On two subgroups of U(n), useful for quantum computing

As two basic building blocks for any quantum circuit, we consider the 1-qubit PHASOR circuit Phi(theta) and the 1-qubit NEGATOR circuit N(theta). Both are roots of the IDENTITY circuit. Indeed: both (NO) and N(0) equal the 2 x 2 unit matrix. Additionally, the NEGATOR is a root of the classical NOT gate. Quantum circuits (acting on w qubits) consisting of controlled PHASORs are represented by matrices from ZU(2(w)); quantum circuits consisting of controlled NEGATORs are represented by matrices from XU(2(w)). Here, ZU(n) and XU(n) are subgroups of the unitary group U(n): the group XU(n) consists of all n x n unitary matrices with all 2n line sums (i.e. all n row sums and all n column sums) equal to 1 and the group ZU(n) consists of all n x n unitary diagonal matrices with first entry equal to 1. Any U(n) matrix can be decomposed into four parts: U = exp(i alpha) Z(1)XZ(2), where both Z(1) and Z(2) are ZU(n) matrices and X is an XU(n) matrix. We give an algorithm to find the decomposition. For n = 2(w) it leads to a four-block synthesis of an arbitrary quantum computer

Basic parameters governing the behaviour of cement-treated clays

AbstractAlthough extensive research has been conducted on the mechanical behaviour of Portland cement-treated soft clays, there has been less emphasis on the correlation of the observed behaviour with clay mineralogy. In this study, experimental results from the authors have been combined with the data found in the literature to investigate the effect of parameters such as curing time, cement content, moisture content, liquidity index, and mineralogy on the mechanical properties of cement-treated clays. The findings show that undrained shear strength and sensitivity of cemented clays still continue to increase after relatively long curing times; expressions are proposed to predict the strength and sensitivity with time. This parametric study also indicates the relative importance of the activity of the soil, as well as the water–cement ratio, to the mechanical properties of cementitious admixtures. Two new empirical parameters are introduced herein. Based on the results of unconfined compression, undrained triaxial, and oedometer tests on cement-enhanced clays, expressions that use these parameters to predict undrained shear strength, yield stress, and the slope of the compression line are proposed. The observed variations in the mechanical behaviour with respect to mineralogy and the important effect of curing time are explained in terms of the pozzolanic reactions. The possible limitations of applying Abrams׳ law to cement–admixed clays are also discussed