Expanding the reach of quantum optimization with fermionic embeddings

TL;DR

This work casts quadratic optimization over orthogonal and special orthogonal groups as a quantum-embeddable problem by mapping orthogonal matrices to fermionic Gaussian states through the Pin/Spin double cover. The authors introduce a quantum relaxation in the form of two-body fermionic Hamiltonians H and \widetilde{H}, whose ground states encode relaxed solutions to the LNCG objective, and they prove that these relaxations respect the convex hull structure of $O(n)$ and $SO(n)$ via a PSD-lift perspective. To recover feasible rotations, they propose two rounding routes: edge-based convex-rounding using a quantum Gram matrix and vertex-based rounding from single-vertex marginals, with the edge-rounding generally outperforming vertex rounding in numerical tests. Numerical experiments on random SO(3) group-synchronization instances show that quantum-rounding methods maintain higher approximation ratios than classical SDP-based rounding, even under noise, and quasi-adiabatic state preparation can yield high-quality rounded solutions without exact ground-state preparation. The results illuminate a promising bridge between quantum information tools and continuous-variable optimization, suggesting potential quantum advantages for structured nonconvex problems and guiding future extensions to unitary groups and complexity analyses.

Abstract

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.

Figures (4)

• Figure 1: A cartoon description of our quantum encoding of the LNCG problem, compared to the classical formulation. (Left) The description of the problem that we consider, which is described by a graph $([m], E)$ and $n \times n$ matrices $C_{uv}$ for each edge $(u, v) \in E$. We wish to assign elements of $\mathrm{O}(n)$ or $\mathrm{SO}(n)$ to each vertex such that Eq. \ref{['eq:summary_obj']} is maximized. (Center top) The standard classical relaxation optimizes an $mn \times mn$ PSD matrix $M$ via semidefinite programming. (Right top) The classical rounding procedure, which returns a collection of orthogonal matrices from $M$. (Center bottom) Our quantum formulation of the problem as a two-body fermionic Hamiltonian. On each vertex we place a $d$-dimensional Hilbert space, with interaction terms $H_{uv}$ on the edges constructed from each $C_{uv}$. The classical solution to the LNCG problem lies as a subset of this full Hilbert space. (Right bottom) Our proposed quantum rounding protocols. One protocol requires knowledge of the two-body reduced density matrices across edges, while the other uses the one-body reduced density matrices on each vertex.
• Figure 2: Approximation ratios for solutions obtained from rounding the maximum eigenvector of the relaxed Hamiltonian $\widetilde{H}$. Violin plots show the distribution of approximation ratios over 50 randomly generated instance, and with the median being indicated by the center marker. CR refers to rounding according to the $\mathop{\mathrm{conv}}\nolimits\mathrm{SO}(n)$-based scheme (Section \ref{['sec:conv_rounding']}), while VR denotes the vertex-marginal rounding scheme (Section \ref{['sec:vertex_rounding']}). The classical SDP solution was rounded by the standard randomized algorithm bandeira2016approximating, and we report the best solution over 1000 rounding trials. (Left) Varying the number of vertices $m$ in the graph (random 3-regular graphs). Note that the number of qubits required here is $2m$. (Right) Varying the noise strength parameter $\sigma$ which defines the problem via $C_{uv} = g_u g_v^\mathsf{T} + \sigma W_{uv}$.
• Figure 3: Demonstration of a typical instance of adiabatic state preparation for preparing relaxed quantum solutions. The initial state is the product of Gaussian states corresponding to the rounded solution of the classical SDP. As the total evolution time $T$ increases, the evolution becomes more adiabatic, indicated by the convergence of the relaxed value to the maximum eigenvalue (in units of the original problem's optimal value). The rounded solutions of course can never exceed the original problem's optimal value.
• Figure 4: Approximation ratios for solutions obtained from rounding the "adiabatically" evolved state $| \psi(T) \rangle$ with fixed $T = 1$ for all $m, \sigma$. Problem instances and visualization is the same as in Figure \ref{['fig:eigen_group-sync']}. We also include the maximum eigenvalue of the relaxed Hamiltonian and the energy of the prepared unrounded state, to demonstrate how far $| \psi(T) \rangle$ is from the exact maximum eigenvector. (Left) Varying the number of vertices $m$ in the graph. Note that the number of qubits required here is $2m$. (Right) Varying the noise strength parameter $\sigma$.

Theorems & Definitions (14)

• Lemma B.1
• proof
• Lemma B.2
• proof
• Theorem E.1
• proof
• Theorem F.1: bandeira2016approximating
• Theorem F.2
• Lemma F.3: Adapted from bandeira2016approximating
• Lemma F.4: Adapted from bandeira2016approximating