An Algebra of Reversible Quantum Computing

0.53 MB·English

by  Yong Wang

AnEnglishDraftofDr.YongWang
An Algebra of Reversible Quantum
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Computing
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Yong Wang
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SchoolofComputerScienceandTechnology,
7 BeijingUniversityofTechnology,Beijing,China
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Abstract. Based on the axiomatization of reversible computing RACP, we generalize it to quantum re-
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s versible computing which is called qRACP. By use of the framework of quantum configuration, we show
c that structural reversibility and quantum state reversibility must be satisfied simultaneously in quantum
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reversible computation. RACP and qRACP has the same axiomatization modulo the so-called quantum

1 forward-reverse bisimularity, that is, classical reversible computing and quantum reversible computing are
v unified.
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6 Keywords: Reversible Computation; Process Algebra; Algebra of Communicating Processes; Axiomatiza-
2 tion; Quantum Computing
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v 1. Introduction
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Reversiblecomputationhasgainedmoreandmoreattentioninmanyapplicationareas,suchasthemodeling
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a ofbiochemicalsystems,programdebuggingandtesting,andalsoquantumcomputing.Quantumcomputing
is one of the most important application areas of reversible computation.
There are several research works on reversible computation. Abramsky maps functional programs into
reversible automata [8]; Danos and Krivine’s reversible RCCS [9] uses the concept of thread to reverse a
CCS [3][4] process; Boudol and Castellani [10][11] compare three different non-interleaving models for CCS;
Phillips and Ulidowski’s CCSK [12] formulates a procedure for converting operators of standard algebraic
process calculi such as CCS into reversible operators, while preserving their operational semantics. While
RACP is a new reversible axiomatical algebra, which is an axiomatical refinement to CCSK.
In quantum process algebra, qACP [17] [18] is an axiomatization of quantum processes, which is a quan-
tum generalization of process algebra ACP. qACP still uses the concept of a quantum process configuration
⟨p,(cid:37)⟩,whichisusuallyconsistedofaprocesstermpandstateinformation(cid:37)ofall(public)quantuminforma-
tion variables. In qACP, quantum operations are chosen to describe transformations of quantum states, and
Correspondence and offprint requests to: Yong Wang, Pingleyuan 100, Chaoyang District, Beijing, China. e-mail:
[email protected]
2 YongWang
behave as the atomic actions of a pure quantum process. Quantum measurements are treated as quantum
operations, so probabilistic bisimulations are avoided.
As qACP is a quantum generalization of ACP, we generalize RACP in quantum reversible computing
in this paper, which is called qRACP. RACP and qRACP has the same axiomatization modulo the so-
called quantum forward-reverse bisimularity, that is, classical reversible computing and quantum reversible
computing are unified.
The paper is organized as follows. In section 2, some basic concepts about basic linear algebra, basic
quantum mechanics, equational logic, structural operational semantics and process algebra ACP are intro-
duced. In section 3, we extend classical structural operational semantics to support quantum processes. The
BRQPA is introduced in section 4, ARQCP is introduced in section 5, recursion is introduced in section 6,
andabstractionisintroducedinsection7.Classicalreversiblecomputingandquantumreversiblecomputing
are unified in section 8. An application of qRACP is introduced in section 9. The extensions of qRACP is
discussed in section 10. Finally, we conclude this paper in section 11.
2. Preliminaries
For convenience of the reader, we introduce some basic concepts about basic linear algebra, basic quantum
mechanics (Please refer to [14] for details), equational logic, structural operational semantics and process
algebra ACP (Please refer to [6] and [5] for more details).
2.1. Basic Linear Algebra
Definition 2.1.1 (Hilbert space).AnisolatedphysicalsystemisassociatedwithaHilbertspace,whichis
calledthestatespaceofthesystem.Afinite-dimensionalHilbertspaceisacomplexvectorspaceH together
with an inner product, which is a mapping ⟨⋅∣⋅⟩ ∶ H×H → C satisfying: (1)⟨ϕ∣ϕ⟩ ≥ 0 with equality if and
only if ∣ϕ⟩=0; (2)⟨ϕ∣ψ⟩=⟨ψ∣ϕ⟩∗; (3) ⟨ϕ∣λ ψ +λ ψ ⟩=λ ⟨ϕ∣ψ ⟩+λ ⟨ϕ∣ψ ⟩, where C is the set of complex
1 1 2 2 1 1 2 2
numbers, and λ∗ denotes the conjugate of λ (λ∈C).
√
Definition 2.1.2 (Orthonormal basis). For any vector ∣ψ⟩ in H, the length ∣∣ψ∣∣= ⟨ψ∣ψ⟩. A vector
∣ψ⟩ with ∣∣ψ∣∣ = 1 is called a unit vector in its state space. An orthonormal basis of a Hilbert space H is a
basis {∣i⟩} with
1 if i=j,
⟨i∣j⟩={
0 otherwise.
Definition 2.1.3 (Trace of a linear operator). The trace of a linear operator A on H is defined as
tr(A)=∑⟨i∣A∣i⟩.
i
Definition 2.1.4 (Tensor products). The state space of a composite system is the tensor product of
the state space of its components. Let H and H be two Hilbert spaces, then their tensor product H ⊗H
1 2 1 2
consists of linear vectors ∣ψ ψ ⟩=∣ψ ⟩⊗∣ψ ⟩, where ψ ∈H and ψ ∈H .
1 2 1 2 1 1 2 2
For two linear operator A on Hilbert space H , A on Hilbert space H , A ⊗A is defined as
1 1 2 2 1 2
(A ⊗A )∣ψ ψ ⟩=A ∣ψ ⟩⊗A ∣ψ ⟩
1 2 1 2 1 1 2 2
where ∣ψ ⟩∈H and ∣ψ ⟩∈H .
1 1 2 2
Let ∣ϕ⟩=∑ α ∣ϕ ϕ ⟩∈H ⊗H and ∣ψ⟩=∑ β ∣ψ ψ ⟩∈H ⊗H . Then the inner product of ∣ϕ⟩ and
i i 1i 2i 1 2 j i 1j 2j 1 2
∣ψ⟩ is defined as follows.
⟨ϕ∣ψ⟩=∑α∗β ⟨ϕ ∣ψ ⟩⟨ϕ ∣ψ ⟩.
i j 1i 1j 2i 2j
i,j
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2.2. Basic Quantum Mechanics
Definition 2.2.1 (Density operator). A mixed state of quantum system is represented by a density
operator. A density operator in H is a linear operator (cid:37) satisfying:(1) (cid:37) is positive, that is, ⟨ψ∣(cid:37)∣ψ⟩≥0 for
all ∣ψ⟩; (2) tr((cid:37))=1. Let D(H) denote the set of all positive operators on H.
