3,222 research outputs found
Average case quantum lower bounds for computing the boolean mean
We study the average case approximation of the Boolean mean by quantum
algorithms. We prove general query lower bounds for classes of probability
measures on the set of inputs. We pay special attention to two probabilities,
where we show specific query and error lower bounds and the algorithms that
achieve them. We also study the worst expected error and the average expected
error of quantum algorithms and show the respective query lower bounds. Our
results extend the optimality of the algorithm of Brassard et al.Comment: 18 page
Quantum vs Classical Proofs and Subset Verification
We study the ability of efficient quantum verifiers to decide properties of
exponentially large subsets given either a classical or quantum witness. We
develop a general framework that can be used to prove that QCMA machines, with
only classical witnesses, cannot verify certain properties of subsets given
implicitly via an oracle. We use this framework to prove an oracle separation
between QCMA and QMA using an "in-place" permutation oracle, making the first
progress on this question since Aaronson and Kuperberg in 2007. We also use the
framework to prove a particularly simple standard oracle separation between
QCMA and AM.Comment: 23 pages, presentation and notation clarified, small errors fixe
The Value of Help Bits in Randomized and Average-Case Complexity
"Help bits" are some limited trusted information about an instance or
instances of a computational problem that may reduce the computational
complexity of solving that instance or instances. In this paper, we study the
value of help bits in the settings of randomized and average-case complexity.
Amir, Beigel, and Gasarch (1990) show that for constant , if instances
of a decision problem can be efficiently solved using less than bits of
help, then the problem is in P/poly. We extend this result to the setting of
randomized computation: We show that the decision problem is in P/poly if using
help bits, instances of the problem can be efficiently solved with
probability greater than . The same result holds if using less than
help bits (where is the binary entropy function),
we can efficiently solve fraction of the instances correctly with
non-vanishing probability. We also extend these two results to non-constant but
logarithmic . In this case however, instead of showing that the problem is
in P/poly we show that it satisfies "-membership comparability," a notion
known to be related to solving instances using less than bits of help.
Next we consider the setting of average-case complexity: Assume that we can
solve instances of a decision problem using some help bits whose entropy is
less than when the instances are drawn independently from a particular
distribution. Then we can efficiently solve an instance drawn from that
distribution with probability better than .
Finally, we show that in the case where is super-logarithmic, assuming
-membership comparability of a decision problem, one cannot prove that the
problem is in P/poly by a "black-box proof.
Spatial isolation implies zero knowledge even in a quantum world
Zero knowledge plays a central role in cryptography and complexity. The seminal work of Ben-Or et al. (STOC 1988) shows that zero knowledge can be achieved unconditionally for any language in NEXP, as long as one is willing to make a suitable physical assumption: if the provers are spatially isolated, then they can be assumed to be playing independent strategies. Quantum mechanics, however, tells us that this assumption is unrealistic, because spatially-isolated provers could share a quantum entangled state and realize a non-local correlated strategy. The MIP* model captures this setting. In this work we study the following question: does spatial isolation still suffice to unconditionally achieve zero knowledge even in the presence of quantum entanglement? We answer this question in the affirmative: we prove that every language in NEXP has a 2-prover zero knowledge interactive proof that is sound against entangled provers; that is, NEXP ⊆ ZK-MIP*. Our proof consists of constructing a zero knowledge interactive PCP with a strong algebraic structure, and then lifting it to the MIP* model. This lifting relies on a new framework that builds on recent advances in low-degree testing against entangled strategies, and clearly separates classical and quantum tools. Our main technical contribution is the development of new algebraic techniques for obtaining unconditional zero knowledge; this includes a zero knowledge variant of the celebrated sumcheck protocol, a key building block in many probabilistic proof systems. A core component of our sumcheck protocol is a new algebraic commitment scheme, whose analysis relies on algebraic complexity theory
- …