Sample-Optimal Quantum Process Tomography with Non-Adaptive Incoherent Measurements

TL;DR

This paper establishes near-optimal bounds on the number of copies needed for non-adaptive quantum process tomography, showing the precise scaling with input/output dimensions and error tolerance.

Contribution

It extends previous results to provide tight bounds on the sample complexity for non-adaptive quantum process tomography, including necessary and sufficient conditions.

Findings

Upper bound of (d_in^3 d_out^3 / \u03b5^2) copies for learning quantum channels

Lower bound of (d_in^3 d_out^3 / b5^2) copies for any strategy with incoherent measurements

Bounds hold even with ancilla assistance

Abstract

How many copies of a quantum process are necessary and sufficient to construct an approximate classical description of it? We extend the result of Surawy-Stepney, Kahn, Kueng, and Guta (2022) to show that $\tilde{\mathcal{O}}(d_{\text{in}}^3d_{\text{out}}^3/\varepsilon^2)$ copies are sufficient to learn any quantum channel $C^{d_{\text{in}}\times d_{\text{in}}} \rightarrow C^{d_{\text{out}}\times d_{\text{out}}}$ to within $\varepsilon$ in diamond norm. Moreover, we show that $\Omega(d_{\text{in}}^3 d_{\text{out}}^3/\varepsilon^2)$ copies are necessary for any strategy using incoherent non-adaptive measurements. This lower bound applies even for ancilla-assisted strategies.