Definition 2.2.2 (Unitary operator). The evolution of a closed quantum system is described by a
unitary operator on its state space. A unitary operator is a linear operator U on a Hilbert space H with
U†U =I , where I is the identity operator on H and U† is the adjoint of U.
H H
Definition 2.2.3 (Quantum measurement). A quantum measurement consists of a collection of
measurement operators {M }, where m is the measurement outcomes and satisfies
m
∑M†M =I .
m m H
m
Definition2.2.4(Quantumoperation(superoperator)).Theevolutionofanopenquantumsystem
can be described by a quantum operation. A quantum operation on a Hilbert space H is a linear operator
S from the space of linear operators on H into itself satisfying: (1) tr[S((cid:37))]≤tr((cid:37)) for each (cid:37)∈D(H); (2)
for any extra Hilbert space H , (I ⊗S(A)) is positive if A is a positive operator on H ⊗H, where I is
R R R R
the identity operator in H . If tr[S((cid:37))]=tr((cid:37)) for all (cid:37)∈D(H), then S is said to be trace-preserving.
R
Definition 2.2.5 (Relation between quantum operation and unitary operator). Let U be a
unitary operator on the Hilbert space H, and S((cid:37))=U(cid:37)U† for any (cid:37)∈D(H). Then S is a trace-preserving
quantum operation.
Definition 2.2.6 (Relation between quantum operation and measurement operator). Let
{M } be a quantum measurement on the Hilbert space H. For each m, let S ((cid:37)) = M (cid:37)M† for any
m m m m
(cid:37)∈D(H). The S is a quantum operation and is not necessarily trace-preserving.
m
In this paper, we refer to quantum operations all as reversible quantum operations. Quantum measure-
ments are treated as a special kind of quantum operations, in [13], quantum measurements can also be
reversible.
2.3. Equational Logic
We introduce some basic concepts about equational logic briefly, including signature, term, substitution,
axiomatization, equality relation, model, term rewriting system, rewrite relation, normal form, termination,
weak confluence and several conclusions. These concepts are coming from [5], and are introduced briefly as
follows. About the details, please see [5].
Definition 2.3.1 (Signature). A signature Σ consists of a finite set of function symbols (or operators)
f,g,⋯,whereeachfunctionsymbol f hasanarity ar(f),beingitsnumberofarguments.Afunctionsymbol
a,b,c,⋯ of arity zero is called a constant, a function symbol of arity one is called unary, and a function
symbol of arity two is called binary.
Definition 2.3.2 (Term). Let Σ be a signature. The set T(Σ) of (open) terms s,t,u,⋯ over Σ is
defined as the least set satisfying: (1)each variable is in T(Σ); (2) if f ∈ Σ and t ,⋯,t ∈ T(Σ), then
1 ar(f)
f(t ,⋯,t ∈T(Σ)). A term is closed if it does not contain variables. The set of closed terms is denoted
1 ar(f)
by T(Σ).
Definition 2.3.3 (Substitution). Let Σ be a signature. A substitution is a mapping σ from variables
to the set T(Σ) of open terms. A substitution extends to a mapping from open terms to open terms: the
term σ(t) is obtained by replacing occurrences of variables x in t by σ(x). A substitution σ is closed if
σ(x)∈T(Σ) for all variables x.
Definition 2.3.4 (Axiomatization). An axiomatization over a signature Σ is a finite set of equations,
called axioms, of the form s=t with s,t∈T(Σ).
Definition 2.3.5 (Equality relation).AnaxiomatizationoverasignatureΣinducesabinaryequality
relation = on T(Σ) as follows. (1)(Substitution) If s=t is an axiom and σ a substitution, then σ(s)=σ(t).
(2)(Equivalence)Therelation=isclosedunderreflexivity,symmetry,andtransitivity.(3)(Context)Therela-
tion=isclosedundercontexts:ift=uandf isafunctionsymbolwithar(f)>0,thenf(s ,⋯,s ,t,s ,⋯,s )=
1 i−1 i+1 ar(f)
f(s ,⋯,s ,u,s ,⋯,s ).
1 i−1 i+1 ar(f)
Definition 2.3.6 (Model). Assume an axiomatization E over a signature Σ, which induces an equality
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relation =. A model for E consists of a set M together with a mapping φ∶T(Σ)→M. (1)(M,φ) is sound
for E if s=t implies φ(s)≡φ(t) for s,t∈T(Σ); (2)(M,φ) is complete for E if φ(s)≡φ(t) implies s=t for
s,t∈T(Σ).
Definition 2.3.7 (Term rewriting system). Assume a signature Σ. A rewrite rule is an expression
s→twiths,t∈T(Σ),where:(1)theleft-handsidesisnotasinglevariable;(2)allvariablesthatoccuratthe
right-hand side t also occur in the left-hand side s. A term rewriting system (TRS) is a finite set of rewrite
rules.
Definition 2.3.8 (Rewrite relation). A TRS over a signature Σ induces a one-step rewrite relation
→ on T(Σ) as follows. (1)(Substitution) If s → t is a rewrite rule and σ a substitution, then σ(s) → σ(t).
(2)(Context) The relation → is closed under contexts: if t → u and f is a function symbol with ar(f) > 0,
then f(s ,⋯,s ,t,s ,⋯,s )→f(s ,⋯,s ,u,s ,⋯,s ). The rewrite relation →∗ is the reflexive
1 i−1 i+1 ar(f) 1 i−1 i+1 ar(f)
transitive closure of the one-step rewrite relation →: (1) if s → t, then s →∗ t; (2) t →∗ t; (3) if s →∗ t and
t→∗u, then s→∗u.
Definition 2.3.9 (Normal form). A term is called a normal form for a TRS if it cannot be reduced
by any of the rewrite rules.
Definition 2.3.10 (Termination). A TRS is terminating if it does not induce infinite reductions
t →t →t →⋯.
0 1 2
Definition2.3.11(Weakconfluence).ATRSisweaklyconfluentifforeachpairofone-stepreductions
s→t and s→t , there is a term u such that t →∗u and t →∗u.
1 2 1 2
Theorem 2.3.1 (Newman’s lemma). If a TRS is terminating and weakly confluent, then it reduces
each term to a unique normal form.
Definition 2.3.12 (Commutativity and associativity).AssumeanaxiomatizationE.Abinaryfunc-
tionsymbolf iscommutativeifE containsanaxiomf(x,y)=f(y,x)andassociativeifE containsanaxiom
f(f(x,y),z)=f(x,f(y,z)).
