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A Theory of Computation Based on Quantum Logic Mingsheng Ying State Key Laboratory of Intelligent Technology and Systems Department of Computer Science and Technology Tsinghua University Email: yingmsh@tsinghua.edu.cn

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I. Background ---- Non-classical Logics An axiomatization of a mathematical theory An axiomatization of a mathematical theory consists of a system of fundamental notions as well as a set of axioms about these notions The mathematical theory is then the set of The mathematical theory is then the set of theorems which can be derived from the axioms

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One needs a certain logic to provide tools for reasoning in the derivation of these theorems from the axioms One needs a certain logic to provide tools for reasoning in the derivation of these theorems from the axioms

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A. Heyting (1963), Axiomatic Projective Geometry, North-Holland, Amsterdam, 1963 In elementary axiomatics logic was used in an unanalyzed form In elementary axiomatics logic was used in an unanalyzed form In the studies for foundations of mathematics beginning in the early of twentieth century, it had been realized that a major part of mathematics has to exploit the full power of classical (Boolean) logic, the strongest one in the family of existing logics. In the studies for foundations of mathematics beginning in the early of twentieth century, it had been realized that a major part of mathematics has to exploit the full power of classical (Boolean) logic, the strongest one in the family of existing logics.

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A few mathematicians, including the big names L. E. J. Brouwer, H. Poincare, L. A few mathematicians, including the big names L. E. J. Brouwer, H. Poincare, L. Kronecker and H. Weyl, took some kind of constructive position which is in more or less explicit opposition to certain forms of Kronecker and H. Weyl, took some kind of constructive position which is in more or less explicit opposition to certain forms of mathematical reasoning used by the majority of the mathematical community mathematical reasoning used by the majority of the mathematical community

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Some of them even endeavored to establish so- called constructive mathematics, the part of mathematics that could be rebuilt on constructivist principles Some of them even endeavored to establish so- called constructive mathematics, the part of mathematics that could be rebuilt on constructivist principles The logic employed in the development of constructive mathematics is intuitionistic logic which is weaker than classical logic The logic employed in the development of constructive mathematics is intuitionistic logic which is weaker than classical logic

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Many logics different from classical logic and intuitionistic logic have been invented in the last century Many logics different from classical logic and intuitionistic logic have been invented in the last century Question: Question: Whether we are able to establish some mathematical theories based on other non-classical logics besides intuitionistic logic?

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J. B. Rosser and A. R. Turquette, Many-Valued Logics, North-Holland, Amsterdam, 1952 "The fact that it is thus possible to generalize the ordinary two-valued logic so as not only to cover the case of many-valued statement calculi, but of many-valued quantification theory as well, naturally suggests the possibility of further extending our treatment of many-valued logic to cover the case of many- valued sets, equality, numbers, etc.

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Since we now have a general theory of many valued predicate calculi, there is little doubt about the possibility of successfully developing such extended many-valued theories.... we shall consider their careful study one of the major unsolved problems of many-valued logic."

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A. Mostowski, Thirty Years of Foundational Studies Acta Philosophica Fennica, 1965 J. Lukasiewicz (1920’s) hoped that there would be some non-classical logics which can be properly used in mathematics as non-Euclidean geometry does Most of non-classical logics invented so far have not been really used in mathematics, and intuitionistic logic seems that unique one of non-classical logics which still has an opportunity to carry out the Lukasiewicz's project

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J. Dieudonne, The current trend of pure mathematics, Advances in Mathematics 27(1978)235-255 Mathematical logicians have been developing a variety of non-classical logics such as second-order logic, modal logic and many- valued logic, but these logics are completely useless for mathematicians working in other research areas

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The theory of computation is a mathematical theory The theory of computation is a mathematical theory Quantum logic is a non-classical logic Quantum logic is a non-classical logicQuestion: we develop a theory of (in particular) Should we develop a theory of computation based on quantum logic?

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II. Backgroud ---- Quantum Computation The idea of quantum computation came from the studies The idea of quantum computation came from the studies of connections between physics and computation. The first step toward it was the understanding of the The first step toward it was the understanding of the thermodynamics of classical computation. C. H. Bennet (1973) noted that a logically reversible C. H. Bennet (1973) noted that a logically reversible operation need not dissipate any energy and found that a logically reversible Turing machine is a theoretical possibility.

