Same Spectrum, Harder First-Order Training: Phase-Locking in Single- versus Multi-Qubit Re-Uploading Circuits

Key points

• Angle-encoded variational quantum circuits implement truncated Fourier series.1
• Because its state lives on the Bloch sphere, all parameters of the serial circuit move the state, at each input, along one complex direction, which we call phase-locking.1
• We prove that this costs no Fourier direction but leaves no direction to spare: however many parameters it has, the serial circuit has at most $2E+2$ tangent directions, one more than its $2E+1$ Fourier coefficients need, while a two-qubit control with the same encoding budget and model class generically has at least $6E+6$.1
• On an exact loss with flat-spectrum targets the gap grows with $E$; at $E=32$ the serial circuit converges in 3, 53 and 83 runs under gradient descent, Adam and Levenberg--Marquardt, the control in 99, 100 and 100.1

Sources (1)

• [1]Same Spectrum, Harder First-Order Training: Phase-Locking in Single- versus Multi-Qubit Re-Uploading Circuits
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 15, 12:36 PM

  For one-dimensional inputs, a single-qubit re-uploading (serial) circuit represents every function that a multi-qubit circuit with the same encoding budget $E$ can; we find that this exchange of width for depth is not free in first-order training.

  Angle-encoded variational quantum circuits implement truncated Fourier series.

Extractive summary: sentences quoted from the sources.