Nonlinear Hamiltonians and Boolean satisfiability

TL;DR

This paper explores how nonlinear Hamiltonians in quantum systems can efficiently solve complex Boolean satisfiability problems, potentially outperforming classical algorithms.

Contribution

It introduces specific nonlinear Hamiltonians that enable efficient quantum solutions to NP-hard and #P-hard problems like SAT and #SAT.

Findings

Nonlinear Hamiltonians can distinguish between zero and non-zero solutions efficiently.

The approach solves NP-hard problems such as 3SAT using nonlinear quantum dynamics.

It demonstrates the potential of mean field nonlinear models for quantum computation of complex problems.

Abstract

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 Schr\"odinger 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 $\langle \sigma^z \rangle \sigma^z$ nonlinearity can be used to efficiently determine whether $s = 0$ or $s = 1$, solving the UNIQUE SAT problem. Here $\langle A \rangle := \langle \psi | A |\psi \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 \sigma^x \rangle \sigma^y - \langle \sigma^y \rangle \sigma^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 \sigma^y \rangle \langle \sigma^z \rangle \sigma^x - \langle \sigma^x \rangle \langle \sigma^z \rangle \sigma^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.