Definition 2.3.13 (Convergence). A pair of terms s and t is said to be convergent if there exists a
term u such that s→∗u and t→∗u.
Axiomatizations can give rise to TRSs that are not weakly confluent, which can be remedied by Knuth-
Bendix completion[1]. It determines overlaps in left hand sides of rewrite rules, and introduces extra rewrite
rules to join the resulting right hand sides, witch are called critical pairs.
Theorem 2.3.2. A TRS is weakly confluent if and only if all its critical pairs are convergent.
2.4. Structural Operational Semantics
The concepts about structural operational semantics include labelled transition system (LTS), transition
system specification (TSS), transition rule and its source, source-dependent, conservative extension, fresh
operator,panthformat,congruence,bisimulation,etc.Theseconceptsarecomingfrom[5],andareintroduced
briefly as follows. About the details, please see [6]. Also, to support reversible computation, we introduce a
new kind of bisimulation called forward-reverse bisimulation (FR bisimulation) which first occurred in [12].
We assume a non-empty set S of states, a finite, non-empty set of transition labels A and a finite set of
predicate symbols.
Definition 2.4.1 (Labeled transition system). A transition is a triple (s,a,s′) with a ∈ A, or a
pair (s, P) with P a predicate, where s,s′ ∈S. A labeled transition system (LTS) is possibly infinite set of
transitions. An LTS is finitely branching if each of its states has only finitely many outgoing transitions.
Definition 2.4.2 (Transition system specification). A transition rule ρ is an expression of the form
H, with H a set of expressions t —→a t′ and tP with t,t′ ∈ T(Σ), called the (positive) premises of ρ, and π
π
an expression t—→a t′ or tP with t,t′ ∈T(Σ), called the conclusion of ρ. The left-hand side of π is called the
source of ρ. A transition rule is closed if it does not contain any variables. A transition system specification
(TSS) is a (possible infinite) set of transition rules.
Definition 2.4.3 (Proof). A proof from a TSS T of a closed transition rule H consists of an upwardly
π
branching tree in which all upward paths are finite, where the nodes of the tree are labelled by transitions
such that: (1) the root has label π; (2) if some node has label l, and K is the set of labels of nodes directly
above this node, then (a) either K is the empty set and l∈H, (b) or K is a closed substitution instance of
l
a transition rule in T.
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Definition 2.4.4 (Generated LTS). We define that the LTS generated by a TSS T consists of the
transitions π such that ∅ can be proved from T.
π
Definition 2.4.5. A set N of expressions t↛a and t¬P (where t ranges over closed terms, a over A and
P over predicates) hold for a set S of transitions, denoted by S ⊧N, if: (1) for each t↛a∈N we have that
t—→a t′∉S for all t′∈T(Σ); (2) for each t¬P ∈N we have that tP ∉S.
Definition 2.4.6 (Three-valued stable model).Apair⟨C,U⟩ofdisjointsetsoftransitionsisathree-
valued stable model for a TSS T if it satisfies the following two requirements: (1) a transition π is in C if
and only if T proves a closed transition rule N where N contains only negative premises and C∪U ⊧N; (2)
π
a transition π is in C∪U if and only if T proves a closed transition rule N where N contains only negative
π
premises and C ⊧N.
Definition 2.4.7 (Ordinal number). The ordinal numbers are defined inductively by: (1) 0 is the
smallest ordinal number; (2) each ordinal number α has a successor α+1; (3) each sequence of ordinal
number α<α+1<α+2<⋯ is capped by a limit ordinal λ.
Definition 2.4.8 (Positive after reduction).ATSSispositiveafterreductionifitsleastthree-valued
stable model does not contain unknown transitions.
Definition 2.4.9 (Stratification). A stratification for a TSS is a weight function φ which maps tran-
sitions to ordinal numbers, such that for each transition rule ρ with conclusion π and for each closed substi-
tution σ: (1) for positive premises t—→a t′ and tP of ρ, φ(σ(t)—→a σ(t′))≤φ(σ(π)) and φ(σ(t)P ≤φ(σ(π))),
respectively; (2) for negative premise t↛a and t¬P of ρ, φ(σ(t)—→a t′)<φ(σ(π)) for all closed terms t′ and
φ(σ(t)P <φ(σ(π))), respectively.
Theorem 2.4.1. If a TSS allows a stratification, then it is positive after reduction.
Definition 2.4.10 (Process graph). A process (graph) p is an LTS in which one state s is elected to
be the root. If the LTS contains a transition s—→a s′, then p—→a p′ where p′ has root state s′. Moreover, if the
LTS contains a transition sP, then pP. (1) A process p is finite if there are only finitely many sequences
0
p —a→1 p —a→2 ⋯ —a→k P . (2) A process p is regular if there are only finitely many processes p such that
0 1 k 0 k
p —a→1 p —a→2 ⋯—a→k P .
0 1 k
Definition 2.4.11 (Reverse transition).There aretwo processesp and p′,twotransitions p—→a p′ and
p′ —a—[m↠] p, the transition p′ —a—[m↠] p is called reverse transition of p—→a p′, and the transition p—→a p′ is called
forward transition. If p —→a p′ then p′ —a—[m↠] p, the forward transition p —→a p′ is reversible. Where a[m] is a
kind of special action constant a[m]∈A×K, K∈N, called the histories of an action a, and m∈K.
Definition 2.4.12 (Bisimulation). A bisimulation relation B is a binary relation on processes such
that: (1) if pBq and p—→a p′ then q—→a q′ with p′Bq′; (2) if pBq and q—→a q′ then p—→a p′ with p′Bq′; (3) if pBq
and pP, then qP; (4) if pBq and qP, then pP. Two processes p and q are bisimilar, denoted by p↔q, if there
is a bisimulation relation B such that pBq.
Definition 2.4.13 (Forward-reverse bisimulation). A forward-reverse (FR) bisimulation relation B
is a binary relation on processes such that: (1) if pBq and p —→a p′ then q —→a q′ with p′Bq′; (2) if pBq and
q —→a q′ then p—→a p′ with p′Bq′; (3)if pBq and p—a—[m↠] p′ then q —a—[m↠] q′ with p′Bq′; (4) if pBq and q —a—[m↠] q′
then p—a—[m↠] p′ with p′Bq′; (5) if pBq and pP, then qP; (6) if pBq and qP, then pP. Two processes p and q
are FR bisimilar, denoted by p↔frq, if there is a FR bisimulation relation B such that pBq.
