Results 1  10
of
30
Quantum MerlinArthur Proof Systems: Are Multiple Merlins More Helpful to Arthur?
, 2008
"... This paper introduces quantum “multipleMerlin”Arthur proof systems in which Arthur receives multiple quantum proofs that are unentangled with each other. Although classical multiproof systems are obviously equivalent to classical singleproof systems (i.e., usual MerlinArthur proof systems), it ..."
Abstract

Cited by 40 (7 self)
 Add to MetaCart
This paper introduces quantum “multipleMerlin”Arthur proof systems in which Arthur receives multiple quantum proofs that are unentangled with each other. Although classical multiproof systems are obviously equivalent to classical singleproof systems (i.e., usual MerlinArthur proof systems), it is unclear whether or not quantum multiproof systems collapse to quantum singleproof systems (i.e., usual quantum MerlinArthur proof systems). This paper presents a necessary and sufficient condition under which the number of quantum proofs is reducible to two. It is also proved that, in the case of perfect soundness, using multiple quantum proofs
The power of unentanglement
, 2008
"... The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than o ..."
Abstract

Cited by 27 (3 self)
 Add to MetaCart
The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. • We give a protocol by which a verifier can be convinced that a 3Sat formula of size n is satisfiable, with constant soundness, given Õ ( √ n) unentangled quantum witnesses with O (log n) qubits each. Our protocol relies on Dinur’s version of the PCP Theorem and is inherently nonrelativizing. • We show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k) = QMA(2) for all k ≥ 2. • We give evidence that QMA(2) ⊆ PSPACE, by showing that this would follow from “strong amplification ” of QMA(2) protocols. • We prove the nonexistence of “perfect disentanglers” for simulating multiple Merlins with one.
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 21 (3 self)
 Add to MetaCart
(Show Context)
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 CopyProtection and Quantum Money
"... Forty years ago, Wiesner proposed using quantum states to create money that is physically impossible to counterfeit, something that cannot be done in the classical world. However, Wiesner’s scheme required a central bank to verify the money, and the question of whether there can be unclonable quantu ..."
Abstract

