PP is as Hard as the Polynomial-Time Hierarchy

Toda, Seinosuke

doi:10.1137/0220053

Cited by 588 publications

(379 citation statements)

References 27 publications

Supporting

Mentioning

373

Contrasting

Unclassified

Order By: Relevance

“…If this is the case, then SampP = SampBQP would imply P #P = BPP NP , which in turn would imply PH = BPP NP by Toda's Theorem [9]. Or to put it differently: we could rule out a polynomial-time classical algorithm to sample the output distribution of a quantum computer, under the sole assumption that the polynomial hierarchy is infinite.…”

Section: Our Resultsmentioning

confidence: 99%

The Equivalence of Sampling and Searching

Aaronson

2011

Computer Science – Theory and Applications

View full text Add to dashboard Cite

In a sampling problem, we are given an input x ∈ {0, 1} n , and asked to sample approximately from a probability distribution D x over poly (n)-bit strings. In a search problem, we are given an input x ∈ {0, 1} n , and asked to find a member of a nonempty set A x with high probability. (An example is finding a Nash equilibrium.) In this paper, we use tools from Kolmogorov complexity to show that sampling and search problems are "essentially equivalent." More precisely, for any sampling problem S, there exists a search problem R S such that, if C is any "reasonable" complexity class, then R S is in the search version of C if and only if S is in the sampling version. What makes this nontrivial is that the same R S works for every C.As an application, we prove the surprising result that SampP = SampBQP if and only if FBPP = FBQP. In other words, classical computers can efficiently sample the output distribution of every quantum circuit, if and only if they can efficiently solve every search problem that quantum computers can solve.

show abstract

Section: Our Resultsmentioning

confidence: 99%

The Equivalence of Sampling and Searching

Aaronson

2011

Computer Science – Theory and Applications

View full text Add to dashboard Cite

show abstract

“…Clearly NP ⊆ P #P but it was not clear if Σ p 2 ⊆ P #P . However, Toda [71] proved the somewhat surprising result that PH ⊆ P #P . It is not know if this containments is proper.…”

Section: #Pmentioning

confidence: 99%

Classifying Problems into Complexity Classes

Gasarch

2014

Advances in Computers

View full text Add to dashboard Cite

A fundamental problem in computer science is, stated informally: Given a problem, how hard is it?. We measure hardness by looking at the following question: Given a set A whats is the fastest algorithm to determine if "x ∈ A?" We measure the speed of an algorithm by how long it takes to run on inputs of length n, as a function of n. For example, sorting a list of length n can be done in roughly n log n steps.Obtaining a fast algorithm is only half of the problem. Can you prove that there is no better algorithm? This is notoriously difficult; however, we can classify problems into complexity classes where those in the same class are of roughly the same complexity.In this chapter we define many complexity classes and describe natural problems that are in them. Our classes go all the way from regular languages to various shades of undecidable. We then summarize all that is known about these classes.

show abstract

“…This conjecture is relevant because of the continuing interest in lower bounds for the permanent. Computing the permanent is #P-hard [26] and is hard for the entire polynomial-time hierarchy [24]. Schrijver [20] was the first to give an interesting lower bound for the permanent in the form per(Ã) ≥ ∏1≤i≤n 1≤ j≤n (1 − a i, j ), whereÃ is the matrix whose (i, j) th entry is a i, j (1−a i, j ).…”

Section: Permanent Of Doubly Stochastic Matricesmentioning

confidence: 99%

Quest for Negative Dependency Graphs

Lü

Mohr

Székely

2012

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

The Lovász Local Lemma is a well-known probabilistic technique commonly used to prove the existence of rare combinatorial objects. We explore the lopsided (or negative dependency graph) version of the lemma, which, while more general, appears infrequently in literature due to the lack of settings in which the additional generality has thus far been needed. We present a general framework (matchings in hypergraphs) from which many such settings arise naturally. We also prove a seemingly new generalization of Cayley's formula, which helps defining negative dependency graphs for extensions of forests into spanning trees. We formulate open problems regarding partitions and doubly stochastic matrices that are likely amenable to the use of the lopsided local lemma.

show abstract

PP is as Hard as the Polynomial-Time Hierarchy

Cited by 588 publications

References 27 publications

The Equivalence of Sampling and Searching

The Equivalence of Sampling and Searching

Classifying Problems into Complexity Classes

Quest for Negative Dependency Graphs

Contact Info

Product

Resources

About