Robust Polynomials and Quantum Algorithms

Buhrman, Harry; Newman, Ilan; Röhrig, Hein; Wolf, Ronald de

doi:10.1007/s00224-006-1313-z

Cited by 51 publications

(49 citation statements)

References 17 publications

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“…Together with inequality (14), this leads to the bound on the minimal evolution time T , as stated in the main result (8).…”

Section: Runtime Lower Boundmentioning

confidence: 56%

“…In quantum query algorithms two classes of noise models have been considered. One class of noise model considers coherent errors [8,9], whereas the other class models errors in terms of either dephasing or bit-flip errors [4][5][6][7]10]. For the latter class, the quadratic speedup of Grover's algorithm vanishes and the runtime assumes a linear scaling [4].…”

Section: Introductionmentioning

confidence: 99%

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Runtime of unstructured search with a faulty Hamiltonian oracle

Temme

2014

Phys. Rev. A

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We show that it is impossible to obtain a quantum speedup for a faulty Hamiltonian oracle. The effect of dephasing noise to this continuous-time oracle model has first been investigated by Shenvi, Brown, and Whaley [Phys. Rev. A 68, 052313 (2003).]. The authors consider a faulty oracle described by a continuous-time master equation that acts as dephasing noise in the basis determined by the marked item. The analysis focuses on the implementation with a particular driving Hamiltonian. A universal lower bound for this oracle model, which rules out a better performance with a different driving Hamiltonian, has so far been lacking. Here, we derive an adversary-type lower bound which shows that the evolution time T has to be at least in the order of N , i.e., the size of the search space, when the error rate of the oracle is constant. This means that quadratic quantum speedup vanishes and the runtime assumes again the classical scaling. For the standard quantum oracle model this result was first proven by Regev and

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“…Together with inequality (14), this leads to the bound on the minimal evolution time T , as stated in the main result (8).…”

Section: Runtime Lower Boundmentioning

confidence: 56%

Section: Introductionmentioning

confidence: 99%

Runtime of unstructured search with a faulty Hamiltonian oracle

Temme

2014

Phys. Rev. A

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“…It is still unknown what worst-case success probability in computing f ⊗k can be achieved in general, when the number of queries allowed is αR 2 (f )k for α ≫ 1. The corresponding question in the quantum query model was settled by [BNRdW07]. As mentioned earlier, O(R 2 (f )k log k) queries always suffice to compute f ⊗k with high success probability; work of [FRPU94] implies that we cannot do better than this by using a bounded-error randomized algorithm for f in a black-box fashion.…”

Section: Dpts For Dynamic Interactionmentioning

confidence: 99%

Improved direct product theorems for randomized query complexity

Drucker

2012

comput. complex.

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The "direct product problem" is a fundamental question in complexity theory which seeks to understand how the difficulty of computing a function on each of k independent inputs scales with k. We prove the following direct product theorem (DPT) for query complexity: if every T -query algorithm has success probability at most 1 − ε in computing the Boolean function f on input distribution µ, then for α ≤ 1, every αεT k-query algorithm has success probability at most (2 αε (1 − ε)) k in computing the k-fold direct product f ⊗k correctly on k independent inputs from µ. In light of examples due to Shaltiel, this statement gives an essentially optimal tradeoff between the query bound and the error probability. Using this DPT, we show that for an absolute constant α > 0, the worst-case success probability of any αR 2 (f )k-query randomized algorithm for f ⊗k falls exponentially with k. The best previous statement of this type, due to Klauck,Špalek, and de Wolf, required a query bound of O(bs(f )k).Our proof technique involves defining and analyzing a collection of martingales associated with an algorithm attempting to solve f ⊗k . Our method is quite general and yields a new XOR lemma and threshold DPT for the query model, as well as DPTs for the query complexity of learning tasks, search problems, and tasks involving interaction with dynamic entities. We also give a version of our DPT in which decision tree size is the resource of interest.

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“…The proof details in [33] show that the procedure is still expected to make progress, and with high probability finds all differences after O(n) queries. 8 This algorithm implies that we can compute, with O(n) queries and error probability ε, any Boolean function f : {0, 1} n → {0, 1} on ε-bounded-error inputs: just compute x and output f (x).…”

Section: Theorem 43 (Bnrw)mentioning

confidence: 99%

“…Even more surprisingly, the only way we know how to construct such robust polynomials is via the connection with quantum algorithms. Based on the quantum search algorithm for bounded-error inputs mentioned in Section 2.3, Buhrman et al [33] showed the following:…”

Section: Robust Polynomialsmentioning

confidence: 99%