AM with Multiple Merlins

We introduce and study a new model of interactive proofs: AM(k), or Arthur-Merlin with k non-communicating Merlins. Unlike with the better-known MIP, here the assumption is that each Merlin receives an independent random challenge from Arthur. One motivation for this model (which we explore in detail) comes from the close analogies between it and the quantum complexity class QMA(k), but the AM(k) model is also natural in its own right. We illustrate the power of multiple Merlins by giving an AM(2) protocol for 3SAT, in which the Merlins' challenges and responses consist of only n^{1/2+o(1)} bits each. Our protocol has the consequence that, assuming the Exponential Time Hypothesis (ETH), any algorithm for approximating a dense CSP with a polynomial-size alphabet must take n^{(log n)^{1-o(1)}} time. Algorithms nearly matching this lower bound are known, but their running times had never been previously explained. Brandao and Harrow have also recently used our 3SAT protocol to show quasipolynomial hardness for approximating the values of certain entangled games. In the other direction, we give a simple quasipolynomial-time approximation algorithm for free games, and use it to prove that, assuming the ETH, our 3SAT protocol is essentially optimal. More generally, we show that multiple Merlins never provide more than a polynomial advantage over one: that is, AM(k)=AM for all k=poly(n). The key to this result is a subsampling theorem for free games, which follows from powerful results by Alon et al. and Barak et al. on subsampling dense CSPs, and which says that the value of any free game can be closely approximated by the value of a logarithmic-sized random subgame.Comment: 48 page

The Power of Unentanglement

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? 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 m is satisfiable, with constant soundness, given Õ (√m) unentangled quantum witnesses with O(log m) qubits each. Our protocol relies on the existence of very short PCPs. * We show that assuming a weak version of the 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 prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one

Testing product states, quantum Merlin-Arthur games and tensor optimisation

We give a test that can distinguish efficiently between product states of n quantum systems and states which are far from product. If applied to a state psi whose maximum overlap with a product state is 1-epsilon, the test passes with probability 1-Theta(epsilon), regardless of n or the local dimensions of the individual systems. The test uses two copies of psi. We prove correctness of this test as a special case of a more general result regarding stability of maximum output purity of the depolarising channel. A key application of the test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain several structural results that had been previously conjectured, including the fact that efficient soundness amplification is possible and that two Merlins can simulate many Merlins: QMA(k)=QMA(2) for k>=2. Building on a previous result of Aaronson et al, this implies that there is an efficient quantum algorithm to verify 3-SAT with constant soundness, given two unentangled proofs of O(sqrt(n) polylog(n)) qubits. We also show how QMA(2) with log-sized proofs is equivalent to a large number of problems, some related to quantum information (such as testing separability of mixed states) as well as problems without any apparent connection to quantum mechanics (such as computing injective tensor norms of 3-index tensors). As a consequence, we obtain many hardness-of-approximation results, as well as potential algorithmic applications of methods for approximating QMA(2) acceptance probabilities. Finally, our test can also be used to construct an efficient test for determining whether a unitary operator is a tensor product, which is a generalisation of classical linearity testing.Comment: 44 pages, 1 figure, 7 appendices; v6: added references, rearranged sections, added discussion of connections to classical CS. Final version to appear in J of the AC

Advancing Methods of Diet Analysis: A Case Study Using Degraded Merlin (Falco columbarius) Prey Remains

Prey remains have long been used as a mechanism to approach diet analyses. As understanding diet is key to comprehending ecosystem dynamics, prey remains identification requires a unique methodological approach to determine diversity within a sample. With the advancement of technology, molecular protocols designed for species-specific identification have improved to incredible accuracy and precision. Yet, the visual identification method has remained a predominant technique within diet studies. With entry-level observers, we matched visual identifications with molecular-based methods to quantify the accuracy of the visual identification method. This study determined what fraction of visually identified prey remains could be correctly identified to a high degree of certainty. Using the mitochondrial DNA of \u3e 40-year-old Merlin (Falco columbarius) feather samples, we found that the correct identification of visually identified “high” certainty samples was 41.7%. Furthermore, visually identified samples with a “medium to low” certainty plummeted to 19.0%. This study reveals that correct identification of visually identified samples is significantly lower than previously considered but that certainty level has a significant role in correct identification. Similarly, visual identification can provide rapid determination of separate taxa and the number of species in a sample. It is critical to assess prey remains using multiple techniques to procure definitive identification of individual prey items. Anecdotally, I found that the primers AWF2-R4 and AWF4-R6 targeting regions within the cytochrome c oxidase subunit-1 gene are effective for degraded (i.e. \u3e 40 years old) feather samples of Passeriformes and Charadriiformes

Rational proofs

We study a new type of proof system, where an unbounded prover and a polynomial time verifier interact, on inputs a string x and a function f, so that the Verifier may learn f(x). The novelty of our setting is that there no longer are "good" or "malicious" provers, but only rational ones. In essence, the Verifier has a budget c and gives the Prover a reward r ∈ [0,c] determined by the transcript of their interaction; the prover wishes to maximize his expected reward; and his reward is maximized only if he the verifier correctly learns f(x). Rational proof systems are as powerful as their classical counterparts for polynomially many rounds of interaction, but are much more powerful when we only allow a constant number of rounds. Indeed, we prove that if f ∈ #P, then f is computable by a one-round rational Merlin-Arthur game, where, on input x, Merlin's single message actually consists of sending just the value f(x). Further, we prove that CH, the counting hierarchy, coincides with the class of languages computable by a constant-round rational Merlin-Arthur game. Our results rely on a basic and crucial connection between rational proof systems and proper scoring rules, a tool developed to elicit truthful information from experts.United States. Office of Naval Research (Award number N00014-09-1-0597

