A measurement-driven quantum algorithm for SAT: Performance guarantees via spectral gaps and measurement parallelization

TL;DR

The paper analyzes a measurement-driven quantum SAT solver that encodes SAT into a rotated Hamiltonian $H(\theta)$ and uses alternating projections via clause-check measurements. It proves an exponential trade-off between the Hamiltonian gap $\Delta(\theta)$ and the per-cycle success probability, controlled by the rotation angle $\theta$, and provides a general runtime bound that improves with a tunable $\theta$ and possible amplitude amplification. The authors introduce a parallelization scheme based on perfect hash families to reduce measurements to $\mathcal{O}(\ln n)$ layers and develop rigorous readout routines capable of extracting a solution for both unique and multiple-solution instances. They show that for certain restricted inputs (e.g., Unate-SAT) and appropriate $\theta$, the algorithm's runtime collapses from exponential to polynomial, aligning with known classical tractability. Open questions include establishing non-trivial lower bounds on the spectral gap and extending the framework to MAX-SAT, with implications for potential super-quadratic quantum advantages under favorable gap behavior.

Abstract

The Boolean satisfiability problem (SAT) is of central importance in both theory and practice. Yet, most provable guarantees for quantum algorithms rely exclusively on Grover-type methods that cap the possible advantage at only quadratic speed-ups, making the search for approaches that surpass this quadratic barrier a key challenge. In this light, this work presents a rigorous worst-case runtime analysis of a recently introduced measurement-driven quantum SAT solver. Importantly, this quantum algorithm does not exclusively rely on Grover-type methods and shows promising numerical performance. Our analysis establishes that the algorithm's runtime depends on an exponential trade-off between two key properties: the spectral gap of the associated Hamiltonian and the success probability of the driving measurements. We show that this trade-off can be systematically controlled by a tunable rotation angle. Beyond establishing a worst-case runtime expression, this work contributes significant algorithmic improvements. First, we develop a new readout routine that efficiently finds a solution even for instances with multiple satisfying assignments. Second, a measurement parallelization scheme, based on perfect hash families, is introduced. Third, we establish an amplitude-amplified version of the measurement-driven algorithm. Finally, we demonstrate the practical utility of our framework: By suitably scheduling the algorithm's parameters, we show that its runtime collapses from exponential to polynomial on a special class of SAT instances, consistent with their known classical tractability. A problem we leave open is to establish a non-trivial lower bound on the spectral gap as a function of the rotation angle. Resolving this directly translates into an improved worst-case runtime, potentially realizing a super-quadratic quantum advantage.

Figures (3)

• Figure 1: Illustration of (a) the unrotated and (b) the rotated settings. The top panels show the locations of the corresponding states in the XZ-projection of the Bloch sphere. The bottom panels illustrate, for two projectors, how convergence proceeds in each setting. Using an orthogonal encoding ($\theta=\frac{\pi}{2}$), we converge to the ground space at $(0,0)$ with a single pass of each clause check. In the non-orthogonal setting with $\theta \neq \frac{\pi}{2}$, we slowly converge towards the ground space.
• Figure 2: Algorithmic primitive of the measurement-driven approach. We sequentially perform measurements $\{C_i(\theta), P_i(\theta)\}$, with the measurement outcomes associated with $C_i(\theta)$ driving the state in the desired direction. Whenever we encounter an undesired outcome, we restart the procedure. The procedure stops once we are sufficiently close to our target state.
• Figure 3: Decision process for the proposed readout. If the estimate upon performing a $Z$-measurement on one of the qubits excludes for sure either $-\sin(\theta)$ or $\sin(\theta)$, then decide for the corresponding assignment variable. To be precise: if $-\sin(\theta)$ is excluded, then propagate TRUE on corresponding variable, if $\sin(\theta)$ is excluded, then propagate for FALSE on variable $x_i$. In the orange-colored case, we are not yet in a clear regime. Therefore, we need more shots to minimize the error bar. In the pink-colored regime, we can be relatively sure that TRUE is the right choice. Therefore, fix the variable.

Theorems & Definitions (52)

• Lemma 1: Overlap between the initial state and any rotated state
• proof
• Lemma 2: Commutation relations
• proof
• Proposition 3: Alternating projections from Ref. escalante2011alternating (see Eq. (3.8)), adjusted to our setting
• Lemma 4: Cycle bound
• proof
• Lemma 5: Success probability
• proof
• Theorem 6: State preparation cost