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 a new representation of orthogonal matrices as fermionic quantum states, which we achieve through the well-known double covering of the orthogonal group. 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 \emph{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大类难以高效量子表示的困难优化问题。这类问题属于小非交换格罗滕迪克问题（LNCG）的实例，即二元二次规划在连续非交换变量上的推广。本研究为此类LNCG问题建立了到费米子哈密顿量的自然嵌入，从而能够用量子信息工具研究这一经典问题。该嵌入通过正交矩阵作为费米子量子态的新表示实现，这是借助正交群的双重覆盖性质完成的。相应地，嵌入后的LNCG哈密顿量是一个二体费米子模型。确定该哈密顿量的极值态可为原问题提供外部近似，这相当于经典半定松弛的量子类比。特别地，当在\emph{特殊}正交群上优化时，我们的量子松弛遵循基于旋转矩阵凸包结构的额外强约束条件。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，我们提出了舍入程序，通过量子态的适当测量得到正交矩阵。数值实验表明，这种舍入后的量子松弛能够产生高质量的近似解。