A quantum annealer (QA, [1]) is a specific kind of quantum device designed to deal with optimization problems by means of a process inspired by classical simulated annealing [2]. These computers leverage quantum mechanics to efficiently explore solution spaces in an attempt to find the optimum value of a given objective function. Today, advances in quantum technologies have contributed to the building of intermediate-scale QAs that implement quantum annealing for programmable use. There are different platforms for building quantum annealers, such as optical tweezers [3] or superconducting integrated circuits [4], with the latter being the most recognized to date. Additionally, several companies are working on this technology and building their own devices, such as NEC ¹ , Qilimanjaro ² , and D-Wave Systems ³ .

However, all the progress made in recent years was born in the noisy intermediate-scale quantum (NISQ) era [5], when quantum devices presented great limitations in solving optimization problems efficiently, even when these problems are small- or medium-sized. As a consequence, the community as a whole is striving to devise schemes and mechanisms to address the current limitations and take advantage of the potential that quantum computing currently offers. Among the most common strategies are the design and implementation of advanced hybrid resolution schemes, which leverage the best of both computing paradigms [6]. Other alternatives include the implementation of error mitigation strategies [7] or making use of reliable quantum simulators [8].

As part of these activities, an interesting strategy studied in works such as [9–11] involves initiating the quantum evolution with a state that is an educated guess of the problem’s solution. It is interesting to note that similar strategies have also been explored in gate-based quantum computing, with the warm-start initialization of quantum approximate optimization algorithms (QAOAs) being particularly well known [12–14].

In this article, we focus on a specific strategy called reverse annealing (RA, [15]), which was introduced by D-Wave Systems in 2017. The motivation behind the implementation of RA lies in the limitations of D-Wave’s annealers to conduct local refinement of states previously found either through a forward anneal or by means of a classical technique. Thus, RA allows quantum local search by annealing backward from a previously specified state and then forward to a new one.

Today, many practical studies employ RA to improve a solution previously found by other solving schemes [16, 17]. A significant amount of research has also been published studying theoretical aspects of RA, such as its sensitivity to proper parameterization, mainly regarding the RA Schedule [18, 19]. Despite the existence of this buoyant research activity, the full potential of RA is yet to be revealed, as some crucial aspects of its performance still require a deeper analysis.

On this basis and bringing to the fore the proven potential of RA, this paper is committed to extending the frontiers of this promising mechanism, exploring its application in situations not yet studied by the community. To achieve this, the analysis provided in this work is driven by experimentation focused on answering two key research questions (RQ).

As mentioned before, RA was conceived as a local refinement mechanism. For this reason, in order to solve a problem using this paradigm, the QA must be initialized with a known (classical) solution. RA explores the local space around that solution for bitstrings with even lower energy. Despite the existence of intense research activity around RA, no one has theorized about the possible benefits related to the transfer of knowledge (ToK, [20]) under the umbrella of this paradigm. So far, all studies feed the annealer with an existing solution to the same problem, with this existing solution usually obtained either by a forward annealing process executed immediately before the RA [21] or by a classical optimization method [22]. However, it would be reasonable to think that, in a real application context where the problems faced by the industry share analogies, a previous solution may be a good starting point for the resolution of a new (or perhaps not so new, as we argue) problem. At the time of writing, the use of solutions from different problems has remained unexplored. In other words, the influence of solving a specific problem through the RA paradigm by using as input a solution from another similar but not exactly the same problem has not been studied. In this context, the first of the research questions is:

RQ1 : Is RA a paradigm that can benefit from knowledge transfer between problem instances with similar characteristics?

In relation to the analysis of RA behavior, several studies have tried to provide insights about the requirements that input solutions must meet, determining in many cases that RA is efficient when it is fed by a solution close enough to the optimum of the problem to be solved. However, the term close is usually ambiguously used [23], with studies translating this closeness into energy [24], while others into the composition of the solution itself, that is, the difference in terms of the Hamming distance between the input solution and the optimum one [25]. Thus, the second RQ posed in this paper aims to shed light on the characteristics that an input solution should have to increase the probability of success:

RQ2 : Can we infer the characteristics that an input solution should meet to help increase the probability of succeeding in an RA process?

