Model averaging: A shrinkage perspective

Model averaging (MA), a technique for combining estimators from a set of candidate models, has attracted increasing attention in machine learning and statistics. In the existing literature, there is an implicit understanding that MA can be viewed as a form of shrinkage estimation that draws the response vector towards the subspaces spanned by the candidate models. This paper explores this perspective by establishing connections between MA and shrinkage in a linear regression setting with multiple nested models. We first demonstrate that the optimal MA estimator is the best linear estimator with monotone non-increasing weights in a Gaussian sequence model. The Mallows MA, which estimates weights by minimizing the Mallows' $C_p$, is a variation of the positive-part Stein estimator. Motivated by these connections, we develop a novel MA procedure based on a blockwise Stein estimation. Our resulting Stein-type MA estimator is asymptotically optimal across a broad parameter space when the variance is known. Numerical results support our theoretical findings. The connections established in this paper may open up new avenues for investigating MA from different perspectives. A discussion on some topics for future research concludes the paper

Iterative quantum state transfer along a chain of nuclear spin qubits

Transferring quantum information between two qubits is a basic requirement for many applications in quantum communication and quantum information processing. In the iterative quantum state transfer (IQST) proposed by D. Burgarth et al. [Phys. Rev. A 75, 062327 (2007)], this is achieved by a static spin chain and a sequence of gate operations applied only to the receiving end of the chain. The only requirement on the spin chain is that it transfers a finite part of the input amplitude to the end of the chain, where the gate operations accumulate the information. For an appropriate sequence of evolutions and gate operations, the fidelity of the transfer can asymptotically approach unity. We demonstrate the principle of operation of this transfer scheme by implementing it in a nuclear magnetic resonance quantum information processor.Comment: Version for submission. Comments are welcom

Effect of system level structure and spectral distribution of the environment on the decoherence rate

Minimizing the effect of decoherence on a quantum register must be a central part of any strategy to realize scalable quantum information processing. Apart from the strength of the coupling to the environment, the decoherence rate is determined by the the system level structure and by the spectral composition of the noise trace that the environment generates. Here, we discuss a relatively simple model that allows us to study these different effects quantitatively in detail. We evaluate the effect that the perturbation has on an NMR system while it performs a Grover search algorithm.Comment: Generalizations are added. Comments are welcom

On optimality of Mallows model averaging

In the past decades, model averaging (MA) has attracted much attention as it has emerged as an alternative tool to the model selection (MS) statistical approach. Hansen [\emph{Econometrica} \textbf{75} (2007) 1175--1189] introduced a Mallows model averaging (MMA) method with model weights selected by minimizing a Mallows' $C_p$ criterion. The main theoretical justification for MMA is an asymptotic optimality (AOP), which states that the risk/loss of the resulting MA estimator is asymptotically equivalent to that of the best but infeasible averaged model. MMA's AOP is proved in the literature by either constraining weights in a special discrete weight set or limiting the number of candidate models. In this work, it is first shown that under these restrictions, however, the optimal risk of MA becomes an unreachable target, and MMA may converge more slowly than MS. In this background, a foundational issue that has not been addressed is: When a suitably large set of candidate models is considered, and the model weights are not harmfully constrained, can the MMA estimator perform asymptotically as well as the optimal convex combination of the candidate models? We answer this question in a nested model setting commonly adopted in the area of MA. We provide finite sample inequalities for the risk of MMA and show that without unnatural restrictions on the candidate models, MMA's AOP holds in a general continuous weight set under certain mild conditions. Several specific methods for constructing the candidate model sets are proposed. Implications on minimax adaptivity are given as well. The results from simulations back up our theoretical findings

Quantitative complementarity between local and nonlocal character of quantum states in a three-qubit system

Local or nonlocal character of quantum states can be quantified and is subject to various bounds that can be formulated as complementarity relations. Here, we investigate the local vs. nonlocal character of pure three-qubit states by a four-way interferometer. The complete entanglement in the system can be measured as the entanglement of a specific qubit with the subsystem consisting of the other two qubits. The quantitative complementarity relations are verified experimentally in an NMR quantum information processor.Comment: 10 pages, 10 figure

Speedup of quantum state transfer by three- qubit interactions: Implementation by nuclear magnetic resonance

Universal quantum information processing requires single-qubit rotations and two-qubit interactions as minimal resources. A possible step beyond this minimal scheme is the use of three-qubit interactions. We consider such three-qubit interactions and show how they can reduce the time required for a quantum state transfer in an XY spin chain. For the experimental implementation, we use liquid-state nuclear magnetic resonance (NMR), where three-qubit interactions can be implemented by sequences of radio-frequency pulses.Comment: Comments are welcome to [email protected] or [email protected]. More experimental results are adde