In a previous post, we discussed how quantum annealing is an efficient technique for finding global minima in complex optimization problems by gradually guiding the system toward its lowest energy state. This approach has proven to be highly effective in navigating vast solution spaces and escaping local minima. Now, we will take a step further by introducing reverse annealing, a powerful variation that enhances the search process by starting from a known solution and refining it through local exploration. This method allows for a more targeted optimization, making it particularly useful when an initial good solution is already available and needs to be improved [1].
In quantum annealing, we start with an initial Hamiltonian (Hin), where the minimum energy state is a superposition of all possible states. This is followed by a transition to a final Hamiltonian, whose minimum corresponds to the solution of the optimization problem formulated as a QUBO (Hf). One aspect of quantum annealing those merits particular attention is the schedule. The schedule defines the time-dependent evolution of the system’s Hamiltonian, dictating how the quantum state transitions from the initial superposition to the final ground state, defined by A(s), B(s) in:
Ht = A(s)Hin + B(s)Hf (1)
In the conventional quantum annealing process, the schedule is comprised of a gradual transition that occurs under the conditions: A(0) ≫ B(0) y A(1) ≪ B(1). The standard quantum annealing schedule allows us to find the minimum by searching the entire state space. However, sometimes we know an approximation to the solution in advance, and it will be more useful to search in the neighbourhood of this solution. In this context, reverse annealing appears as an alternative that allows for this local exploration. The generalisation of quantum annealing for local searches comprises alterations to the schedule. In the case of reverse annealing, the starting and ending point is the Hamiltonian of the problem (Hf), given that both the approximate and the desired solution are minima of this Hamiltonian. During the annealing process, the Hamiltonian of the transverse field (Hin) is introduced to permit the state, via effects such as tunneling, to escape the local minimum and identify the optimal solution.

Reverse annealing presents a significant advantage in the context of portfolio optimization by effectively combining classical and quantum techniques to refine existing solutions and achieve superior results [2]. In portfolio optimization, the objective is to find the optimal allocation of assets that maximizes return while minimizing risk. This problem is highly complex due to the large solution space and the presence of numerous local minima, which often causes classical optimization methods to get stuck in suboptimal configurations. Classical approaches, such as greedy local search or genetic algorithms, can quickly generate a good initial solution but are limited in their ability to escape these local minima. Reverse annealing addresses this challenge by starting from a known classical solution and using a quantum annealer to explore the surrounding solution space more effectively. This hybrid approach leverages the speed and efficiency of classical methods for initial solution generation and the unique capabilities of quantum tunneling to refine these solutions further, exploring beyond the reach of conventional algorithms. The ability to start from a classical solution reduces the time and computational resources required for the quantum annealer to converge on a superior solution. In practical terms, this means that reverse annealing can significantly improve the quality of the optimized portfolio by finding allocations that better balance risk and return.
References
[1] Nicholas Chancellor. Modernizing quantum annealing using local searches. New Journal of Physics, 19(2):023024, 2017.
[2] Davide Venturelli and Alexei Kondratyev. Reverse quantum annealing approach to portfolio optimization problems. Quantum Machine Intelligence, 1(1):17–30, 2019.

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