Definition 2.4.14 (Congruence). Let Σ be a signature. An equivalence relation B on T(Σ) is a
congruence if for each f ∈Σ, if s Bt for i∈{1,⋯,ar(f)}, then f(s ,⋯,s )Bf(t ,⋯,t ).
i i 1 ar(f) 1 ar(f)
Definition 2.4.15 (Panth format). A transition rule ρ is in panth format if it satisfies the following
three restrictions: (1) for each positive premise t—→a t′ of ρ, the right-hand side t′ is single variable; (2) the
source of ρ contains no more than one function symbol; (3) there are no multiple occurrences of the same
variable at the right-hand sides of positive premises and in the source of ρ. A TSS is said to be in panth
format if it consists of panth rules only.
Theorem 2.4.2. If a TSS is positive after reduction and in panth format, then the bisimulation equiva-
lence that it induces is a congruence.
Definition 2.4.16 (Branching bisimulation).AbranchingbisimulationrelationBisabinaryrelation
onthecollectionofprocessessuchthat:(1)ifpBqandp—→a p′ theneithera≡τ andp′Bqorthereisasequence
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of (zero or more) τ-transitions q —→τ ⋯ —→τ q such that pBq and q —→a q′ with p′Bq′; (2) if pBq and q —→a q′
0 0 0
then either a ≡ τ and pBq′ or there is a sequence of (zero or more) τ-transitions p —→τ ⋯ —→τ p such that
0
p Bq and p —→a p′ with p′Bq′; (3) if pBq and pP, then there is a sequence of (zero or more) τ-transitions
0 0
τ τ
q —→ ⋯ —→ q such that pBq and q P; (4) if pBq and qP, then there is a sequence of (zero or more) τ-
0 0 0
τ τ
transitions p—→⋯—→p such that p Bq and p P. Two processes p and q are branching bisimilar, denoted by
0 0 0
p↔ q, if there is a branching bisimulation relation B such that pBq.
b
Definition 2.4.17 (Branching forward-reverse bisimulation). A branching forward-reverse (FR)
bisimulation relation B is a binary relation on the collection of processes such that: (1) if pBq and p —→a p′
then either a≡τ and p′Bq or there is a sequence of (zero or more) τ-transitions q—→τ ⋯—→τ q such that pBq
0 0
and q —→a q′ with p′Bq′; (2) if pBq and q —→a q′ then either a≡τ and pBq′ or there is a sequence of (zero or
0
more) τ-transitions p—→τ ⋯—→τ p such that p Bq and p —→a p′ with p′Bq′; (3) if pBq and pP, then there is a
0 0 0
τ τ
sequence of (zero or more) τ-transitions q—→⋯—→q such that pBq and q P; (4) if pBq and qP, then there
0 0 0
is a sequence of (zero or more) τ-transitions p—→τ ⋯—→τ p such that p Bq and p P; (5) if pBq and p—a—[m↠] p′
0 0 0
then either a≡τ and p′Bq or there is a sequence of (zero or more) τ-transitions q——↠τ ⋯——↠τ q such that pBq
0 0
and q —a—[m↠] q′ with p′Bq′; (6) if pBq and q—a—[m↠] q′ then either a≡τ and pBq′ or there is a sequence of (zero
0
or more) τ-transitions p——↠τ ⋯——↠τ p such that p Bq and p —a—[m↠] p′ with p′Bq′; (7) if pBq and pP, then there
0 0 0
τ τ
is a sequence of (zero or more) τ-transitions q——↠⋯——↠q such that pBq and q P; (8) if pBq and qP, then
0 0 0
τ τ
there is a sequence of (zero or more) τ-transitions p ——↠ ⋯ ——↠ p such that p Bq and p P. Two processes
0 0 0
p and q are branching FR bisimilar, denoted by p↔frq, if there is a branching FR bisimulation relation B
b
such that pBq.
Definition 2.4.18 (Rooted branching bisimulation). A rooted branching bisimulation relation B
is a binary relation on processes such that: (1) if pBq and p —→a p′ then q —→a q′ with p′↔ q′; (2) if pBq and
b
q —→a q′ then p—→a p′ with p′↔ q′; (3) if pBq and pP, then qP; (4) if pBq and qP, then pP. Two processes p
b
and q are rooted branching bisimilar, denoted by p↔ q, if there is a rooted branching bisimulation relation
rb
B such that pBq.
Definition2.4.19(Rootedbranchingforward-reversebisimulation).Arootedbranchingforward-
reverse (FR) bisimulation relation B is a binary relation on processes such that: (1) if pBq and p—→a p′ then
q—→a q′ with p′↔frq′; (2) if pBq and q—→a q′ then p—→a p′ with p′↔frq′; (3) if pBq and p—a—[m↠] p′ then q—a—[m↠] q′
b b
with p′↔frq′; (4) if pBq and q —a—[m↠] q′ then p—a—[m↠] p′ with p′↔frq′; (5) if pBq and pP, then qP; (6) if pBq
b b
and qP, then pP. Two processes p and q are rooted branching FR bisimilar, denoted by p↔frq, if there is
rb
a rooted branching FR bisimulation relation B such that pBq.
Definition 2.4.20 (Lookahead). A transition rule contains lookahead if a variable occurs at the left-
hand side of a premise and at the right-hand side of a premise of this rule.
Definition 2.4.21 (Patience rule). A patience rule for the i-th argument of a function symbol f is a
panth rule of the form
τ
x —→y
i .
τ
f(x ,⋯,x )—→f(x ,⋯,x ,y,x ,⋯,x )
1 ar(f) 1 i−1 i+1 ar(f)
Definition 2.4.22 (RBB cool format). A TSS T is in RBB cool format if the following requirements
are fulfilled. (1) T consists of panth rules that do not contain lookahead. (2) Suppose a function symbol f
occurs at the right-hand side the conclusion of some transition rule in T. Let ρ∈T be a non-patience rule
with source f(x ,⋯,x ). Then for i∈{1,⋯,ar(f)}, x occurs in no more than one premise of ρ, where
1 ar(f) i
a
this premise is of the form x P or x —→y with a≠τ. Moreover, if there is such a premise in ρ, then there is
i i
a patience rule for the i-th argument of f in T.
Theorem 2.4.3.IfaTSSispositiveafterreductionandinRBBcoolformat,thentherootedbranching
bisimulation equivalence that it induces is a congruence.
Definition 2.4.23 (Conservative extension). Let T and T be TSSs over signatures Σ and Σ ,
0 1 0 1
AnAlgebraofReversibleQuantumComputing 7
respectively. The TSS T ⊕T is a conservative extension of T if the LTSs generated by T and T ⊕T
0 1 0 0 0 1
contain exactly the same transitions t—→a t′ and tP with t∈T(Σ ).
