A Theory of Computation Based A Theory of Computation Based on Quantum Logicon Quantum Logic

Mingsheng YingMingsheng Ying

State Key Laboratory of Intelligent Technology anState Key Laboratory of Intelligent Technology and Systems d Systems

Department of Computer Science and TechnologyDepartment of Computer Science and Technology

Tsinghua UniversityTsinghua University

Email: yingmsh@tsinghua.edu.cnEmail: yingmsh@tsinghua.edu.cn

I. Background ---- Non-I. Background ---- Non-classical Logics classical Logics

An axiomatization of a mathematical theoryAn axiomatization of a mathematical theory

consists of a system of fundamental notions consists of a system of fundamental notions

as well as a set of axioms about these 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 theorems which can be derived from the

axiomsaxioms

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

A. Heyting (1963), A. Heyting (1963), Axiomatic Projective Axiomatic Projective GeometryGeometry, North-Holland, Amsterdam, 19, North-Holland, Amsterdam, 196363

In elementary axiomatics logic was used in an unIn elementary axiomatics logic was used in an unanalyzed formanalyzed form

In the studies for foundations of mathematics beIn the studies for foundations of mathematics beginning in the early of twentieth century, it had bginning in the early of twentieth century, it had been realized that a major part of mathematics has een realized that a major part of mathematics has to exploit the full power of classical (Boolean) loto exploit the full power of classical (Boolean) logic, the strongest one in the family of existing logic, the strongest one in the family of existing logics. gics.

A few mathematicians, including the big names A few mathematicians, including the big names L. E. J. Brouwer, H. Poincare, L.L. E. J. Brouwer, H. Poincare, L.

Kronecker and H. Weyl, took some kind of constKronecker and H. Weyl, took some kind of constructive position which is in more or less explicit ructive position which is in more or less explicit opposition to certain forms ofopposition to certain forms of

mathematical reasoning used by the majority of tmathematical reasoning used by the majority of the mathematical communityhe mathematical community

Some of them even endeavored to establish so-caSome of them even endeavored to establish so-called constructive mathematics, the part of mathelled constructive mathematics, the part of mathematics that could be rebuilt on constructivist prinmatics that could be rebuilt on constructivist principlesciples

The logic employed in the development of constrThe logic employed in the development of constructive mathematics is intuitionistic logic which iuctive mathematics is intuitionistic logic which is weaker than classical logics weaker than classical logic

Many logics different from classical logic and intMany logics different from classical logic and intuitionistic logic have been invented in the last ceuitionistic logic have been invented in the last century ntury

Question: Question:

Whether we are able to establish some Whether we are able to establish some mathematical theories based on other non-classical mathematical theories based on other non-classical logics besides intuitionistic logic?logics besides intuitionistic logic?

J. B. Rosser and A. R. Turquette, J. B. Rosser and A. R. Turquette, Many-Valued Many-Valued LogicsLogics, , North-Holland, Amsterdam, 1952North-Holland, Amsterdam, 1952

"The fact that it is thus possible to generalize "The fact that it is thus possible to generalize the ordinary two-valued logic so as not only tothe ordinary two-valued logic so as not only tocover the case of many-valued statement cover the case of many-valued statement calculi, but of many-valued quantification calculi, but of many-valued quantification theory as well, naturally suggests the theory as well, naturally suggests the possibility of further extending our treatment of possibility of further extending our treatment of many-valued logic to cover the case of many-many-valued logic to cover the case of many-valued sets, equality, numbers, etc. valued sets, equality, numbers, etc.

Since we now have a general theory of manySince we now have a general theory of many

valued predicate calculi, there is little doubt valued predicate calculi, there is little doubt

about the possibility of successfully about the possibility of successfully

developing such extended many-valued developing such extended many-valued

theories. ... we shall consider their carefultheories. ... we shall consider their careful

study one of the major unsolved problems of study one of the major unsolved problems of

many-valued logic."many-valued logic."

