← 研踪 PaperTrailarXiv:2611.15794 · v1
On Multi-Constrained Path Finding utilizing Grover Search and Decoded Quantum Interferometry
首次提交 2026-11-12 02:08
版本更新 2026-11-12 02:08
雷达观测 2026-11-13 14:16
摘要
Finding a path whose totals under several additive metrics all fall inside prescribed ranges is weakly NP-complete. We ask what two quantum algorithms can do for it, against strong classical baselines. For directed acyclic graphs (DAGs) we give a Grover search whose state preparation builds the uniform superposition over all source-to-destination paths with one qubit per arc and no ancilla. We prove it correct, give its exact gate count, and show that measured strings form an error- detecting code whose distance is a girth-like graph parameter. We complete the oracle and count its gates: one search on our largest instance, 136 arc qubits and 262,144 paths, needs about 4.4 x 10^5 Toffoli gates, while classical dynamic programming decides the same instance with 18,754 labels. Grover search needs O(sqrt(N/M)) oracle calls for M good paths among N . Classical meet-in-the-middle matches this exponent in the worst case, and known quantum algorithms with quantum memory beat it. For a staged fabric with node-independent linear weights, maximizing the number of satisfied metrics is max-LINSAT, and with power- of-index weights it is Optimal Polynomial Intersection, where no known polynomial-time classical algorithm matches decoded quantum interferometry (DQI). We verify this translation in simulation and run two small binary instances on IBM Heron processors, which keep 67 % and 15 % of the ideal gain over uniform sampling.
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v1 · 2026-11-12 02:08 · 最新观测版本
On Multi-Constrained Path Finding utilizing Grover Search and Decoded Quantum Interferometry
Finding a path whose totals under several additive metrics all fall inside prescribed ranges is weakly NP-complete. We ask what two quantum algorithms can do for it, against strong classical baselines. For directed acyclic graphs (DAGs) we give a Grover search whose state preparation builds the uniform superposition over all source-to-destination paths with one qubit per arc and no ancilla. We prove it correct, give its exact gate count, and show that measured strings form an error- detecting code whose distance is a girth-like graph parameter. We complete the oracle and count its gates: one search on our largest instance, 136 arc qubits and 262,144 paths, needs about 4.4 x 10^5 Toffoli gates, while classical dynamic programming decides the same instance with 18,754 labels. Grover search needs O(sqrt(N/M)) oracle calls for M good paths among N . Classical meet-in-the-middle matches this exponent in the worst case, and known quantum algorithms with quantum memory beat it. For a staged fabric with node-independent linear weights, maximizing the number of satisfied metrics is max-LINSAT, and with power- of-index weights it is Optimal Polynomial Intersection, where no known polynomial-time classical algorithm matches decoded quantum interferometry (DQI). We verify this translation in simulation and run two small binary instances on IBM Heron processors, which keep 67 % and 15 % of the ideal gain over uniform sampling.
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