0
Definition 2.4.24 (Source-dependency).Thesource-dependentvariablesinatransitionruleofρare
defined inductively as follows: (1) all variables in the source of ρ are source-dependent; (2) if t —→a t′ is a
premise of ρ and all variables in t are source-dependent, then all variables in t′ are source-dependent. A
transition rule is source-dependent if all its variables are. A TSS is source-dependent if all its rules are.
Definition 2.4.25 (Freshness).LetT andT beTSSsoversignaturesΣ andΣ ,respectively.Aterm
0 1 0 1
in T(T ⊕T ) is said to be fresh if it contains a function symbol from Σ ∖Σ . Similarly, a transition label
0 1 1 0
or predicate symbol in T is fresh if it does not occur in T .
1 0
Theorem 2.4.4. Let T and T be TSSs over signatures Σ and Σ , respectively, where T and T ⊕T
0 1 0 1 0 0 1
are positive after reduction. Under the following conditions, T ⊕T is a conservative extension of T . (1) T
0 1 0 0
is source-dependent. (2) For each ρ∈T , either the source of ρ is fresh, or ρ has a premise of the form t—→a t′
1
or tP, where t∈T(Σ ), all variables in t occur in the source of ρ and t′, a or P is fresh.
0
2.5. Process Algebra – ACP
ACP[5] is a kind of process algebra which focuses on the specification and manipulation of process terms by
use of a collection of operator symbols. In ACP, there are several kind of operator symbols, such as basic
operators to build finite processes (called BPA), communication operators to express concurrency (called
PAP), deadlock constants and encapsulation enable us to force actions into communications (called ACP),
liner recursion to capture infinite behaviors (called ACP with linear recursion), the special constant silent
step and abstraction operator (called ACP with guarded linear recursion) allows us to abstract away from
τ
internal computations.
Bisimulationorrootedbranchingbisimulationbasedstructuraloperationalsemanticsisusedtoformally
provideeachprocesstermusedtheaboveoperatorsandconstantswithaprocessgraph.Theaxiomatization
ofACP(accordingtheaboveclassificationofACP,theaxiomatizationsareE ,E ,E ,E +RDP
BPA PAP ACP ACP
(Recursive Definition Principle) + RSP (Recursive Specification Principle), E + RDP + RSP + CFAR
ACPτ
(Cluster Fair Abstraction Rule) respectively) imposes an equation logic on process terms, so two process
terms can be equated if and only if their process graphs are equivalent under the semantic model.
ACPcanbeusedtoformallyreasonaboutthebehaviors,suchasprocessesexecutedsequentiallyandcon-
currently by use of its basic operator, communication mechanism, and recursion, desired external behaviors
by its abstraction mechanism, and so on.
ACPisorganizedbymodulesandcanbeextendedwithfreshoperatorstoexpressmorepropertiesofthe
specificationforsystembehaviors.Theseextensionsarerequiredboththeequationallogicandthestructural
operational semantics to be extended. Then the extension can use the whole outcomes of ACP, such as its
concurrency, recursion, abstraction, etc.
3. Structural Operational Semantics Extended to Support Reversible Quantum
Processes
We extend the above classical reversible structural operational semantics to support quantum processes,
about the details, please refer to [17] [18]. Here, we use α,β to denote quantum operations in contrast to
classical actions a,b.
Definition 3.1 (Quantum process configuration).Aquantumprocessconfigurationisdefinedtobe
a pair ⟨p,(cid:37)⟩, where p is a process (graph) called structural part of the configuration, and (cid:37)∈D(H) specifies
the current state of the environment, which is called its quantum part.
Definition 3.2 (Quantum process graph). A quantum process (graph) ⟨p,(cid:37)⟩ is an LTS in which one
state s is elected to be the root. If the LTS contains a transition s—α→s′, then ⟨p,(cid:37)⟩—α→⟨p′,(cid:37)′⟩ where ⟨p′,(cid:37)′⟩
hasrootstates′.Moreover,iftheLTScontainsatransitionsP,then⟨p,(cid:37)⟩P.(1)Aquantumprocess⟨p ,(cid:37) ⟩
0 0
is finite if and only if the process p is finite. (2) A quantum process ⟨p ,(cid:37) ⟩ is regular if and only if the
0 0 0
process p is regular.
0
Definition 3.3 (Quantum transition system specification). A quantum process transition rule ρ
8 YongWang
is an expression of the form H, with H a set of expressions ⟨t,(cid:37)⟩—α→⟨t′,(cid:37)′⟩ and ⟨t,(cid:37)⟩P with t,t′∈T(Σ) and
π
(cid:37),(cid:37)′ ∈D(H), called the (positive) premises of ρ, and t—α→t′, called structural part of H and denoted as H .
s
And π an expression ⟨t,(cid:37)⟩ —α→ ⟨t′,(cid:37)′⟩ or ⟨t,(cid:37)⟩P with t,t′ ∈ T(Σ) and (cid:37),(cid:37)′ ∈ D(H), called the conclusion of
ρ, and t—α→t′, called structural part of π and denoted as π . The left-hand side of π is called the source of
s
ρ. Hs is called the structural part of ρ and denoted as ρ . A quantum process transition rule ρ is closed if
πs s
and only its structural part ρ is closed. A quantum transition system specification (QTSS) is a (possible
s
infinite) set of transition rules.
Definition 3.4 (Quantum reverse transition). There are two quantum processes ⟨p,(cid:37)⟩ and ⟨p′,(cid:37)′⟩,
two transitions ⟨p,(cid:37)⟩ —α→ ⟨p′,(cid:37)′⟩ and ⟨p′,(cid:37)′⟩ —α—[m↠] ⟨p,(cid:37)⟩, the transition ⟨p′,(cid:37)′⟩ —α—[m↠] ⟨p,(cid:37)⟩ is called reverse
transition of p—→a p′, and the transition ⟨p,(cid:37)⟩—α→⟨p′,(cid:37)′⟩ is called forward transition. If ⟨p,(cid:37)⟩—α→⟨p′,(cid:37)′⟩ then
⟨p′,(cid:37)′⟩ —α—[m↠] ⟨p,(cid:37)⟩, the forward transition ⟨p,(cid:37)⟩ —α→ ⟨p′,(cid:37)′⟩ is reversible. Where α[m] is a kind of special
quantumoperationconstantα[m]∈A×K,K∈N,calledthehistoriesofanquantumoperationα,andm∈K.
