Quantum Computing: Computation in Coherence

Quantum Computing: Computation in Coherence Spaces Renata H. S. Reiser1 , Antônio Carlos R. Costa1 , Rafael B. Amaral1 1

Escola de Informática – Universidade Católica de Pelotas (UCPEL) Rua Félix da Cunha, 412 - 96010-000 – Caixa Postal 402 –Pelotas – RS – Brazil {reiser, rocha, rafaelbba}@ucpel.tche.br

Abstract. The quantum version of the Geometric Machine Model, shorten to qGM, is an abstract machine model, based on Girard’s coherence space, capable of modelling quantum parallelism (synchronous computation) and measurement operation (non-deterministic computations) on a (possibly infinite) global shared quantum memory. The processes of the qGM are inductively constructed in the Coherence Space of Quantum Processes. The qGM memory, supporting the Coherence Space of Quantum States, is conceived as a set of memory values characterized as normalized complex numbers and labelled by positions of a transfinite set of memory positions. Thus, the quantum version of the GM model operates with quantum control and provides semantic interpretations of algorithms of quantum computing. In this work, we make use of the domain-theoretic linear operators to obtain the related compound parallel processes and measurements performed over array structures. The domain-theoretic semantics shed light on the adequacy of the chosen representation of quantum states and the performance benefit of the parallel structuring of quantum circuits. Resumo. Este trabalho apresenta a versão quântica do modelo de Máquina Geométrica (qGM). O modelo qGM está baseado nos espaços coerentes introduzidos por Girard, capazes de modelar computaço˜ es s´ıncronas (paralelismo) e não-determin´ısticas (medidas) sobre uma memória quântica global e compartilhada. Os processos no modelo qGM são indutivamente constru´ıdos no espaço coerente de processos quânticos. A memória do modelo qGM, obtida a partir do espaço coerente de estados quânticos, e´ concebida como um conjunto de valores de memória (caracterizados como números complexos normalizados) rotulados por pontos de um conjunto transfinito de posiço˜ es de memória. A versão quântica do modelo GM apresentada neste trabalho opera sobre controle quântico e permite uma semântica denotacional para algoritmos da computaça˜ o quântica. Neste trabalho, fazemos uso de operadores dom´ınio-teorético lineares, para obter processos paralelos e mediço˜ es realizadas sobre estruturas de array. A semântica dom´ınio-teorética deixa clara a adequaça˜ o da representaça˜ o escolhida para estados quânticos e o benef´ıcio de performance da estruturaça˜ o paralela dos circuitos quânticos.

1. Introduction It seems that the application of both category theoretic techniques and domain theory, specially denotational semantics, is providing a new perspective of interpretation to the quantum theory and quantum computing [Abramsky and Coecke 2004,

Abramsky and Duncan 2004]. The development of semantic models has improved the algorithmic analysis and the semantic interpretation associated with the quantum data and quantum control in quantum programming languages [Selinger 2004a]. This allows quantum computing to be defined in a high level of abstraction by structured and well-defined languages independently from the technological advances. Thus, the study of new algorithm development techniques based on the quantum approach is viable, despite the fact that quantum computers are not yet available outside research laboratories. Recently, there have been some proposals for higher-order quantum programming languages, based on the linear versions of the lambda calculus [Tonder 2004]. In [Altenkirch and Grattage 2005], the QML semantics is inspired by the denotational semantics of classical reversible computations. A formal theory of continuous normed cones is applied in [Selinger 2004b]. Lastly, the substantial research that has been done by Girard in his study of quantum coherent spaces can not be forgotten [Girard 2004]. The use of techniques of denotational semantics provides interpretations not only for the encoding of data using quantum bits, but also for the high order data structures dealing with quantum control, as the case of the “black box” which can only be tested via observing its input/output behavior. Based on such approach, quantum computation can be understood as a transformation of information encoded in the state of a quantum physical system. In this work superpositions and linearity, directly related to a spatial distribution (orthogonal condition) and temporal construction (parallel computations) of memory and processes in a quantum model, are studied. By addressing this question, the current work has as its main goal the quantum extension of the Geometric Machine model, shortened to GM model. The advantages of Girard’s Coherence Spaces [Girard 1986, Girard 1987] are considered to obtain the domain-theoretic structure of the qGM model. Following the methodology proposed by Scott [Scott 1971], the coherence space of quantum processes D2∞ is built over the coherence space of elementary quantum processes, which are obtained synchronizing single coherent subsets of tokens (actions labelled by the positions of a geometric space associated to an instant of computational time). The completion procedure of such space D2∞ ensures interpretations of infinite quantum computations. Each coherent set in D2∞ provides the description of a distributed process representing unitary transformations and measurement operations, which are obtained by the algebraic constructors (sequential or parallel products, deterministic or non-deterministic sums). In this text, basic aspects of coherence spaces are considered in Sect. 2 in order to describe the qGM memory states, quantum control and quantum processes, in Sect. 3 and 4. Then, final remarks and the Conclusion are presented.

