Quantum 2-SAT on low dimensional systems is $\mathsf{QMA}_1$-complete: Direct embeddings and black-box simulation

TL;DR

This work addresses the question of the local-dimension threshold at which quantum satisfiability becomes $QMA_1$-hard. It introduces a direct embedding strategy using a novel clock construction within a 2D circuit-Hamiltonian framework, and proves a $oldsymbol{oldsymbol{QMA_1 ext{-}hard}}$ result for $(2,5)$-QSAT, alongside a $oldsymbol{QMA_1}$-hardness result for $(3,d)$-QSAT on a 1D line with constant $d$. A key technical contribution is the Nullspace Connection Lemma, coupled with Unitary Labelled Graphs, which yields a simplified, analytic gap analysis and supports a black-box simulation approach to 1D QSAT hardness. The paper also exhibits a 1D frustration-free construction with a unique entangled ground state for a $(2,4)$-line, illustrating that low-dimensional quantum Hamiltonians can encode complex computations even without frustration. Overall, the results push the boundary of qubit-based hardness in QSAT, provide new analytic tools for Hamiltonian analysis, and raise questions about further dimension reductions and hardness in 1D settings. It advances both the theoretical understanding of QSAT complexity and the toolbox for studying quantum Hamiltonians in low dimensions.

Abstract

Despite the fundamental role the Quantum Satisfiability (QSAT) problem has played in quantum complexity theory, a central question remains open: At which local dimension does the complexity of QSAT transition from "easy" to "hard"? Here, we study QSAT with each constraint acting on a $k$-dimensional and $l$-dimensional qudit pair, denoted $(k,l)$-QSAT. Our first main result shows that, surprisingly, QSAT on qubits can remain $\mathsf{QMA}_1$-hard, in that $(2,5)$-QSAT is $\mathsf{QMA}_1$-complete. In contrast, $2$-SAT on qubits is well-known to be poly-time solvable [Bravyi, 2006]. Our second main result proves that $(3,d)$-QSAT on the 1D line with $d\in O(1)$ is also $\mathsf{QMA}_1$-hard. Finally, we initiate the study of 1D $(2,d)$-QSAT by giving a frustration-free 1D Hamiltonian with a unique, entangled ground state. Our first result uses a direct embedding, combining a novel clock construction with the 2D circuit-to-Hamiltonian construction of [Gosset, Nagaj, 2013]. Of note is a new simplified and analytic proof for the latter (as opposed to a partially numeric proof in [GN13]). This exploits Unitary Labelled Graphs [Bausch, Cubitt, Ozols, 2017] together with a new "Nullspace Connection Lemma", allowing us to break low energy analyses into small patches of projectors, and to improve the soundness analysis of [GN13] from $Ω(1/T^6)$ to $Ω(1/T^2)$, for $T$ the number of gates. Our second result goes via black-box reduction: Given an arbitrary 1D Hamiltonian $H$ on $d'$-dimensional qudits, we show how to embed it into an effective null-space of a 1D $(3,d)$-QSAT instance, for $d\in O(1)$. Our approach may be viewed as a weaker notion of "simulation" (à la [Bravyi, Hastings 2017], [Cubitt, Montanaro, Piddock 2018]). As far as we are aware, this gives the first "black-box simulation"-based $\mathsf{QMA}_1$-hardness result, i.e. for frustration-free Hamiltonians.

Figures (8)

• Figure 1: Graphical representations of constraints from the circuit Hamiltonian of \ref{['thm:generic-clock']}. The $X$-axis goes from left to right, the $Y$-axis from top to bottom; both start counting at index $1$. Dots: Each dot represents a clock state (here a $5\times 4$ subspace of clock states is depicted). Edges: Transitions between two clock states. Red edges: Unitary transitions. Dashed edges: Conditional transitions (with control $\ketbra00$ horizontally, $\ketbra11$ vertically), i.e. transitions which only take place if the computation/control register reads $0$ (respectively, $1$). Dashed arrows: Indicate that edges continue in that direction on a larger clock space when combining gadgets. Hatched/shaded area: Penalized clock states, which are thus not in the Hamiltonian's nullspace.
• Figure 2: Graphical representation of the Hamiltonian $H_V$ implementing the $V$ gate with control $c$ and target $t$. Dashed edges: $\ketbra00_{c}\otimes (h_{3,4})_X + \ketbra11_{c}\otimes (h_{3,4})_Y$. Note that technically there are also dashed edges $(3,4)$ in rows $6$ and $7$, but these are not drawn, since the identity transitions on the same edges (drawn as solid edges) also act here. Red edges: $h_{4,5}(B)_{t,X} + h_{4,5}(\sigma^Z)_{t,Y}$, i.e. $B$ applied on red edges along the $X$-axis/horizontally, $\sigma^Z$ on red edges on the $Y$-axis/vertically.
• Figure 3: Decomposition of $H_V = \ketbra00_c \otimes H_{V,0} + \ketbra11_c \otimes H_{V,1}$.
• Figure 4: Graphical representation of the Hamiltonian $H_U$ implementing a single-qubit unitary $U$ on computational qubit $z$.
• Figure 5: Graphical representation of $H_{\mathrm{diag}} + H_{\mathrm{gate}}+H_{\mathrm{link}}$ from \ref{['eq:Hdiag', 'eq:Htrans1', 'eq:Htrans2']} for $T=2$. Blue zigzag edges represent the new edges from $H_{\mathrm{link}}$ not present in $H_{\mathrm{gate}}$.

Theorems & Definitions (66)

• theorem 1.1
• theorem 1.2
• theorem 1.3
• theorem 1.4
• theorem 1.5
• definition 2.1: $\mathrm{QMA}_1\xspace$
• definition 2.2: $(k,l)$-QSAT
• definition 2.3: Unitary Labelled Graph (ULG) BCO17
• definition 2.4: Simple ULG BCO17
• lemma 2.5: Kitaev's Geometric Lemma KSV02 as stated in GN13