A. Mostowski, Thirty Years of Foundational Studies A. Mostowski, Thirty Years of Foundational Studies Acta Philosophica FennicaActa Philosophica Fennica, 1965, 1965

J. Lukasiewicz (1920’s) hoped that there would be J. Lukasiewicz (1920’s) hoped that there would be some non-classical logics which can be properly used some non-classical logics which can be properly used in mathematics as non-Euclidean geometry doesin mathematics as non-Euclidean geometry does

Most of non-classical logics invented so far have not Most of non-classical logics invented so far have not been really used in mathematics, and intuitionistic been really used in mathematics, and intuitionistic logic seems that unique one of non-classical logics logic seems that unique one of non-classical logics which still has an opportunity to carry out the which still has an opportunity to carry out the Lukasiewicz's projectLukasiewicz's project

J. Dieudonne, The current trend of pure mathematics,J. Dieudonne, The current trend of pure mathematics,Advances in MathematicsAdvances in Mathematics 27(1978)235-255 27(1978)235-255

Mathematical logicians have been developing Mathematical logicians have been developing

a variety of non-classical logics such as a variety of non-classical logics such as

second-order logic, modal logic and many-second-order logic, modal logic and many-

valued logic, but these logics are completely valued logic, but these logics are completely

useless for mathematicians working in other useless for mathematicians working in other

research areasresearch areas

• The theory of computation is a The theory of computation is a mathematical theorymathematical theory

• Quantum logic is a non-classical logicQuantum logic is a non-classical logic

Question: Question:

(in particular) Should we develop a theory of we develop a theory of

computation based on quantum logic?computation based on quantum logic?

II. Backgroud ---- Quantum II. Backgroud ---- Quantum ComputationComputation The idea of quantum computation came from the studiesThe idea of quantum computation came from the studiesof connections between physics and computation. 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. 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 operation need not dissipate any energy and found that a logically reversible Turing machine is a theoretical logically reversible Turing machine is a theoretical possibility.possibility.

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 model of Turing machine. His construction is the first quantum mechanical description of computer, but it is not a quantum mechanical description of computer, but it is not a real quantum computer. 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 machine may exist in an intrinsically quantum state, but at the end of each computation step the tape of the machine the end of each computation step the tape of the machine always goes back to one of its classical states. always goes back to one of its classical states.

Quantum computers were first envisaged by R. P. Quantum computers were first envisaged by R. P. Feynman (1982). Feynman (1982).

He conceived that no classical Turing machine He conceived that no classical Turing machine could simulate certain quantum phenomena without could simulate certain quantum phenomena without an exponential slowdown, and so he realized that an exponential slowdown, and so he realized that quantum mechanical effects should offer quantum mechanical effects should offer something genuinely new to computation. 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 universal quantum simulator, he did not give a concrete design of such a simulator. concrete design of such a simulator.

Feymann’s ideas were elaborated and formalized by D. Feymann’s ideas were elaborated and formalized by D. Deutsch (1985). 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.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 quantum parallelism parallelism

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 perform certain types of computations that classical computers can only perform very inefficiently. computers can only perform very inefficiently.

One of the most striking advances was made byOne of the most striking advances was made byP. W. Shor (1994)P. W. Shor (1994)

He discovered a polynomial-time algorithm on He discovered a polynomial-time algorithm on quantum computers for prime factorization of quantum computers for prime factorization of which the best known algorithm on classical which the best known algorithm on classical computers is exponential. computers is exponential.

L. K. Grover (1996) offered another apt L. K. Grover (1996) offered another apt killer of quantum computationkiller of quantum computation

He found a quantum algorithm for He found a quantum algorithm for searching a single item in an unsorted searching a single item in an unsorted database in square root of the time it woulddatabase in square root of the time it wouldtake on a classical computer. take on a classical computer.

Both prime factorization and database search are central Both prime factorization and database search are central problems in computer scienceproblems in computer science

The quantum algorithms for them are highly faster than The quantum algorithms for them are highly faster than the classical onesthe 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. 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.exciting and rapidly growing field of research.

The studies of quantum computation The studies of quantum computation

may be roughly divided into four categories:may be roughly divided into four categories:

(1)(1) Physical implementationsPhysical implementations

(2)(2) Physical modelsPhysical models

(3)(3) Mathematical modelsMathematical models

(4)(4) Algorithms and complexityAlgorithms and complexity

(5)(5) Quantum programming, quantum softwareQuantum programming, quantum software

Question:Question:

What is the logical foundation of quantum What is the logical foundation of quantum computation?computation?

