Results 1  10
of
26
Complexity Limitations on Quantum Computation
 Journal of Computer and System Sciences
, 1997
"... We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP.  BQP is low for PP, i.e., PP BQP = PP.  There exists a relativized world where P = ..."
Abstract

Cited by 95 (3 self)
 Add to MetaCart
We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP.  BQP is low for PP, i.e., PP BQP = PP.  There exists a relativized world where P = BQP and the polynomialtime hierarchy is infinite.  There exists a relativized world where BQP does not have complete sets.  There exists a relativized world where P = BQP but P 6= UP " coUP and oneway functions exist. This gives a relativized answer to an open question of Simon.
An oracle builder’s toolkit
, 2002
"... We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and ..."
Abstract

Cited by 47 (11 self)
 Add to MetaCart
(Show Context)
We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SPgenerics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SPgenerics, ULIN ∩ coULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ coNP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩coNP/1 ̸ ⊆ (NP∩coNP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.
The Role of Relativization in Complexity Theory
 Bulletin of the European Association for Theoretical Computer Science
, 1994
"... Several recent nonrelativizing results in the area of interactive proofs have caused many people to review the importance of relativization. In this paper we take a look at how complexity theorists use and misuse oracle results. We pay special attention to the new interactive proof systems and progr ..."
Abstract

Cited by 40 (9 self)
 Add to MetaCart
Several recent nonrelativizing results in the area of interactive proofs have caused many people to review the importance of relativization. In this paper we take a look at how complexity theorists use and misuse oracle results. We pay special attention to the new interactive proof systems and program checking results and try to understand why they do not relativize. We give some new results that may help us to understand these questions better.
Inverting Onto Functions
, 1996
"... We look at the hypothesis that all honest onto polynomialtime computable functions have a polynomialtime computable inverse. We show this hypothesis equivalent to several other complexity conjectures including ffl In polynomial time, one can find accepting paths of nondeterministic polynomialtim ..."
Abstract

Cited by 34 (6 self)
 Add to MetaCart
(Show Context)
We look at the hypothesis that all honest onto polynomialtime computable functions have a polynomialtime computable inverse. We show this hypothesis equivalent to several other complexity conjectures including ffl In polynomial time, one can find accepting paths of nondeterministic polynomialtime Turing machines that accept \Sigma . ffl Every total multivalued nondeterministic function has a polynomialtime computable refinement. ffl In polynomial time, one can compute satisfying assignments for any polynomialtime computable set of satisfiable formulae. ffl In polynomial time, one can convert the accepting computations of any nondeterministic Turing machine that accepts SAT to satisfying assignments. We compare these hypotheses with several other important complexity statements. We also examine the complexity of these statements where we only require a single bit instead of the entire inverse. 1 Introduction Understanding the power of nondeterminism has been one of the pri...
Two queries
 In CCC
, 1999
"... We consider the question whether two queries to SAT are as powerful as one query. We show that if P NP�℄� P NP�℄then Locally either NP�coNP or NP has polynomialsize circuits. ..."
Abstract

Cited by 31 (6 self)
 Add to MetaCart
(Show Context)
We consider the question whether two queries to SAT are as powerful as one query. We show that if P NP�℄� P NP�℄then Locally either NP�coNP or NP has polynomialsize circuits.
NP Might Not Be As Easy As Detecting Unique Solutions
, 1997
"... We construct an oracle A such that P A = \PhiP A and NP A = EXP A : This relativized world has several amazing properties: ffl The oracle A gives the first relativized world where one can solve satisfiability on formulae with at most one assignment yet P 6= NP. ffl The oracle A is the fi ..."
Abstract

Cited by 22 (6 self)
 Add to MetaCart
We construct an oracle A such that P A = \PhiP A and NP A = EXP A : This relativized world has several amazing properties: ffl The oracle A gives the first relativized world where one can solve satisfiability on formulae with at most one assignment yet P 6= NP. ffl The oracle A is the first where P A = UP A 6= NP A = coNP A : ffl The construction gives a much simpler proof than Fenner, Fortnow and Kurtz of a relativized world where all NPcomplete sets are polynomialtime isomorphic. It is the first such computable oracle. ffl Relative to A we have a collapse of \PhiEXP A ` ZPP A ` P A /poly. We also create a different relativized world where there exists a set L in NP that is NP complete under reductions that make one query to L but not under traditional manyone reductions. This contrasts with the result of Buhrman, Spaan and Torenvliet showing that these two completeness notions for NEXP coincide.
Separation of NPcompleteness notions
 SIAM JOURNAL ON COMPUTING
, 2001
"... We use hypotheses of structural complexity theory to separate various NPcompleteness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is ¡ P Tcomplete but not ¡ P ttcomplete. We provide fairly thorough analyses of the hypotheses that we introduce. ..."
Abstract

Cited by 18 (11 self)
 Add to MetaCart
We use hypotheses of structural complexity theory to separate various NPcompleteness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is ¡ P Tcomplete but not ¡ P ttcomplete. We provide fairly thorough analyses of the hypotheses that we introduce.
Easy sets and hard certificate schemes
 Acta Informatica
, 1997
"... Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for al ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for all NP certificate schemes, infinitely often have easy acceptance certificates. In particular, we provide equivalent characterizations of these classes in terms of relative generalized Kolmogorov complexity, showing that they are robust. We also provide structural conditions—regarding immunity and class collapses—that put upper and lower bounds on the sizes of these two classes. Finally, we provide negative results showing that some of our positive claims are optimal with regard to being relativizable. Our negative results are proven using a novel observation: we show that the classical “wide spacing ” oracle construction technique yields instant nonbiimmunity results. Furthermore, we establish a result that improves upon Baker, Gill, and Solovay’s classical result that NP = P = NP ∩ coNP holds in some relativized world.
A General Method to Construct Oracles Realizing Given Relationships between Complexity Classes
, 1994
"... We present a method to prove oracle theorems of the following type. ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
We present a method to prove oracle theorems of the following type.
The Isomorphism Conjecture Holds and Oneway Functions Exist Relative to an Oracle
 Journal of Computer and System Sciences
, 1994
"... In this paper we demonstrate an oracle relative to which there are oneway functions but every paddable 1lidegree collapses to an isomorphism type, thus yielding a relativized failure of the JosephYoung Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which t ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
(Show Context)
In this paper we demonstrate an oracle relative to which there are oneway functions but every paddable 1lidegree collapses to an isomorphism type, thus yielding a relativized failure of the JosephYoung Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which the Isomorphism Conjecture (IC) is true but oneway functions exist, which answers an open question of Fenner, Fortnow, and Kurtz [FFK92]. Thus, there are now relativizations realizing every one of the four possible states of affairs between the IC and the existence of oneway functions. 1 Introduction Berman and Hartmanis [BH76, BH77] showed that if two languages A and B are equivalent to one another under polynomialtime manytoone reductions and if they are both paddable then they are polynomialtime isomorphic. After surveying all of the thenknown NPcomplete languages and discovering that each was indeed paddable, they posed: The Isomorphism Conjecture (IC) Every NPcomplete lan...