通过排序网络实现排列问题的紧凑QUBO公式
Succinct QUBO formulations for permutation problems by sorting networks
AI总结：
本文提出一种使用排序网络的紧凑QUBO公式，仅需O(n log²n)个二进制变量，优于传统排列矩阵编码，同时支持排列操作和顺序检查。
摘要：
二次无约束二进制优化（QUBO）是一个标准的NP难优化问题。最近，由于QUBO可直接转化为Ising模型，而量子退火设备基于此模型，QUBO重新受到关注。我们引入一种使用比较交换网络的排列QUBO公式，仅需O(n log²n)个二进制变量。这一方法的核心特征是统一性：每个排列对应一个唯一的变量赋值，从而实现无偏采样。我们的构造还允许额外约束，包括固定点和奇偶性。此外，它提供了一种支持排列乘法和求逆操作的排列表示，并使检查排列顺序成为可能。这可用于均匀生成给定顺序的排列或例如与指定排列相容的排列。到目前为止，这是首次将无偏比较交换网络与QUBO编码联系起来的结果。虽然使用排列矩阵可以实现类似功能，但我们的方法产生的QUBO更小且更稀疏。我们预计此方法在需要无偏采样受限排列的重要领域中具有实际应用价值，包括密码学和组合设计。
英文摘要：
Quadratic Unconstrained Binary Optimization (QUBO) is a standard NP-hard optimization problem. Recently, it has gained renewed interest through quantum computing, as QUBOs directly reduce to the Ising model, on which quantum annealing devices are based. We introduce a QUBO formulation for permutations using compare-exchange networks, with only $O(n \log^2 n)$ binary variables. This is a substantial improvement over the standard permutation matrix encoding, which requires $n^2$ variables and has a much denser interaction graph. A central feature of our approach is uniformity: each permutation corresponds to a unique variable assignment, enabling unbiased sampling. Our construction also allows additional constraints, including fixed points and parity. Moreover, it provides a representation of permutations that supports the operations multiplication and inversion, and also makes it possible to check the order of a permutation. This can be used to uniformly generate permutations of a given order or, for example, permutations that commute with a specified permutation. To our knowledge, this is the first result linking oblivious compare-exchange networks with QUBO encodings. While similar functionality can be achieved using permutation matrices, our method yields QUBOs that are both smaller and sparser. We expect this method to be practically useful in areas where unbiased sampling of constrained permutations is important, including cryptography and combinatorial design.
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