2. Coherence Spaces and Domain-theoretic Linear Functions A web W = (W, ≈W ) is a pair consisting of a set W with a symmetric and reflexive relation ≈W , called coherence relation. A subset of this web with pairwise coherent elements is called a coherent subset. The collection of coherent subsets of the web W, ordered by the inclusion relation, is called a coherence space, W ≡ (Coh(W), ⊆). A dom-linear function is a continuous function in the sense of domain theory, which also satisfies the stability and linearity properties that assure the existence of the least approximation in the image set [Troelstra 1992, Abramsky and Jung 1994]. Let A

and B be coherence spaces. A dom-linear function f : A → B is given by its linear trace, ltr(f ) = {(α, β) | β ∈ f (α)} ⊆ A×B. Let A ( B = (A×B, ≈( ) be the web of linear traces and (α, β) ≈( (α0 , β 0 ) ↔ ((α ≈A α0 → β ≈B β 0 ) and (β = β 0 → α = α0 )). The family of coherent subsets of the web A ( B, ordered by inclusion relation, defines the domain A ( B ≡ (Coh(A ( B), ⊆) of the linear traces of functions from A to B.

3. Quantum memory states in the qGM The notions of memory state and memory position are considered to summarize the main aspects of machine states of the qGM Model. 3.1. Intuitive notion of the qGM state machine According to [Stoll 1961], the ordering of the natural numbers ℵ, as dictated by their roles as ordinals, coincides with that in their original role as members of the natural number sequence. The order type of 0, 1, 2, . . . is indicated by ω, with successors ω + 1, after ω + 2, and so on, so that the order type 2ω denotes the set of transfinite ordinal numbers from 0 to 2ω. Considering the sequence of subsets Q = {0.0, 0.1}, Q2 = {0.0, 1.0, 0.1, 1.1}, Q3 = {00.0, 01.0, 10.0, 11.0, 00.1, 01.1, 10.1, 11.1}, .. . Qn = {0n .0, 0n−1 1.0, . . . , 0n−m 1m .0, . . . , 1n .0, 0n .1, 0n−1 1.1, . . . , 0n−m 1m .1, . . . , 1n .1},

and assuming the next two equivalences related to possible infinite words which result from the concatenation of the symbol 0: .0 ≡ 0.0 ≡ 0n .0 . . . and .1 ≡ 0.1 ≡ 0n .1 . . . denoting the first and the ω-th transfinite memory position, respectively, the diadic expansion of the transfinite set 2ω, given by the expression ∪n∈ℵ Qn = Qω = {.0, . . . , 1n .0, . . . , .1, . . . , 1n .1, . . .},

