Quadratic programming over orthogonal matrices encompasses a broad class of hard optimization problems that do not have an efficient quantum representation. Such problems are instances of the little noncommutative Grothendieck problem (LNCG), a generalization of binary quadratic programs to continuous, noncommutative variables. In this work, we establish a natural embedding for this class of LNCG problems onto a fermionic Hamiltonian, thereby enabling the study of this classical problem with the tools of quantum information. This embedding is accomplished by identifying the orthogonal group with its double cover, which can be represented as fermionic quantum states. Correspondingly, the embedded LNCG Hamiltonian is a two-body fermion model. Determining extremal states of this Hamiltonian provides an outer approximation to the original problem, a quantum analogue to classical semidefinite relaxations. In particular, when optimizing over the special orthogonal group our quantum relaxation obeys additional, powerful constraints based on the convex hull of rotation matrices. The classical size of this convex-hull representation is exponential in matrix dimension, whereas our quantum representation requires only a linear number of qubits. Finally, to project the relaxed solution back into the feasible space, we propose rounding procedures which return orthogonal matrices from appropriate measurements of the quantum state. Through numerical experiments we provide evidence that this rounded quantum relaxation can produce high-quality approximations.

翻译：正交矩阵上的二次规划涵盖了一类不具备高效量子表示的困难优化问题。此类问题属于小非交换Grothendieck问题（LNCG）的实例，它是二元二次规划到连续非交换变量的推广。本研究为这类LNCG问题建立了到费米子哈密顿量的自然嵌入，从而能够借助量子信息工具研究这一经典问题。该嵌入通过将正交群与其双重覆盖（可表示为费米子量子态）等同来实现。相应地，嵌入后的LNCG哈密顿量为二体费米子模型。确定此哈密顿量的极值态可为原问题提供外近似，这是经典半定松弛的量子模拟。特别地，当在特殊正交群上进行优化时，我们的量子松弛基于旋转矩阵凸包呈现额外且强大的约束。该凸包表示的经典规模随矩阵维度呈指数增长，而我们的量子表示仅需线性数量的量子比特。最后，为将松弛解投影回可行空间，我们提出能从量子态的适当测量中返回正交矩阵的舍入程序。数值实验表明，这种舍入量子松弛可产生高质量的近似解。