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73
Complexity Measures and Decision Tree Complexity: A Survey
 Theoretical Computer Science
, 2000
"... We discuss several complexity measures for Boolean functions: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial. We survey the relations and biggest gaps known between these measures, and show how they give bounds for the decision tr ..."
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Cited by 122 (15 self)
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We discuss several complexity measures for Boolean functions: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial. We survey the relations and biggest gaps known between these measures, and show how they give bounds for the decision tree complexity of Boolean functions on deterministic, randomized, and quantum computers. 1 Introduction Computational Complexity is the subfield of Theoretical Computer Science that aims to understand "how much" computation is necessary and sufficient to perform certain computational tasks. For example, given a computational problem it tries to establish tight upper and lower bounds on the length of the computation (or on other resources, like space). Unfortunately, for many, practically relevant, computational problems no tight bounds are known. An illustrative example is the well known P versus NP problem: for all NPcomplete problems the current upper and lower bounds lie exponentially ...
Succinct Quantum Proofs for Properties of Finite Groups
 In Proc. IEEE FOCS
, 2000
"... In this paper we consider a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NPtype proof. Specifically, we consider quantum proofs for properties of blackbox groups, which are finite g ..."
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Cited by 64 (3 self)
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In this paper we consider a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NPtype proof. Specifically, we consider quantum proofs for properties of blackbox groups, which are finite groups whose elements are encoded as strings of a given length and whose group operations are performed by a group oracle. We prove that for an arbitrary group oracle there exist succinct (polynomiallength) quantum proofs for the Group NonMembership problem that can be checked with small error in polynomial time on a quantum computer. Classically this is impossibleit is proved that there exists a group oracle relative to which this problem does not have succinct proofs that can be checked classically with bounded error in polynomial time (i.e., the problem is not in MA relative to the group oracle constructed). By considering a certain subproblem of the Group NonMembership problem we obtain a simple proof that there exists an oracle relative to which BQP is not contained in MA. Finally, we show that quantum proofs for nonmembership and classical proofs for various other group properties can be combined to yield succinct quantum proofs for other group properties not having succinct proofs in the classical setting, such as verifying that a number divides the order of a group and verifying that a group is not a simple group.
Limitations of Quantum Advice and OneWay Communication
 Theory of Computing
, 2004
"... Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones. ..."
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Cited by 50 (15 self)
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Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones.
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 ..."
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Cited by 47 (10 self)
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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.
Quantum computing, postselection, and probabilistic polynomialtime
, 2004
"... I study the class of problems efficiently solvable by a quantum computer, given the ability to “postselect” on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic PolynomialTime. Using this result, I show that several simple ..."
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Cited by 39 (12 self)
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I study the class of problems efficiently solvable by a quantum computer, given the ability to “postselect” on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic PolynomialTime. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us solve PPcomplete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold, and Spielman that PP is closed under intersection, as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum computing can yield new and simpler proofs of major results about classical computation.
Quantum algorithms for solvable groups
 In Proceedings of the 33rd ACM Symposium on Theory of Computing
, 2001
"... ABSTRACT In this paper we give a polynomialtime quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, r ..."
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Cited by 38 (1 self)
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ABSTRACT In this paper we give a polynomialtime quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, reduce to computing orders of solvable groups and therefore admit polynomialtime quantum algorithms as well. Our algorithm works in the setting of blackbox groups, wherein none of these problems have polynomialtime classical algorithms. As an important byproduct, our algorithm is able to produce a pure quantum state that is uniform over the elements in any chosen subgroup of a solvable group, which yields a natural way to apply existing quantum algorithms to factor groups of solvable groups. 1.
Determining Acceptance Possibility for a Quantum Computation is Hard for PH
, 1997
"... It is shown that determining whether a quantum computation has a nonzero probability of accepting is at least as hard as the polynomial time hierarchy. This hardness ..."
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Cited by 34 (3 self)
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It is shown that determining whether a quantum computation has a nonzero probability of accepting is at least as hard as the polynomial time hierarchy. This hardness
Quantum ArthurMerlin games
 Computational Complexity
"... Abstract This paper studies quantum ArthurMerlin games, whichare a restricted form of quantum interactive proof system in which the verifier's messages are given by unbiased coinflips. The following results are proved. ffl For onemessage quantum ArthurMerlin games, whichcorrespond to the complex ..."
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Cited by 33 (2 self)
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Abstract This paper studies quantum ArthurMerlin games, whichare a restricted form of quantum interactive proof system in which the verifier's messages are given by unbiased coinflips. The following results are proved. ffl For onemessage quantum ArthurMerlin games, whichcorrespond to the complexity class QMA, completeness and soundness errors can be reduced exponentially without increasing the length of Merlin's message. Previous constructions for reducing error required a polynomial increase in the length of Merlin's message. Applications of this fact include a proof that logarithmic length quantum certificates yield no increase in powerover BQP and a simple proof that QMA ` PP. ffl In the case of three or more messages, quantum ArthurMerlin games are equivalent in power to ordinary quantum interactive proof systems. In fact, for any languagehaving a quantum interactive proof system there exists a threemessage quantum ArthurMerlin game in whichArthur's only message consists of just a single coinflip that achieves perfect completeness and soundness errorexponentially close to 1/2. ffl Any language having a twomessage quantum ArthurMerlin game is contained in BP \Delta PP. This gives somesuggestion that three messages are stronger than two in
Quantum multiprover interactive proof systems with limited prior entanglement
 Journal of Computer and System Sciences
"... This paper gives the first formal treatment of a quantum analogue of multiprover interactive proof systems. In quantum multiprover interactive proof systems there can be two natural situations: one is with prior entanglement among provers, and the other does not allow prior entanglement among prov ..."
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Cited by 32 (5 self)
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This paper gives the first formal treatment of a quantum analogue of multiprover interactive proof systems. In quantum multiprover interactive proof systems there can be two natural situations: one is with prior entanglement among provers, and the other does not allow prior entanglement among provers. This paper focuses on the latter situation and proves that, if provers do not share any prior entanglement each other, the class of languages that have quantum multiprover interactive proof systems is equal to NEXP. It implies that the quantum multiprover interactive proof systems without prior entanglement have no gain to the classical ones. This result can be extended to the following statement of the cases with prior entanglement: if a language L has a quantum multiprover interactive proof system allowing at most polynomially many prior entangled qubits among provers, L is necessarily in NEXP. Another interesting result shown in this paper is that, in the case the prover does not have his private qubits, the class of languages that have singleprover quantum interactive proof systems is also equal to NEXP. Our results are also of importance in the sense of giving exact correspondances between quantum and classical complexity classes, because there have been known only a few results giving such correspondances.