Open Project Winter 2025 Submission

Technical Overview

This project implements a comprehensive pipeline for Quantum State Tomography (QST), evolving from standard linear inversion techniques to machine learning-enhanced reconstruction using Classical Shadows. The primary goal is to reconstruct a quantum density matrix $\rho$ given a set of measurement outcomes, enforcing physical constraints such as Hermiticity, positive semi-definiteness, and unit trace.

Methodology

Standard Tomography (Baseline)

• Measurement: Utilized Pauli-projective measurements ($X, Y, Z$ bases).
• Reconstruction: Implemented Linear Inversion using the generalized Born rule.
• Validation: Verified against analytical ground-truth states ($|0\rangle, |1\rangle, |+\rangle$) using Frobenius norm error.

Machine Learning Approach (Classical Shadows)

• Architecture: A Transformer-based neural network (ShadowReconstructor) processes sequences of random Pauli measurements (Shadows).
• Physical Constraints: The model outputs a lower-triangular matrix $L$ to construct $\rho$ via Cholesky decomposition: $$ \rho = \frac{L L^\dagger}{\text{Tr}(L L^\dagger)} $$ This strictly enforces $\rho \succeq 0$ and $\text{Tr}(\rho) = 1$.
• Training: Supervised learning on synthetic datasets generated via Qiskit Aer, optimizing a custom loss function combining real and imaginary reconstruction errors.

Workflow

1. Data Generation: Synthetic quantum states and measurement shots are generated using src/data_gen.py.
2. Training: The ShadowReconstructor model is trained using src/train.py, which minimizes reconstruction loss while monitoring Quantum Fidelity.
3. Scalability Analysis: Assignment_3 benchmarking scripts assess how fidelity and runtime degrade as the qubit count $n$ increases.

Results

Mathematical Definitions

We evaluate reconstruction quality using Quantum Fidelity ($F$) and Trace Distance ($T$):

$$ F(\rho, \sigma) = \left( \text{Tr} \sqrt{ \sqrt{\rho} \sigma \sqrt{\rho} } \right)^2 $$

$$ T(\rho, \sigma) = \frac{1}{2} || \rho - \sigma ||_1 = \frac{1}{2} \text{Tr} \left[ \sqrt{(\rho - \sigma)^\dagger (\rho - \sigma)} \right] $$

Numerical Results

Table 1: Baseline Linear Inversion (Single Qubit)

| State | Frobenius Error ($|| \rho_{true} - \rho_{recon} ||_F$) | | :--- | :--- | | $|0\rangle$ | $0.0170$ | | $|1\rangle$ | $0.0300$ | | $|+\rangle$ | $0.0146$ |

Table 2: Machine Learning Reconstruction (Classical Shadows) Training results after 10 Epochs on 1000 random states.

Metric	Final Value	Convergence Behavior
Average Fidelity	98.72%	Rose from 80.17% in Epoch 1
Trace Distance	0.0770	Dropped from 0.3642 in Epoch 1
Loss (MSE)	0.0030	Stabilized after Epoch 8
Inference Latency	5.63 ms	Per-sample average on CPU

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Final Reflection

Scaling Limits Observed

Our scalability study (Assignment 3) revealed the "Exponential Wall" characteristic of statevector tomography:

1. Runtime Cliff ($n \approx 14$): For full statevector reconstruction, the parameter space grows as $4^n$. While operations remain instant for low qubit counts, runtime doubles with every added qubit beyond $n=12$. Matrix multiplication becomes computationally infeasible on standard hardware beyond $n=14$.
2. Fidelity Concentration: As $n$ grows, the volume of the Hilbert space explodes. A randomly initialized model has an exponentially vanishing overlap with the target state, leading to "Barren Plateaus" where gradients vanish, making optimization difficult without specific initialization strategies.

Future Improvements

1. Tensor Networks: To overcome the $4^n$ memory bottleneck, we should replace the dense Cholesky reconstruction with Matrix Product States (MPS). This would allow us to reconstruct states with low entanglement entropy for $n > 50$.
2. Noise-Robust Loss: The current MSE loss assumes perfect measurements (finite-shot noise only). Integrating a noise model (e.g., depolarizing channel) directly into the loss function would improve resilience against hardware errors.
3. Attention Mechanisms: The current Transformer averages over shadows. Implementing a deeper attention mechanism could allow the model to weight "informative" measurements higher than noisy ones, reducing the required shot count.