Moving in this direction is especially interesting for ToK because it helps to identify which solution or solutions from the sampling pool are the most promising to feed RA and to set the analysis framework for future work in this area. For clarification purposes and following the nomenclature of ToK, the term source refers to the problem or task whose knowledge will be leveraged to solve a target task [26].

To properly guide the experimentation of this paper, the well-known knapsack problem (KP, [27]) has been used as a benchmarking problem. We have chosen the KP because:

• It has been extensively used for benchmarking purposes in QC-oriented studies [28–30].

Two main instances were used in the planned experimentation, composed of 14 and 16 items, respectively. We call these seed instances parent-instances. From each of them, 16 new cases with similar characteristics have been created, hereinafter named descendant-instances. With all this, the main objective of the experimentation is to analyze, given a target-instance, the impact of feeding RA with solutions obtained from similar yet different tasks.

It is worth noting that the size of the instances was not chosen randomly. This selection was made after thoroughly analyzing the current state of the art in the conjunction between QA and KP. Additionally, our motivation has been to construct instances that are, on the one hand, large enough so that D-Wave’s QA would not always solve them optimally through the forward annealing process and, on the other hand, small enough to be efficiently embedded in the quantum computer.

Lastly, we have used the Hamming distance as a reference measure to evaluate the difference between the two solutions. This measure stands out particularly for its simplicity and efficiency in calculation. Additionally, the Hamming distance is especially suited for measuring the similarity among binary strings, which are often used in quantum computing. It has also been frequently used to compare DNA sequences and error-correction codes.

As mentioned in the introduction, a QA is a specialized quantum device engineered to tackle optimization problems through a process inspired by classical simulated annealing. More specifically, a QA operates on the principle of adiabatic computation, where an initial, easily prepared Hamiltonian is gradually evolved from its ground state to the ground state of a final, problem-specific Hamiltonian. If this evolution is sufficiently slow, the adiabatic theorem ensures that the system stays in the ground state throughout the computation. In QA, the adiabatic theorem is deliberately relaxed, permitting the process to evolve more quickly than the adiabatic limit would normally permit. Consequently, transitions to higher energy states often occur during the evolution. Despite this, the computational model can also be considered universal [34]. The D-Wave annealer, which is the one used in this study, is based on an Ising Hamiltonian, which limits the kind of problems that can be executed on the computer. Nevertheless, this type of QA is as well-suited for solving combinatorial optimization problems as the one addressed in this article [35].

The RA method is a specific type of quantum annealing process designed to perform local refinement of good states found elsewhere [15]. To achieve this, RA resorts to a time-dependent Hamiltonian: H t = A s t H 0 + B s t H 1 (1) in which s ( t ) ∈ [ 0,1 ] , A ( s ) , and B ( s ) are time-dependent amplitudes that must satisfy A ( 0 ) ≫ B ( 0 ) and A ( 1 ) ≪ B ( 1 ) . Additionally, H 0 defines the initial Hamiltonian, while the final Hamiltonian H 1 corresponds to the unconstrained optimization problem, with a ground state that represents the computational solution. In contrast to forward annealing, RA reverses the time-evolution procedure by starting in an eigenstate of H 1 and evolving under Equation 1 in the opposite direction. Thus, H ( t ) = H ( 1 ) at t = 0, and the process is divided into three different steps in which the Hamiltonian.

1. First, it evolves backward to a previously specified point s p in the control schedule in a time t r = t 1 coined ramp-time.

Therefore, the RA Schedule can be defined as shown in the following Equation 2 [18] s t = 1 + s p − 1 t 1 t , 0 ≤ t ≤ t 1 , s p , t 1 ≤ t ≤ t 2 , s p + 1 − s p T − t 2 t − t 2 , t 2 ≤ t ≤ T . . (2)

Before starting with the description of the experimentation, it is important to note that, at present, no metric accurately indicates how similar any two problems are to each other (in order to make them good candidates for ToK). The existence of such a metric would be very advantageous for this purpose. More specifically, similarity should be measured in terms of overlap in the energy landscape of a unified search space [36]. However, in the real world, it is not possible to determine this overlap beforehand without having executed and, therefore, solved the problem. Given this circumstance, similarity becomes a matter of intuition and is subject to the domain knowledge acquired by the expert. This situation does not detract an iota of value because, in real industrial contexts, most problems have a repetitive nature: a pool of recurring customers in routing problems, production of similar materials with stable task typologies and machines, etc. In such situations, ToK is a promising strategy to resort to past solutions in order to improve the results or speed up the computation of a new yet similar task.