V. Vedral and M. B. Plenio (1998) already V. Vedral and M. B. Plenio (1998) already advocated that quantum computers require advocated that quantum computers require quantum logic, something fundamentally different quantum logic, something fundamentally different to classical Boolean logic. to classical Boolean logic.

III. Quantum LogicIII. 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. 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. 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.space formalism of quantum mechanics.

G. Birkhoff and J. von Neumann, G. Birkhoff and J. von Neumann, The logic of quantum mechanics, The logic of quantum mechanics, Annals of MathematicsAnnals of Mathematics, 37(1936)823-843, 37(1936)823-843

““what logical structure one may hope to find in physicalwhat logical structure one may hope to find in physicaltheories which, like quantum mechanics, do not conform totheories which, like quantum mechanics, do not conform toclassical logic. Our main conclusion, based on admittedlyclassical logic. Our main conclusion, based on admittedlyheuristic arguments, is that heuristic arguments, is that one can reasonably expect to one can reasonably expect to find a calculus of propositions which is formally find a calculus of propositions which is formally indistinguishable from the calculus of linear subspaces [of indistinguishable from the calculus of linear subspaces [of Hilbert space]Hilbert space] with respect to set products, linear sums, and with respect to set products, linear sums, and orthogonal complements – and resembles the usual calculus orthogonal complements – and resembles the usual calculus of propositions with respect to 'and', 'or', and 'not'.”of propositions with respect to 'and', 'or', and 'not'.”

Question:Question:

What is the algebraic structures of the set of What is the algebraic structures of the set of

all closed subspaces of a Hilbert space with all closed subspaces of a Hilbert space with

the inclusion relation as an orderingthe inclusion relation as an ordering ？？

Sasaki’s Theorem (1957):Sasaki’s Theorem (1957):

(1) The set of all closed subspaces of a (1) The set of all closed subspaces of a

Hilbert space with the inclusion relation is a Hilbert space with the inclusion relation is a

complete orthomodular lattice; complete orthomodular lattice;

(2) It is a modular lattice if and only if the (2) It is a modular lattice if and only if the

Hilbert space is finite-dimensional. Hilbert space is finite-dimensional.

The algebraic counterpart of classical logic:The algebraic counterpart of classical logic:

Boolean algebraBoolean algebra

Two understandings of quantum logic: Two understandings of quantum logic:

（（ 11 ） ） The theory of The theory of orthomodular latticesorthomodular lattices（（ 22 ） ） A logic whose set of truth values A logic whose set of truth values is an is an orthomodular latticeorthomodular lattice

Orthomodular lawOrthomodular law::

bbaaba )(

IV. A Theory of Computation IV. A Theory of Computation Based on Quantum LogicBased on Quantum Logic

Question: Question:

How to develop How to develop a theory of computation a theory of computation

based on quantum logic?based on quantum logic?

Classical logic is applied as the deduction Classical logic is applied as the deduction

tool in almost all mathematical theories. Ittool in almost all mathematical theories. It

should be noted that what is used in these should be noted that what is used in these

theories is the deductive (proof-theoretical) theories is the deductive (proof-theoretical)

aspect of classical logicaspect of classical logic

The proof theory of non-classical logics is The proof theory of non-classical logics is much more complicated than that of much more complicated than that of classical logic, and it is not an easy task to classical logic, and it is not an easy task to conduct reasoning in the realm of the conduct reasoning in the realm of the proof theory of non-classical logicsproof theory of non-classical logics

It is the case even for the simplest non-It is the case even for the simplest non-

classical logics, three-valued logicsclassical logics, three-valued logics

H. Hodes, Three-valued logics - an introduction, aH. Hodes, Three-valued logics - an introduction, acomparison of various logical lexica, and some philosophical remarcomparison of various logical lexica, and some philosophical remarks, ks, Annals of Pure and Applied LogicAnnals of Pure and Applied Logic, 43(1989)99-145, 43(1989)99-145

