates the quadratic character of f(x). We design a classical algorithm of complexity O(d 2 p d+ε) and also show that the quantum quer ..."
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Cited by 3 (2 self)
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We consider the problem of recovering a hidden monic polynomial f(X) of degree d ≥ 1 over a finite field Fp of p elements given a black box which, for any x ∈ Fp, evaluates the quadratic character of f(x). We design a classical algorithm of complexity O(d 2 p d+ε) and also show that the quantum query complexity of this problem is O(d). Some of our results extend those of Wim van Dam, Sean Hallgren and Lawrence Ip obtained in the case of a linear polynomial f(X) = X +s (with unknown s); some are new even in this case. 1 1
IS GROVER’S ALGORITHM A QUANTUM HIDDEN SUBGROUP ALGORITHM?
, 2006
"... Abstract. The arguments given in this paper suggest that Grover’s and Shor’s algorithms are more closely related than one might at first expect. Specifically, we show that Grover’s algorithm can be viewed as a quantum algorithm which solves a nonabelian hidden subgroup problem (HSP). But we then go ..."
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Cited by 1 (1 self)
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Abstract. The arguments given in this paper suggest that Grover’s and Shor’s algorithms are more closely related than one might at first expect. Specifically, we show that Grover’s algorithm can be viewed as a quantum algorithm which solves a nonabelian hidden subgroup problem (HSP). But we then go on to show that the standard nonabelian quantum hidden subgroup (QHS) algorithm can not find a solution to this particular HSP. This leaves open the question as to whether or not there is some modification of the standard nonabelian QHS algorithm which is equivalent to Grover’s algorithm. Contents