Exponential Complexity of the Quantum Adiabatic Algorithm for certain Satisfiability Problems

We determine the complexity of several constraint satisfaction problems using the quantum adiabatic algorithm in its simplest implementation. We do so by studying the size dependence of the gap to the first excited state of “typical” instances. We find that at large sizes N, the complexity increases exponentially for all models that we study. We also compare our results against the complexity of the analogous classical algorithm WalkSAT and show that the harder the problem is for the classical algorithm the harder it is also for the quantum adiabatic algorithm.

💡 Research Summary

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This paper investigates the computational complexity of the quantum adiabatic algorithm (QAA) when applied to three representative constraint‑satisfaction problems: locked 1‑in‑3 SAT, locked 2‑in‑4 SAT, and 3‑regular 3‑XOR SAT. The authors adopt the simplest possible driver Hamiltonian, a uniform transverse‑field term, and study the scaling of the minimum spectral gap ΔE_min between the ground state and first excited state, because the adiabatic runtime T scales as the inverse square of this gap (T ∝ ΔE_min⁻²).

Locked problems are defined by two properties: every variable participates in at least two clauses, and any two satisfying assignments cannot be connected by a single bit flip. Consequently, typical random instances possess a unique satisfying assignment (USA), which eliminates ground‑state degeneracy and makes the gap a reliable indicator of algorithmic cost. The three models are described in detail. In locked 1‑in‑3 SAT each clause contains three bits and is satisfied when exactly one bit equals 1; in locked 2‑in‑4 SAT each clause contains four bits and is satisfied when exactly two bits are 0 and two are 1, with an overall bit‑flip symmetry; in 3‑regular 3‑XOR SAT each clause contains three bits and is satisfied when the XOR of the bits matches a prescribed value. The latter model is at the satisfiability threshold (M = N) and, despite being solvable in polynomial time by Gaussian elimination over GF(2), is known to be hard for generic algorithms.

To estimate ΔE_min for system sizes beyond the reach of exact diagonalization (N > 24), the authors employ continuous‑time quantum Monte Carlo using the stochastic series expansion (SSE) method. For each size N they generate roughly 50 random instances, compute the gap ΔE(s) at a set of interpolation parameters s∈