# Quantum Circuit Synthesis Using Fuzzy-Logic-Assisted Genetic Algorithms
## 3 Related Work  
Several exact analytical compilation techniques have been proposed in the current literature [32; 33; 34; 35], each using different compilation techniques and mathematical frameworks [36; 37], with different goals and objectives. Many of these have also been incorporated into industry standard compilation frameworks [38]. Additionally, exact compilation techniques often incorporate staged compilation, where an initial compiled circuit is further optimized using dedicated depth reduction techniques [39; 40]. However, exact techniques are limited by two prevailing factors. First, exact compilers generate very deep circuits that are unsuitable for NISQ hardware, which have very short decoherence times. Second, attempts to produce optimal depth circuits, while adhering to qubit topologies, quickly become computationally infeasible for all but the smallest of qubit sizes using these methods. To overcome these limitations, approximate methods trade-off compilation error to produce circuits with reduced depths and/or fewer gates while also trying to be scalable to a higher number of qubits. These compilers use various types of optimization techniques, such as numerical instantiation [41; 42; 43; 11; 44; 45], SAT-solvers [46] and other methods [47]. We \begin{table}
\begin{tabular}{c c} \hline \hline
**Boolean Operator** & **Fuzzy Equivalent** \\ \hline \(AND(a,b)\) & \(\min(a,b)\) \\ \(OR(a,b)\) & \(\max(a,b)\) \\ \(NOT(a)\) & \(1-a\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fuzzy equivalent of Boolean operators. refer the interested reader to a detailed survey of many such compilation methods that have been presented in [48]. GA-based compilation methods are a subcategory of approximate compilation. Several GA-based techniques have been proposed which target circuit compilation, generation, discovery and related tasks. For instance, a simple three-qubit quantum teleportation circuit was evolved using a GA in [49], while [50] introduced an encoding scheme to represent quantum circuits as 1-dimensional array chromosomes. [PERSON] et al. developed a GA-based circuit synthesis technique that analyzes circuit costs in terms of number of gates used and utilizes predetermined quantum operation primitives to generate low-cost quantum circuits [51]. The Q-PACE framework introduced in [52] leveraged Genetic Programming (GP) [53] to evolve optimized quantum circuits, programs, and algorithms. [PERSON] et al. proposed a software flow for quantum circuit synthesis with runtime adaptation of GA parameters based on operator performance [54]. [PERSON] et al. utilized a Non-dominated Sorting Genetic Algorithm (NSGA) for multi-objective circuit synthesis, with objectives of circuit error and cost reduction, incorporating unique genetic operators specifically designed for quantum circuits [54]. Another novel technique utilized evolutionary methods that leverage quantum computers to produce optimized circuits [55]. The work in [56] introduced the use of island GAs [57], which promote diverse evolution trajectories by running separate, mutually exclusive instances of GAs, occasionally allowing crossover between these instances to enhance the search for optimal solutions. The GA4 QCO framework [58] proposed a comprehensive set of crossover and mutation methods, along with an encoding scheme that represents circuits as 2-dimensional arrays of gates, allowing for a wide range of genetic operations and allows the user to set ad hoc fitness functions. GAs have also been used in tasks adjacent to quantum compilation, such as discovering unitary matrices [59], optimizing specific algorithms [60], and minimizing routing overhead in circuits [61]. Additionally, GA-based techniques for quantum state preparation have been developed to produce efficient circuits that generate desired quantum states, as demonstrated by [PERSON] et al. in [62] and the work using the NSGA-II algorithm in [63]. One more area of research where GAs have seen a lot of interest in is variational quantum circuits.