用量子算法生成既有旋律又有和声的音乐,实现理论加速。
HHL with a Coherent Fourier Oracle: A Proof-of-Concept Quantum Architecture for Joint Melody-Harmony Generation
- 将量子线性求解器HHL与傅里叶和声预言机结合,实现音符与和弦联合采样。
- 8小节音乐生成中97%和弦进行被判定为强或可接受,语法正确。
- 首次验证了量子音乐生成中保持相干性的关键架构可行。
量子算法中具备理论加速优势的极为稀少,其中最突出的是用于求解稀疏线性系统的HHL算法。本文将其应用于编码旋律偏好:系统矩阵融合Narmour蕴含-实现关系与Krumhansl-Kessler调性稳定性,其解向量即为基于音乐认知加权的音符对分布。HHL的关键限制在于经典读出会抵消量子加速,因此必须保持相干消费。为此提出一个相干傅里叶和声预言机:通过酉算子直接对HHL振幅向量施加和弦转移权重,使单次测量同时选择旋律音与双和弦进行。采用两音/两和弦(2/2)块结构控制联合状态空间的指数增长,避免经典模拟失效。为生成更长乐句,以经典链式方式连接多个块,前一块输出决定下一块输入,作为容错硬件出现前的临时方案。四块链生成8个音符对应8个和弦,每块边界均满足语法规则。独立基于规则的和声验证表明97%的生成和弦进行被评为强或可接受。核心动机在于:HHL具备对经典线性求解器的指数级加速潜力;本工作证明,实现该加速所必需的相干HHL+预言机流程在机械上是可行的。代表性输出已提供在线音频试听。
原文摘要 · Abstract (English)
Quantum algorithms with a proven theoretical speedup over classical computation are rare. Among the most prominent is the Harrow-Hassidim-Lloyd (HHL) algorithm for solving sparse linear systems. Here, HHL is applied to encode melodic preference: the system matrix encodes Narmour implication-realisation and Krumhansl-Kessler tonal stability, so its solution vector is a music-cognition-weighted note-pair distribution. The key constraint of HHL is that reading its output classically cancels the quantum speedup; the solution must be consumed coherently. This motivates a coherent Fourier harmonic oracle: a unitary that applies chord-transition weights directly to the HHL amplitude vector, so that a single measurement jointly selects both melody notes and a two-chord progression. A two-note/two-chord (2/2) block is used to contain the exponential growth of the joint state space that would otherwise make classical simulation of larger blocks infeasible. For demonstrations of longer passages, blocks are chained classically - each block's collapsed output conditions the next -- as a temporary workaround until fault-tolerant hardware permits larger monolithic circuits. A four-block chain produces 8 notes over 8 chords with grammatically valid transitions at every block boundary. Independent rule-based harmony validation confirms that 97% of generated chord progressions are rated strong or acceptable. The primary motivation is that HHL carries a proven exponential speedup over classical linear solvers; this work demonstrates that a coherent HHL+oracle pipeline - the prerequisite for that speedup to be realised in a musical setting - is mechanically achievable. Audio realisations of representative outputs are made available for listening online.
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