[论文解读] Quantum Annealing Based Binary Compressive Sensing with Matrix Uncertainty

Ramin Ayanzadeh, Seyedahmad Mousavi+2 more|arXiv (Cornell University)|Jan 1, 2019

Sparse and Compressive Sensing Techniques参考文献 35被引用 14

一句话总结

本文提出一种基于量子退火的二值压缩感知方法，适用于矩阵不确定性问题，将问题形式化为自旋模型（Ising model），并映射至QUBO以通过量子退火器求解。该方法采用交替最小化策略，迭代优化测量矩阵的不确定性，实现比经典方法更宽松约束下的稀疏信号恢复，已在D-Wave 2000Q系统上得到验证。

ABSTRACT

Compressive sensing is a novel approach that linearly samples sparse or compressible signals at a rate much below the Nyquist-Shannon sampling rate and outperforms traditional signal processing techniques in acquiring and reconstructing such signals. Compressive sensing with matrix uncertainty is an extension of the standard compressive sensing problem that appears in various applications including but not limited to cognitive radio sensing, calibration of the antenna, and deconvolution. The original problem of compressive sensing is NP-hard so the traditional techniques, such as convex and nonconvex relaxations and greedy algorithms, apply stringent constraints on the measurement matrix to indirectly handle this problem in the realm of classical computing. We propose well-posed approaches for both binary compressive sensing and binary compressive sensing with matrix uncertainty problems that are tractable by quantum annealers. Our approach formulates an Ising model whose ground state represents a sparse solution for the binary compressive sensing problem and then employs an alternating minimization scheme to tackle the binary compressive sensing with matrix uncertainty problem. This setting only requires the solution uniqueness of the considered problem to have a successful recovery process, and therefore the required conditions on the measurement matrix are notably looser. As a proof of concept, we can demonstrate the applicability of the proposed approach on the D-Wave quantum annealers; however, we can adapt our method to employ other modern computing phenomena -like adiabatic quantum computers (in general), CMOS annealers, optical parametric oscillators, and neuromorphic computing.

研究动机与目标

• 为解决具有矩阵不确定性的二值压缩感知问题的NP难性质，采用量子退火方法。
• 减少经典凸松弛方法对测量矩阵施加的严格约束依赖。
• 为量子退火器开发一个良好定义且可计算的公式，仅需解的唯一性要求。
• 在测量矩阵不完整或部分未知的实际场景中实现可行的信号恢复。
• 在当前一代D-Wave量子退火器上展示方法的可行性，并验证其在其他量子与神经形态平台上的可扩展性。

提出的方法

• 将二值压缩感知问题形式化为无约束二值二次优化（QUBO）问题。
• 将QUBO映射为自旋模型，其基态对应于稀疏信号解。
• 采用交替最小化策略：固定不确定性参数d，通过量子退火求解x；固定x，利用闭式解更新d。
• 使用闭式更新 d* = (G^T G + γI)^{-1} G^T c 在每次迭代中优化不确定性参数。
• 通过量子比特链技术将自旋模型编译至D-Wave的Chimera拓扑，以模拟全连接性。
• 将该方法适配至其他平台，如CMOS退火器、光学参量振荡器及神经形态系统。

实验结果

研究问题

• RQ1 量子退火能否有效求解具有矩阵不确定性的NP难二值压缩感知问题？
• RQ2 与经典凸松弛方法相比，所提方法是否能在测量矩阵约束更宽松的情况下实现信号恢复？
• RQ3 在矩阵不确定性条件下，结合自旋模型映射的交替最小化策略是否能成功收敛至稀疏解？
• RQ4 尽管存在硬件限制（如稀疏连接性），该方法在当前D-Wave量子退火器上的表现如何？
• RQ5 该方法在多大程度上可推广至其他量子与非量子退火平台？

主要发现

• 所提方法仅需解的唯一性要求，即可通过量子退火成功恢复稀疏信号，显著放宽了经典方法的约束。
• 交替最小化策略能收敛至满足终止条件 ||A x + d A x - y||_2 ≤ r 的解。
• 该方法适用于D-Wave 2000Q系统，该系统通过量子比特链限制，可模拟约64量子比特的全连接量子退火器。
• 该方法可适配至其他新兴计算平台，包括绝热量子计算机、CMOS退火器、光学参量振荡器及神经形态系统。
• 自旋模型公式可直接映射至量子硬件，在许多情况下避免了复杂嵌入启发式算法的需求。
• d的闭式更新确保了矩阵不确定性参数的高效优化，且无需额外优化开销。