and satisfying the inclusion Q ⊆ Q2 ⊆ . . . ⊆ Qn ⊆ Qn+1 ⊆ . . . ⊆ Qω , characterizes the memory positions (addresses) of the qGM memory machine. The qGM model uses such sequence to label its transfinite global memory, shared by synchronized processes distributed over an enumerable set of GM-models, each one operating over a quantum system of 1-qubit. Consider the set of complex numbers as the set of memory values, whose elements are represented by pairs of real numbers in the polar form: α = (ρ, θ) ∈ C ⊂ R+ × R and take f : Qω → C to be a state of the qGM memory, so that every αn ∈ f is a complex number labelled by a position index n ∈ Qω , and so that the normalization condition is held X X f : Qω → C ⇒ |α|2 = (ρ)2 ≤ 1. (1) n∈Qω ,(n,α)∈f

n∈Qω ,(n,(ρ,θ))∈f

That is, using complex numbers as memory values, labelled by the position indexes in the transfinite set Qω , it is possible to represent in the memory of the qGM model a quantum system of ω qubits. An intuitive illustration of a (deterministic) machine state s is given by the matrix structure presented in Figure 1(a). Therefore, to represent a quantum state of a n-qubits system we need to consider the first 2n−1 -th columns. In the first column of

s (a) .0 .1

0 α0.0 α0.1

1 α1.0 α1.1

10 α10.0 α10.1

11 α11.0 α11.1

|0i ... . . . (b) .0 .1 ...

0 (1, 0) (0, 0)

|1i ... . . . (c) .0 .1 ...

0 (0, 0) (1, 0)

... ... ...

Figure 1. Illustration of a (deterministic) qGM state and its interpretation.

the matrix structure, the classical and quantum superpositions of memory states related to the 1-qubit system can be represented. In Figures 1(b) and 1(c), representations of the classical states |0i and |1i are presented, respectively. The concept of a superposition of states is a novelty introduced by quantum mechanics. If a system may be in any one of two pure states |0i and |1i, we must consider that it may also be in any one of many superpositions of |0i and |1i. Superpositions must be considered when one cannot distinguish between possible paths leading to the current state of a quantum system. In such case, the resulting state is some compound of the states that result from each of the possible paths. Since different classical states are orthogonal, the claim implies that no non-trivial superpositions can be observed in classical systems [Lehmann 2007]. 3.2. Coherence space of quantum memory states Quantum machine states are conceived as coherent subsets of linear traces of strict, continuous, stable and linear functions from Qω to C. Definition 3.1 Let Q = {0, 1} be the set of basic states. Consider the flat coherence space of memory positions, Qω ≡ (Coh(Qω , =), ⊆), and of memory values, C ≡ (Coh(C, =), ⊆), to define the Qω ( C as the coherence space modelling (deterministic) machine states. Based on the normalization condition expressed by Equation 1, for all (deterministic) quantum machine states interpreted by the linearP trace of the linear function f from Qω to ω n C, if f ∈ Q ( C then it holds that ∀(α ) ∈ f , ωn=0 | αn |≤ 1. Thus, a coherent subset f in Qω ( C provides a (partial) representation for a normalized vector in the Hilbert space l2 (H). As a generalization, the non-deterministic machine states are modelled as families of deterministic machine states, with one trace for each deterministic state component of the non-deterministic state: Definition 3.2 Let S = (Coh(Qω ( C), ≈S ) be the web given by the set of all coherent subsets of Qω ( C together with the trivial (i.e., universal) coherence relation ≈S . The collection of all coherent subsets of S, ordered by inclusion, defines the coherence space that models the non-deterministic machine states, denoted by S ≡ (Coh(S), ⊆). According to Definition 3.2, an object in S gives representation to an enumerable subset of vectors in the Hilbert space satisfying the normalization restriction.

4. Processes in the qGM model The coherent space Q2∞ [Reiser et al. 2002] provides an ordered structure over which quantum processes with classical and quantum control structures may be represented. The domain D2∞ is constructed by a succession of levels, each a coherence subspace D∞+n representing all the possible quantum gates of n qubits, and combined according