““Of course three-valued logics will be Of course three-valued logics will be

somewhat more complicated than classical somewhat more complicated than classical

two-valued logic. In fact, proof-theoretically two-valued logic. In fact, proof-theoretically

they are at least twice as complicated: .... But they are at least twice as complicated: .... But

model-theoretically they are only 50 percentmodel-theoretically they are only 50 percent

more complicated,.…”more complicated,.…”

And much worse, some non-classical logics were And much worse, some non-classical logics were introduced only in a semantic way, and the axiointroduced only in a semantic way, and the axiomatizations of some among them are still to be fmatizations of some among them are still to be found, and some of them may be not (finitely) axiound, and some of them may be not (finitely) axiomatizableomatizable

Our experience in studying classical mathematicOur experience in studying classical mathematics may be not suited, or at least cannot directly aps may be not suited, or at least cannot directly apply, to develop mathematics based on non-classiply, to develop mathematics based on non-classical logicscal logics

We employ a We employ a semantical analysis semantical analysis

approachapproach to establish a theory of to establish a theory of

computation based on quantum logiccomputation based on quantum logic

The semantical analysis approach:The semantical analysis approach:

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

Orthomodular lattice-valued (quantum) autOrthomodular lattice-valued (quantum) automataomata

Let be an orthomodular Let be an orthomodular lattice. Then an -valued automaton is a 5-tuple lattice. Then an -valued automaton is a 5-tuple

wherewhere ：：

1,0,,,,,L

,,,, TIQM

1. is a finite set of states, 1. is a finite set of states,

2. is the input alphabet, a finite set of input 2. is the input alphabet, a finite set of input

symbols,symbols,

3. is the ( -valued) set of initial states,3. is the ( -valued) set of initial states,

4. is the ( -valued) set of final states,4. is the ( -valued) set of final states,

5. is the ( -valued) transition 5. is the ( -valued) transition

relation.relation.

Q

LQI : LQT :

LQQ :

Orthomodular lattice-valued (quantum) rOrthomodular lattice-valued (quantum) recognizabilityecognizability

Definition 1Definition 1.. (Non-commutative version) (Non-commutative version)

)))(,((:)(Re MrecAAMAg

Definition 2Definition 2. (Commutative version). (Commutative version)

)))(

)(())(,((:)(Re

MrecAAatom

MatomAMAgC

Takeuti commutator:Takeuti commutator:

denotes itself and denotes denotes itself and denotes

}}1,0{:|{)( )( ff

1 1

Theorem 1 Theorem 1 (Nondeterministic => Deterministic)(Nondeterministic => Deterministic) ：：

1. 1.

2. The following two statements are equivalent:2. The following two statements are equivalent:

(i) is a Boolean algebra(i) is a Boolean algebra ；；(ii) for any and the following (ii) for any and the following

holdsholds （（ universally)universally)M

)()(|)(

srecsrec MM

)()(|)(

srecsrec MM

s

3. 3.

In particularIn particular ，， if is the Sasaki arrow, if is the Sasaki arrow,

thenthen

)()())((|)(

srecsrecMatom MM

))()((

))((|

)(srecsrec

Matom

MM

Theorem 2Theorem 2 （（ Pumping Lemma)Pumping Lemma) ：：

Let be the Sasaki arrowLet be the Sasaki arrow ，， thenthen

))])(0(1||||

)(,,(

||)[(

)0()(Re|

*

*

Awuvivnuv

uvwswvu

nsAss

nAgC

i

Theoem 3Theoem 3 (Kleene Theorem): (Kleene Theorem):

1. 1.

2. The following two statements are equivalent:2. The following two statements are equivalent:

(i) is a Boolean algebra(i) is a Boolean algebra ；；(ii) for any and the following holds(ii) for any and the following holds （（ universunivers

ally)ally)

))(()(| MkLssrecM

))(()(| MkLssrecM

M s

3. 3.

In particularIn particular ，， if is the Sasaki arrow, if is the Sasaki arrow,

thenthen

)())(())((| srecMkLsMatom M

))())(((

))((|

srecMkLs

Matom

M

Orthomodular lattice-valued pushdown autOrthomodular lattice-valued pushdown automataomata

Orthomodular lattice-valued Turing machiOrthomodular lattice-valued Turing machinesnes

……………… ………………..