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P. A. Benioff (1980) constructed a quantum mechanical P. A. Benioff (1980) constructed a quantum mechanical model of Turing machine. His construction is the first quantum mechanical description of computer, but it is not a real quantum computer. In P. A. Benioff's model between computation steps the In P. A. Benioff's model between computation steps the machine may exist in an intrinsically quantum state, but at the end of each computation step the tape of the machine always goes back to one of its classical states.

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Quantum computers were first envisaged by R. P. Feynman (1982). Quantum computers were first envisaged by R. P. Feynman (1982). He conceived that no classical Turing machine He conceived that no classical Turing machine could simulate certain quantum phenomena without an exponential slowdown, and so he realized that quantum mechanical effects should offer something genuinely new to computation. Although R. P. Feynman proposed the idea of Although R. P. Feynman proposed the idea of universal quantum simulator, he did not give a concrete design of such a simulator.

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Feymann’s ideas were elaborated and formalized by D. Feymann’s ideas were elaborated and formalized by D. Deutsch (1985). Deutsch described the first true quantum Turing machine. Deutsch described the first true quantum Turing machine. In his machine, the tape is able to exist in quantum states. D. Deutsch introduced the technique of quantum D. Deutsch introduced the technique of quantumparallelism He proposed that quantum computers might be able to He proposed that quantum computers might be able to perform certain types of computations that classical computers can only perform very inefficiently.

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One of the most striking advances was made by One of the most striking advances was made by P. W. Shor (1994) He discovered a polynomial-time algorithm on He discovered a polynomial-time algorithm on quantum computers for prime factorization of which the best known algorithm on classical computers is exponential.

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L. K. Grover (1996) offered another apt L. K. Grover (1996) offered another apt killer of quantum computation He found a quantum algorithm for He found a quantum algorithm for searching a single item in an unsorted database in square root of the time it would take on a classical computer.

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Both prime factorization and database search are central Both prime factorization and database search are central problems in computer science The quantum algorithms for them are highly faster than The quantum algorithms for them are highly faster than the classical ones P. W. Shor and L. K. Grover's works stimulated an P. W. Shor and L. K. Grover's works stimulated an intensive investigation on quantum computation. After that, quantum computation has been an extremely After that, quantum computation has been an extremely exciting and rapidly growing field of research.

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The studies of quantum computation The studies of quantum computation may be roughly divided into four categories: (1) Physical implementations (2) Physical models (3) Mathematical models (4) Algorithms and complexity (5) Quantum programming, quantum software

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Question: What is the logical foundation of quantum computation? V. Vedral and M. B. Plenio (1998) already V. Vedral and M. B. Plenio (1998) already advocated that quantum computers require quantum logic, something fundamentally different to classical Boolean logic.

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III. Quantum Logic Quantum logic was introduced by G. Birkhoff and J. von Quantum logic was introduced by G. Birkhoff and J. von Neumann (1936) as the logic of quantum mechanics. They realized that quantum mechanical systems are not They realized that quantum mechanical systems are not governed by classical logical laws. Their proposed logic stems from von Neumann's Hilbert Their proposed logic stems from von Neumann's Hilbert space formalism of quantum mechanics.

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G. Birkhoff and J. von Neumann, The logic of quantum mechanics, Annals of Mathematics, 37(1936)823-843 “what logical structure one may hope to find in physical theories which, like quantum mechanics, do not conform to classical logic. Our main conclusion, based on admittedly heuristic arguments, is that one can reasonably expect to find a calculus of propositions which is formally indistinguishable from the calculus of linear subspaces [of Hilbert space] with respect to set products, linear sums, and orthogonal complements – and resembles the usual calculus of propositions with respect to 'and', 'or', and 'not'.”

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Question: What is the algebraic structures of the set of all closed subspaces of a Hilbert space with the inclusion relation as an ordering ？

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Sasaki’s Theorem (1957): (1) The set of all closed subspaces of a Hilbert space with the inclusion relation is a complete orthomodular lattice; (2) It is a modular lattice if and only if the Hilbert space is finite-dimensional.

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The algebraic counterpart of classical logic: Boolean algebra Boolean algebra Two understandings of quantum logic: （ 1 ） The theory of orthomodular lattices （ 2 ） A logic whose set of truth values is an orthomodular lattice

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Orthomodular law : Orthomodular law :

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IV. A Theory of Computation Based on Quantum Logic Question: How to develop a theory of computation based on quantum logic?

