Computational complexity of the homology problem with orientable filtration: MA-completeness

TL;DR

The paper proves that the homology decision problem for simplicial complexes with a uniform orientable filtration is MA-complete, introducing a new orientability notion that leverages upward and downward adjacency in filtrations. Containment in MA is achieved via a fixed-node Hamiltonian construction that converts the combinatorial Laplacian into a stoquastic operator and a polynomial-time randomized verifier operating on non-negative harmonic states. MA-hardness is established by a gadget-based reduction from an MA-hard stoquastic SAT problem, encoding the n-qubit Hilbert space into the harmonic space of a gadgetized clique complex while preserving the filtration’s orientability. A detailed spectral-sequence analysis of the gadget complex demonstrates the existence and persistence of harmonics corresponding to YES instances and the spectral gap in NO instances, tying topology, persistent homology, and quantum computing into a unified framework. The results open avenues for quantum-assisted topological data analysis under carefully designed filtrations and pose open questions about derandomization and extensions to broader projector sets.

Abstract

We show the existence of an MA-complete homology problem for a certain subclass of simplicial complexes. The problem is defined through a new concept of orientability of simplicial complexes that we call a "uniform orientable filtration", which is related to sign-problem freeness in homology. The containment in MA is achieved through the design of new, higher-order random walks on simplicial complexes associated with the filtration. For the MA-hardness, we design a new gadget with which we can reduce from an MA-hard stoquastic satisfiability problem. Therefore, our result provides the first natural MA-complete problem for higher-order random walks on simplicial complexes, combining the concepts of topology, persistent homology, and quantum computing.

Figures (14)

• Figure 1: Example of states that are utilized in the fixed-node construction. In this figure, we are assuming that the simplices are filtrated in a cyclic way as in Figure \ref{['fig:orientable_filtration']}. (a) A simplicial complex with a 1-dimensional hole. (b) Center triangles fill the cycle. Therefore, the red edges possess a badness of "non-cocycleness" (c) There is no cycle. The red edges possess a badness of "non-cycleness".
• Figure 2: Intuitive gadget construction for $h_i=\frac{1}{2}(\ket{x}-\ket{y})(\bra{x}-\bra{y})$.
• Figure 3: An example of an orientable filtration ${X}_d^0\subseteq \cdots \subseteq {X}_d^4$ with $d=1$. Edges with the same color belong to the same $\tilde{X}_1^i$. Arrows indicate the orientation that makes simplices in different subsets induce opposite orientations on the triangles. The dashed edges are not the elements in $\hat{X}_1^i$. They are regarded as "internal" edges, and the adjacent triangles can be effectively regarded as a single cell. Here, $f^{0,1}=1,f^{1,0}=1,f^{1,2}=2,f^{2,3}=1,f^{3,2}=2$ and so on.
• Figure 4: Examples for non-negative ground states in the 1-dimensional case. The arrows on the edge indicate the given orientation. (a) The down Laplacian for the colored vertex does not have a non-negative ground state, while (b) has a non-negative ground state. The up Laplacian for the colored triangle in (c) has a non-negative ground state, while (d) does not.
• Figure 5: An example of a harmonic state associated with the uniform orientable filtration. Here, we are considering an unweighted setting. The coefficients are determined by the contributions of coboundaries on the shared cofaces, which can be calculated with eq. \ref{['eq:relative_amplitude']}.

Theorems & Definitions (59)

• Theorem 1: Main (Informal statement of Theorem \ref{['thm:main']})
• Definition 1: non-negative state
• Definition 2: Fixed-node Hamiltonian ten1995proofbravyi2023rapidly
• Remark
• Lemma 1
• Definition 3: Clique complex
• Definition 4: Faces and cofaces
• Definition 5: Internal simplices of $X_d^i$
• Definition 6: Orientable $d$-filtration
• Definition 7: Uniform orientable $d$-filtration