to classical and quantum control structures, in order to produce non-deterministic and parallel computations. Starting with D∞ , each level D∞+n is distinguished by its elementary quantum processes, which are concurrent elementary processes performed in a synchronized way. The coherence space D2∞ , which provides representations for quantum computations operating on the whole infinite memory of the model, is given by the completion procedure. Classical computations may be modelled as restrictions of quantum computations, the ones based only on classical elementary process. 4.1. Classical elementary processes A classical elementary process may be defined as a transition between two deterministic states, performed over a single memory position in a single unit of computational time (uct). It is a function d(k) satisfying: Proposition 1 Let A ≡ [Qω ( C] ( C be the coherence space of the so-called computational actions. If d, pr(i.k) ∈ A, with pr(k.i) (s) = s(k.i), then the function d(k.i) ∈ [Qω ( C] → [Qω ( C] given by (l.j) pr (s) if l 6= k ∨ i 6= j (k.i) d (s)(l.i) = d(s) if l = k ∧ i = j, is a dom-linear function. Proof. 1 Consider s, s0 , z, z 0 ∈ [Qω ( C], k.i, l.j ∈ Qω and s0 = d(k.i) (s), z 0 = d(l.j) (z). We show that, for all (s, s0 ), (z, z 0 ) in the graph of the function d(k.i) , the following properties hold. 1. If s ≈( z then s0 ≈( z 0 : Suppose s ≈( z. If i 6= k or l 6= k then d(k.i) (s)(l.j) = s(l.j) ≈( z(l.j) = d(k.i) (z)(l.j). In the other case, if i = k and k = j then s0 (l.j) = dk.i (s)(l.j) = d(s) ≈( d(z) = d(k.i) (z)(l.j) = z 0 (l.j). Thus, s0 ≈( z 0 . 2. If s0 = z 0 then s = z: Suppose s0 = z 0 . When k 6= j and i 6= j and d ∈ A is a linear function, s(j.l) = s0 (l.j) = z 0 (j.l) = z(l.j). Therefore, s = z. Otherwise, if k = l and i = j, (s)0 (l.j) = d(k.i) (s)(l.j) = d(s) = d(z) = d(k.i) (z)(l.j). Thus, s ⊆ z which means s = z. Therefore, d(k.i) is linear.♦ The set of classical elementary processes is given by the collection of all linear functions d(i.k) in D∞ ≡ [Qω ( C] → [Qω ( C] verifying Proposition 1. Its elements can simultaneously read from distinct memory positions, but they can only write to exactly one memory position. Thus, an action d(k.i) can be interpreted as transition from state s to s0 , performing one of the two attributions in each memory position: s0 (l.j) := s(l.j) if l 6= k ∨ i 6= j, and s0 (k.i) := d(s)) if l = k ∧ i = j. 4.2. Coherence space of quantum boolean tests In order to represent the quantum boolean test usually called “quantum if” [Altenkirch and Grattage 2005], which allows for the definition of general quantum control structures, we consider initially the set B of boolean values and the coherence space of boolean values B ≡ (Coh(B), ⊆), which is the collection of all coherent subsets of classical boolean tests of the discrete web B ≡ (B, =), ordered by the inclusion