V. ImplicationsV. Implications

V1. Logical Distributivity and Nondeterminism V1. Logical Distributivity and Nondeterminism

in Computationin Computation

M. O. Rabin and D. Scott, Finite automata and M. O. Rabin and D. Scott, Finite automata and

their decision problems, their decision problems, IBM Journal of Research IBM Journal of Research

and Developmentand Development, 3(1959)114-125, 3(1959)114-125

A.M. Turing Award (1976): Michael O. Rabin,A.M. Turing Award (1976): Michael O. Rabin, Dana S. Scott Dana S. Scott

CitationCitation: For their joint paper “Finite Automata : For their joint paper “Finite Automata and Their Decision Problem,” which introduced tand Their Decision Problem,” which introduced the idea of nondeterministic machines, which has he idea of nondeterministic machines, which has proved to be an enormously valuable concept. Tproved to be an enormously valuable concept. Their (Scott & Rabin) classic paper has been a conheir (Scott & Rabin) classic paper has been a continuous source of inspiration for subsequent wortinuous source of inspiration for subsequent wor

k in thisk in this field field

IsIs （（ Nondeterministic => DeterministiNondeterministic => Deterministicc ）） valid in any logical framework?valid in any logical framework?

No!No!

““Logical distributivity” implies “NondeterLogical distributivity” implies “Nondeterminism in computation”minism in computation”

V2. A Re-examination of Church-Turing ThesisV2. A Re-examination of Church-Turing Thesis

Church-Turing Thesis:Church-Turing Thesis:

Every ‘function which would naturally be Every ‘function which would naturally be

regarded as computable’ can be computed regarded as computable’ can be computed

by a Turing machineby a Turing machine

Why we accept Church-Turing Thesis?Why we accept Church-Turing Thesis?

A. Some vastly dissimilar formalisms are all A. Some vastly dissimilar formalisms are all

computationally equivalent:computationally equivalent:

Turing machines, Post systems, mu-recursive Turing machines, Post systems, mu-recursive

functions, lambda-calculus, combinatory logic, …functions, lambda-calculus, combinatory logic, …

B. Turing machine is equivalent to many B. Turing machine is equivalent to many

modified versions that would seem off-hand modified versions that would seem off-hand

to have increased computing power:to have increased computing power:

Two-way infinite tape, multi-tape, Two-way infinite tape, multi-tape,

nondeterministic, multidimensional, nondeterministic, multidimensional,

multi-head,…multi-head,…

D. Deutsch, Quantum theory, the Church-Turing D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, principle and the universal quantum computer, Proceedings of the Royal Society of LondonProceedings of the Royal Society of London, A400 (1985) , A400 (1985) 97-11797-117

D. Deutsch argued that underlying the D. Deutsch argued that underlying the

Church-Turing thesis there is an implicit Church-Turing thesis there is an implicit

physical assertionphysical assertion

Question: Question: What is it?What is it?

(The physical version of) (The physical version of) Church-Turing Church-Turing Principle:Principle:

Every finitely realizable physical system can be Every finitely realizable physical system can be perfectly simulated by a universal model perfectly simulated by a universal model computing machine operating by finite meanscomputing machine operating by finite means

Question:Question: Is it related to some fundamental Is it related to some fundamental principle in principle in PhysicsPhysics??

Theorem 1 Theorem 1 (generalization for Turing machines, (generalization for Turing machines,

Nondeterministic => Deterministic): Nondeterministic => Deterministic):

1. The following two statements are equivalent:1. The following two statements are equivalent:

(1) The lattice of truth values is a Boolean algebra;(1) The lattice of truth values is a Boolean algebra;

(2) Nondeterministic => Deterministic, universally.(2) Nondeterministic => Deterministic, universally.

2. Commutativity => (Nondeterministic => 2. Commutativity => (Nondeterministic =>

Deterministic, locally)Deterministic, locally)

B. N.: Commutativity of Projection B. N.: Commutativity of Projection

Operators Operators Distributivity in the Lattice of Distributivity in the Lattice of

Subspaces of a Hilbert SpaceSubspaces of a Hilbert Space

QuestionQuestion ：： What is the implication of this What is the implication of this

theoremtheorem??

