The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics. The language is sound and complete: a stabilizer ZX-diagram can be transformed into another one if and only if these two diagrams represent the same quantum evolution or quantum state. We show that the stabilizer ZX-calculus can be simplified, removing unnecessary equations while keeping only the ...

Let A be an isotropic, sub-gaussian m × n matrix. We prove that the process Zx := ‖Ax‖ 2 −√m ‖x‖ 2 has sub-gaussian increments, that is, ‖Zx−Zy‖ψ2 ≤ C‖x−y‖2 for any x, y ∈ R. Using this, we show that for any bounded set T ⊆ R, the deviation of ‖Ax‖2 around its mean is uniformly bounded by the Gaussian complexity of T . We also prove a local version of this theorem, which allows for unbounded se...

A regularized algorithm is presented for localized in vivo shimming. The technique uses first- (X,Y,Z), second- (Z(2), ZX, ZY, X(2)-Y(2), XY), and third-order (Z(3)) shim coils, and is robust when applied to arbitrarily-shaped, as well as off-center, regions of interest (ROIs). A single-shot spiral pulse sequence is used for rapid field map acquisition, and a least-squares calculation of the sh...

We consider Volterra Gaussian processes on [0, T ], where T > 0 is a fixed time horizon. These are processes of type Xt = R t 0 zX(t, s)dWs, t ∈ [0, T ], where zX is a square-integrable kernel, and W is a standard Brownian motion. An example is fractional Brownian motion. By using classical techniques from operator theory, we derive measure-preserving transformations of X, and their inherently ...

In this study, the photocatalytic activity of highly dispersed silver (Ag) embedded into ZX-Bi zeolite during methanol (MeOH) photooxidation under visible light and at room temperature was investigated. The size dispersion Agn?+ subnanoclusters (<1 nm) studied by high-resolution transmission electron microscopy (HRTEM), diffuse-reflectance UV–vis, in situ Fourier transform infrared (FTIR), etc....

In this paper, we propose a general scheme to analyze the gradient vanishing phenomenon, also known as barren plateau in training quantum neural networks with ZX-calculus. More precisely, extend plateaus theorem from unitary 2-design circuits any parameterized under certain reasonable assumptions. The main technical contribution of paper is representing integrations ZX-diagrams and computing th...

In this paper we exploit the utility of triangle symbol which has a complicated expression in terms spider diagrams ZX-calculus, and its role within ZX-representation AND-gates particular. First, derive nest identities are key importance to recent developments quantum circuit optimisation T-count reduction Then, using same rule set, prove completeness theorem for Boolean circuits (QBCs) whose r...