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The challenge of quantum computing is to combine error resilience with universal computation. Diagonal gates such as the transversal T gate play an important role in implementing a universal set of quantum operations. This paper introduces a framework that describes the process of preparing a code state, applying a diagonal physical gate, measuring a code syndrome, and applying a Pauli correction that may depend on the measured syndrome (the average logical channel induced by an arbitrary diagonal gate). It focuses on CSS codes, and describes the interaction of code states and physical gates in terms of generator coefficients determined by the induced logical operator. The interaction of code states and diagonal gates depends very strongly on the signs of Z -stabilizers in the CSS code, and the proposed generator coefficient framework explicitly includes this degree of freedom. The paper derives necessary and sufficient conditions for an arbitrary diagonal gate to preserve the code space of a stabilizer code, and provides an explicit expression of the induced logical operator. When the diagonal gate is a quadratic form diagonal gate (introduced by Rengaswamy et al.), the conditions can be expressed in terms of divisibility of weights in the two classical codes that determine the CSS code. These codes find application in magic state distillation and elsewhere. When all the signs are positive, the paper characterizes all possible CSS codes, invariant under transversal Z -rotation through π / 2 l , that are constructed from classical Reed-Muller codes by deriving the necessary and sufficient constraints on l . The generator coefficient framework extends to arbitrary stabilizer codes but there is nothing to be gained by considering the more general class of non-degenerate stabilizer codes. 

A diagonal physical gate U has 2^n diagonal entries, each indexed by a binary vector v of length n. A CSS codespace C on n qubits is specified by two classical code C1 and C2, where C2 provides the X-stabilizers and the dual of C1 provides the Z-stabilizers. We proved U preserves C if and only if entries indexed by the same coset of C2 in C1 (same X-logical) are identical. 

• To compute protein pKas, a continuum dielectric Poisson-Boltzmann model defined on a molecular domain and a solvent domain is used for computing the related electrostatic free energies (top left). • The PB model in its boundary integral form is accurately solved on the triangulated molecular surface (e.g. BPTI) accelerated by a fast Treecode algorithm (top right). • The method obtains the intrinsic pKa and then computes the protonation probability for a given pH including site-site interactions by going through an energy driven titrating procedure. Comparison with experimental results are provided (bottom left and right).