Results 1  10
of
10
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. ..."
Abstract

Cited by 59 (15 self)
 Add to MetaCart
(Show Context)
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.
The learnability of quantum states
 quantph/0608142
, 2006
"... Traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n. But using ideas from computational learning theory, we show that “for most practical purposes ” one can learn a state using a number of measurements that grows only linearly w ..."
Abstract

Cited by 22 (2 self)
 Add to MetaCart
Traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n. But using ideas from computational learning theory, we show that “for most practical purposes ” one can learn a state using a number of measurements that grows only linearly with n. Besides possible implications for experimental physics, our learning theorem has two applications to quantum computing: first, a new simulation of quantum oneway communication protocols, and second, the use of trusted classical advice to verify untrusted quantum advice. 1
Quantum information and the PCP theorem
 In FOCS
, 2005
"... We show how to encode 2n (classical) bits a1,...,a2 n by a single quantum state Ψ 〉 of size O(n) qubits, such that: for any constant k and any i1,...,ik ∈ {1,...,2 n}, the values of the bits ai1,...,aik can be retrieved from Ψ 〉 by a oneround ArthurMerlin interactive protocol of size polynomial ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
We show how to encode 2n (classical) bits a1,...,a2 n by a single quantum state Ψ 〉 of size O(n) qubits, such that: for any constant k and any i1,...,ik ∈ {1,...,2 n}, the values of the bits ai1,...,aik can be retrieved from Ψ 〉 by a oneround ArthurMerlin interactive protocol of size polynomial in n. This shows how to go around HolevoNayak’s Theorem, using ArthurMerlin proofs. We use the new representation to prove the following results: 1. Interactive proofs with quantum advice: We show that the class QIP/qpoly contains all languages. That is, for any language L (even nonrecursive), the membership x ∈ L (for x of length n) can be proved by a polynomialsize quantum interactive proof, where the verifier is a polynomialsize quantum circuit with working space initiated with some quantum state ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. 2. PCP with only one query: We show that the membership x ∈ SAT (for x of length n) can be proved by a logarithmicsize quantum state Ψ〉, together with a polynomialsize classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state Ψ 〉 the verifier only needs to read one block of the classical proof. While the first result is a straight forward consequence of the new representation, the second requires an additional machinery of quantum lowdegreetest that may be interesting in its own right.
An application of quantum finite automata to interactive proof systems
 in Proc. 9th International Conference on Implementation and Application of Automata, LNCS, Vol.3317
, 2004
"... Abstract: Quantum finite automata have been studied intensively since their introduction in late 1990s as a natural model of a quantum computer with finitedimensional quantum memory space. This paper seeks their direct application to interactive proof systems in which a mighty quantum prover commun ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
(Show Context)
Abstract: Quantum finite automata have been studied intensively since their introduction in late 1990s as a natural model of a quantum computer with finitedimensional quantum memory space. This paper seeks their direct application to interactive proof systems in which a mighty quantum prover communicates with a quantumautomaton verifier through a common communication cell. Our quantum interactive proof systems are juxtaposed to DworkStockmeyer’s classical interactive proof systems whose verifiers are twoway probabilistic automata. We demonstrate strengths and weaknesses of our systems and further study how various restrictions on the behaviors of quantumautomaton verifiers affect the power of quantum interactive proof systems.
Oracles are subtle but not malicious
 In Proc. IEEE Conference on Computational Complexity
, 2006
"... Theoretical computer scientists have been debating the role of oracles since the 1970’s. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from trying to solve those problems. First, we give an ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
(Show Context)
Theoretical computer scientists have been debating the role of oracles since the 1970’s. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from trying to solve those problems. First, we give an oracle relative to which PP has linearsized circuits, by proving a new lower bound for perceptrons and lowdegree threshold polynomials. This oracle settles a longstanding open question, and generalizes earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More importantly, it implies the first nonrelativizing separation of “traditional ” complexity classes, as opposed to interactive proof classes such as MIP and MAEXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does not have circuits of size n k for any fixed k. We present an alternative proof of this fact, which shows that PP does not even have quantum circuits of size n k with quantum advice. To our knowledge, this is the first nontrivial lower bound on quantum circuit size. Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean circuits in ZPP NP. We show that the NP queries in this algorithm cannot be parallelized by any relativizing technique, by giving an oracle relative to which ZPP NP   and even BPP NP   have linearsize circuits. On the other hand, we also show that the NP queries could be parallelized if P = NP. Thus, classes such as ZPP NP inhabit a “twilight zone, ” where we need to distinguish between relativizing and blackbox techniques. Our results on this subject have implications for computational learning theory as well as for the circuit minimization problem. 1
A Full Characterization of Quantum Advice
"... We prove the following surprising result: given any quantum state ρ on n qubits, there exists a local Hamiltonian H on poly (n) qubits (e.g., a sum of twoqubit interactions), such that any ground state of H can be used to simulate ρ on all quantum circuits of fixed polynomial size. In terms of comp ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
We prove the following surprising result: given any quantum state ρ on n qubits, there exists a local Hamiltonian H on poly (n) qubits (e.g., a sum of twoqubit interactions), such that any ground state of H can be used to simulate ρ on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly ⊆ QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly ⊆ PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice. Proving our main result requires combining a large number of previous tools—including a result of Alon et al. on learning of realvalued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on ‘QMA+ superverifiers’—and also creating some new ones. The main new tool is a socalled majoritycertificates lemma, which is related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set S of Boolean functions on n variables, any function f ∈ S can be expressed as the pointwise majority of m = O (n) functions f1,..., fm ∈ S, such that each fi is the unique function in S compatible with O (log S) input/output constraints.
Security notions for quantum publickey cryptography
 IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences (Japanese Edition), J90A(5):367–375
, 2007
"... ..."
(Show Context)
Language Recognition by Nonconstructive Finite Automata
"... Abstract. Nonconstructive finite automata were first considered by R. Freivalds in [1]. We prove tight upper bound for amount of nonconstructivity that can be needed to recognize a language. We prove some theorems about saving amount of the nonconstructive help needed by encoding that information i ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Nonconstructive finite automata were first considered by R. Freivalds in [1]. We prove tight upper bound for amount of nonconstructivity that can be needed to recognize a language. We prove some theorems about saving amount of the nonconstructive help needed by encoding that information in automata. We also show that nonconstructive probabilistic automata can be more concise than nonconstructive deterministic automata. 1