MQPA

What is / and hov is MQPA (Molecular Quantum Particle Algorithm) represents in mathe and how itt’s applied in fidelity calculations :

\[ F(\rho, \sigma) = \left( \text{Tr} \left[ \sqrt{ \sqrt{\rho} \sigma \sqrt{\rho} } \right] \right)^2 \quad \text{or for pure states:} \quad F = |\langle \psi_{\text{real}} | \psi_{\text{MQPA}} \rangle|^2 \]

ψₘ𝓺ₚₐ

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MQPA Formula: What Is \(\psi_{\text{MQPA}}\)?

1. Formal Representation of MQPA Quantum State

\[ |\psi_{\text{MQPA}}(\vec{x}, \boldsymbol{\theta})\rangle = U_{\text{MQPA}}(\vec{x}, \boldsymbol{\theta}) |0\rangle^{\otimes n} \]

Where:

• \(\vec{x}\) = classical molecular/environmental input features
  
• \(\boldsymbol{\theta}\) = trainable variational parameters
  
• \(U_{\text{MQPA}}\) = parameterized unitary composed of:
  • Quantum encoding (angle or amplitude)
  • Entangling layers (e.g., CZ, CNOT)
  • Variational evolution (e.g., RealAmplitudes, VQE ansatz)
  • Symmetry-preserving layers (for particle conservation)
• \(|0\rangle^{\otimes n}\) = initial quantum state of n qubits

###fidelity equation:

\[ F = |\langle \psi_{\text{real}} | \psi_{\text{MQPA}}(\vec{x}, \boldsymbol{\theta}) \rangle|^2 \]

-when we’’re com paring:

• \(\psi_{\text{real}}\): reference state (e.g., from DFT, experimental ground truth, or simulator)
• \(\psi_{\text{MQPA}}\): learned/predicted state from MQPA circuit

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2. What is \(U_{\text{MQPA}}\) Exactly?

\[ U_{\text{MQPA}}(\vec{x}, \boldsymbol{\theta}) = U_{\text{evolution}}(\boldsymbol{\theta}) \cdot U_{\text{encoding}}(\vec{x}) \]

Where:

a. Encoding Layer

\[ U_{\text{encoding}}(\vec{x}) = \bigotimes_i R_y(x_i) \quad \text{or} \quad U_{\text{encoding}}(\vec{x}) = ZZFeatureMap(\vec{x}) \]

b. Variational Evolution (Trainable Layers)

\[ U_{\text{evolution}}(\boldsymbol{\theta}) = \prod_{l=1}^{L} \left( \prod_{i=1}^{n} R_y(\theta_{l,i}) \cdot \text{Entangle} \right) \]

Entangle = a layer of CNOT, CZ, or custom Hamiltonian-based gates.

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3. When MQPA Is Measured

After applying \(U_{\text{MQPA}}\), we measure:

• observables \(O_j\)
• or collapse the full statevector \(\psi_{\text{MQPA}}\)

Then compare with ground truth using:

Fidelity:

\[ F = |\langle \psi_{\text{real}} | \psi_{\text{MQPA}} \rangle|^2 \]

or Density Matrices:

\[ F(\rho, \sigma) = \left( \text{Tr} \left[ \sqrt{ \sqrt{\rho} \sigma \sqrt{\rho} } \right] \right)^2 \quad \text{where } \rho = |\psi_{\text{real}}\rangle\langle\psi_{\text{real}}| ,\quad \sigma = |\psi_{\text{MQPA}}\rangle\langle\psi_{\text{MQPA}}| \]

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PLain Summary

• \(\psi_{\text{MQPA}}\) is a quantum state produced by applying a quantum circuit to the input vector \(\vec{x}\)
• That circuit encodes molecular/environmental data and evolves it via trainable parameters \(\boldsymbol{\theta}\)
• The result is a quantum state approximating the real-world particle behavior
• We compare \(\psi_{\text{MQPA}}\) to known reference states (simulation, DFT, or experimental data) using fidelity
• If \(F \to 1\), MQPA is a high-fidelity simulation of reality

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