relation. The coherence space Qω ( B models classical boolean tests performed over the transfinite memory positions of a quantum state. Now, let Ct = (Coh(Qω ( B), =) be the web given by the set of all coherent subsets of Qω ( Ct together with the trivial coherence relation. The collection of all coherent subsets of Ct , ordered by inclusion, defines the coherence space that models the control test defining a non-deterministic choice between two alternative control flows, denoted by Ct ≡ (Coh(Ct ), ⊆). So: Definition 4.1 The domain S ( Ct of the linear traces of functions from S to Ct defines the coherence space of quantum boolean tests. Also, we note that non-determinism enforces a non-traditional treatment of tests. For each classical boolean test bk ∈ Qω ( B capable of testing a memory position of a deterministic state f = (αn )n∈Qω Qω ( Ct , we consider two forms for boolean tests bk ∈ Ct on memory positions of a non-deterministic state F = (f )m∈N ∈ S: 1. a universal form bk∀ (F) = bk∀ (f )m∈ω ≡ ∀m . b(q k )m∈ω and 2. an existential form bk∃ (F) = bk∃ (f )m∈ω ≡ ∃m . b(q k )m∈ω . If F is a deterministic singleton set, both forms coincide with the classical test bk . 4.3. Quantum elementary processes A quantum elementary process may be described as a transition between states of quantum superpositions, performed over a quantum memory position in a single unit of computational time (uct). Thus, the synchronization of linear functions in [Qω ( C] → [Qω ( C] interprets either a unitary transformation or a projection operation, related to the orthogonality and coplanarity conditions, respectively. Consider d(k.i) and e(l.j) as elementary processes in D∞ . The binary relation denoted by d(k.i) ⊥ e(j.i) provides interpretation for computational actions performed over two orthogonal subspaces. Therefore, in the dual construction, the incoherence relation d(k.i) 6⊥ e(j.i) provides interpretations for computational actions performed over two non-orthogonal subspaces. Definition 4.2 Let L⊥ ∞ ≡ (D∞ , ./) be a web defined by the subset of elementary processes D∞ with the coherence relation given by (k.i) d = e(k.j) , ou (k.i) (k.j) d ./ e ⇔ (k.i) d ⊥ e(k.j) (i 6= j). In this case, L./ ∞ = (Coh(L∞ ), ⊆) denotes the domain of unitary transformations. Based on Definition 4.2, ∅, {d(k.0) }, {e(k.1) } and {d(k.0) , e(k.1) } are the only forms of coherent subsets on the web L∞ . In addition, we have ∅ ⊆ {d(k.0) } ⊆ {d(k.0) , e(k.1) } and ∅ ⊆ {e(k.1) } ⊆ {d(k.0) , e(k.1) }, which means ∅ is the bottom object, {d(k.0) } and {e(k.1) } are partial objects and {d(k.0) , e(k.1) } is a total object in L./ ∞ (see examples of functions in Figs. 2, 3, 4 and 5).

Figure 2. {e(k.1) } ⊆ L∞

Figure 3. {d(k.0) } ⊆ L∞

Figure 4. ∅ ⊆ L∞

Figure 5. {d(k.0) , e(k.1) } ⊆ L∞

Thus, the coherence relation ./ models the orthogonality relation and essentially says that two elementary processes are coherent if they can perform computational actions which do not conflict in the positions k.0 and k.1 of the k th -qubit of the quantum memory. Definition 4.3 Let L ∞ ≡ (D∞ , ) be a web defined by the subset of elementary processes D∞ with the coherence relation given by (k.i) d = e(k.j) , or (k) (k.i) (l) (l.j) d =d e =e ⇔ d(k.i) 6⊥ e(k.j) (i = j). L ∞ = (Coh(L∞ ), ⊆) denotes the coherence space of projection functions. The coherence space L ∞ gives interpretation for non-orthogonal computational actions representing projection functions. In such case, the coherent subsets are given by one of the forms: ∅, {d(k.1) } , {e(k.1) } , and {d(k.1) , e(k.1) } (see example functions in Figs. 6 and 7). One observes that {d(k.1) , e(k.1) } needs to be defined for both representations, because a compound function is non-linear function (see example in Fig. 8).

Figure 6. {d(k.1) } ⊆ L ∞

Figure 7. {e(k.1) } ⊆ L ∞

Definition ` 4.4 The coherence ` space generated by the direct (amalgamated) sum D∞ ≡ ./ ./ L∞ L∞ ≡ (Coh(L∞ L∞ ), ⊆) defines the domain of elementary quantum processes, ` S ./ ‘ ˙ D∞ = {0} × where the web (L L ), ≈ ) is given by the disjoint union D ∞ ∞ ∞ S ‘ D∞ {1} × D∞ and the coherence relation ≈ is given by:

Figure 8. Non-linear function

(α, a) ≈ (β, b) ⇔ ‘

α = β = 0 and α = β = 1 and

a ≈L.∞/ b, or a ≈L∞ b.