Cited by 20 (5 self)
 Add to MetaCart
(Show Context)
Forty years ago, Wiesner proposed using quantum states to create money that is physically impossible to counterfeit, something that cannot be done in the classical world. However, Wiesner’s scheme required a central bank to verify the money, and the question of whether there can be unclonable quantum money that anyone can verify has remained open since. One can also ask a related question, which seems to be new: can quantum states be used as copyprotected programs, which let the user evaluate some function f, but not create more programs for f? This paper tackles both questions using the arsenal of modern computational complexity. Our main result is that there exist quantum oracles relative to which publiclyverifiable quantum money is possible, and any family of functions that cannot be efficiently learned from its inputoutput behavior can be quantumly copyprotected. This provides the first formal evidence that these tasks are achievable. The technical core of our result is a “ComplexityTheoretic NoCloning Theorem,” which generalizes both the standard NoCloning Theorem and the optimality of Grover search, and might be of independent interest. Our security argument also requires explicit constructions of quantum tdesigns. Moving beyond the oracle world, we also present an explicit candidate scheme for publiclyverifiable quantum money, based on random stabilizer states; as well as two explicit schemes for copyprotecting the family of point functions. We do not know how to base the security of these schemes on any existing cryptographic assumption. (Note that without an oracle, we can only hope for security under some computational assumption.)
MerlinArthur games and stoquastic complexity. Arxiv: quantph/0611021
, 2006
"... MA is a class of decision problems for which ‘yes’instances have a proof that can be efficiently checked by a classical randomized algorithm. We prove that MA has a natural complete problem which we call the stoquastic kSAT problem. This is a matrixvalued analogue of the satisfiability problem in ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
(Show Context)
MA is a class of decision problems for which ‘yes’instances have a proof that can be efficiently checked by a classical randomized algorithm. We prove that MA has a natural complete problem which we call the stoquastic kSAT problem. This is a matrixvalued analogue of the satisfiability problem in which clauses are kqubit projectors with nonnegative matrix elements, while a satisfying assignment is a vector that belongs to the space spanned by these projectors. We also study the minimum eigenvalue problem for local stoquastic Hamiltonians that was introduced in Ref. [1], stoquastic LHMIN. A new complexity class StoqMA is introduced so that stoquastic LHMIN is StoqMAcomplete. We show that MA ⊆ StoqMA ⊆ SBP ∩ QMA. Lastly, we consider the average minimum eigenvalue problem for local stoquastic Hamiltonians that depend on a random or ‘quenched disorder ’ parameter, stoquastic AVLHMIN. We prove that stoquastic AVLHMIN is complete for the complexity class AM, the class of decision problems for which yesinstances have a randomized interactive proof with twoway communication between prover and verifier. Stoquastic kSAT and AVLHMIN are the first nontrivial examples of a MAcomplete and an AMcomplete problem respectively. 1
Why philosophers should care about computational complexity
 In Computability: Gödel, Turing, Church, and beyond (eds
, 2012
"... One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed casethat onewouldbe wrong. In particular, I arguethat computational complexity theory—the field that ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
(Show Context)
One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed casethat onewouldbe wrong. In particular, I arguethat computational complexity theory—the field that studies the resources (such as time, space, and randomness) needed to solve computational problems—leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume’s problem of induction, Goodman’s grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing
Quantum Money from Hidden Subspaces
"... Forty years ago, Wiesner pointed out that quantum mechanics raises the striking possibility of money that cannot be counterfeited according to the laws of physics. We propose the first quantum money scheme that is (1) publickey—meaning that anyone can verify a banknote as genuine, not only the bank ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
(Show Context)
Forty years ago, Wiesner pointed out that quantum mechanics raises the striking possibility of money that cannot be counterfeited according to the laws of physics. We propose the first quantum money scheme that is (1) publickey—meaning that anyone can verify a banknote as genuine, not only the bank that printed it, and (2) cryptographically secure, under a “classical ” hardness assumption that has nothing to do with quantum money. Our scheme is based on hidden subspaces, encoded as the zerosets of random multivariate polynomials. A main technical advance is to show that the “blackbox ” version of our scheme, where the polynomials are replaced by classical oracles, is unconditionally secure. Previously, such a result had only been known relative to a quantum oracle (and even there, the proof was never published). Even in Wiesner’s original setting—quantum money that can only be verified by the bank— we are able to use our techniques to patch a major security hole in Wiesner’s scheme. We give the first privatekey quantum money scheme that allows unlimited verifications and that remains unconditionally secure, even if the counterfeiter can interact adaptively with the bank. Our money scheme is simpler than previous publickey quantum money schemes, including a knotbased scheme of Farhi et al. The verifier needs to perform only two tests, one in the standard basis and one in the Hadamard basis—matching the original intuition for quantum money, based on the existence of complementary observables. Our security proofs use a new variant of Ambainis’s quantum adversarymethod, and several other tools that might be of independent interest. 1
On Perfect Completeness for QMA
"... Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with onesided error, has been an open problem for years. This note helps to explain why the problem is difficult, by using ideas from real analysis to give a “quantum oracle ” relative to which QMA = QMA1. As a byproduct, we fin ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with onesided error, has been an open problem for years. This note helps to explain why the problem is difficult, by using ideas from real analysis to give a “quantum oracle ” relative to which QMA = QMA1. As a byproduct, we find that there are facts about quantum complexity classes that are classically relativizing but not quantumly relativizing, among them such “trivial” containments as BQP ⊆ ZQEXP. 1
QMA/qpoly ⊆ PSPACE/poly: DeMerlinizing quantum protocols
 In TwentyFirst Annual IEEE Conference on Computational Complexity
, 2006
"... This paper introduces a new technique for removing existential quantifiers over quantum states. Using this technique, we show that there is no way to pack an exponential number of bits into a polynomialsize quantum state, in such a way that the value of any one of those bits can later be proven wit ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
This paper introduces a new technique for removing existential quantifiers over quantum states. Using this technique, we show that there is no way to pack an exponential number of bits into a polynomialsize quantum state, in such a way that the value of any one of those bits can later be proven with the help of a polynomialsize quantum witness. We also show that any problem in QMA with polynomialsize quantum advice, is also in PSPACE with polynomialsize classical advice. This builds on our earlier result that BQP/qpoly ⊆ PP/poly, and offers an intriguing counterpoint to the recent discovery of Raz that QIP/qpoly = ALL. Finally, we show that QCMA/qpoly ⊆ PP/poly and that QMA/rpoly = QMA/poly. 1