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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. 

Binary Chirps (BCs) are 2^m dimensional complex vectors employed in deterministic compressed sensing and in random/unsourced multiple access in wireless networks. The vectors are obtained by exponentiating codewords from a 2nd order Reed-Muller code defined over Z4, the ring of integers modulo 4. We doubled the size of the BC codebook, without compromising performance in wireless multiple access. 

We consider autocorrelation-based low-complexity decoders for identifying Binary Chirp codewords from noisy signals in N = 2^m dimensions. The underlying algebraic structure enables dimensionality reduction from N complex to m binary dimensions, which can be used to reduce decoding complexity, when decoding is successively performed in the m binary dimensions. Existing low-complexity decoders suffer from poor performance in scenarios with strong noise. This is problematic especially in a vector quantization scenario, where quantization noise power cannot be controlled in the system. We construct two improvements to existing algorithms; a geometrically inspired algorithm based on successive projections, and an algorithm based on adaptive decoding order selection. When combined with a breadth-first list decoder, these algorithms make it possible to approach the performance of exhaustive search with low complexity. 

Encoding the scale information explicitly into the representation learned by a convolutional neural network (CNN) is beneficial for many computer vision tasks especially when dealing with multiscale inputs. We study, in this paper, a scaling-translation-equivariant (ST-equivariant) CNN with joint convolutions across the space and the scaling group, which is shown to be both sufficient and necessary to achieve equivariance for the regular representation of the scaling-translation group ST. To reduce the model complexity and computational burden, we decompose the convolutional filters under two pre-fixed separable bases and truncate the expansion to low-frequency components. A further benefit of the truncated filter expansion is the improved deformation robustness of the equivariant representation, a property which is theoretically analyzed and empirically verified. Numerical experiments demonstrate that the proposed scaling-translation-equivariant network with decomposed convolutional filters (ScDCFNet) achieves significantly improved performance in multiscale image classification and better interpretability than regular CNNs at a reduced model size. 

Encoding the scale information explicitly into the representation learned by a convolutional neural network (CNN) is beneficial for many computer vision tasks especially when dealing with multiscale inputs. We study, in this paper, a scaling-translation-equivariant (ST-equivariant) CNN with joint convolutions across the space and the scaling group, which is shown to be both sufficient and necessary to achieve equivariance for the regular representation of the scaling-translation group ST. To reduce the model complexity and computational burden, we decompose the convolutional filters under two pre-fixed separable bases and truncate the expansion to low-frequency components. A further benefit of the truncated filter expansion is the improved deformation robustness of the equivariant representation, a property which is theoretically analyzed and empirically verified. Numerical experiments demonstrate that the proposed scaling-translation-equivariant network with decomposed convolutional filters (ScDCFNet) achieves significantly improved performance in multiscale image classification and better interpretability than regular CNNs at a reduced model size.