🤖 AI Summary
This work addresses the challenges of implementing Grover search for implication-based model checking, particularly the requirement of minority-state conditions and the difficulty of executing highly entangling operations on near-term quantum hardware. To overcome these limitations, the authors propose a simplified Eigenmarking scheme that employs only a single ancillary qubit and a universal two-qubit controlled-phase gate (CCZ), reducing the original multi-controlled phase rotation to a doubly controlled operation. This approach substantially diminishes reliance on high-entanglement states while preserving effective amplitude amplification and unsatisfiability detection capabilities. Consequently, it significantly alleviates hardware demands and enhances scalability. Simulation results demonstrate that the proposed method achieves markedly superior performance, with a minimum relative local winning rate of W = 3.17 and discriminability of D = 0.769, outperforming both conventional marking (W = 0.67, D = 0.19) and fine-grained marking (W = 0.28, D = 0.55).
📝 Abstract
Targeting entailment model checking, a recent study has pioneered an idea of Eigenmarking search, an improvement over Grover search using extra qubits. The extra qubits condition the quantum state evolution such that the answer states (if exist) are always in the minority. The minority criteria is essential to Grover probability-amplitude amplification and consequently the effectiveness of Grover search. In addition to enforce the minority criteria, Eigenmarking also employs complementary states (through well-orchestrated phase rotation) for easy identification of a no-answer case (related to a no-violation case in the context of model checking). Eigenmarking search has been shown effective in two-qubit simulations.
The three Eigenmarking schemes have been previously proposed. Two schemes require two extra qubits. One scheme (called ``subtle marking'') requires one extra qubit with a multiple-qubit-controlled phase rotation. Our study refines the mechanism using only one extra qubit with only two-qubit-controlled phase rotation, commonly known as \texttt{ccz}, regardless of how many qubits the input has. Using a multiple-qubit-controlled phase rotation (as in subtle marking) associates with highly entangled states. Highly entangled states in a real quantum hardware are difficult (or in some cases may even be unachievable) particularly in a scaled up scenario involving many qubits. Our proposed new Eigenmarking scheme has lightened the burden for the hardware requirement. The new Eigenmarking search has been experimented in two-qubit-system simulations and shown viable, achieving the minimal relative local winning margin of W=3.17 and the worst-case distinguishability of D=0.769 (cf. W=0.67; D=0.19 from conventional marking and W=0.28; D=0.55 from subtle marking).