# Tailored Quantum Alternating Operator Ansatzes for Circuit Fault Diagnostics
## 1 Introduction  
We introduce and characterize several different quantum alternating operator ansatz (QAOA) [1] approaches to solving the combinational circuit fault diagnostic (CCFD) problem, a combinatorial optimization problem of importance for diagnosing faults in circuits. In particular, we identify different constrained spaces of interest and explore various constructions of mixing and cost operators that allow us to evolve within each constrained subspace, allowing the protocol to focus on bringing the support of the system's wavefunction to an optimal or approximately optimal state within this subspace that can be significantly smaller than the space of the physical qubits needed to run the system. There are many such relevant subspaces, and through a different selection of mixing and phase separation operators, even the same subspaces can be explored in very different ways. After describing QAOAs for optimization problems, we delineate the stuck-at-fault model of CCFD problems considered in this paper and how they can be cast as optimization problems. We then develop several different ansatzes for solving this optimization problem, beginning with the least constrained (measured as the resulting size of the constrained space maintained) and ending with the most constrained. The most constrained ansatz has a superpolynomially smaller constrained space. For circuits with a logarithmic or lower minimum fault explanation, it can be modified to have a constrained space thatgrows subexponentially in the number of wires while the most generic approach grows exponentially. The methods used for constructing ansatzes in this domain can be useful for constructing more tailored approaches for other domains, which is an interesting avenue for future quantum algorithms. The results from running each ansatz under several parameter selections indicate that ansatzes more closely tailored to a problem can have better performance than more generic ansatzes. The relative success of the simple parameter selection methods used in this manuscript are of interest to researchers focusing on QAOAs for other domains. For many classes of combinatorial optimization problems that are NP-hard, the features of specific instances can make them, in practice, more or less accessible to different algorithms. The features of typical CCFD instances, such as a planar-like topology that is often amendable to embedding on local two-dimensional lattice quantum architectures, make them a strong candidate for quantum algorithms utilized on NISQ devices. Given the real-life applications of CCFD problems, they have become problems of interest for leveraging the potential quantum advantage for industrial purposes [2]. Our results in this manuscript support that tailored ansatzes should be explored on NISQ quantum computers to solve CCFD problems. The family of scalable, dense, and highly connected circuits we introduce is a useful testbed for such devices. Even between different QAOA algorithms, the features of particular instances can affect which approaches are more or less promising. For example, we consider an ansatz that is similar to a recently studied CQA approach [3] that shows promise in the regime where an instance has higher degeneracy in the solution space and higher minimum fault explanations, while another approach maintains a subexponential constrained space when there is an upper bound on the minimum fault explanation. This suggests that the scaling of the minimum fault explanation for the instances of interest is important for determining the best ansatz to utilize. However, there are still more considerations for implementing these approaches on quantum systems, including the preparation of the initial state and considering the best implementation of the required unitaries through a specific gate set.