Quantum Machine Learning

• When you train a neural network to imitate a quantum gate, does the representation of its output decide whether the learned gate composes?

Each block answers a piece of the question and feeds the next — foundations at the top, the answer at the bottom.

flowchart TD
  Q(["Does the output representation<br/>decide if a learned gate composes?"]):::q
  D["exact data<br/>Qiskit statevectors"]
  P["probability mode<br/>trains on Born-rule p(b)"]
  U["unitary mode<br/>trains on (Re, Im) amplitudes"]
  C["composition test<br/>apply the gate U^k"]
  NZ["noisy channel<br/>density matrix vs populations"]
  Q --> D
  D --> P --> C
  D --> U --> C
  C -->|"phase squared away"| FAIL(["probability mode diverges"]):::a
  C -->|"phase kept"| HOLD(["unitary mode holds ~0.998"]):::a
  D --> NZ
  classDef q fill:#1a1206,stroke:#f0a83e,color:#f3c98a;
  classDef a fill:#f0a83e,stroke:#f0a83e,color:#10141a;

Each stage of the calculation. Click one to expand its full write-up and validation figure.

Click any figure to enlarge. Use ← → to move through the set.

The Qiskit ground-truth setup: random input-state preparation, the target gates (CNOT and RY(0.7)), and the composed circuit U³.The core result: a network trained on Born-rule probabilities diverges from the true gate under composition, while one trained on full amplitudes (unitary mode) holds to 0.998–0.9997 fidelity at depth 6.Extended to an unknown noisy channel: the full density-matrix representation composes, while a populations-only (coherence-discarded) representation diverges.The learned gate's coherent error benchmarked against superconducting and trapped-ion decoherence models versus exact simulation, as a function of circuit depth.

• The barrier is the information in the target, not the network. Training on probabilities squares away phase, so U² diverges from the truth; training on amplitudes keeps phase and composition holds to ~0.998.
• It is a representation problem, provably so: feed the probability model tomographically complete (X/Y/Z-basis) data and it recovers phase after all — the wall is what the target encodes, not the architecture.
• The same logic extends to open systems: under noise the density matrix composes while its diagonal alone does not.