Definition 3.5 (Quantum forward-reverse bisimulation). A quantum forward-reverse (FR) bisim-
ulation relation B is a binary relation on processes such that: (1) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and ⟨p,(cid:37)⟩ —α→ ⟨p′,(cid:37)′⟩ then
⟨q,ς⟩ —α→ ⟨q′,ς′⟩ with ⟨p′,(cid:37)′⟩B⟨q′,ς′⟩; (2) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and ⟨q,ς⟩ —α→ ⟨q′,ς′⟩ then ⟨p,(cid:37)⟩ —α→ ⟨p′,(cid:37)′⟩ with
⟨p′,(cid:37)′⟩B⟨q′,ς′⟩; (3) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and ⟨p,(cid:37)⟩ —α—[m↠] ⟨p′,(cid:37)′⟩ then ⟨q,ς⟩ —α—[m↠] ⟨q′,ς′⟩ with ⟨p′,(cid:37)′⟩B⟨q′,ς′⟩; (4)
if ⟨p,(cid:37)⟩B⟨q,ς⟩ and ⟨q,ς⟩ —α—[m↠] ⟨q′,ς′⟩ then ⟨p,(cid:37)⟩ —α—[m↠] ⟨p′,(cid:37)′⟩ with ⟨p′,(cid:37)′⟩B⟨q′,ς′⟩; (5) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and
⟨p,(cid:37)⟩P, then ⟨q,ς⟩P; (6) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and ⟨q,ς⟩P, then ⟨p,(cid:37)⟩P. Two quantum process ⟨p,(cid:37)⟩ and ⟨q,ς⟩ are
FR bisimilar, denoted by ⟨p,(cid:37)⟩↔fr⟨q,ς⟩, if there is a FR bisimulation relation B such that ⟨p,(cid:37)⟩B⟨q,ς⟩.
Definition 3.6 (Relation between quantum FR bisimulation and classical FR bisimulation).
For two quantum processes, ⟨p,(cid:37)⟩↔fr⟨q,ς⟩ , with (cid:37) = ς, if and only if p↔frq and (cid:37)′ = ς′, where (cid:37) evolves
into (cid:37)′ after execution of p and ς evolves into ς′ after execution of q.
Definition3.7(Quantumbranchingforward-reversebisimulation).Aquantumbranchingforward-
reverse (FR) bisimulation relation B is a binary relation on the collection of quantum processes such that:
(1) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and ⟨p,(cid:37)⟩—α→⟨p′,(cid:37)′⟩ then either α≡τ and ⟨p′,(cid:37)′⟩B⟨q,ς⟩ or there is a sequence of (zero or
more) τ-transitions ⟨q,ς⟩—→τ ⋯—→τ ⟨q ,ς ⟩ such that ⟨p,(cid:37)⟩B⟨q ,ς ⟩ and ⟨q ,ς ⟩—α→⟨q′,ς′⟩ with ⟨p′,(cid:37)′⟩B⟨q′,ς′⟩;
0 0 0 0 0 0
(2) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and ⟨q,ς⟩—α→⟨q′,ς′⟩ then either α≡τ and ⟨p,(cid:37)⟩B⟨q′,ς′⟩ or there is a sequence of (zero or
more)τ-transitions⟨p,(cid:37)⟩—→τ ⋯—→τ ⟨p ,(cid:37) ⟩suchthat⟨p ,(cid:37) ⟩B⟨q,ς⟩and⟨p ,(cid:37) ⟩—α→⟨p′,(cid:37)′⟩with⟨p′,(cid:37)′⟩B⟨q′,ς′⟩;
0 0 0 0 0 0
τ τ
(3) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and ⟨p,(cid:37)⟩P, then there is a sequence of (zero or more) τ-transitions ⟨q,ς⟩—→⋯—→⟨q ,ς ⟩
0 0
such that ⟨p,(cid:37)⟩B⟨q ,ς ⟩ and ⟨q ,ς ⟩P; (4) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and ⟨q,ς⟩P, then there is a sequence of (zero or
0 0 0 0
τ τ
more) τ-transitions ⟨p,(cid:37)⟩ —→ ⋯ —→ ⟨p ,(cid:37) ⟩ such that ⟨p ,(cid:37) ⟩B⟨q,ς⟩ and ⟨p ,(cid:37) ⟩P; (5) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and
0 0 0 0 0 0
⟨p,(cid:37)⟩—α—[m↠] ⟨p′,(cid:37)′⟩ then either α≡τ and ⟨p′,(cid:37)′⟩B⟨q,ς⟩ or there is a sequence of (zero or more) τ-transitions
⟨q,ς⟩——↠τ ⋯——↠τ ⟨q ,ς ⟩ such that ⟨p,(cid:37)⟩B⟨q ,ς ⟩ and ⟨q ,ς ⟩—α—[m↠] ⟨q′,ς′⟩ with ⟨p′,(cid:37)′⟩B⟨q′,ς′⟩; (6) if ⟨p,(cid:37)⟩B⟨q,ς⟩
0 0 0 0 0 0
and ⟨q,ς⟩ —α—[m↠] ⟨q′,ς′⟩ then either α ≡ τ and ⟨p,(cid:37)⟩B⟨q′,ς′⟩ or there is a sequence of (zero or more) τ-
transitions ⟨p,(cid:37)⟩——↠τ ⋯——↠τ ⟨p ,(cid:37) ⟩ such that ⟨p ,(cid:37) ⟩B⟨q,ς⟩ and ⟨p ,(cid:37) ⟩—α—[m↠] ⟨p′,(cid:37)′⟩ with ⟨p′,(cid:37)′⟩B⟨q′,ς′⟩; (7)
0 0 0 0 0 0
τ τ
if⟨p,(cid:37)⟩B⟨q,ς⟩and⟨p,(cid:37)⟩P,thenthereisasequenceof(zeroormore)τ-transitions⟨q,ς⟩——↠⋯——↠⟨q ,ς ⟩such
0 0
that ⟨p,(cid:37)⟩B⟨q ,ς ⟩ and ⟨q ,ς ⟩P; (8) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and ⟨q,ς⟩P, then there is a sequence of (zero or more)
0 0 0 0
τ τ
τ-transitions ⟨p,(cid:37)⟩ ——↠ ⋯ ——↠ ⟨p ,(cid:37) ⟩ such that ⟨p ,(cid:37) ⟩B⟨q,ς⟩ and ⟨p ,(cid:37) ⟩P. Two quantum processes ⟨p,(cid:37)⟩
0 0 0 0 0 0
and ⟨q,ς⟩ are branching FR bisimilar, denoted by ⟨p,(cid:37)⟩↔fr⟨q,ς⟩, if there is a branching FR bisimulation
b
relation B such that ⟨p,(cid:37)⟩B⟨q,ς⟩.
Definition 3.8 (Relation between quantum branching FR bisimulation and classical branch-
ing FR bisimulation). For two quantum processes, ⟨p,(cid:37)⟩↔fr⟨q,ς⟩, with (cid:37) = ς, if and only if p↔frq and
b b
(cid:37)′=ς′, where (cid:37) evolves into (cid:37)′ after execution of p and ς evolves into ς′ after execution of q.
