Abstract
We give a quantum algorithm to exactly solve certain problems in combinatorial optimization, including weighted MAX-2-SAT as well as problems where the objective function is a weighted sum of products of Ising variables, all terms of the same degree D; this problem is called weighted MAX-ED-LIN2. We require that the optimal solution be unique for odd D and doubly degenerate for even D; however, we expect that the algorithm still works without this condition and we show how to reduce to the case without this assumption at the cost of an additional overhead. While the time required is still exponential, the algorithm provably outperforms Grover's algorithm assuming a mild condition on the number of low energy states of the target Hamiltonian. The detailed analysis of the runtime dependence on a tradeoff between the number of such states and algorithm speed: fewer such states allows a greater speedup. This leads to a natural hybrid algorithm that finds either an exact or approximate solution.
Highlights
While quantum algorithms are useful for many problems involving linear algebra, there are few proven speedups for combinatorial optimization problems
The most basic such speedup is Grover’s algorithm14, which gives a quadratic speedup over a brute-force search. For a problem such as finding the ground state of an Ising model on N spins, this can lead to a speedup from a brute force time O∗(2N ) (where O∗(·) is big-O notation up to polylogarithmic factors, in this case polynomials in N ) to O∗(2N/2)
Does that algorithm require exponential space, but it is not known how to give a Grover speedup of this algorithm, so that no quantum algorithm is known taking time O∗(2cN/2) for any c < 1. This algorithm is specific to constraint satisfaction problems where each constraint only involves a pair of variables, rather than a triple or more
Summary
We require that the optimal solution be unique for odd D and doubly degenerate for even D; we expect that the algorithm still works without this condition and we show how to reduce to the case without this assumption at the cost of an additional overhead. The detailed analysis of the runtime depends on a tradeoff between the number of such states and algorithm speed: having fewer such states allows a greater speedup. This leads to a natural hybrid algorithm that finds either an exact or approximate solution
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