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Classical logic is applied as the deduction Classical logic is applied as the deduction tool in almost all mathematical theories. It should be noted that what is used in these theories is the deductive (proof-theoretical) aspect of classical logic

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The proof theory of non-classical logics is much more complicated than that of classical logic, and it is not an easy task to conduct reasoning in the realm of the proof theory of non-classical logics The proof theory of non-classical logics is much more complicated than that of classical logic, and it is not an easy task to conduct reasoning in the realm of the proof theory of non-classical logics It is the case even for the simplest non- classical logics, three-valued logics It is the case even for the simplest non- classical logics, three-valued logics

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H. Hodes, Three-valued logics - an introduction, a comparison of various logical lexica, and some philosophical remarks, Annals of Pure and Applied Logic, 43(1989)99-145 “ Of course three-valued logics will be somewhat more complicated than classical two-valued logic. In fact, proof-theoretically they are at least twice as complicated:.... But model-theoretically they are only 50 percent more complicated,.…”

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And much worse, some non-classical logics were introduced only in a semantic way, and the axiomatizations of some among them are still to be found, and some of them may be not (finitely) axiomatizable And much worse, some non-classical logics were introduced only in a semantic way, and the axiomatizations of some among them are still to be found, and some of them may be not (finitely) axiomatizable Our experience in studying classical mathematics may be not suited, or at least cannot directly apply, to develop mathematics based on non-classical logics Our experience in studying classical mathematics may be not suited, or at least cannot directly apply, to develop mathematics based on non-classical logics

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We employ a semantical analysis We employ a semantical analysis approach to establish a theory of computation based on quantum logic

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The semantical analysis approach: transforms our intended conclusions in mathematics, which are usually expressed as implication formulas in our logical language, into certain inequalities in the truth-value lattice by truth valuation rules, and then we demonstrate these inequalities in an algebraic way and conclude that the original conclusions are semantically valid

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Orthomodular lattice-valued (quantum) automata Let be an orthomodular lattice. Then an -valued automaton is a 5-tuple where ：

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1. is a finite set of states, 2. is the input alphabet, a finite set of input symbols, 3. is the ( -valued) set of initial states, 4. is the ( -valued) set of final states, 5. is the ( -valued) transition relation.

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Orthomodular lattice-valued (quantum) recognizability Definition 1. (Non-commutative version) Definition 1. (Non-commutative version)

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Definition 2. (Commutative version)

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Takeuti commutator: denotes itself and denotes denotes itself and denotes

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Theorem 1 (Nondeterministic => Deterministic) ： 1. 2. The following two statements are equivalent: (i) is a Boolean algebra ； (ii) for any and the following holds （ universally)

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3. In particular ， if is the Sasaki arrow, then

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Theorem 2 （ Pumping Lemma) ： Let be the Sasaki arrow ， then

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Theoem 3 (Kleene Theorem): 1. 2. The following two statements are equivalent: (i) is a Boolean algebra ； (ii) for any and the following holds （ universally)

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3. In particular ， if is the Sasaki arrow, then

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Orthomodular lattice-valued pushdown automata Orthomodular lattice-valued pushdown automata Orthomodular lattice-valued Turing machines Orthomodular lattice-valued Turing machines ………………. ……………….

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V. Implications V1. Logical Distributivity and Nondeterminism in Computation M. O. Rabin and D. Scott, Finite automata and their decision problems, IBM Journal of Research and Development, 3(1959)114-125

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A.M. Turing Award (1976): Michael O. Rabin, Dana S. Scott A.M. Turing Award (1976): Michael O. Rabin, Dana S. Scott Citation: For their joint paper “Finite Automata and Their Decision Problem,” which introduced the idea of nondeterministic machines, which has proved to be an enormously valuable concept. Their (Scott & Rabin) classic paper has been a continuous source of inspiration for subsequent work in this field Citation: For their joint paper “Finite Automata and Their Decision Problem,” which introduced the idea of nondeterministic machines, which has proved to be an enormously valuable concept. Their (Scott & Rabin) classic paper has been a continuous source of inspiration for subsequent work in this field

