[Paper Review] The Flow of Information in Interactive Quantum Protocols: the Cost of Forgetting

Mathieu Laurière, Dave Touchette|arXiv (Cornell University)|Jan 9, 2017

Quantum Computing Algorithms and Architecture13 references3 citations

TL;DR

This paper provides a new operational characterization of quantum information cost (QIC) for protocols with classical inputs, defining it as the sum of classical information transmission cost and the cost of forgetting quantum information. It proves that forgetting is essential for quantum speedups—specifically, any protocol that avoids forgetting cannot achieve sublinear communication for Disjointness, thereby establishing that information erasure is a necessary feature for significant quantum advantage in interactive protocols.

ABSTRACT

In the context of two-party interactive quantum communication protocols, we study a recently defined notion of quantum information cost (QIC), which possesses most of the important properties of its classical analogue. Although this definition has the advantage to be valid for fully quantum inputs and tasks, its interpretation for classical tasks remained rather obscure. Also, the link between this new notion and other notions of information cost for quantum protocols that had previously appeared in the literature was not clear, if existent at all. We settle both these issues: for quantum communication with classical inputs, we provide an alternate characterization of QIC in terms of information about the input registers, avoiding any reference to the notion of a purification of the classical input state. We provide an exact operational interpretation of this alternative characterization as the sum of the cost of transmitting information about the classical inputs and the cost of forgetting information about these inputs. To obtain this characterization, we prove a general lemma, the Information Flow Lemma, assessing exactly the transfer of information in general interactive quantum processes. Furthermore, we clarify the link between QIC and IC of classical protocols by simulating quantumly classical protocols. Finally, we apply these concepts to argue that any quantum protocol that does not forget information solves Disjointness on n-bits in Omega (n) communication, completely losing the quadratic quantum speedup. This provides a specific sense in which forgetting information is a necessary feature of interactive quantum protocols. We also apply these concepts to prove that QIC at zero-error is exactly n for the Inner Product function, and n (1 - o(1)) for a random Boolean function on n+n bits.

Motivation & Objective

• To resolve ambiguity in the interpretation of quantum information cost (QIC) for classical tasks by eliminating reliance on input state purification.
• To provide an operational, information-theoretic interpretation of QIC in terms of information transmission and forgetting costs.
• To clarify the relationship between QIC and prior notions of information cost in quantum protocols, including classical IC and other quantum variants.
• To establish that forgetting information is a necessary condition for achieving quantum speedups in bounded-round protocols.
• To derive tight lower bounds on QIC for fundamental functions like Inner Product and random Boolean functions.

Proposed method

• Introduce the Information Flow Lemma, a general tool for precisely tracking quantum information transfer in interactive protocols.
• Define a new characterization of QIC using mutual information between input registers and protocol transcripts, avoiding purification of classical inputs.
• Operationalize QIC as the sum of classical information cost (CIC) and the cost of forgetting (CRIC), quantified via von Neumann entropy changes.
• Use isometric invariance of von Neumann entropy to relate entropy of output registers to input information content.
• Apply the framework to simulate classical protocols quantumly while preserving classical information cost, enabling comparison between classical and quantum settings.
• Leverage Rényi entropy and concentration bounds to prove high-probability lower bounds on QIC for random Boolean functions.

Experimental results

Research questions

• RQ1 Can quantum information cost (QIC) be characterized without relying on the notion of input purification, especially for classical inputs?
• RQ2 What is the operational meaning of QIC in terms of information transmission and erasure in quantum protocols?
• RQ3 How does QIC relate to classical information cost (IC) and other prior quantum information cost notions?
• RQ4 Is forgetting information a necessary condition for achieving quantum speedups in bounded-round protocols?
• RQ5 What are the exact values of QIC for the Inner Product function and random Boolean functions at zero error?

Key findings

• QIC for the Inner Product function under uniform input distribution is exactly n qubits at zero error.
• For a random Boolean function on n+n bits, QIC is n(1−o(1)) with high probability under uniform input distribution.
• Any quantum protocol that does not forget information cannot solve Disjointness in sublinear communication, implying that forgetting is necessary for quantum speedup.
• The QIC of a protocol is operationally equivalent to the sum of classical information cost (CIC) and the cost of forgetting (CRIC).
• QIC and classical IC are almost equivalent in the sense that quantum simulation preserves classical IC, enabling direct comparison.
• The limit of classical IC for random functions approaches n(1−o(1)) as error ε→0, and a similar bound holds for QIC under the same conditions.