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

Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem

We study the approximation of the smallest eigenvalue of a Sturm-Liouville problem in the classical and quantum settings. We consider a univariate Sturm-Liouville eigenvalue problem with a nonnegative function $q$ from the class $C^2([0,1])$ and study the minimal number n(\e) of function evaluations or queries that are necessary to compute an \e-approximation of the smallest eigenvalue. We prove that n(\e)=\Theta(\e^{-1/2}) in the (deterministic) worst case setting, and n(\e)=\Theta(\e^{-2/5}) in the randomized setting. The quantum setting offers a polynomial speedup with {\it bit} queries and an exponential speedup with {\it power} queries. Bit queries are similar to the oracle calls used in Grover's algorithm appropriately extended to real valued functions. Power queries are used for a number of problems including phase estimation. They are obtained by considering the propagator of the discretized system at a number of different time moments. They allow us to use powers of the unitary matrix $\exp(\tfrac12 {\rm i}M)$, where $M$ is an $n\times n$ matrix obtained from the standard discretization of the Sturm-Liouville differential operator. The quantum implementation of power queries by a number of elementary quantum gates that is polylog in $n$ is an open issue.Comment: 33 page

Average Case Tractability of Non-homogeneous Tensor Product Problems

We study d-variate approximation problems in the average case setting with respect to a zero-mean Gaussian measure. Our interest is focused on measures having a structure of non-homogeneous linear tensor product, where covariance kernel is a product of univariate kernels. We consider the normalized average error of algorithms that use finitely many evaluations of arbitrary linear functionals. The information complexity is defined as the minimal number n(h,d) of such evaluations for error in the d-variate case to be at most h. The growth of n(h,d) as a function of h^{-1} and d depends on the eigenvalues of the covariance operator and determines whether a problem is tractable or not. Four types of tractability are studied and for each of them we find the necessary and sufficient conditions in terms of the eigenvalues of univariate kernels. We illustrate our results by considering approximation problems related to the product of Korobov kernels characterized by a weights g_k and smoothnesses r_k. We assume that weights are non-increasing and smoothness parameters are non-decreasing. Furthermore they may be related, for instance g_k=g(r_k) for some non-increasing function g. In particular, we show that approximation problem is strongly polynomially tractable, i.e., n(h,d)\le C h^{-p} for all d and 0<h<1, where C and p are independent of h and d, iff liminf |ln g_k|/ln k >1. For other types of tractability we also show necessary and sufficient conditions in terms of the sequences g_k and r_k

Qubit Complexity of Continuous Problems

The number of qubits used by a quantum algorithm will be a crucial computational resource for the foreseeable future. We show how to obtain the classical query complexity for continuous problems. We then establish a simple formula for a lower bound on the qubit complexity in terms of the classical query complexityComment: 6 pages, 2 figure

Evaluation of advance statements in psychiatric care.

Background: An advance statement in psychiatric care is a statement of a person's preferences for treatment, should he or she lose capacity to make treatment decisions in the future. The underlying principle for implementing these instruments is the promotion of patients' self-determination and autonomy.;Objective: To evaluate whether use of advance statements by patients with severe mental illness leads to lower rates of compulsory readmission to hospital.;Design: Randomised controlled trial. Setting Two inner city psychiatric hospitals in North London.;Participants: One hundred and fifty six in-patients about to be discharged from compulsory treatment under the Mental Health Act were recruited. To be included, participants had to be 18 years old and over, with mental capacity, able to read and write English and on section 2, 3 or 4 of the Mental Health Act.;Intervention: The preference for care group and the control group both received standard psychiatric care plus a number of standardised questionnaires at baseline and a year after discharge from section. In addition to that the preference for care group received the psychiatric advance statement at baseline.;Outcome measures: The main outcome measure was the rate of compulsory re-admission. Other outcome measures involved: the patients' self-efficacy and satisfaction with psychiatric services, their mental health status assessment, their views about the usefulness of the advance statements, assessment of the content of the statement and the views of mental health professionals in relation to the usefulness of the statement.;Results: Fifteen patients (19%) in the intervention group and 16 (21%) in the control group were readmitted compulsorily within 1 year of discharge. There was no difference in the numbers of compulsory readmissions, numbers of patients readmitted voluntarily, self-efficacy or satisfaction with psychiatric services. Patients with severe and enduring mental health problems were capable of drawing up advance statements with their views in relation to signs of lapses and relapses, and their preferences and refusals on certain aspects of their treatment and needs whilst hospitalised. Patients did not use the advance statements as an opportunity to refuse all subsequent treatment. Although 40% of patients did not find the advance statements useful, this may have occurred because the professionals involved in their care did not refer to or take account of them. Most mental health professionals who returned questionnaires did not find the advance statements useful in the management of the patients.;Conclusion: Users' advance statements for psychiatric care had little observable impact on the outcome of care at twelve months. Even if rates of compulsory treatment were not affected, one cannot rule out possible beneficial effects such as improvement of therapeutic alliance and communication with mental health professionals. Thus, the impact of advance statements on other aspects of care requires further study