[Paper Review] Quantum Online Algorithms with Respect to Space Complexity

Kamil Khadiev, Aliya Khadieva+1 more|arXiv (Cornell University)|Sep 25, 2017

Quantum Computing Algorithms and Architecture1 references3 citations

TL;DR

This paper introduces quantum online algorithms and demonstrates their superiority over classical counterparts in both sublogarithmic and polylogarithmic space complexity. It shows that quantum algorithms can achieve better competitive ratios with less memory and, when using shared EPR-pairs, require only half the advice qubits compared to classical algorithms, as demonstrated on the Paging Problem with optimal performance using n/2 advice qubits.

ABSTRACT

Online algorithm is a well-known computational model. We introduce quantum online algorithms and investigate them with respect to a competitive ratio in two points of view: space complexity and advice complexity. We start with exploring a model with restricted memory and show that quantum online algorithms can be better than classical ones (deterministic or randomized) for sublogarithmic space (memory), and they can be better than deterministic online algorithms without restriction for memory. Additionally, we consider polylogarithmic space case and show that in this case, quantum online algorithms can be better than deterministic ones as well.

Motivation & Objective

• To investigate quantum online algorithms under space and advice complexity constraints.
• To compare quantum online algorithms with classical deterministic and randomized algorithms in terms of competitive ratio.
• To explore whether quantum models can simulate classical algorithms with equivalent or better efficiency.
• To analyze the impact of shared entangled qubits (EPR-pairs) on advice complexity in online computation.
• To apply the quantum model to well-known problems like the Paging Problem and the (n,k,r,w)-Parity for Number of Hats problem.

Proposed method

• The paper introduces a model of quantum online algorithms with both quantum and classical components, and defines pure quantum online algorithms that use only quantum states.
• It analyzes space complexity by restricting memory to s bits, showing that single-qubit quantum algorithms outperform classical algorithms in sublogarithmic space.
• For advice complexity, it proposes two models: one with private qubits and another with shared EPR-pairs, using quantum communication between adviser and algorithm.
• It applies the quantum advantage from EPR-pair entanglement to reduce advice qubit usage by a factor of two, based on the protocol from [BW92].
• It proves that quantum online algorithms can simulate classical randomized algorithms with the same competitive ratio using at most s classical bits and one qubit of memory.
• It applies the theoretical framework to the (n,k,r,w)-Parity for Number of Hats problem and the Paging Problem, showing optimal results with minimal advice.

Experimental results

Research questions

• RQ1 Can quantum online algorithms achieve better competitive ratios than classical deterministic or randomized algorithms under sublogarithmic space constraints?
• RQ2 Can quantum online algorithms simulate classical randomized algorithms with comparable or better memory efficiency?
• RQ3 Does using shared entangled qubits (EPR-pairs) reduce the number of required advice qubits in online algorithms compared to classical advice?
• RQ4 What is the minimal advice qubit requirement for quantum online algorithms in the Paging Problem, and can it be halved compared to classical advice?
• RQ5 How does the competitive ratio of quantum online algorithms compare to classical ones in polylogarithmic space settings?

Key findings

• Quantum online algorithms with a single qubit can outperform any classical deterministic or randomized algorithm in sublogarithmic space, even when classical algorithms have no memory restriction.
• For the (n,k,r,w)-Parity for Number of Hats problem, a quantum online algorithm with one advice qubit achieves optimal performance, while classical algorithms require more advice or have worse competitive ratios.
• Quantum online algorithms can simulate classical randomized algorithms with the same competitive ratio using at most s classical bits and one additional qubit, proving quantum models have at least the same computational power.
• When using shared EPR-pairs, quantum online algorithms require only ⌈b/2⌉ advice qubits instead of b bits, effectively halving the advice cost compared to classical algorithms.
• For the Paging Problem, an optimal quantum online algorithm using shared entangled qubits requires only n/2 advice qubits, compared to n bits in the classical optimal case.
• The paper establishes that quantum online algorithms with advice can achieve the same or better competitive ratios than classical algorithms, with reduced space and advice complexity.