Composability of global phase invariant distance and its application to approximation error management

Many quantum algorithms can be written as a composition of unitaries, some of which can be exactly synthesized by a universal fault-tolerant gate set like Clifford+T, while others can be approximately synthesized. A quantum compiler synthesizes each approximately synthesizable unitary up to some approximation error, such that the error of the overall unitary remains bounded by a certain amount. In this paper we consider the case when the errors are measured in the global phase invariant distance. Apart from deriving a relation between this distance and the Frobenius norm, we show that this distance composes. If a unitary is written as a composition (product and tensor product) of other unitaries, we derive bounds on the error of the overall unitary as a function of the errors of the composed unitaries. Our bound is better than the sum-of-error bound, derived by Bernstein- Vazirani(1997), for the operator norm. This builds the intuition that working with the global phase invariant distance might give us a lower resource count while synthesizing quantum circuits. Next we consider the following problem. Suppose we are given a decomposition of a unitary, that is, the unitary is expressed as a composition of other unitaries. We want to distribute the errors in each component such that the resource-count (specifically T-count) is optimized. We consider the specific case when the unitary can be decomposed such that the $R_z(\theta)$ gates are the only approximately synthesizable component. We prove analytically that for both the operator norm and global phase invariant distance, the error should be distributed equally among these components(given some approximations) . The optimal number of T-gates obtained by using the global phase invariant distance is less. Furthermore, for approx-QFT the error due to pruning of rotation gates is less when measured in this distance.

On an invariant distance induced by the Szegő Kernel

In this paper we introduce a new distance by means of the so-called Szegő kernel and examine some basic properties and its relationship with the so-called Skwarczyński distance. We also examine the relationship between this distance, and the so-called Bergman distance and Szegő distance.

A New Hybrid Forecasting Model Based on SW-LSTM and Wavelet Packet Decomposition: A Case Study of Oil Futures Prices

The crude oil futures prices forecasting is a significant research topic for the management of the energy futures market. In order to optimize the accuracy of energy futures prices prediction, a new hybrid model is established in this paper which combines wavelet packet decomposition (WPD) based on long short-term memory network (LSTM) with stochastic time effective weight (SW) function method (WPD-SW-LSTM). In the proposed framework, WPD is a signal processing method employed to decompose the original series into subseries with different frequencies and the SW-LSTM model is constructed based on random theory and the principle of LSTM network. To investigate the prediction performance of the new forecasting approach, SVM, BPNN, LSTM, WPD-BPNN, WPD-LSTM, CEEMDAN-LSTM, VMD-LSTM, and ST-GRU are considered as comparison models. Moreover, a new error measurement method (multiorder multiscale complexity invariant distance, MMCID) is improved to evaluate the forecasting results from different models, and the numerical results demonstrate that the high-accuracy forecast of oil futures prices is realized.

Model independent feature attributions: Shapley values that uncover non-linear dependencies

Shapley values have become increasingly popular in the machine learning literature, thanks to their attractive axiomatisation, flexibility, and uniqueness in satisfying certain notions of ‘fairness’. The flexibility arises from the myriad potential forms of the Shapley value game formulation. Amongst the consequences of this flexibility is that there are now many types of Shapley values being discussed, with such variety being a source of potential misunderstanding. To the best of our knowledge, all existing game formulations in the machine learning and statistics literature fall into a category, which we name the model-dependent category of game formulations. In this work, we consider an alternative and novel formulation which leads to the first instance of what we call model-independent Shapley values. These Shapley values use a measure of non-linear dependence as the characteristic function. The strength of these Shapley values is in their ability to uncover and attribute non-linear dependencies amongst features. We introduce and demonstrate the use of the energy distance correlations, affine-invariant distance correlation, and Hilbert–Schmidt independence criterion as Shapley value characteristic functions. In particular, we demonstrate their potential value for exploratory data analysis and model diagnostics. We conclude with an interesting expository application to a medical survey data set.

Spacetime colour reconnection in Herwig 7

We present a model for generating spacetime coordinates in the Monte Carlo event generator Herwig 7, and perform colour reconnection by minimizing a boost-invariant distance measure of the system. We compare the model to a series of soft physics observables. We find reasonable agreement with the data, suggesting that pp-collider colour reconnection may be able to be applied in larger systems.

Complexity Synchronization of Energy Volatility Monotonous Persistence Duration Dynamics

A new concept named volatility monotonous persistence duration (VMPD) dynamics is introduced into the research of energy markets, in an attempt to describe nonlinear fluctuation behaviors from a new perspective. The VMPD sequence unites the maximum fluctuation difference and the continuous variation length, which is regarded as a novel indicator to evaluate risks and optimize portfolios. Further, two main aspects of statistical and nonlinear empirical research on the energy VMPD sequence are observed: probability distribution and autocorrelation behavior. Moreover, a new nonlinear method named the cross complexity-invariant distance (CID) FuzzyEn (CCF) which is composed of cross-fuzzy entropy and complexity-invariant distance is firstly proposed to study the complexity synchronization properties of returns and VMPD series for seven representative energy items. We also apply the ensemble empirical mode decomposition (EEMD) to resolve returns and VMPD sequence into the intrinsic mode functions, and the degree that they follow the synchronization features of the initial sequence is investigated.