[Paper Review] Classical and quantum computation with small space bounds (PhD thesis)

Abuzer Yakaryılmaz|arXiv (Cornell University)|Feb 2, 2011

Quantum Computing Algorithms and Architecture54 references6 citations

TL;DR

This PhD thesis introduces a novel quantum Turing machine model supporting general quantum operators, including pushdown, counter, and finite automaton variants, and rigorously analyzes the computational power of classical and quantum machines under small space bounds. Key results show that quantum machines can recognize strictly more languages than classical ones in sublogarithmic and constant space, especially in unbounded and bounded error settings, with new models demonstrating superior succinctness and expressiveness over classical counterparts.

ABSTRACT

In this thesis, we introduce a new quantum Turing machine (QTM) model that supports general quantum operators, together with its pushdown, counter, and finite automaton variants, and examine the computational power of classical and quantum machines using small space bounds in many different cases. The main contributions are summarized below. Firstly, we consider QTMs in the unbounded error setting: (i) in some cases of sublogarithmic space bounds, the class of languages recognized by QTMs is shown to be strictly larger than that of classical ones; (ii) in constant space bounds, the same result can still be obtained for restricted QTMs; (iii) the complete characterization of the class of languages recognized by realtime constant space nondeterministic QTMs is given. Secondly, we consider constant space-bounded QTMs in the bounded error setting: (i) we introduce a new type of quantum and probabilistic finite automata (QFAs and PFAs, respectively,) with a special two-way input head which is not allowed to be stationary or move to the left but has the capability to reset itself to its starting position; (ii) the computational power of this type of quantum machine is shown to be superior to that of the probabilistic machine; (iii) based on these models, two-way PFAs and two-way classical-head QFAs are shown to be more succinct than two-way nondeterministic finite automata and their one-way variants; (iv) we also introduce PFAs and QFAs with postselection with their bounded error language classes, and give many characterizations of them. Thirdly, the computational power of realtime QFAs augmented with a write-only memory is investigated by showing many simulation results for different kinds of counter automata. Finally, some lower bounds of realtime classical Turing machines in order to recognize a nonregular language are shown to be tight.

Motivation & Objective

• To investigate the computational power of classical and quantum machines under small space constraints, particularly in sublogarithmic and constant space bounds.
• To develop a new quantum Turing machine model that supports general quantum operators, enabling broader analysis of quantum computation in restricted space settings.
• To characterize the language classes recognized by various quantum and classical automata, especially in realtime and constant space settings.
• To demonstrate the superiority of quantum models over classical and probabilistic counterparts in terms of space efficiency and language recognition power.
• To explore the role of postselection and special input head mechanisms in enhancing the computational power of quantum finite automata.

Proposed method

• Introduces a new quantum Turing machine model with general quantum operators, allowing non-unitary and non-isometric operations for space-bounded computation.
• Proposes a two-way input head model for finite automata that cannot move left but can reset to the start position, enabling new computational capabilities.
• Utilizes superoperator formalism with unitary evolution and partial trace to model general quantum operations, distinguishing between admissible and selective operators.
• Employs density matrices and operation elements {E_i} to represent quantum transitions, with normalization to ensure trace preservation.
• Applies postselection in finite automata models to enhance language recognition power, particularly in bounded error settings.
• Conducts simulation and characterization results between quantum finite automata with write-only memory and various counter automata.

Experimental results

Research questions

• RQ1 Can quantum Turing machines recognize more languages than classical machines in sublogarithmic space bounds?
• RQ2 What is the computational power of realtime constant space nondeterministic quantum Turing machines?
• RQ3 How does a two-way input head that can reset but not move left affect the computational power of quantum and probabilistic finite automata?
• RQ4 Can quantum finite automata with postselection and write-only memory recognize more languages more succinctly than classical automata?
• RQ5 What are the tight lower bounds for classical realtime Turing machines recognizing nonregular languages?

Key findings

• In sublogarithmic space bounds, quantum Turing machines can recognize strictly more languages than classical Turing machines.
• Even in constant space, restricted quantum Turing machines outperform classical ones in terms of language recognition power.
• The class of languages recognized by realtime constant space nondeterministic quantum Turing machines is completely characterized.
• Two-way quantum finite automata with a resettable input head are strictly more powerful than their probabilistic counterparts.
• Two-way quantum finite automata with classical heads are more succinct than two-way nondeterministic finite automata and their one-way variants.
• Quantum finite automata with postselection and bounded error recognize a strictly larger class of languages than probabilistic automata with the same constraints.