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

Unbounded-error One-way Classical and Quantum Communication Complexity

This paper studies the gap between quantum one-way communication complexity $Q(f)$ and its classical counterpart $C(f)$, under the {\em unbounded-error} setting, i.e., it is enough that the success probability is strictly greater than 1/2. It is proved that for {\em any} (total or partial) Boolean function $f$, $Q(f)=\lceil C(f)/2 \rceil$, i.e., the former is always exactly one half as large as the latter. The result has an application to obtaining (again an exact) bound for the existence of $(m,n,p)$-QRAC which is the $n$-qubit random access coding that can recover any one of $m$ original bits with success probability $\geq p$. We can prove that $(m,n,>1/2)$-QRAC exists if and only if $m\leq 2^{2n}-1$. Previously, only the construction of QRAC using one qubit, the existence of $(O(n),n,>1/2)$-RAC, and the non-existence of $(2^{2n},n,>1/2)$-QRAC were known.Comment: 9 pages. To appear in Proc. ICALP 200

Observation of quantum interference as a function of Berry's phase in a complex Hadamard optical network

Emerging models of quantum computation driven by multi-photon quantum interference, while not universal, may offer an exponential advantage over classical computers for certain problems. Implementing these circuits via geometric phase gates could mitigate requirements for error correction to achieve fault tolerance while retaining their relative physical simplicity. We report an experiment in which a geometric phase is embedded in an optical network with no closed-loops, enabling quantum interference between two photons as a function of the phase.Comment: Comments welcom

On Hausdorff dimension of the set of closed orbits for a cylindrical transformation

We deal with Besicovitch's problem of existence of discrete orbits for transitive cylindrical transformations $T_\varphi:(x,t)\mapsto(x+\alpha,t+\varphi(x))$ where $Tx=x+\alpha$ is an irrational rotation on the circle \T and \varphi:\T\to\R is continuous, i.e.\ we try to estimate how big can be the set D(\alpha,\varphi):=\{x\in\T:|\varphi^{(n)}(x)|\to+\infty\text{as}|n|\to+\infty\}. We show that for almost every $\alpha$ there exists $\varphi$ such that the Hausdorff dimension of $D(\alpha,\varphi)$ is at least $1/2$. We also provide a Diophantine condition on $\alpha$ that guarantees the existence of $\varphi$ such that the dimension of $D(\alpha,\varphi)$ is positive. Finally, for some multidimensional rotations $T$ on \T^d, $d\geq3$, we construct smooth $\varphi$ so that the Hausdorff dimension of $D(\alpha,\varphi)$ is positive.Comment: 32 pages, 1 figur

Operator renewal theory and mixing rates for dynamical systems with infinite measure

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates $L^n$ of the transfer operator. This was previously an intractable problem. Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points. In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of $\sum_{j=1}^nL^j$) for the class of systems under consideration. In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for $L^n$ and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.Comment: Preprint, August 2010. Revised August 2011. After publication, a minor error was pointed out by Kautzsch et al, arXiv:1404.5857. The updated version includes minor corrections in Sections 10 and 11, and corresponding modifications of certain statements in Section 1. All main results are unaffected. In particular, Sections 2-9 are unchanged from the published versio

Quantum Commuting Circuits and Complexity of Ising Partition Functions

Instantaneous quantum polynomial-time (IQP) computation is a class of quantum computation consisting only of commuting two-qubit gates and is not universal in the sense of standard quantum computation. Nevertheless, it has been shown that if there is a classical algorithm that can simulate IQP efficiently, the polynomial hierarchy (PH) collapses at the third level, which is highly implausible. However, the origin of the classical intractability is still less understood. Here we establish a relationship between IQP and computational complexity of the partition functions of Ising models. We apply the established relationship in two opposite directions. One direction is to find subclasses of IQP that are classically efficiently simulatable in the strong sense, by using exact solvability of certain types of Ising models. Another direction is applying quantum computational complexity of IQP to investigate (im)possibility of efficient classical approximations of Ising models with imaginary coupling constants. Specifically, we show that there is no fully polynomial randomized approximation scheme (FPRAS) for Ising models with almost all imaginary coupling constants even on a planar graph of a bounded degree, unless the PH collapses at the third level. Furthermore, we also show a multiplicative approximation of such a class of Ising partition functions is at least as hard as a multiplicative approximation for the output distribution of an arbitrary quantum circuit.Comment: 36 pages, 5 figure

Quantum Algorithm for Molecular Properties and Geometry Optimization

It is known that quantum computers, if available, would allow an exponential decrease in the computational cost of quantum simulations. We extend this result to show that the computation of molecular properties (energy derivatives) could also be sped up using quantum computers. We provide a quantum algorithm for the numerical evaluation of molecular properties, whose time cost is a constant multiple of the time needed to compute the molecular energy, regardless of the size of the system. Molecular properties computed with the proposed approach could also be used for the optimization of molecular geometries or other properties. For that purpose, we discuss the benefits of quantum techniques for Newton's method and Householder methods. Finally, global minima for the proposed optimizations can be found using the quantum basin hopper algorithm, which offers an additional quadratic reduction in cost over classical multi-start techniques.Comment: 6 page

Quantum walk approach to search on fractal structures

We study continuous-time quantum walks mimicking the quantum search based on Grover's procedure. This allows us to consider structures, that is, databases, with arbitrary topological arrangements of their entries. We show that the topological structure of the database plays a crucial role by analyzing, both analytically and numerically, the transition from the ground to the first excited state of the Hamiltonian associated with different (fractal) structures. Additionally, we use the probability of successfully finding a specific target as another indicator of the importance of the topological structure.Comment: 15 pages, 14 figure

Humanism in the Age of COVID-19: Renewing Focus on Communication and Compassion

The global COVID-19 pandemic has become one of the largest clinical and operational challenges faced by emergency medicine, and our EDs continue to see increased volumes of infected patients, many of whom are not only ill, but acutely aware and fearful of their circumstances and potential mortality. Given this, there may be no more important time to focus on staff-patient communication and expression of compassion.However, many of the techniques usually employed by emergency clinicians to provide comfort to patients and their families are made more challenging or impossible by the current circumstances. Geriatric ED patients, who are at increased risk of severe disease, are particularly vulnerable to the effects of isolation.Despite many challenges, emergency clinicians have at their disposal a myriad of tools that can still be used to express compassion and empathy to their patients. Placing emphasis on using these techniques to maximize humanism in the care of COVID-19 patients during this crisis has the potential to bring improvements to ED patient care well after this pandemic has passed

Quantum Weakly Nondeterministic Communication Complexity

We study the weakest model of quantum nondeterminism in which a classical proof has to be checked with probability one by a quantum protocol. We show the first separation between classical nondeterministic communication complexity and this model of quantum nondeterministic communication complexity for a total function. This separation is quadratic.Comment: 12 pages. v3: minor correction