Constructs quantum filtered oracle for Valiant-Vazirani theorem reducing SAT to UNIQUE SAT, enabling polynomial-time NP solution via torsion nonlinearity in noise-free limit but not #P.
Notes on nonlinear quantum algorithms
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abstract
Recenty Abrams and Lloyd have proposed a fast algorithm that is based on a nonlinear evolution of a state of a quantum computer. They have explicitly used the fact that nonlinear evolutions in Hilbert spaces do not conserve scalar products of states, and applied a description of separated systems taken from Weinberg's nonlinear quantum mechanics. On the other hand it is known that violation of orthogonality combined with the Weinberg-type description generates unphysical, arbitrarily fast influences between noninteracting systems. It was not therefore clear whether the algorithm is fast because arbitrarily fast unphysical effects are involved. In these notes I show that this is not the case. I analyze both algorithms proposed by Abrams and Lloyd on concrete, simple models of nonlinear evolution. The description I choose is known to be free of the unphysical influences (therefore it is not the Weinberg one). I show, in particular, that the correct local formalism allows even to simplify the algorithm.
fields
quant-ph 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Nonlinear Hamiltonians on ancilla qubits enable efficient solution of UNIQUE SAT with ⟨σ^z⟩σ^z, 3SAT with ⟨σ^x⟩σ^y - ⟨σ^y⟩σ^x, and #SAT with ⟨σ^y⟩⟨σ^z⟩σ^x - ⟨σ^x⟩⟨σ^z⟩σ^y nonlinearity.
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Quantum algorithm for Valiant-Vazirani reduction
Constructs quantum filtered oracle for Valiant-Vazirani theorem reducing SAT to UNIQUE SAT, enabling polynomial-time NP solution via torsion nonlinearity in noise-free limit but not #P.
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Nonlinear Hamiltonians and Boolean satisfiability
Nonlinear Hamiltonians on ancilla qubits enable efficient solution of UNIQUE SAT with ⟨σ^z⟩σ^z, 3SAT with ⟨σ^x⟩σ^y - ⟨σ^y⟩σ^x, and #SAT with ⟨σ^y⟩⟨σ^z⟩σ^x - ⟨σ^x⟩⟨σ^z⟩σ^y nonlinearity.