Definition 3.9 (Quantum rooted branching forward-reverse bisimulation). A quantum rooted
branchingforward-reverse(FR)bisimulationrelationB isabinaryrelationonquantumprocessessuchthat:
(1) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and ⟨p,(cid:37)⟩ —α→ ⟨p′,(cid:37)′⟩ then ⟨q,ς⟩ —α→ ⟨q′,ς′⟩ with ⟨p′,(cid:37)′⟩↔fr⟨q′,ς′⟩; (2) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and
b
AnAlgebraofReversibleQuantumComputing 9
⟨q,ς⟩ —α→ ⟨q′,ς′⟩ then ⟨p,(cid:37)⟩ —α→ ⟨p′,(cid:37)′⟩ with ⟨p′,(cid:37)′⟩↔fr⟨q′,ς′⟩; (3) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and ⟨p,(cid:37)⟩ —α—[m↠] ⟨p′,(cid:37)′⟩ then
b
⟨q,ς⟩ —α—[m↠] ⟨q′,ς′⟩ with ⟨p′,(cid:37)′⟩↔fr⟨q′,ς′⟩; (4) if ⟨p,(cid:37)⟩B⟨q,ς⟩ and ⟨q,ς⟩ —α—[m↠] ⟨q′,ς′⟩ then ⟨p,(cid:37)⟩ —α—[m↠] ⟨p′,(cid:37)′⟩
b
with⟨p′,(cid:37)′⟩↔fr⟨q′,ς′⟩;(5)if⟨p,(cid:37)⟩B⟨q,ς⟩and⟨p,(cid:37)⟩P,then⟨q,ς⟩P;(6)if⟨p,(cid:37)⟩B⟨q,ς⟩and⟨q,ς⟩P,then⟨p,(cid:37)⟩P.
b
Two quantum processes ⟨p,(cid:37)⟩ and ⟨q,ς⟩ are rooted branching FR bisimilar, denoted by ⟨p,(cid:37)⟩↔fr⟨q,ς⟩, if
rb
there is a rooted branching FR bisimulation relation B such that ⟨p,(cid:37)⟩B⟨q,ς⟩.
Definition 3.10 (Relation between quantum rooted branching FR bisimulation and classical
rooted branching FR bisimulation). For two quantum processes, ⟨p,(cid:37)⟩↔fr⟨q,ς⟩, with (cid:37)=ς, if and only
rb
if p↔frq and (cid:37)′=ς′, where (cid:37) evolves into (cid:37)′ after execution of p and ς evolves into ς′ after execution of q.
rb
Definition 3.11 (Congruence).LetΣbeasignatureandD(H)bethestatespaceoftheenvironment.
An equivalence relation B on ⟨t ∈ T(Σ),(cid:37) ∈ D(H)⟩ is a congruence, i.e., for each f ∈ Σ, if ⟨s ,(cid:37) ⟩B⟨t ,ς ⟩
i i i i
for i ∈ {1,⋯,ar(f)}, then f(⟨s ,(cid:37) ⟩,⋯,⟨s ,(cid:37) ⟩)Bf(⟨t ,ς ⟩,⋯,⟨t ,ς ). An equivalence rela-
1 1 ar(f) ar(f) 1 1 ar(f) ar(f)
tion B on ⟨t ∈ T(Σ),(cid:37) ∈ D(H)⟩ is a congruence, if for each f ∈ Σ, s Bt for i ∈ {1,⋯,ar(f)}, and
i i
f(s ,⋯,s )Bf(t ,⋯,t ).
1 ar(f) 1 ar(f)
Definition 3.12 (Quantum conservative extension). Let T and T be QTSSs over signature Σ
0 1 0
andD(H ),andΣ andD(H ),respectively.TheQTSST ⊕T isaconservativeextensionofT iftheLTSs
0 1 1 0 1 0
generated by T and T ⊕T contain exactly the same transitions ⟨t,(cid:37)⟩—α→⟨t′,(cid:37)′⟩ and ⟨t,(cid:37)⟩P with t∈T(Σ )
0 0 1 0
and (cid:37)∈D(H ), and T ⊕T =⟨Σ ∪Σ ,D(H ⊗H )⟩.
0 0 1 0 1 0 1
Definition 3.13 (Relation between quantum conservative extension and classical conserva-
tiveextension).TheQTSST ⊕T isaquantumconservativeextensionofT withtransitions⟨t,(cid:37)⟩—α→⟨t′,(cid:37)′⟩
0 1 0
and⟨t,(cid:37)⟩P,ifitscorrespondingTSST′⊕T′ isaconservativeextensionofT′ withtransitionst—α→t′ andtP.
0 1 0
4. BRQPA – Basic Reversible Quantum Process Algebra
In the following, the variables x,x′,y,y′,z,z′ range over the collection of process terms, the variables υ,ω
range over the set A of quantum operations, α,β ∈ A, s,s′,t,t′ are closed items, τ is the special constant
silent step, δ is the special constant deadlock. We define a kind of special action constant α[m] ∈ A×K
where K∈N, called the histories of a quantum operation α, denoted by α[m],α[n],⋯ where m,n∈K. Let
A=A∪{A×K}.
BRQPA includes three kind of operators: the execution of quantum operation α, the choice composition
operator + and the sequential composition operator ⋅. Each finite process can be represented by a closed
term that is built from the set A of atomic actions or histories of an atomic action, the choice composition
operator+,andthesequentialcompositionoperator⋅.ThecollectionofallbasicprocesstermsiscalledBasic
Reversible Quantum Process Algebra (BRQPA), which is abbreviated to BRQPA.
4.1. Transition Rules of BRQPA
We give the forward transition rules under transition system specification (QTSS) for BRQPA as follows.
