The Complexity of Local Stoquastic Hamiltonians on 2D Lattices

TL;DR

The paper proves that the $2$-Local Stoquastic Hamiltonian problem on a $2$D square lattice is $StoqMA$-complete by building a comprehensive reduction pipeline. It starts from general StoqMA circuits, converts them to nearest-neighbour form, then to spatially sparse circuits, and finally uses stoquastic perturbation gadgets (including subdivision, cross, fork, and triangle gadgets) to reduce locality while maintaining stoquasticity. Through geometrical gadgetry and planar embeddings, the authors show one can map spatially sparse stoquastic interactions to planar graphs and then to a $2$D lattice, preserving the ground-state structure up to small perturbative errors; this establishes $StoqMA$-completeness for the lattice case. The work also examines Pauli-decomposed stoquastic Hamiltonians, identifying a Pauli-term-wise complete family and discussing extents and limitations of such reductions. Overall, the results connect physically natural lattice geometries with the intrinsic hardness of stoquastic ground-state computation, advancing our understanding of where stoquastic quantum problems sit in the complexity landscape.

Abstract

We show the 2-Local Stoquastic Hamiltonian problem on a 2D square lattice is StoqMA-complete. We achieve this by extending the spatially sparse circuit construction of Oliveira and Terhal, as well as the perturbative gadgets of Bravyi, DiVincenzo, Oliveira, and Terhal. Our main contributions demonstrate StoqMA circuits can be made spatially sparse and that geometrical, stoquastic-preserving, perturbative gadgets can be constructed.

Figures (16)

• Figure 1: A pictorial representation of an interaction edge. The labels $u$/$v$ either represent single qubits or a set of qubits. The term $A_uB_v$ represents the local interaction between $u$ and $v$.
• Figure 2: Workflow of the required circuit modifications. We take generic (long-range) StoqMA circuits to ones comprised of only nearest-neighbour gates. A subsequent mapping takes such circuits to the spatially sparse construction. Here, $M \coloneqq n+w+m+p$ is the total number of qubits in the circuit. Additionally, $m'>m$ and $p'>p$ are the number of ancilla qubits required for the spatially sparse construction. The important parameters at each stage are: the completeness parameter, the soundness parameter and the gate count.
• Figure 3: A flow diagram of the complexity of the Local Stoquastic Hamiltonian problem. Arrows represent modifications/reductions to the problem. Grey boxes represent the results of prior work. Green boxes represent the results of this work.
• Figure 4: A visual representation of the modified Feynman-Kitaev construction. Each gate in the verification sequence is applied to a row of qubits in succession. After each round of gate applications, a Swap gate sequence is applied between rows of qubits from right to left. The time cursor is shown in green. The small circles represent qubits. The dashed boxes represent one of the non-trivial gates in the verification sequence. Qubits that have no dashed box are assumed to be acted on trivially.
• Figure 5: The Subdivision gadget.

Theorems & Definitions (41)

• Definition 1: Local Hamiltonian
• Definition 2: Stoquastic Hamiltonian
• Definition 3: Stoquastic Verification Circuit
• Definition 4: StoqMA($\alpha$,$\beta$)
• Remark 1: Merlin's message
• Theorem 1
• Theorem 1
• Theorem 1
• Definition 5: Semi-Classical Verification Circuit
• Definition 6: MAq BDOT06