Each elementary process U(k) ≡ {d(k.0) , e(k.1) } ∈ D∞ can simultaneously read from all distinct quantum memory positions in the global shared quantum memory S (concurrent reading), but it can only write to exactly one quantum memory position (exclusive (k) writing) in Qω . In addition, pri ≡ {id(k.i) , zero(k.i) } ∈ D∞ (for i ∈ {0, 1}) provides interpretation for domain-linear projection functions. ¯∞ ≡ Definition 4.5 The domain of parallel quantum processes is given by D ¯ ∞ ), ⊆). Its web D ¯ ∞ ≡ (Coh(D∞ ), ≈) is constructed by taking each token as (Coh(D a coherent subset of elementary processes and by applying the coherence relation ( U(k) = V(l) , or T (2) U(k) ≈D¯ ∞ V(l) ⇔ ΥD∞ (U(k) ) ΥD∞ (V(l) ) = ∅, where the memory position function ΥD∞ : D∞ → ℘(Qω ) is given by ΥD∞ (x) = {k | U(k) ∈ x} ¯ ∞ ), and In this case, ∅ ∈ Coh(D

S

k∈Qw {d

k}

(3)

¯ ∞ ). ∈ Coh(D

¯∞ ≡ Definition 4.6 The domain interpreting quantum measurements is given by D ¯ ∞ ), ⊆). Its web D ¯ ∞ ≡ (Coh(D∞ ), ≈) is constructed by taking each token as a (Coh(D coherent subset of elementary quantum processes and by applying the coherence relation U(k) = V(l)T , or (k) (l) U ≈D¯ ∞ V ⇔ (k) ΥD∞ (U ) ΥD∞ (V(l) ) 6= ∅, ¯ ∞ ) = {∅} ∪ {U(k) }U∈L . In this case, Coh(D ∞ ` ¯ ` ¯⊥ ¯ ∞ and Consider P∞ ≡ D∞ D∞ D∞ as the amalgamated sum of D∞ , D ¯ ⊥ , in order to construct sequential and control quantum processes. The direct product D ∞Q P∞ P∞ is the coherence space of sequential products of two (parallel, non-deterministic or elementary) processes, whose execution is performed in 2 uct. The coherence Q space P∞ B P∞ of the controlled processes performed over (parallel, measurements or elementary) quantum processes, in 2 uct, is defined as the direct product between B and Q P∞ P∞ . We can put all the above together, in order to obtain the domain a Y a Y a a ¯∞ ¯⊥ . D∞+1 = P∞ (P∞ P∞ ) (P∞ P∞ ), where P∞ = D∞ D D ∞ B

Figure 9. Methodology of construction of the domain D2∞

The coherence space D∞+1 encompasses the first step of the construction of the ordered structure of the qGM model and provides the representations for all qGM computational processes performed in at most 2uct. Following the methodology resumed in Fig. 9 and described in [Reiser 2002], analogous descriptions can be obtained in the levels above, resulting in the domain of quantum processes D2∞ .

5. Conclusion The most basic notion of this work is the definition of the coherence relation as the admissibility of parallelism between basic operations (elementary quantum processes). That relation defines the web over which the coherence space of the whole set of quantum processes (unitary transformations and measurements) is step-wise and systematically build. Over the set of the compatible points of such graph, the strict coherence models quantum parallelism. In the dual construction, justified by the presence of involutive negation in the complementary graph, the incoherence interprets the condition that models measurement operations, through the non-determinism induced by the conflict of memory access. The other constructors, the sequential product and the deterministic sum, are defined by the endofunctors in the CospLin category, which means, the category whose objects are coherence spaces and related morphisms are linear functions. The ordered structure of this model is formalized by the coherence space D2∞ of all processes, constructed by levels from the coherence space D∞ of the elementary quantum processes, following the Scott’s methodology. In this sense, each level is identified by a subspace D∞+n , which reconstructs all the objects from the level before, preserving their properties and relations, and drives the construction of the new

objects. Compatible with the algebraic-theoretic approach to computational processes, the relationship between the levels is expressed by linear functions and the completion procedure guarantees the existence of the least fixed point of the recursive equations, defined by the infinite composition of these morphisms.

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