Quantum heuristics for linear optimization over large separable operators

TL;DR

This work addresses the challenge of optimizing over separable quantum objects, which is NP-hard and suffers from exponential dimension growth. It proposes a hybrid approach that combines semidefinite programming for small instances with a quantum co-processor-driven dimension-reduction technique to produce feasible separable solutions for large problems. The authors develop a see-saw based heuristic, derive reduced problems through ansatz-based projections, and show how to compute reduced-data matrices via quantum measurements, enabling separable-ground-state approximations for Hamiltonians up to 28 qubits. They define a ground-space entanglement measure based on the separable ground energy and demonstrate the method on the one-dimensional Ising model, including analyses of ansatz-size effects and performance relative to full ground-state calculations. The results indicate practical scalability gains and provide a path toward entanglement-aware characterization of ground spaces in large quantum systems.

Abstract

Optimizing over separable quantum objects is challenging for two key reasons: determining separability is NP-hard, and the dimensionality of the problem grows exponentially with the number of qubits. We address both challenges by introducing a heuristic algorithm that leverages a quantum co-processor to significantly reduce the problem's dimensionality. We then numerically demonstrate that see-saw-type optimization performs well in lower-dimensional settings. A notable feature of our approach is that it yields feasible solutions, not just bounds on the optimal value, in contrast to many outer-approximation-based methods. We apply our method to the problem of finding separable states with minimal energy for a given Hamiltonian and use this to define an entanglement measure for its ground space. Finally, we demonstrate how our approach can approximate the separable ground energy of Hamiltonians up to 28 qubits.

Figures (4)

• Figure 1: The circuit for the Hadamard test which approximates the inner product between two states $\ket{\psi}$ and $\ket{\phi}$ given many samples. Here, the unitary $U$ maps the state $\ket{\psi}$ to the state $\ket{\phi}$. When we set $b = 0$, the circuit approximates $\Re(\braket{\psi}{\phi})$, and when we set $b = 1$, it approximates $\Im(\braket{\psi}{\phi})$.
• Figure 2: A depiction of the approximate ground space entanglement measure as a function of the transverse field $h$, for $H_{\mathrm{Ising}}$ with $12$ qubits (left) and $14$ qubits (right) setting $J = 1$ and $g = 0$. Here we set $\mathcal{A}$ to be the first half of the qubits and $\mathcal{B}$ to be the second half of the qubits. Note that we are able to compute $\lambda_{\max}$ and $\lambda_{\min}$ for these graphs.
• Figure 3: The error generated by the use of our heuristic as we vary the number of ansatz states for the Ising Hamiltonian on 12 qubits (with $J = 1, g = 0, h = 1.3$). Here $\hat{\alpha}_L$ and $\tilde{\alpha}_L$ denote the maximum and the mean of the $\hat{\alpha}_{L,i}$ values, respectively, computed using $10$ randomly chosen reference states.
• Figure 4: Comparing the calculations (when possible) and approximations to the ground energy and separable ground energy for the Ising Hamiltonian ($J = 1, g = 0, h = 1.4$) for up to $28$ qubits.

Theorems & Definitions (2)

• Definition 4.1: Separable ground energy
• Definition 4.2: Ground space entanglement measure, and an approximation