# Tailored Quantum Alternating Operator Ansatzes for Circuit Fault Diagnostics
## 4 QAOA Approaches to CFD
### Approach 4: Transverse Field on Faults with an Oracle Circuit Simulator  
Rather than representing the wire and fault bits explicitly, in our next approach, we focus on the space of valid fault configurations (of dimensions \(2^{n-n_{q}}\)). Notice that for any invalid fault configuration, it can be made a valid configuration by simply flipping the fault bit for the output bits which are incorrect.  
Consider a fault configuration \(f=(f_{1},\ldots,f_{n-n_{q}})\) over non-output wires. The energy of the state is dependent on the number of nonzero fault bits \(P(f)=\sum_{i=1}^{n-n_{q}}f_{i}\) as well as the number of _implied_ faults needed on the outputs to make this a valid configuration \(Q(f)=\sum_{i=n-n_{o}}^{n}\left(w_{i}-S(f,i)\right)^{2}\), where \(S(f,i)\) is the value placed on the output wire \(w_{i}\) when simulating the circuit with the given inputs and the fault configuration \(f\) over the non-output wires.  
Then, \(R(f)=P(f)+Q(f)\) counts the total number of faults implied by the fault configuration \(f\) in this constrained space. By constructing an oracle that simulates the circuit and then uses this information to find the proper fault count, we have a phase separation operator with the same cost function as in Section 4.3:  
\[U_{c}(\alpha_{t})=\sum_{x\in\{0,1\}^{n-n_{o}}}e^{ia_{R}(x)}\ket{x}\bra{x}.\]  
As in QAOA1, our starting state will be \(\ket{\psi}=\ket{+}^{\otimes n-n_{o}}=\frac{1}{\sqrt{2^{n-n_{o}}}}\sum_{f\in\{0,1\}^{n-n_{o}}}\ket{f}\). and we use the one-local Pauli X mixing operators:  
\[U_{d}(\beta_{t})=\prod_{i=1}^{n}e^{-i\beta_{t}\left(\sigma_{t}^{x}\right)}.\]  
To see how \(U_{c}(t)\) can be implemented in practice, we consider a protocol where ancilla qubits are used for the computation of the faults needed on the outputs. Let \(\ket{\psi(r_{i-1})}\in\mathbb{C}^{2^{n-n_{o}}}\) be the wave function at the beginning of round \(r_{i}\), with the initial wave function defined as shown above. We show how \(\ket{\psi(r_{i})}\) is then generated over the current round. For \(r_{i}\in[r_{1},r_{1}]\), we have  
\[\ket{v(r_{i})} =U_{circ}\ket{0}_{n}\ket{\psi(r_{i-1})}\ket{0}_{n_{o}}\] \[=\sum_{x\in\{0,1\}^{n}=-n_{o}}^{n}\bra{x}\ket{\psi}f(x)_{n}\ket{x}_ {n-n_{o}}g(x)_{n_{o}} (\text{Simulate Circuit})\] \[\ket{\mu(r_{i})} =U_{c}(\alpha_{i})\ket{v(r_{i})}\] (Apply Phases) \[=\prod_{j=n+1}^{2n}e^{i\alpha_{i}(1-c_{i}^{x})/2}\sum_{x\in\{0,1 \}^{n}}\ket{f(x)}_{n}\ket{x}_{n-n_{o}}g(x)_{n_{o}}\] \[=\sum_{x\in\{0,1\}^{n-n_{o}}}\left(\sum_{j=n+1}^{2n}e^{i\alpha_{i} x_{j}}\right)\ket{f(x)}_{n}\ket{x}_{n-n_{o}}g(x)_{n_{o}}\] \[\ket{0}_{n}\ket{\phi(r_{i})}\ket{0}_{n_{o}} =U_{circ}^{t}\ket{\mu(r_{i})}\] (Undo Simulation) \[\ket{\psi(r_{i})} =\prod_{j=1}^{n-n_{o}}e^{-i\beta_{i}\sigma_{t}^{x}}\ket{\phi(r_{i })}\] (Apply Mixing)  
Here, \(U_{circ}\) implements the simulation of the faulty circuit such that \(f(x)\) is the valid configuration over the wire bits that \(\ket{x}\) specifies from the non-output fault bits, while \(g(x)\) are the required faults on the output bits such that \(x\oplus g(x)\oplus f(x)\) is a valid configuration over the \(2n\) bits.