Alfons Laarman

Alexis de Colnet, Floris Geerts, Rihan Hai et al.

Strongly simulating a quantum circuit, that is, computing an output amplitude, amounts to summing the circuit's Feynman paths, a weighted count over assignments to the Boolean ``path'' variables. The circuit's gates induce correlations among these variables, forming a graph whose structure determines the hardness of the simulation task. This sum-of-powers viewpoint underlies recent simulators built on knowledge-representation tools from artificial intelligence, namely binary decision diagrams and weighted model counting. We show that the structural quantity most accurately governing the difficulty is the rank-width of the path-variable graph, and we give an algorithm that evaluates the amplitude in time that is exponential only in this rank-width and polynomial in the circuit size. Rank-width can be far smaller than the widths that control competing methods: as corollaries, our algorithm reproduces a recent decision-diagram simulation breakthrough as a special case and matches the Markov--Shi tensor-network contraction bound. To complement this, we exhibit circuit families on which our algorithm provably beats both competing methods. The new method applies to every circuit built from Hadamard and diagonal gates, in particular to circuits over Clifford+T. In practical terms, general-purpose decision-diagram and model-counting tools can serve as the workhorse, with our specialized algorithm dispatched to exploit a small rank-width of the associated graph when it is present.

Alexis de Colnet, Alfons Laarman, Joon Hyung Lee

We prove the existence of two thresholds regarding the compilability of random 2-CNF formulas to OBDDs. The formulas are drawn from $\mathcal{F}_2(n,δn)$, the uniform distribution over all 2-CNFs with $δn$ clauses and $n$ variables, with $δ\geq 0$ a constant. We show that, with high probability, the random 2-CNF admits OBDDs of size polynomial in $n$ if $0 \leq δ< 1/2$ or if $δ> 1$. On the other hand, for $1/2 < δ< 1$, with high probability, the random $2$-CNF admits only OBDDs of size exponential in $n$. It is no coincidence that the two ``compilability thresholds'' are $δ= 1/2$ and $δ= 1$. Both are known thresholds for other CNF properties, namely, $δ= 1$ is the satisfiability threshold for 2-CNF while $δ= 1/2$ is the treewidth threshold, i.e., the point where the treewidth of the primal graph jumps from constant to linear in $n$ with high probability.

Samuel González-Castillo, Joon Hyung Lee, Alfons Laarman

We study PRODSAT-QSAT($k$): given rank-one $k$-local projectors, determine whether a quantum $k$-SAT instance admits a satisfying product state. We present a CDCL-style refutation framework that searches a finite partition of each qubit's Bloch sphere while a sound theory solver checks region feasibility using a geometric overapproximation of the projection amplitudes for each constraint. When the theory solver proves that no state in a region can satisfy a constraint, it produces a sound conflict clause that blocks that region; accumulated blocking clauses can yield a global result of product-state unsatisfiability (UN-PRODSAT). We formalise the problem, prove the soundness of the clause-learning rule, and describe a practical algorithm and implementation.

Wei-Jia Huang, Christophe Chareton, Yu-Fang Chen et al.

Equivalence checking of quantum circuits is a central verification task in quantum computing, ensuring the correctness of circuit optimizations, hardware mappings, and compilation pipelines. Among the primary symbolic methods for this purpose, the path-sum formalism provides a compact representation with powerful reduction rules that yield a canonical form for the classically simulable Clifford fragment, but confluence fails beyond the Clifford fragment. We introduce a new weighted model counting (WMC) encoding for path-sums and combine it with the existing path-sum reductions to obtain a verifier that is both complete and efficient. Our method applies reductions whenever possible and invokes the WMC-based decision procedure on the residual path-sum, yielding a complete semantic check up to a global phase. We implement the approach and evaluate it on standard benchmarks. Results show that the hybrid method outperforms either component in isolation and competes with state-of-the-art tools.

Arend-Jan Quist, Marc Farreras Bartra, Alexis de Colnet et al.

Tensor networks and circuits are widely used data structures to represent pseudo-Boolean functions. These two formalisms have been studied primarily in separate communities, and this paper aims to establish equivalences between them. We show that some classes of tensor networks that are appealing in practice correspond to classes of circuits with specific properties that have been studied in knowledge compilation as \emph{tractable circuits}. In particular, we prove that matrix product states (tensor trains) coincide with nondeterministic edge-valued decision diagrams and that tree tensor networks exactly correspond to structured-decomposable circuits. These correspondences enable direct transfer of structural and algorithmic results; for example, canonicity and tractability guarantees known for circuits yield analogous guarantees for the associated tensor networks, and vice versa.