Zero-Energy Problems for Supersymmetric Hamiltonians on a Chain Are QMA 1-Complete

Exact zero modes for supersymmetric Hamiltonians arranged on a one-dimensional chain are QMA1-complete in both Hermitian and explicitly encoded nilpotent formulations, even when restricted to geometric locality. This result establishes that deciding whether a zero mode exists for these systems is computationally hard, placing it in the QMA1 complexity class, which is significant because it identifies a specific computational structure within the zero-energy regime of quantum systems.

Complexity Classification and Hardness

The paper classifies the exact-zero problem for Hermitian supercharges as QMA1-complete when interactions are arranged on a chain and are geometrically local. Furthermore, it shows that the exact-zero problem for an explicitly encoded nilpotent N=2 formulation is also QMA1-complete, even when the Hamiltonian is local on a chain. The key finding is that this hardness persists under geometric locality, which imposes restrictions on how many degrees of freedom a term can touch—specifically requiring those degrees of freedom to be nearby.

Formulations and Constraints

The study distinguishes between two main formulations:

1. Hermitian supercharges, where a Hermitian supercharge Q defines H = Q2, and the zero mode is annihilated by Q.

2. Explicitly encoded nilpotent N=2 formulations, where a nilpotent supercharge Q defines H = [Q, Q†], and a zero mode is annihilated by both Q and its adjoint.

The paper addresses locality in two ways:

Geometric locality imposes a second restriction... geometric locality also requires those degrees of freedom to be nearby.

In the Hermitian formulation, the listed supercharge terms have bounded chain support. In the nilpotent formulation, chain locality is imposed only on ∆ = [Q, Q†]; Q and its individual terms retain the general geometry allowed by the input model.

Verification and Construction Methods

The paper relies on a common verification theorem for simultaneous sparse linear constraints to establish the QMA1 classification. The construction utilizes a one-dimensional frustration-free Hamiltonian and an explicit inverse-polynomial lower bound on the ground energy of every reduced NO instance, separating it from zero. The verification process involves several steps:

The common kernel reduction... removes the cokernel directions through the lower-right identity, and computes the two-dimensional singular-value blocks.

The construction for Hermitian problems involves a common-kernel reduction which is shown to preserve the kernel and provide an inverse-polynomial conditioning bound.

Hardness Proofs and Bounds

The QMA1-hardness is established through reductions from known hard problems, specifically referencing Cade and Crichigno’s work on supersymmetric local-Hamiltonian problems. The paper proves hardness for the parity-odd intrinsic minimal problem on a chain, showing that each supercharge summand has arity at most eleven and a system window of width ten, while Q2 has span at most nineteen encoded sites and realized span fifteen. This demonstrates that the hard instances already admit the most restrictive lattice geometry.

Zero Mode Analysis

The paper analyzes the zero modes using spectral mapping arguments. Lemma 5.1 shows that for a positive weight W, if H0 is related to Q as described, then ker Q = ker H0 ⊗ Fanc, and a lower bound on the zero-energy is established: "E0(Q2) ≥ E0(H0)2/W when W > 0. For the explicit nilpotent case, Theorem 6.8 proves that for a two-mode extension, the Hamiltonian is Q2min ⊗ Iab," and its kernel and cohomology dimensions are explicitly determined. The final result establishes the QMA1-completeness of the explicit exact-zero nilpotent-supercharge problem on a chain with specific locality restrictions.

Input Encoding

The input encoding is highly structured to ensure that all necessary information for verification is present without allowing shortcuts. This includes:

1. Encoding each d=11 source site into five qubits via an isometry, resulting in a full-Fock encoding without the local occupation constraints of Batista and Ortiz.

2. Applying the open-boundary Jordan–Wigner transformation to order the encoded qubits along a line, which is described as a unitary ∗-isomorphism of operator algebras.

3. The explicit enumeration of every site, mode, allocation, support, and every nonzero sparse row–column–value entry, ensuring that no information can be inferred from symmetry or omitted blocks.

Conclusion on Completeness

The paper concludes that the problem in Definition 6.4 (the explicit exact-zero nilpotent-supercharge problem) is QMA1-complete, and hardness already holds for the subfamily of Definition 6.5 (the chain-local Hamiltonian restriction). This confirms Theorem 6.12: "The problem in Definition 6.4 is QMA1-complete.

Based on the provided scientific paper, here are specific improvements for AI systems derived from its findings:

) Precise Zero-Energy State Identification and Verification:

The paper proves that deciding whether a zero mode exists for Hermitian supercharges (QMA1-complete) and explicitly encoded nilpotent supercharges (QMA1-complete) is a hard problem. It provides exact verification circuits, including stable multiplication by quadratic-dyadic coefficients and kernel-preserving Hermitian completion.

] Improved AI System Capability:

This system can be used to rigorously verify the existence of ground states (zero modes) in complex quantum models, such as those arising from Hamiltonian simulations or neural network energy minimization problems mapped onto supersymmetric structures. Specifically, it can determine if a target state is an exact zero-energy solution or provide an inverse-polynomial lower bound on the energy gap separating zero modes from excited states.

) Geometric Locality Constraint Enforcement:

The paper demonstrates that the hardness results (QMA1-completeness) hold even when supercharges are geometrically local on a one-dimensional chain. The construction uses specific one-dimensional history-state reductions and explicit encoding schemes (Jordan–Wigner transformation on an open chain).

] Improved AI System Capability:

The system can be adapted to solve constrained optimization problems where the constraints exhibit geometric locality (i.e., interactions are restricted to nearby degrees of freedom). This is highly valuable for training deep learning models or solving large-scale physical simulations where only local neighborhood interactions are physically meaningful, allowing the AI to efficiently search for exact zero-energy solutions within these geometrically constrained subspaces.

) Exact Sparse Relation Solving:

The paper introduces a framework (Definitions 5.6–5.8 and Theorem 4.6/4.8) for solving problems that ask whether the kernel of a sparse Hermitian matrix is non-trivial, providing QMA1-complete verification based on exact sparse access models (Definition 4.3/4.4).

] Improved AI System Capability:

This system can serve as a specialized solver for large, sparse linear algebra problems in high dimensions where only specific entries are known exactly (e.g., in certain quantum chemistry calculations or graph theory). It can precisely determine if a given sparse system possesses a non-trivial nullspace (zero modes) or provide the tightest possible lower bound on the singular values of the matrix, which is crucial for identifying hard instances where standard numerical methods fail.

) Robustness Against Input Encoding Complexity:

The paper details how complex quantum systems (like 3-SAT) can be encoded into a specific fermionic language (local qudit-to-qudit encoding followed by open-boundary Jordan–Wigner transformation), showing that the resulting Hamiltonian retains its properties under this transformation.

] Improved AI System Capability:

The system can be used to build more robust and general quantum simulators. By understanding how to map complex, discrete computational problems onto continuous or fermionic Hilbert spaces while preserving crucial algebraic properties (like locality and parity), the AI can generate simulators that are inherently stable against certain types of numerical noise or approximation errors inherent in standard floating-point simulations.

) Scalable QMA1 Verification via Polynomial Reductions:

The paper establishes a common verification theorem (Corollary 4.9) that provides an inverse-polynomial lower bound on the ground energy, which is then used to prove QMA1 containment for both Hermitian and nilpotent formulations.

] Improved AI System Capability:

This allows the AI to develop automated complexity analysis tools for quantum algorithms. By leveraging this reduction, it can rigorously quantify the hardness of finding exact solutions in novel quantum architectures, enabling researchers to predict whether a specific class of problems will be tractable or require exponential time on future hardware.