Lower bounds on the complexity of simulating quantum gates

We give a simple proof of a formula for the minimal time required to simulate a two-qubit unitary operation using a fixed two-qubit Hamiltonian together with fast local unitaries. We also note that a related lower bound holds for arbitrary n-qubit gates.

A geometric approach to quantum circuit lower bounds

What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on the manifold SU(2(n)). The geodesic curves on these manifolds have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size. For each Finsler metric we give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call Pauli geodesics, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.

Quantum states far from the energy eigenstates of any local hamiltonian

What quantum states are possible energy eigenstates of a many-body Hamiltonian? Suppose the Hamiltonian is nontrivial, i.e., not a multiple of the identity, and L local, in the sense of containing interaction terms involving at most L bodies, for some fixed L. We construct quantum states psi which are far away from all the eigenstates E of any nontrivial L-local Hamiltonian, in the sense that parallel topsi-Eparallel to is greater than some constant lower bound, independent of the form of the Hamiltonian.

Multipartite entanglement and quantum state exchange

We investigate multipartite entanglement in relation to the process of quantum state exchange. In particular, we consider such entanglement for a certain pure state involving two groups of N trapped atoms. The state, which can be produced via quantum state exchange, is analogous to the steady-state intracavity state of the subthreshold optical nondegenerate parametric amplifier. We show that, first, it possesses some 2N-way entanglement. Second, we place a lower bound on the amount of such entanglement in the state using a measure called the entanglement of minimum bipartite entropy.

On the distributed compression of quantum information

The problem of distributed compression for correlated quantum sources is considered. The classical version of this problem was solved by Slepian and Wolf, who showed that distributed compression could take full advantage of redundancy in the local sources created by the presence of correlations. Here it is shown that, in general, this is not the case for quantum sources, by proving a lower bound on the rate sum for irreducible sources of product states which is stronger than the one given by a naive application of Slepian-Wolf. Nonetheless, strategies taking advantage of correlation do exist for some special classes of quantum sources. For example, Devetak and Winter demonstrated the existence of such a strategy when one of the sources is classical. Optimal nontrivial strategies for a different extreme, sources of Bell states, are presented here. In addition, it is explained how distributed compression is connected to other problems in quantum information theory, including information-disturbance questions, entanglement distillation and quantum error correction.

A generalisation of the Delogne-Kasa method for fitting hyperspheres

In this paper, we examine the problem of fitting a hypersphere to a set of noisy measurements of points on its surface. Our work generalises an estimator of Delogne (Proc. IMEKO-Symp. Microwave Measurements 1972,117-123) which he proposed for circles and which has been shown by Kasa (IEEE Trans. Instrum. Meas. 25, 1976, 8-14) to be convenient for its ease of analysis and computation. We also generalise Chan's 'circular functional relationship' to describe the distribution of points. We derive the Cramer-Rao lower bound (CRLB) under this model and we derive approximations for the mean and variance for fixed sample sizes when the noise variance is small. We perform a statistical analysis of the estimate of the hypersphere's centre. We examine the existence of the mean and variance of the estimator for fixed sample sizes. We find that the mean exists when the number of sample points is greater than M + 1, where M is the dimension of the hypersphere. The variance exists when the number of sample points is greater than M + 2. We find that the bias approaches zero as the noise variance diminishes and that the variance approaches the CRLB. We provide simulation results to support our findings.

Reduced hepatic extraction of palmitate in steatosis correlated to lower level of liver fatty acid binding protein

Nonalcoholic fatty liver disease is the most common of all liver diseases. The hepatic disposition [H-3]palmitate and its low-molecular-weight metabolites in perfused normal and steatotic rat liver were studied using the multiple indicator dilution technique and a physiologically based slow diffusion/bound pharmacokinetic model. The steatotic rat model was established by administration of 17alpha-ethynylestradiol to female Wistar rats. Serum biochemistry markers and histology of treated and normal animals were assessed and indicated the presence of steatosis in the treatment group. The steatotic group showed a significantly higher alanine aminotransferase-to-aspartate aminotransferase ratio, lower levels of liver fatty acid binding protein and cytochrome P-450, as well as microvesicular steatosis with an enlargement of sinusoidal space. Hepatic extraction for unchanged [H-3]palmitate and production of low-molecular-weight metabolites were found to be significantly decreased in steatotic animals. Pharmacokinetic analysis suggested that the reduced extraction and sequestration for palmitate and its metabolites was mainly attributed to a reduction in liver fatty acid binding protein in steatosis.

Identifying the energy gap: Magnitude and determinants of 5-year weight gain in midage women

Objective: The aims of this study were to estimate average yearly weight gain in midage women and to identify the determinants of weight gain and gaining weight at double the average rate. Research Methods and Procedures: The study sample comprised 8071 participants (45 to 55 years old) in the Australian Longitudinal Study on Women's Health who completed mailed surveys in 1996, 1998, and 2001. Results: On average, the women gained almost 0.5 kg per year [average 2.42 kg (95% confidence interval, 2.29 to 2.54) over 5 years]. In multivariate analyses, variables associated with energy balance (physical activity, sitting time, and energy intake), as well as quitting smoking, menopause/hysterectomy, and baseline BMI category were significantly associated with weight gain, but other behavioral and demographic characteristics were not. After adjustment for all of the other biological and behavioral variables, the odds of gaining weight at about twice the average rate (> 5 kg over 5 years) were highest for women who quit smoking (odds ratio = 2.94; 95% confidence interval, 2.17, 3.96). There were also independent relationships between the odds of gaining > 5 kg and lower levels of habitual physical activity, more time spent sitting, energy intake (but only in women with BMI > 25 at baseline), menopause transition, and hysterectomy. Discussion: The average weight gain equates with an energy imbalance of only about 10 kcal or 40 kJ per day, which suggests that small sustained changes in the modifiable behavioral variables could prevent further weight gain.

Understanding city fringe gentrification: The role of a potential investment gap

Discussion of gentrification has become ‘balkanised’ into a series of competing and intensely-held positions. The dichotomies are between economic and cultural explanations, supply-side and demand-side explanations and structural Marxist and liberal humanist views. Despite the long academic and policy interest in gentrification there is still no clear definition of what it is and why it occurs. However, almost all previous analyses see gentrification as an inner-city phenomenon and so deal with it within framework of inner-city theory and causation. This paper approaches the debate from a somewhat different position. It argues that gentrification, seen as the replacement of lower status and income households by higher status and income households, can occur outside the inner city. It uses clear cases of gentrification on the urban fringe of metropolitan Brisbane in South East Queensland, to explore mechanisms and explanations. The key to this ‘gentrification by the sea’ is a ‘potential investment gap’ between current and potential future property values, based on increasing demand for a limited locational resource – but instead of this being inner-city properties it is waterside land in a regional facing rapid population increase. The paper also draws attention to the inadequate recognition of the roles of the state and the media in previous analyses of gentrification.