NP vs QMA_log(2)

Although it is believed unlikely that \NP-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve \NP-complete problems given a "short" quantum proof; more precisely, \NP\subseteq \QMA_{\log}(2) where \QMA_{\log}(2) denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion \NP\subseteq \QMA_{\log}(2) has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap $\frac{1}{24n^6}$. Moreover, Aaronson {\it et al.} have shown the above inclusion with a constant gap by considering $\widetilde{O}(\sqrt{n})$ witnesses of logarithmic size. However, we still do not know if \QMA_{\log}(2) with a constant gap contains \NP. In this paper, we show that 3-SAT admits a \QMA_{\log}(2) protocol with the gap $\frac{1}{n^{3+\epsilon}}$ for every constant $\epsilon>0$.Comment: 10 pages. Thanks to referees, the main result is now stated in terms of 3-SAT instead of NP. Clearer proofs. To appear in Quantum Information and Computatio

The breeding ecology of the Merlin (<i>Falco columbarius aesalon</i>), with particular reference to north-east Scotland and land-use change

The breeding population of the Merlin in Britain in 1993 and 1994 was estimated at 1300 ± 200 pairs following a survey of around 60% of the range and calculated extrapolation of the remaining suitable habitat. This reflected a recovery, following widespread declines in numbers and range earlier in the 20th century. Over 1000 nest records from the survey were analysed, with habitat and nest site described and quantified, and related to clutch size, successful brood size and productivity. Heather moor or mixed grass-heather moor, and tall conifer plantations were the main core habitats at 88% and 9% of territories respectively. Habitat choice influenced nest site, with 77% of nests on the ground, 2% on crags and 19% in trees. Productivity averaged 2.25 fledged young per pair and was indicative of a stable or increasing population. In north-east Scotland, part of a delimited Merlin study area was afforested with conifers, providing an opportunity to monitor the effects of this land-use change on Merlin breeding ecology. One of the forestry schemes led to public outcry, official objections, Government assessment, judicial Review and an appeal to the European Commission. These events were unprecedented in British forestry history and were seen as a test-case. Despite modifications to the scheme, by leaving approximately 30% of land unpianted, the Merlins declined to zero, as they also did at the other afforested areas. Breeding phenology and clutch size at the afforested areas were similar to comparable adjacent and further afield Merlin study areas, where there was minimal change in habitat management. However, productivity was significantly less and it was concluded that commercially afforested moorland was an inappropriate breeding habitat for Merlin in north-east Scotland. Identifying and quantifying prey remains assessed breeding season diet, with small birds accounting for 95% of numbers and 99% of biomass from 11,225 items. It is reasoned that the majority of the new potential prey resource associated with commercial afforestation was unavailable for Merlin, due to the protection provided by dense thicket plantations. Guidelines for retaining breeding Merlin within commercial forestry schemes in Britain are recommended. These could be requested by conservation planners, adopted by foresters or tested by raptor ecologists, and their use could be a condition within new grant-aided forestry schemes in Britain

A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs

A $(k \times l)$-birthday repetition $\mathcal{G}^{k \times l}$ of a two-prover game $\mathcal{G}$ is a game in which the two provers are sent random sets of questions from $\mathcal{G}$ of sizes $k$ and $l$ respectively. These two sets are sampled independently uniformly among all sets of questions of those particular sizes. We prove the following birthday repetition theorem: when $\mathcal{G}$ satisfies some mild conditions, $val(\mathcal{G}^{k \times l})$ decreases exponentially in $\Omega(kl/n)$ where $n$ is the total number of questions. Our result positively resolves an open question posted by Aaronson, Impagliazzo and Moshkovitz (CCC 2014). As an application of our birthday repetition theorem, we obtain new fine-grained hardness of approximation results for dense CSPs. Specifically, we establish a tight trade-off between running time and approximation ratio for dense CSPs by showing conditional lower bounds, integrality gaps and approximation algorithms. In particular, for any sufficiently large $i$ and for every $k \geq 2$, we show the following results: - We exhibit an $O(q^{1/i})$-approximation algorithm for dense Max $k$-CSPs with alphabet size $q$ via $O_k(i)$-level of Sherali-Adams relaxation. - Through our birthday repetition theorem, we obtain an integrality gap of $q^{1/i}$ for $\tilde\Omega_k(i)$-level Lasserre relaxation for fully-dense Max $k$-CSP. - Assuming that there is a constant $\epsilon > 0$ such that Max 3SAT cannot be approximated to within $(1-\epsilon)$ of the optimal in sub-exponential time, our birthday repetition theorem implies that any algorithm that approximates fully-dense Max $k$-CSP to within a $q^{1/i}$ factor takes $(nq)^{\tilde \Omega_k(i)}$ time, almost tightly matching the algorithmic result based on Sherali-Adams relaxation.Comment: 45 page