NP-hard but no longer hard to solve? Using quantum computing to tackle optimization problems

Abstract

In the last decade, public and industrial research funding has moved quantum computing from the early promises of Shor’s algorithm through experiments to the era of noisy intermediate scale quantum devices (NISQ) for solving real-world problems. It is likely that quantum methods can efficiently solve certain (NP-) hard optimization problems where classical approaches fail. In our perspective, we examine the field of quantum optimization, that is, solving optimization problems using quantum computers. We provide an entry point to quantum optimization for researchers from each topic, optimization or quantum computing, by demonstrating advances and obstacles with a suitable use case. We give an overview on problem formulation, available algorithms, and benchmarking. Although we show a proof-of-concept rather than a full benchmark between classical and quantum methods, this gives an idea of the current quality and capabilities of quantum computers for optimization problems. All observations are incorporated in a discussion on some recent quantum optimization breakthroughs, current status, and future directions.

A thermodynamic approach to optimization in complex quantum systems

“…In appendix B, we complement our study by investigating the effect of the parameter α renormalizing the auxiliary system Hamiltonian (see equation ( 5)) on the cooling protocol. Furthermore, we consider the effect of freezing the system dynamics by reducing the interaction strength J appearing in equation (7). We also show that the long time behavior of the system undergoing repeated collision with the auxiliary bath is practically independent of the chosen initial state.…”

“…In the main text we have seen that the optimal cooling effect obtained with repeated collisions with the A system is obtained for f ≃ 0.6, see equations ( 5)- (7). In this appendix we compare the behavior of the auxiliary system with (J ̸ = 0 in equation ( 7)) or without interaction (J = 0) with the problem Hamiltonian.…”

“…While the critical value of f for the Hamiltonian H A is f c = 1/2, such a value is renormalized by the interaction with the system through the interaction Hamiltonian, and the value of f required to induce disorder in the auxiliary system becomes larger, f ≃ 0.6, see the discussion in appendix A. We have checked that adding a term of the type σ y i Σ y i to the Hamiltonian (7) does not lead to any substantial change to the results shown in figure 4. Comparing the results of figure 4 with those of figure 2, we see that that connecting the system to the external cooler improves significantly the optimization of the problem's energy.…”

“…neural networks [2], protein folding [3], socioeconomic behavior such as drug and tobacco use [4] and trading models in stock markets [5]. For reviews and perspectives on quantum computing applied to combinatorial optimization, see [6][7][8].…”

“…Their thickness represents their random relative magnitude. The gray broken lines represent the interaction between the spins in the system and in the bath, as given by J in equation (7). At the end of every collisional stroke the exhaust auxiliary chain is wasted and replaced by a new chain.…”