A Full Characterization of Quantum Advice

We prove the following surprising result: given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly is contained in 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 real-valued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new ones. The main new tool is a so-called majority-certificates lemma, which is closely 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 in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm in S, such that each fi is the unique function in S compatible with O(log|S|) input/output constraints.Comment: We fixed two significant issues: 1. The definition of YQP machines needed to be changed to preserve our results. The revised definition is more natural and has the same intuitive interpretation. 2. We needed properties of Local Hamiltonian reductions going beyond those proved in previous works (whose results we'd misstated). We now prove the needed properties. See p. 6 for more on both point

Modified melting crystal model and Ablowitz-Ladik hierarchy

This paper addresses the issue of integrable structure in a modified melting crystal model of topological string theory on the resolved conifold. The partition function can be expressed as the vacuum expectation value of an operator on the Fock space of 2D complex free fermion fields. The quantum torus algebra of fermion bilinears behind this expression is shown to have an extended set of "shift symmetries". They are used to prove that the partition function (deformed by external potentials) is essentially a tau function of the 2D Toda hierarchy. This special solution of the 2D Toda hierarchy can be characterized by a factorization problem of \ZZ\times\ZZ matrices as well. The associated Lax operators turn out to be quotients of first order difference operators. This implies that the solution of the 2D Toda hierarchy in question is actually a solution of the Ablowitz-Ladik (equivalently, relativistic Toda) hierarchy. As a byproduct, the shift symmetries are shown to be related to matrix-valued quantum dilogarithmic functions.Comment: latex2e, 33 pages, no figure; (v2) accepted for publicatio

Factor Graphs for Quantum Probabilities

A factor-graph representation of quantum-mechanical probabilities (involving any number of measurements) is proposed. Unlike standard statistical models, the proposed representation uses auxiliary variables (state variables) that are not random variables. All joint probability distributions are marginals of some complex-valued function $q$, and it is demonstrated how the basic concepts of quantum mechanics relate to factorizations and marginals of $q$.Comment: To appear in IEEE Transactions on Information Theory, 201

Optimal split of orders across liquidity pools: a stochastic algorithm approach

Evolutions of the trading landscape lead to the capability to exchange the same financial instrument on different venues. Because of liquidity issues, the trading firms split large orders across several trading destinations to optimize their execution. To solve this problem we devised two stochastic recursive learning procedures which adjust the proportions of the order to be sent to the different venues, one based on an optimization principle, the other on some reinforcement ideas. Both procedures are investigated from a theoretical point of view: we prove a.s. convergence of the optimization algorithm under some light ergodic (or "averaging") assumption on the input data process. No Markov property is needed. When the inputs are i.i.d. we show that the convergence rate is ruled by a Central Limit Theorem. Finally, the mutual performances of both algorithms are compared on simulated and real data with respect to an "oracle" strategy devised by an "insider" who knows a priori the executed quantities by every venues

Limitations of semidefinite programs for separable states and entangled games

Semidefinite programs (SDPs) are a framework for exact or approximate optimization that have widespread application in quantum information theory. We introduce a new method for using reductions to construct integrality gaps for SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory, where previously no $\omega(1)$-round integrality gaps were known: the set of separable (i.e. unentangled) states, or equivalently, the $2 \rightarrow 4$ norm of a matrix, and the set of quantum correlations; i.e. conditional probability distributions achievable with local measurements on a shared entangled state. In both cases no-go theorems were previously known based on computational assumptions such as the Exponential Time Hypothesis (ETH) which asserts that 3-SAT requires exponential time to solve. Our unconditional results achieve the same parameters as all of these previous results (for separable states) or as some of the previous results (for quantum correlations). In some cases we can make use of the framework of Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not only the SoS hierarchy. Our hardness result on separable states also yields a dimension lower bound of approximate disentanglers, answering a question of Watrous and Aaronson et al. These results can be viewed as limitations on the monogamy principle, the PPT test, the ability of Tsirelson-type bounds to restrict quantum correlations, as well as the SDP hierarchies of Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.Comment: 47 pages. v2. small changes, fixes and clarifications. published versio

Random Forests and Networks Analysis

D. Wilson~\cite{[Wi]} in the 1990's described a simple and efficient algorithm based on loop-erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph. This algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking, in recent works~\cite{AG1,AG2,ACGM1,ACGM2} we focused on applications of spanning rooted forests on finite graphs. The resulting main conclusions are reviewed in this paper by collecting related theorems, algorithms, heuristics and numerical experiments. A first foundational part on determinantal structures and efficient sampling procedures is followed by four main applications: 1) a random-walk-based notion of well-distributed points in a graph 2) how to describe metastable dynamics in finite settings by means of Markov intertwining dualities 3) coarse graining schemes for networks and associated processes 4) wavelets-like pyramidal algorithms for graph signals.Comment: Survey pape

On the classical r-matrix structure of the rational BC(n) Ruijsenaars-Schneider-van Diejen system

In this paper, we construct a quadratic r-matrix structure for the classical rational BC(n) Ruijsenaars-Schneider-van Diejen system with the maximal number of three independent coupling parameters. As a byproduct, we provide a Lax representation of the dynamics as well.Comment: 36 page