# Tailored Quantum Alternating Operator Ansatzes for Circuit Fault Diagnostics
## 4 QAOA Approaches to CFD
### Size of the Relevant Constrained Space and Summaries for the Approaches  
In each of the approaches delineated in Section 3, the size of the constrained space of interest which the evolution of the wave function is limited to differs. In Table 2 and Figure 2, each approach is summarized by the type of mixing operator, the type of cost function, the size of the constrained space, and the complexity of each round.  
While the approaches in Sections 4.3 and 4.4 differ greatly in the type of mixing and phase separation operators used, they constrain the system to the same subspace. Consider the exploration of this space under these two different approaches. The latter begins with a uniform superposition over all valid fault configurations, and the neighborhood of each computational basis state under the action of each Pauli X operator in the mixer is a state that is a bit flip away, similar to a high-dimensional hypercube. The former begins with a single feasible state, and the neighborhood of each state under the action of the mixing operator is dependent on the available transformations around each gate to move it to a new fault configuration.  
Moreover, while the cost functions associated with these two approaches are _equivalent_, the resulting phase separation operators for implementing these cost functions are very different. For example, while each call to the cost function runs in \(\mathcal{O}(1)\) for the fault counting in the former, each call to the cost function in the latter will require simulating the corresponding circuit, which has a particular depth \(C_{p}\). For example, for the distribution  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Approach** & **Mixing Operator** & **Cost Function** & **Size of Constrained Space** & **Round Complexity** \\ \hline
1 & Pauli X & Ising & \(\mathcal{O}(2^{n})\) & \(\mathcal{O}(1)\) \\
2 & XY mixer and X & Ising & \(\mathcal{O}(2^{n-n_{o}}(n+n_{o})^{n_{o}})\) & \(\mathcal{O}(1)\) \\
3 & Gate-based & Pauli Z & \(\mathcal{O}(2^{n-n_{o}})\) & \(\mathcal{O}(1)\) \\
4 & Pauli X & Circuit Oracle & \(\mathcal{O}(2^{n-n_{o}})\) & \(\mathcal{O}(C_{p})\) \\
5 & XY mixer & Circuit Oracle & \(\mathcal{O}(n^{n_{o}})\) & \(\mathcal{O}(C_{p})\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Depending on our selection of mixing and phase-separating operators, we can constrain the evolution of the wave function to constrained spaces of differing sizes.  
Figure 2: A visual representation of the subspaces that the wave function is kept within during evolution through each QAOA approach.  
of the circuits introduced in Section 5, the depth of the classical circuit grows as \(\mathcal{O}(\sqrt{n})\) if we choose the diameter and depth to be equal. (This relationship holds for the family considered for the experiments in this paper.)  
Approaches 2 and 5 require the preparation of a particle number-conserving state studied in many areas of quantum computation and quantum chemistry [26, 27, 28, 29]. We refer the reader to the appendix of [23] for a discussion of this state in the context of a QAOA with the XY mixer. In general, for a uniform superposition over \(n\) qubits preserving a number \(k\), the state can be constructed with \((^{n}_{k})=\mathcal{O}\Big{(}n^{k}\Big{)}\) CNOT gates [29], which can be prohibitive for a circuit with a large bound on the number of faults. However, the authors of [26] provided a projective measurement method that grows polynomially with \(n\) and independent of \(k\) for large \(n\) values. Alternatively, one can use an approach similar to Approach 3, where the initial state is a computational basis state in the feasible space. The initial state selection for an QAOA remains an important area of current research [23], including warm starts [30, 31]. Moreover, the number of minimum faults for many circuits of interest can be much less than \(n_{o}\), and one could adapt the approaches to utilize this smaller space.