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Is （ Nondeterministic => Deterministic ） valid in any logical framework? Is （ Nondeterministic => Deterministic ） valid in any logical framework? No! No! “Logical distributivity” implies “Nondeterminism in computation” “Logical distributivity” implies “Nondeterminism in computation”

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V2. A Re-examination of Church-Turing Thesis Church-Turing Thesis: Every ‘function which would naturally be regarded as computable’ can be computed by a Turing machine

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Why we accept Church-Turing Thesis? A. Some vastly dissimilar formalisms are all computationally equivalent: Turing machines, Post systems, mu-recursive functions, lambda-calculus, combinatory logic, …

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B. Turing machine is equivalent to many modified versions that would seem off-hand to have increased computing power: Two-way infinite tape, multi-tape, nondeterministic, multidimensional, multi-head,…

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D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proceedings of the Royal Society of London, A400 (1985) 97-117 D. Deutsch argued that underlying the D. Deutsch argued that underlying the Church-Turing thesis there is an implicit physical assertion Question: What is it?

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(The physical version of) Church-Turing Principle: Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means Question: Is it related to some fundamental principle in Physics?

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Theorem 1 (generalization for Turing machines, Nondeterministic => Deterministic): 1. The following two statements are equivalent: (1) The lattice of truth values is a Boolean algebra; (2) Nondeterministic => Deterministic, universally.

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2. Commutativity => (Nondeterministic => Deterministic, locally) B. N.: Commutativity of Projection Operators Distributivity in the Lattice of Subspaces of a Hilbert Space

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Question ： What is the implication of this theorem ? 1. (Nondeterministic Turing machines Deterministic Turing machines, universally) only if the underlying logic is classical (Boolean) logic

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2. Commutativity => (Nondeterministic => Deterministic, locally) The physical interpretation of commutativity: The Heisenberg Uncertainty Principle The Heisenberg Uncertainty Principle

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Suppose and are two (physical) observables in a quantum system. Then Suppose and are two (physical) observables in a quantum system. Then and stand for the respective standard and stand for the respective standard deviations of measurement on and is the commutator between is the commutator betweenand

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If, then may vanish; may vanish; or in other words, and can simultaneously become arbitrarily small.

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A potential physical interpretation for the A potential physical interpretation for the need of commutativity: Some nice properties of Turing machines require the standard deviations of the observables concerning the basic actions in these machines being able to reach simultaneously very small values

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Conjecture: The same thing happens to some other witnesses for the Church-Turing thesis

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This implies: This implies: Some interesting connection might reside between (1) the Heisenberg uncertainty principle, a fundamental principle in Quantum Physics and, (2) the Church-Turing thesis, a fundamental hypothesis in Computer Science

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VI. Open Problems J. P. Crutchfield and C. Moore, Quantum automata and quantum grammar, Theoretical Computer Science 237(2000)275-306. J. P. Crutchfield and C. Moore, Quantum automata and quantum grammar, Theoretical Computer Science 237(2000)275-306. E. Bernstein and U. Vazirani, Quantum complexity theory, SIAM J. Comput. 26 (1997)1411--1473.

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Quantum automata in Crutchfield and Moore’s sense Orthomodular lattice- valued automata ? Quantum automata in Crutchfield and Moore’s sense Orthomodular lattice- valued automata ? Quantum automata in ’s sense Orthomodular lattice- valued Turing machines ? Quantum automata in Bernstein and Vazirani’s sense Orthomodular lattice- valued Turing machines ?

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A. M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6(1957)885-893 A potential way: Gleason’s Theorem: Characterize the set of states on the orthomodular lattice (quantum logic) L(H) for a separable real or complex Hilbert space H

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[1] M. S. Ying, Automata theory based on quantum logic (I), (II), International Journal of Theoretical Physics, 39(2000), 985-995, 2545-2557

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[2] M. S. Ying, A theory of computation based on quantum logic (I), 75 pages, Theoretical Computer Science (accepted) Also see http://xxx.lanl.gov/abs/cs.LO/0403041

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[3] M. S. Ying, Quantum logic and automata theory, preparing for Dov Gabbay, Daniel Lehmann and Kurt Engesser (eds.), Handbook of Quantum Logic, North- Holland (Elsevier), 2006

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[4] D. W. Qiu, Automata theory based on quantum logic: some characterizations, Information and Computation, 109:2(2004) 179-195

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