1. (Nondeterministic Turing machines 1. (Nondeterministic Turing machines

Deterministic Turing machines, universally) Deterministic Turing machines, universally)

only if the underlying logic is classical only if the underlying logic is classical

(Boolean) logic(Boolean) logic

2. Commutativity => (Nondeterministic => 2. Commutativity => (Nondeterministic =>

Deterministic, locally)Deterministic, locally)

The physical interpretation of The physical interpretation of

commutativity: commutativity:

The Heisenberg Uncertainty PrincipleThe Heisenberg Uncertainty Principle

Suppose and are two (physical) observables iSuppose and are two (physical) observables in a quantum system. Then n a quantum system. Then

and stand for the respective standard and stand for the respective standard deviations of measurement on and deviations of measurement on and is the commutator between is the commutator between and and

||B][A,||2

1(B)(A)

A B

)(A )(BA B

BAABBA ],[ AB

If , If ,

then then

may vanish;may vanish;

or in other words, and can or in other words, and can simultaneously become arbitrarily small.simultaneously become arbitrarily small.

BAAB

0||],[|| BA

)()( BA

)(A )(B

A potential physical interpretation for the A potential physical interpretation for the need of commutativity:need of commutativity:

Some nice properties of Turing machines Some nice properties of Turing machines require the standard deviations of the observables require the standard deviations of the observables concerning the basic actions in these machines concerning the basic actions in these machines being able to reach simultaneously very small being able to reach simultaneously very small valuesvalues

Conjecture: Conjecture:

The same thing happens to some other The same thing happens to some other

witnesses for the Church-Turing thesiswitnesses for the Church-Turing thesis

This implies: This implies:

Some Some interesting connectioninteresting connection might reside might reside between between

(1) the Heisenberg uncertainty principle, a (1) the Heisenberg uncertainty principle, a fundamental principle in fundamental principle in Quantum PhysicsQuantum Physics and, and, (2) the Church-Turing thesis, a fundamental (2) the Church-Turing thesis, a fundamental hypothesis in hypothesis in Computer ScienceComputer Science

VI. Open ProblemsVI. Open Problems

J. P. Crutchfield and C. Moore, Quantum aJ. P. Crutchfield and C. Moore, Quantum automata and quantum grammar, utomata and quantum grammar, TheoreticTheoretical Computer Scienceal Computer Science 237(2000)275-306. 237(2000)275-306.

E. Bernstein and U. Vazirani, Quantum complexity theory, SIAM J. Comput. 26 (1997)1411--1473.

Quantum automata in Quantum automata in Crutchfield and MoCrutchfield and Moore’s senseore’s sense Orthomodular lattice-value Orthomodular lattice-valued automata ?d automata ?

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

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

[1] M. S. Ying, Automata theory based on [1] M. S. Ying, Automata theory based on

quantum logic (I), (II), quantum logic (I), (II), International International

Journal of Theoretical PhysicsJournal of Theoretical Physics, 39(2000), , 39(2000),

985-995, 2545-2557985-995, 2545-2557

[2] M. S. Ying, A theory of computation [2] M. S. Ying, A theory of computation

based on quantum logic (I), 75 pages, based on quantum logic (I), 75 pages,

Theoretical Computer Science Theoretical Computer Science (accepted)(accepted)

Also seeAlso see http://xxx.lanl.gov/abs/cs.LO/04030 http://xxx.lanl.gov/abs/cs.LO/040304141

[3] M. S. Ying, Quantum logic and automata [3] M. S. Ying, Quantum logic and automata

theory, preparing for Dov Gabbay, Daniel theory, preparing for Dov Gabbay, Daniel

Lehmann and Kurt Engesser (eds.), Lehmann and Kurt Engesser (eds.),

Handbook of Quantum Logic, Handbook of Quantum Logic, North-North-

Holland (Elsevier), 2006Holland (Elsevier), 2006

[4] D. W. Qiu, Automata theory based on

quantum logic: some characterizations,

Information and Computation, 109:2(2004)

179-195

Thanks!Thanks!