The accurate evaluation of diagonal unitary operators is often the most resourceintensive element of quantum algorithms such as real-space quantum simulation and Grover search. Efficient circuits have been demonstrated in some cases but generally require ancilla registers, which can dominate the qubit resources. In this paper, we give a simple way to construct efficient circuits for diagonal unitaries without ancillas, using a correspondence between Walsh functions and a basis for diagonal operators. This correspondence reduces the problem of constructing the minimal-depth circuit within a given error tolerance, for an arbitrary diagonal unitaryê ( ) if x in the x basis, to that of finding the minimallength Walsh-series approximation to the function f(x). We apply this approach to the quantum simulation of the classical Eckart barrier problem of quantum chemistry, demonstrating that high-fidelity quantum simulations can be achieved with few qubits and low depth. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. New J. Phys. 16 (2014) 033040 J Welch et al. The second equality shows that the functional form is the same whether x is continuous or discrete, the only difference being the number of bits in the expansion of x. This makes Walsh series useful for representing discretely sampled continuous functions. New J. Phys. 16 (2014) 033040 J Welch et al =Û e if that is diagonal in this basis is given in terms of its eigenvalues asˆ= f k f k k . New J. Phys. 16 (2014) 033040 J Welch et al 0 1 j j New J. Phys. 16 (2014) 033040 J Welch et al 5 New J. Phys. 16 (2014) 033040 J Welch et al 7 7, which means that 7 qubits are necessary to represent the series. New J. Phys. 16 (2014) 033040 J Welch et al

Objectives The objective was to measure the variation in missed diagnosis and costs of care for older acute myocardial infarction (AMI) patients presenting to emergency departments (EDs), and to identify the hospital and ED characteristics associated with this variation. Methods Using 2004–2005 Medicare inpatient and outpatient records, the authors identified a cohort of AMI patients age 65 years and older who presented to the ED for initial care. The primary outcome was missed diagnosis of AMI, i.e. AMI hospital admission within 7 days of an ED discharge for a condition suggestive of cardiac ischemia. Costs were defined as Medicare hospital payments for all services associated with and immediately resulting from the ED evaluation. The effect of ED and hospital characteristics on quality and costs were estimated using multilevel models with hospital random effects. Results There were 371,638 AMI patients age 65 and older included in the study, of whom 4,707 were discharged home from their initial ED visits and subsequently admitted to the hospital. The median unadjusted hospital-level missed diagnosis percentage was 0.52% (interquartile range [IQR] 0 to 3.45%). ED characteristics protective of adverse outcomes include higher ED chest pain acuity (adjusted odds ratio [aOR] 0.23, 99% confidence interval [CI] = 0.19 to 0.27), and American Board of Emergency Medicine certification (aOR 0.60, 99% CI = 0.50 to 0.73). Protective hospital characteristics include larger hospital size (aOR 0.46, 99% CI = 0.37 to 0.57), and academic status (aOR 0.74, 99% CI = 0.58 to 0.94). All of these characteristics were associated with higher costs as well. Conclusions The proportion of missed AMI diagnoses and cost of care for patients age 65 years and older presenting to EDs with AMI varies across hospitals. Hospitals with more board-certified emergency physicians and higher average acuity are associated with significantly higher quality. All hospital characteristics associated with better ED outcomes are associated with higher costs.

We present an algorithm for the approximate decomposition of diagonal operators, focusing specifically on decompositions over the Clifford+T basis, that minimizes the number of phase-rotation gates in the synthesized approximation circuit. The equivalent T-count of the synthesized circuit is bounded by kC_0 log_2 (1/ε)+E(n, k), where k is the number of distinct phases in the diagonal n-qubit unitary, ε is the desired precision, C0 is a quality factor of the implementation method (1 < C_0 < 4), and E(n, k) is the total entanglement cost (in T gates). We determine an optimal decision boundary in (n, k, ε)-space where our decomposition algorithm achieves lower entanglement cost than previous state-of-the-art techniques. Our method outperforms state-of-the-art techniques for a practical range of ε values and diagonal operators and can reduce the number of T gates exponentially in n when k far less than 2^ n .

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