[Paper Review] Dense Quantum Coding and a Lower Bound for 1-way Quantum Automata

Andris Ambainis, Nayak, Ashwin+2 more|arXiv (Cornell University)|Apr 18, 1998

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper introduces dense quantum coding, a method to encode m classical bits into n < m qubits such that any individual bit can be retrieved with high probability using quantum measurements. It establishes a lower bound on the number of qubits required for such encodings and applies this to prove an exponential lower bound on the size of 1-way quantum finite automata for certain languages, demonstrating a fundamental advantage of classical over quantum finite automata in this setting.

ABSTRACT

We consider the possibility of encoding m classical bits into much fewer n quantum bits so that an arbitrary bit from the original m bits can be recovered with a good probability, and we show that non-trivial quantum encodings exist that have no classical counterparts. On the other hand, we show that quantum encodings cannot be much more succint as compared to classical encodings, and we provide a lower bound on such quantum encodings. Finally, using this lower bound, we prove an exponential lower bound on the size of 1-way quantum finite automata for a family of languages accepted by linear sized deterministic finite automata.

Motivation & Objective

• To investigate the feasibility and limits of encoding classical information into fewer quantum bits while preserving access to individual bits.
• To identify quantum advantages in encoding that have no classical counterparts.
• To establish a theoretical lower bound on the number of qubits required for quantum random access codes.
• To apply this bound to derive a lower bound on the size of 1-way quantum finite automata for a class of languages.
• To compare the efficiency of quantum and classical finite automata in recognizing specific languages.

Proposed method

• Introduces the concept of quantum random access codes (QRACs), where m classical bits are encoded into n qubits to allow recovery of any single bit with high fidelity.
• Analyzes the trade-off between the number of encoded bits m and the number of qubits n, deriving bounds on the minimum n required for a given success probability.
• Uses quantum information theory techniques, including state distinguishability and fidelity bounds, to establish lower bounds on n.
• Applies the derived lower bounds on QRACs to the model of 1-way quantum finite automata (1QFA), showing that simulating certain deterministic finite automata requires exponentially more states in the quantum model.
• Employs a reduction from the language recognition problem to the QRAC problem to transfer the lower bound.
• Leverages known results on classical finite automata to show that 1QFA cannot simulate linear-sized deterministic automata efficiently.

Experimental results

Research questions

• RQ1 Can m classical bits be encoded into fewer than m qubits such that any individual bit can be recovered with high probability using quantum measurements?
• RQ2 Are there quantum encoding schemes that outperform classical schemes in terms of compression while preserving individual bit access?
• RQ3 What is the minimum number of qubits required to achieve a given success probability in retrieving a single bit from a set of m classical bits?
• RQ4 Can the lower bound on quantum random access codes be used to establish limitations on the size of 1-way quantum finite automata?
• RQ5 How does the state complexity of 1-way quantum finite automata compare to that of classical deterministic finite automata for the same language family?

Key findings

• Non-trivial quantum encodings exist that have no classical counterparts, enabling more efficient access to individual bits than classically possible.
• The number of qubits required to encode m classical bits with high success probability for any single bit is bounded below by a function that grows logarithmically with m.
• A lower bound on the size of 1-way quantum finite automata is derived, showing that for a family of languages recognized by linear-sized deterministic finite automata, 1QFA require exponentially more states.
• The lower bound on quantum random access codes implies that quantum finite automata cannot efficiently simulate classical finite automata for certain languages.
• The paper establishes that quantum encodings cannot be significantly more succinct than classical encodings in terms of the number of qubits needed for reliable bit retrieval.
• The results demonstrate a fundamental separation between classical and quantum finite automata in terms of state complexity for a specific class of languages.