Complexity of Fermionic 2-SAT

TL;DR

The paper introduces Fermionic $k$-SAT, a parity-conserving, fermionic analogue of Quantum $k$-SAT, and provides a detailed complexity landscape. The authors develop a graph-based decomposition into quantum clusters and prove a linear-time classical algorithm for Fermionic $2$-SAT, extend to parity-constrained variants in $O(nm)$, and show NP-completeness for fixed-particle-number PNC Fermionic $2$-SAT. They also establish $QMA_1$-hardness for Fermionic $9$-SAT, elucidating a sharp boundary between tractable and intractable cases. The work ties together cluster-product representations, particle-hole transformations, and efficient cluster-verification techniques to map quantum-fermionic constraints onto tractable classical subproblems and clarifies the role of Gaussian versus non-Gaussian states in satisfying assignments.

Abstract

We introduce the fermionic satisfiability problem, Fermionic $k$-SAT: this is the problem of deciding whether there is a fermionic state in the null-space of a collection of fermionic, parity-conserving, projectors on $n$ fermionic modes, where each fermionic projector involves at most $k$ fermionic modes. We prove that this problem can be solved efficiently classically for $k=2$. In addition, we show that deciding whether there exists a satisfying assignment with a given fixed particle number parity can also be done efficiently classically for Fermionic 2-SAT: this problem is a quantum-fermionic extension of asking whether a classical 2-SAT problem has a solution with a given Hamming weight parity. We also prove that deciding whether there exists a satisfying assignment for particle-number-conserving Fermionic 2-SAT for some given particle number is NP-complete. Complementary to this, we show that Fermionic 9-SAT is QMA$_1$-hard.

Figures (6)

• Figure 1: Several connected subgraphs $G_q^i$ of $G$ with genuinely quantum clauses (orange), the other clauses (black) are classical clauses. The graph $G_{\rm sub}$ is obtained by removing all classical edges and vertices not touching quantum edges (referred to as classical modes), leaving the disconnected quantum clusters $G_q^i$.
• Figure 2: Example of a hidden particle number conserving quantum cluster $G_{q}$ as given in Definition \ref{['def:hiddenPNCcluster']} where the green quantum edges are $\Pi_e^{02,q}$ clauses and the pink quantum edges are $\Pi_e^{1,q}$ clauses. If one does a particle-hole transformation $K_B$ on all the fermionic modes in $B$ (or $K_A$ on $A$), the green $\Pi_e^{02,q}$ clauses become $\tilde{\Pi}_e^{1,q}$ clauses, while the pink $\Pi_e^{1,q}$ clauses become $\tilde{\Pi}_e^{1,q}$ clauses, see Eq. \ref{['eq:ph-edge']}, thus staying of particle-conserving type. Hence, after the particle-hole transformation, all clauses in $G_q$ are particle-number-conserving, and the space of satisfying assignments is then spanned by eigenstates $\ket{\psi}$ of the so-called hidden cluster particle number$\hat{N}_q$, obeying $\hat{N}_q (K_B \ket{\psi})=N_q (K_B \ket{\psi})$. Note that for a hPNC cluster with $B\neq \emptyset$, a satisfying assignment $\ket{\psi}$ will generally not be an eigenstate of cluster particle number in the original basis, but an eigenstate of the cluster parity.
• Figure 3: Satisfying assignments of Fermionic 2-SAT can be taken to be of cluster-product form: ordered products of operators which create satisfying assignments on quantum clusters and classical assignments on classical modes starting from the vacuum state, as in Eq. \ref{['eq:sh']}. The possibly-created states on a hPNC cluster $G_q^i \equiv i \in \mathsf{hPNC}$ (orange clusters) are completely specified by (hidden) cluster particle numbers $N_q^{i}=0,1,\ldots, n_q^i$. The possibly-created states on a non-hPNC cluster $j \in \mathsf{non-hPNC}$ (blue clusters) are completely specified by the cluster parity $P_q^{j}=\pm 1$. The possibly-created states for a classical mode $k \in \mathsf{Class}$ are completely specified by the occupied/unoccupied label $x_k=0,1$. The black edges correspond to classical clauses. Note that these classical clauses can also be internal to a quantum cluster or straddle a quantum cluster and a classical mode.
• Figure 4: Different types of line and loop clusters in the particle-hole-transformed basis. The black edges make up $T_q$ (representing $\Pi^{1,q}_e$-type clauses) and the blue edges are $\Pi^{02,q}_e$-type clauses.
• Figure 5: Satisfying assignments on non-hPNC quantum clusters are such that each mode has to be both empty and occupied, see Corollary \ref{['cor:restricted-PN']} point \ref{['needstobefree2']}. On hPNC clusters, one only has to assign either of two classical states to check whether the instance is satisfiable (apart from a special case when the cluster only has two modes, $n_q=2$), when we do not aim to find an assignment of given parity. The black vertices and edges respectively denote modes that are not in any quantum cluster (i.e., classical modes) and classical clauses.

Theorems & Definitions (39)

• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1: Quantum clusters
• Definition 2.2: Hidden Particle Number Conserving Cluster (hPNC)
• Definition 2.3: Maximal spanning hPNC subgraph of a quantum cluster
• Lemma 3.1: Partner Lemma
• proof
• Corollary 3.2
• proof