Algebraic techniques in designing quantum synchronizable codes

Algebraic and algorithmic frameworks for optimized quantum measurements

Von Neumann projections are the main operations by which information can be extracted from the quantum to the classical realm. They are however static processes that do not adapt to the states they measure. Advances in the field of adaptive measurement have shown that this limitation can be overcome by "wrapping" the von Neumann projectors in a higher-dimensional circuit which exploits the interplay between measurement outcomes and measurement settings. Unfortunately, the design of adaptive measurement has often been ad hoc and setup-specific. We shall here develop a unified framework for desi…

Optimal algorithms for linear algebra by quantum inspiration

Recent results by Harrow et. al. and by Ta-Shma, suggest that quantum computers may have an exponential advantage in solving a wealth of linear algebraic problems, over classical algorithms. Building on the quantum intuition of these results, we step back into the classical domain, and explore its usefulness in designing classical algorithms. We achieve an algorithm for solving the major linear-algebraic problems in time $O(n^{ω+ν})$ for any $ν>0$, where $ω$ is the optimal matrix-product constant. Thus our algorithm is optimal w.r.t. matrix multiplication, and comparable to the state-of-the-ar…

Quantum Algorithms for Boolean Equation Solving and Quantum Algebraic Attack on Cryptosystems

Decision of whether a Boolean equation system has a solution is an NPC problem and finding a solution is NP hard. In this paper, we present a quantum algorithm to decide whether a Boolean equation system FS has a solution and compute one if FS does have solutions with any given success probability. The runtime complexity of the algorithm is polynomial in the size of FS and the condition number of FS. As a consequence, we give a polynomial-time quantum algorithm for solving Boolean equation systems if their condition numbers are small, say polynomial in the size of FS. We apply our quantum algo…

常数穿梭深度的qLDPC码并行代数手术

Parallel algebraic surgery for qLDPC codes with constant shuttling depth

本文提出提升积码的代数手术方法，协同设计逻辑测量、辅助码和原子穿梭，在中性原子处理器上实现常数穿梭深度的并行逻辑测量，显著加速典型逻辑操作并保持竞争性量子比特开销。

A Novel Algebraic Geometry Compiling Framework for Adiabatic Quantum Computations

Adiabatic Quantum Computing (AQC) is an attractive paradigm for solving hard integer polynomial optimization problems. Available hardware restricts the Hamiltonians to be of a structure that allows only pairwise interactions. This requires that the original optimization problem to be first converted -- from its polynomial form -- to a quadratic unconstrained binary optimization (QUBO) problem, which we frame as a problem in algebraic geometry. Additionally, the hardware graph where such a QUBO-Hamiltonian needs to be embedded -- assigning variables of the problem to the qubits of the physical …

基于双矩阵的量子CSS LDPC码用于基于信念传播的解码

Quantum CSS LDPC Codes based on Dyadic Matrices for Belief Propagation-based Decoding

本文提出基于双矩阵的构造方法，用于设计量子CSS LDPC码，以满足CAMEL-ensemble四元信念传播解码器的兼容性条件，从而减少环的影响。