With this in mind, the experiment carried out in this paper uses the well-known KP as a benchmarking problem. In a nutshell, KP consists of a set P composed of n items, describing each item p i by profit ( v i ) and weight ( w i ) , which must be packed into a container with a maximum capacity W . The objective, therefore, is to select a subset of items to be stored that, without exceeding W , maximizes the profit obtained. It is worth noting that the metric used in this study for measuring the quality of a solution corresponds to the energy provided by the quantum annealer, which should be minimized. That is, the less energy, the better the solution.

Aiming to provide an answer to the RQs, we have designed two separate experiments focused on two parent-instances of the KP, which are composed of 14 (s14) and 16 (s16) items, respectively. Both instances have been generated ad hoc for this study with the values v and w randomly selected from {1,2,3,4}. Finally, W = ( ∑ i = 1 n w i ) / 2 .

Taking these cases, each descendant-instance has been generated by applying the following strategy.

• First, the newly generated descendant-instance starts out as an exact copy of the corresponding parent-instance.

Following this strategy, 16 descendant-instances have been created for each parent-instance, equally divided into four categories.

• X_L2L: Lowest-Energy-to-Lower-Energy: X% of the less energetic variables (i.e., items) of the parent-instance are selected and modified so that their energetic impact is even lower.

It is noteworthy that solutions to these problems are represented by a ( n + s ) -long vector, where n is the number of items that compose the instance, and s are slack terms introduced by the penalties. For both s14 and s16, s = 5 , so that the decision variables for each problem are 19 and 21, respectively. In our study, the variables eventually modified correspond only to the first n items.

Systematically, knowledge transfer materializes in our experimentation by taking a descendant-instance as the source-task and a parent-instance as the target-task to be solved. That is, the goal is to solve a parent-instance leveraging the knowledge of a previously solved descendant-instance.

First, all instances have been independently executed 10 times using D-Wave’s Advantage_system6.4 by means of the common forward annealing process. Regarding the parameters, the number of runs was set to 1,000. Furthermore, the Knapsack class belonging to the Qiskit v0.6.0 Optimization Applications open library was employed for generating the KP QUBOs ⁴ . Thus, Table 1 includes.

• The best energy found for each instance across the 10 independent runs.

With all this in mind, we now proceed to answer the two questions posed in the introduction.

In this work, we have preliminarily studied the influence of transferring knowledge between similar tasks through the reverse annealing mechanism implemented by D-Wave. To do that, two research questions have been posed:

RQ1 : Is RA a paradigm that can benefit from knowledge transfer between problem instances with similar characteristics?

RQ2 : Can we infer the characteristics that an input solution should meet to help increase the probability of succeeding in an RA process?

First, answering RQ1 and in view of the positive results obtained, the transfer of knowledge in quantum computing appears to be a promising research avenue that undoubtedly deserves further investigation. As for RA, it is commonly applied as a local refinement procedure right after an optimization process carried out by forward annealing or a classical optimization technique. In this work, RA serves as a mechanism for knowledge transfer, allowing the reuse of a solution obtained in an independent optimization process.

It is important to highlight that, aside from the results regarding performance, knowledge sharing pursues savings in the use of computational resources. These savings are mainly materialized by resorting to previous outcomes when solving a new task under the assumption that industries are likely to face tasks that have much in common with each other. This is especially useful in real-world environments where companies accumulate results from past planning exercises and continually receive new tasks that are sometimes not very distinct from those already executed.

Second, answering RQ2 and according to the results depicted, the closeness in terms of energy is not related to the performance of RA, and only the closeness with respect to the Hamming distance is. This means that the neighborhood must be based on solution codification and not energy, making it a priority to organize the coding of the target problem so that it fits as much as possible with the source problem through a unified search space [36].In light of these positive results, several open research questions should be further studied in subsequent investigations, which can be summarized as

■ Future work of considerable importance will likely focus on advancing toward the generalization of the conclusions drawn from this study. To reach this ambitious goal, the following steps are essential: