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Improved Lower Bounds for Learning Quantum Channels in Diamond Distance

Aadil Oufkir, Filippo Girardi

TL;DR

This work establishes a near-optimal lower bound for learning an unknown quantum channel in diamond distance, showing that any coherent learning strategy needs at least $N = \Omega\left( \dfrac{d_A d_B r}{\varepsilon \log(d_B r / \varepsilon)} \right)$ queries. The authors introduce a general lower-bound framework based on ensembles of channels whose channels are far apart in diamond distance yet share nearly identical Stinespring isometries, and then construct a probabilistic ensemble achieving the desired separation with $\log M = \Omega(d_A d_B r)$. By combining the two steps, they derive an $\varepsilon$-dependent bound that improves on the previous $\Omega(d_A d_B r)$ bound and recovers known lower bounds in special cases (e.g., unitary channels). The results imply a near-optimal $1/\varepsilon$ scaling (up to logs) for many parameter regimes and open questions about exact $\varepsilon$-dependence, coherence advantages, and memory effects in channel-learning tasks.

Abstract

We prove that learning an unknown quantum channel with input dimension $d_A$, output dimension $d_B$, and Choi rank $r$ to diamond distance $\varepsilon$ requires $ Ω\!\left( \frac{d_A d_B r}{\varepsilon \log(d_B r / \varepsilon)} \right)$ queries. This improves the best previous $Ω(d_A d_B r)$ bound by introducing explicit $\varepsilon$-dependence, with a scaling in $\varepsilon$ that is near-optimal when $d_A=rd_B$ but not tight in general. The proof constructs an ensemble of channels that are well-separated in diamond norm yet admit Stinespring isometries that are close in operator norm.

Improved Lower Bounds for Learning Quantum Channels in Diamond Distance

TL;DR

This work establishes a near-optimal lower bound for learning an unknown quantum channel in diamond distance, showing that any coherent learning strategy needs at least queries. The authors introduce a general lower-bound framework based on ensembles of channels whose channels are far apart in diamond distance yet share nearly identical Stinespring isometries, and then construct a probabilistic ensemble achieving the desired separation with . By combining the two steps, they derive an -dependent bound that improves on the previous bound and recovers known lower bounds in special cases (e.g., unitary channels). The results imply a near-optimal scaling (up to logs) for many parameter regimes and open questions about exact -dependence, coherence advantages, and memory effects in channel-learning tasks.

Abstract

We prove that learning an unknown quantum channel with input dimension , output dimension , and Choi rank to diamond distance requires queries. This improves the best previous bound by introducing explicit -dependence, with a scaling in that is near-optimal when but not tight in general. The proof constructs an ensemble of channels that are well-separated in diamond norm yet admit Stinespring isometries that are close in operator norm.
Paper Structure (13 sections, 7 theorems, 68 equations, 2 figures)

This paper contains 13 sections, 7 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Quantum channels and their representations
  4. Channel ensembles with distance constraints
  5. The coherent query model
  6. A general lower bound on channel learning
  7. An ensemble yielding an improved lower bound
  8. Proof of Lemma \ref{['lem:existence-V-tilde']}
  9. Conclusion
  10. A weaker lower bound using existing packing nets
  11. Existence of the quantum channel $\widetilde{\Phi}_0$
  12. Proof of Lemma \ref{['lem:2nd moment']}
  13. Weingarten Calculus

Key Result

Theorem 1

Let $d_A, d_B, r\ge 1$, $M\ge 3$ and $\varepsilon, \eta\in (0, 1/2)$. Consider an ensemble $\{\Phi_i\}_{i=1}^M \in \mathcal{E}(d_A,d_B,r,\varepsilon,\eta)$ of $M$ quantum channels that are $2\varepsilon$-diamond-far and whose Stinespring isometries are $\eta$-operator-norm-close. Any coherent algor uses of $\pazocal{N}$.

Figures (2)

  • Figure 1: Schematic representation of the ensemble $\{\Phi_i\}_{i=1}^M$.
  • Figure 2: Schematic construction of the isometries $V_x$ and of the channels $\Phi_x$, where $U_x\sim {\rm Haar}({\rm U}(rd_B))$, $\widetilde{V}_x=U_xS$, $V_x =\sqrt{1-\varepsilon^2} \ket{0}\otimes \widetilde{V}_0 + \varepsilon \ket{1}\otimes \widetilde{V}_x$, $\Phi_x(\,\cdot\,)=\mathop{\mathrm{Tr}}\nolimits_E[\widetilde{V}_x\,\cdot\, \widetilde{V}_x^\dagger]$.

Theorems & Definitions (12)

  • Theorem 1: (General lower bound)
  • proof
  • Lemma 2
  • proof
  • Theorem 3: (Improved lower bound for channel learning)
  • proof
  • Lemma 4
  • proof
  • Lemma 5: meckes2013spectral
  • proof : Proof of Lemma \ref{['lem:existence-V-tilde']}.
  • ...and 2 more