υ
⟨υ,(cid:37)⟩—→⟨υ[m],υ((cid:37))
⟨x,(cid:37)⟩—→υ ⟨υ[m],(cid:37)′⟩ υ∉y ⟨x,(cid:37)⟩—→υ ⟨x′,(cid:37)′⟩ υ∉y
υ υ
⟨x+y,(cid:37)⟩—→⟨υ[m]+y,(cid:37)′⟩ ⟨x+y,(cid:37)⟩—→⟨x′+y,(cid:37)′⟩
⟨y,(cid:37)⟩—→υ ⟨υ[m],(cid:37)′⟩ υ∉x ⟨y,(cid:37)⟩—→υ ⟨y′,(cid:37)′⟩ υ∉x
υ υ
⟨x+y,(cid:37)⟩—→⟨x+υ[m],(cid:37)′⟩ ⟨x+y,(cid:37)⟩—→⟨x+y′,(cid:37)′⟩
⟨x,(cid:37)⟩—→υ ⟨υ[m],(cid:37)′⟩ ⟨y,(cid:37)⟩—→υ ⟨υ[m],(cid:37)′⟩ ⟨x,(cid:37)⟩—→υ ⟨x′,(cid:37)′⟩ ⟨y,(cid:37)⟩—→υ ⟨υ[m],(cid:37)′⟩
υ υ
⟨x+y,(cid:37)⟩—→⟨υ[m],(cid:37)′⟩ ⟨x+y,(cid:37)⟩—→⟨x′+υ[m],(cid:37)′⟩
10 YongWang
⟨x,(cid:37)⟩—→υ ⟨υ[m],(cid:37)′⟩ ⟨y,(cid:37)⟩—→υ ⟨y′,(cid:37)′⟩ ⟨x,(cid:37)⟩—→υ ⟨x′,(cid:37)′⟩ ⟨y,(cid:37)⟩—→υ ⟨y′,(cid:37)′⟩
υ υ
⟨x+y,(cid:37)⟩—→⟨υ[m]+y′,(cid:37)′⟩ ⟨x+y,(cid:37)⟩—→⟨x′+y′,(cid:37)′⟩
⟨x,(cid:37)⟩—→υ ⟨υ[m],(cid:37)′⟩ ⟨x,(cid:37)⟩—→υ ⟨x′,(cid:37)′⟩
υ υ
⟨x⋅y,(cid:37)⟩—→⟨υ[m]⋅y,(cid:37)′⟩ ⟨x⋅y,(cid:37)⟩—→⟨x′⋅y,(cid:37)′⟩
⟨y,(cid:37)⟩—ω→⟨ω[n],(cid:37)′⟩ ⟨y,(cid:37)⟩—ω→⟨y′,(cid:37)′⟩
,m≠n .
ω ω
⟨υ[m]⋅y,(cid:37)⟩—→⟨υ[m]⋅ω[n],(cid:37)′⟩ ⟨υ[m]⋅y,(cid:37)⟩—→⟨υ[m]⋅y′,(cid:37)′⟩
● Thefirsttransitionrulesaysthateachquantumoperationυcanexecutesuccessfully,andleadstoahistory
υ[m]. Because in reversible computation, there is no successful termination, the forward transition rule
implies a successful execution.
⟨υ,(cid:37)⟩—→υ ⟨υ[m],υ((cid:37))⟩
● The next four transition rules say that s+t can execute only one branch, that is, it can execute either s
or t, but the other branch remains.
● The next four transition rules say that s+t can execute both branches, only by executing the same
quantum operations.
● The last four transition rules say that s⋅t can execute sequentially, that is, it executes s in the first and
leads to a successful history, after successful execution of s, then execution of t follows.
We give the reverse transition rules under transition system specification (QTSS) for BRQPA as follows.
υ[m]
⟨υ[m],(cid:37)⟩——↠⟨υ,υ[m]((cid:37))⟩
⟨x,(cid:37)⟩—υ—[m↠] ⟨υ,(cid:37)′⟩ υ[m]∉y ⟨x,(cid:37)⟩—υ—[m↠] ⟨x′,(cid:37)′⟩ υ[m]∉y
υ[m] υ[m]
⟨x+y,(cid:37)⟩——↠⟨υ+y,(cid:37)′⟩ ⟨x+y,(cid:37)⟩——↠⟨x′+y,(cid:37)′⟩
⟨y,(cid:37)⟩—υ—[m↠] ⟨υ,(cid:37)′⟩ υ[m]∉x ⟨y,(cid:37)⟩—υ—[m↠] ⟨y′,(cid:37)′⟩ υ[m]∉x
υ[m] υ[m]
⟨x+y,(cid:37)⟩——↠⟨x+υ,(cid:37)′⟩ ⟨x+y,(cid:37)⟩——↠⟨x+y′,(cid:37)′⟩
⟨x,(cid:37)⟩—υ—[m↠] ⟨υ,(cid:37)′⟩ ⟨y,(cid:37)⟩—υ—[m↠] ⟨υ,(cid:37)′⟩ ⟨x,(cid:37)⟩—υ—[m↠] ⟨x′,(cid:37)′⟩ ⟨y,(cid:37)⟩—υ—[m↠] ⟨υ,(cid:37)′⟩
υ[m] υ[m]
⟨x+y,(cid:37)⟩——↠⟨υ,(cid:37)′⟩ ⟨x+y,(cid:37)⟩——↠⟨x′+υ,(cid:37)′⟩
⟨x,(cid:37)⟩—υ—[m↠] ⟨υ,(cid:37)′⟩ ⟨y,(cid:37)⟩—υ—[m↠] ⟨y′,(cid:37)′⟩ ⟨x,(cid:37)⟩—υ—[m↠] ⟨x′,(cid:37)′⟩ ⟨y,(cid:37)⟩—υ—[m↠] ⟨y′,(cid:37)′⟩
υ[m] υ[m]
⟨x+y,(cid:37)⟩——↠⟨υ+y′,(cid:37)′⟩ ⟨x+y,(cid:37)⟩——↠⟨x′+y′,(cid:37)′⟩
⟨x,(cid:37)⟩—υ—[m↠] ⟨υ,(cid:37)′⟩ ⟨x,(cid:37)⟩—υ—[m↠] ⟨x′,(cid:37)′⟩
υ[m] υ[m]
⟨x⋅y,(cid:37)⟩——↠⟨υ⋅y,(cid:37)′⟩ ⟨x⋅y,(cid:37)⟩——↠⟨x′⋅y,(cid:37)′⟩
⟨y,(cid:37)⟩—ω—[↠n] ⟨ω,(cid:37)′⟩ ⟨y,(cid:37)⟩—ω—[↠n] ⟨y′,(cid:37)′⟩
,m≠n .
ω[n] ω[n]
⟨υ[m]⋅y,(cid:37)⟩——↠⟨υ[m]⋅ω,(cid:37)′⟩ ⟨υ[m]⋅y,(cid:37)⟩——↠⟨υ[m]⋅y′,(cid:37)′⟩
● The first transition rule says that each history of a quantum operation υ[m] can reverse successfully,
and leads to a quantum operation υ. Similarly, the reverse transition rule implies
⟨υ[m],(cid:37)⟩—υ—[m↠] ⟨υ,υ[m]((cid:37))⟩
a successful reverse.
● The next four transition rules say that s+t can reverse only one branch, that is, it can reverse either s
or t, but the other branch remains.