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Magic state distillation and the Shor factoring algorithm make essential use of logical diagonal gates. We introduce a method of synthesizing CSS codes that realize a target logical diagonal gate at some level l in the Clifford hierarchy. The method combines three basic operations: concatenation, removal of Z-stabilizers, and addition of X-stabilizers. It explicitly tracks the logical gate induced by a diagonal physical gate that preserves a CSS code. The first step is concatenation, where the input is a CSS code and a physical diagonal gate at level l inducing a logical diagonal gate at the same level. The output is a new code for which a physical diagonal gate at level l + 1 induces the original logical gate. The next step is judicious removal of Z-stabilizers to increase the level of the induced logical operator. We identify three ways of climbing the logical Clifford hierarchy from level l to level l + 1 , each built on a recursive relation on the Pauli coefficients of the induced logical operators. Removal of Z-stabilizers may reduce distance, and the purpose of the third basic operation, addition of X-stabilizers, is to compensate for such losses. The approach to logical gate synthesis taken in prior work focuses on the code states, resulting in sufficient conditions for a CSS code to be fixed by a transversal Z-rotation. In contrast, we derive necessary and sufficient conditions by analyzing the action of a transversal diagonal gate on the stabilizer group that determines the code. The power of our approach to logical gate synthesis is demonstrated by two proofs of concept: the [ [ 2 l + 1 − 2 , 2 , 2 ] ] triorthogonal code family and the [ [ 2 m , m r , 2 min { r , m − r } ] ] quantum Reed–Muller code family. 

Discrete radar based on modulo arithmetic 

Spatially-coupled (SC) codes is a class of convolutional LDPC codes that has been well investigated in classical coding theory thanks to their high performance and compatibility with low-latency decoders. We describe toric codes as quantum counterparts of classical two-dimensional spatially-coupled (2D-SC) codes, and introduce spatially-coupled quantum LDPC (SC-QLDPC) codes as a generalization. We use the convolutional structure to represent the parity check matrix of a 2D-SC code as a polynomial in two indeterminates, and derive an algebraic condition that is both necessary and sufficient for a 2D-SC code to be a stabilizer code. This algebraic framework facilitates the construction of new code families. While not the focus of this paper, we note that small memory facilitates physical connectivity of qubits, and it enables local encoding and low-latency windowed decoding. In this paper, we use the algebraic framework to optimize short cycles in the Tanner graph of 2D-SC hypergraph product (HGP) codes that arise from short cycles in either component code. While prior work focuses on QLDPC codes with rate less than 1/10, we construct 2D-SC HGP codes with small memories, higher rates (about 1/3), and superior thresholds.