Victoria S. Ordonez | Searcharxiv

arXiv2026

Quantum algorithm for Valiant-Vazirani reduction

There is growing interest in extensions of the standard model of gate-based quantum computation to include auxiliary degrees of freedom evolving according to a nonlinear Schrödinger equation. By reducing the Boolean satisfiability problem SAT to quantum state discrimination, Abrams and Lloyd argued that the right type of nonlinearity can be used to solve NP and #P problems in polynomial time, at least in an idealized noise-free limit. For practical implementation, however, we are restricted to simulated and emergent nonlinearities, such as that appearing in mean field models for ultracold atoms and similar ensembles. A prominent example is the torsion model, which arises in two-component Bose-Einstein condensates and spin models with all-to-all Ising interaction. But torsion-based state discrimination appears to fall short of solving SAT. Here we close this gap by constructing the filtered oracle of the Valiant-Vazirani theorem, providing a randomized polynomial-time reduction from SAT to UNIQUE SAT, a promise problem where there is at most 1 satisfying assignment. In the noise-free limit, the UNIQUE SAT problem can be solved in polynomial time using torsion nonlinearity. Quantum Valiant-Vazirani reduction is no faster than the efficient classical version, but a fault-tolerant implementation coupled to a nonlinear quantum coprocessor simulating torsion would enable polynomial time solution to NP (but not #P) problems.

quant-ph↗

arXiv2026

Nonlinear Hamiltonians and Boolean satisfiability

We consider an extended model of quantum computation where a scalable fault-tolerant quantum computer is coupled to one or more ancilla qubits that evolve according to a nonlinear Schrödinger equation. Following the approach of Abrams and Lloyd, an efficient quantum circuit evaluating an $n$-bit Boolean function in conjunctive normal form is used to prepare an ancilla encoding its number $s$ of satisfying assignments ($0 \le s \le 2^n$). This is followed by a nonlinear quantum state discrimination gate on the ancilla qubit that is used to learn properties of $s$. Here we consider three types of state discriminators generated by different nonlinear Hamiltonians. First, given a restricted Boolean satisfiability problem with the promise of at most one satisfying assignment ($ 0 \le s \le 1$), we show that a qubit with $\langle σ^z \rangle σ^z$ nonlinearity can be used to efficiently determine whether $s = 0$ or $s = 1$, solving the UNIQUE SAT problem. Here $\langle A \rangle := \langle ψ| A |ψ\rangle $ denotes expectation in the current state. UNIQUE SAT is NP-hard under a randomized polynomial-time reduction (of course any discussion of complexity assumes a scalable, fault-tolerant implementation). Second, for unrestricted satisfiability problems with $ 0 \le s \le 2^n$, a Hamiltonian with $ \langle σ^x \rangle σ^y - \langle σ^y \rangle σ^x$ nonlinearity can be used to efficiently determine whether $s=0$ or $s>0$, thereby solving 3SAT, which is NP-complete. Finally, we show that $ \langle σ^y \rangle \langle σ^z \rangle σ^x - \langle σ^x \rangle \langle σ^z \rangle σ^y $ nonlinearity can be used to efficiently measure $s$ and solve #SAT, which is #P-complete. The nonlinear models are of mean field type and might be simulated with ultracold atoms.

quant-ph↗