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Quantum Channel Testing in Average-Case Distance

Gregory Rosenthal, Hugo Aaronson, Sathyawageeswar Subramanian, Animesh Datta, Tom Gur

Abstract

We study the complexity of testing properties of quantum channels. First, we show that testing identity to any channel $\mathcal N: \mathbb C^{d_{\mathrm{in}} \times d_{\mathrm{in}}} \to \mathbb C^{d_{\mathrm{out}} \times d_{\mathrm{out}}}$ in diamond norm distance requires $Ω(\sqrt{d_{\mathrm{in}}} / \varepsilon)$ queries, even in the strongest algorithmic model that admits ancillae, coherence, and adaptivity. This is due to the worst-case nature of the distance induced by the diamond norm. Motivated by this limitation and other theoretical and practical applications, we introduce an average-case analogue of the diamond norm, which we call the average-case imitation diamond (ACID) norm. In the weakest algorithmic model without ancillae, coherence, or adaptivity, we prove that testing identity to certain types of channels in ACID distance can be done with complexity independent of the dimensions of the channel, while for other types of channels the complexity depends on both the input and output dimensions. Building on previous work, we also show that identity to any fixed channel can be tested with $\tilde O(d_{\mathrm{in}} d_{\mathrm{out}}^{3/2} / \varepsilon^2)$ queries in ACID distance and $\tilde O(d_{\mathrm{in}}^2 d_{\mathrm{out}}^{3/2} / \varepsilon^2)$ queries in diamond distance in this model. Finally, we prove tight bounds on the complexity of channel tomography in ACID distance.

Quantum Channel Testing in Average-Case Distance

Abstract

We study the complexity of testing properties of quantum channels. First, we show that testing identity to any channel in diamond norm distance requires queries, even in the strongest algorithmic model that admits ancillae, coherence, and adaptivity. This is due to the worst-case nature of the distance induced by the diamond norm. Motivated by this limitation and other theoretical and practical applications, we introduce an average-case analogue of the diamond norm, which we call the average-case imitation diamond (ACID) norm. In the weakest algorithmic model without ancillae, coherence, or adaptivity, we prove that testing identity to certain types of channels in ACID distance can be done with complexity independent of the dimensions of the channel, while for other types of channels the complexity depends on both the input and output dimensions. Building on previous work, we also show that identity to any fixed channel can be tested with queries in ACID distance and queries in diamond distance in this model. Finally, we prove tight bounds on the complexity of channel tomography in ACID distance.
Paper Structure (7 sections, 13 theorems, 14 equations, 1 table)

This paper contains 7 sections, 13 theorems, 14 equations, 1 table.

Table of Contents

  1. Introduction
  2. Hardness of channel testing in diamond distance
  3. An average-case analogue of the diamond norm
  4. Channel certification and tomography in ACID distance
  5. Open problems
  6. Preliminaries
  7. Quantum states and transformations

Key Result

Theorem 1.1

For all fixed channels $\mathcal{N}: \mathbb{C}^{d_{\mathrm{in}} \times d_{\mathrm{in}}} \to \mathbb{C}^{d_{\mathrm{out}} \times d_{\mathrm{out}}}$ with $d_{\mathrm{out}} \ge 2$ and all $\varepsilon > 0$, every ancilla-assisted, coherent, adaptive algorithm requires $\Omega(\sqrt{d_{\mathrm{in}}} /

Theorems & Definitions (15)

  • Theorem 1.1: Lower bound for channel certification in diamond distance
  • Theorem 1.2: Lower bound for pure state replacement channel certification in diamond distance
  • Definition 1.3: ACID norm
  • Theorem 1.4: Coherent channel certification
  • Theorem 1.5: Ancilla-free channel certification in ACID distance
  • Theorem 1.6: Ancilla-free channel certification in diamond distance
  • Theorem 1.7: Erasure, unitary, and pure state replacement channel certification
  • Theorem 1.8: Lower bound for the completely depolarizing channel
  • Theorem 1.9: Channel certification using $\chan M$ and $\dual M$
  • Theorem 1.10: Upper bound for coherent channel tomography
  • ...and 5 more