The PRODSAT phase of random quantum satisfiability

TL;DR

The paper analyzes the PRODSAT phase of random k-QSAT by reducing instances to their 2-core via leaf removal and then solving the residual core via complex polynomial equations. The authors prove that the existence of zero-energy product states is governed by a geometric property: a clause-covering dimer configuration on the core is both necessary and sufficient (with probability one) for the existence of PRODSAT states, yielding a threshold \(\alpha_{\rm dc}(k)\) equal to the dimer-covering threshold. They develop a two-pronged proof: (i) a perturbative argument showing existence for small constant terms when a dimer covering exists, and (ii) a non-perturbative argument using Buchberger's algorithm and Hilbert's Nullstellensatz to extend to generic random coefficients. Numerical simulations examine the relationship between dimer coverings, the dimension of the zero-energy space, and the presence of entangled solutions (ENTSAT), finding that in small instances the PRODSAT space can span the kernel, while entanglement emerges at larger sizes. The results illuminate the geometric nature of PRODSAT and the entanglement structure near the PRODSAT-ENTSAT boundary, with implications for phase transitions in random quantum CSPs.

Abstract

The $k$-QSAT problem is a quantum analog of the famous $k$-SAT constraint satisfaction problem. We must determine the zero energy ground states of a Hamiltonian of $N$ qubits consisting of a sum of $M$ random $k$-local rank-one projectors. It is known that product states of zero energy exist with high probability if and only if the underlying factor graph has a clause-covering dimer configuration. This means that the threshold of the PRODSAT phase is a purely geometric quantity equal to the dimer covering threshold. We revisit and fully prove this result through a combination of complex analysis and algebraic methods based on Buchberger's algorithm for complex polynomial equations with random coefficients. We also discuss numerical experiments investigating the presence of entanglement in the PRODSAT phase in the sense that product states do not span the whole zero energy ground state space.

Figures (3)

• Figure 1: Different patterns for $k= 3, m = 4$. (a) Sunflower (b) Loose chain (c) Strong chain (d) Loose cycle (e) Strong cycle. For the strong chain and the strong cycle, each qubit is connected to $k$ clauses, except for the boundary qubits. The sunflower is constructed around one central qubit.
• Figure 2: Example of Mixed Volume Computation. The Newton polytope $P_1$ of the polynomial $f_1 = A xy + B x + Cy + D$ is the square with vertices $\{(1,1), (1,0), (0,1), (0,0)\}$. For $f_2 = Ey^2 + Fx + G$, it is a triangle with vertices $\{(0,2), (1,0), (0,0)\}$. The decomposition of $Vol_n(\lambda_1 P_1 +\lambda_2 P_2) = \lambda_1^2 + 3 \lambda_1\lambda_2 + \lambda_2^2$ is represented on the figure. Then the mixed volume is 3.
• Figure 3: Comparison among $\dim{\ker H_F}, \dim{\rm PRODSAT}$ and $MV$. On the left, $\dim{\ker H_F}$ and $MV$ are compared. On the right, we reinterpret the results in terms of PRODSAT space, UNSAT, and ENTSAT instances. For blue and green, we use $\dim{\rm PRODSAT}\leq MV$. Red and purple follow from the definition of the phases. Orange results are joined with blue since we always have $\dim{\ker H_F} \geq \dim{\rm PRODSAT}$. For $N=5,6$ all the instances with $M=N$ are computed (resp. 252 and 38500). For $7\leq N \leq 10$, 5000 instances are sampled uniformly. For $N=11$, only 500 instances are used.

Theorems & Definitions (54)

• Lemma 1.1
• Proposition 1.2
• Proposition 1.3
• proof : Proof of Theorem \ref{['mainthm']}
• Proposition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Proposition \ref{['lemma_existsol']}