On Coinduction and Quantum Lambda Calculi

Transcription

1 On Coinduction and Quantum Lambda Calculi Yuxin Deng East China Normal University (Joint work with Yuan Feng and Ugo Dal Lago) To appear at CONCUR 15 1

2 Outline Motivation A quantum λ-calculus Coinductive proof techniques Soundness Completeness Summary 2

3 Motivation 3

4 Quantum programming languages Fruitful attempts of language design, e.g. QUIPPER: an expressive functional higher-order language that can be used to program many quantum algorithms and can generate quantum gate representations using trillions of gates. [Green et al. PLDI 13] LIQUi : a modular software architecture designed to control quantum hardware - it enables easy programming, compilation, and simulation of quantum algorithms and circuits. [Wecker and Svore. CoRR 2014] Open problem: Fully abstract denotational semantics wrt operational semantics 4

5 Contextual equivalence An important notion of program equivalence in programming languages. M N if C : C[M] C[N] 5

6 An example in linear PCF f 1 := val(λx.val(0) val(1)) f 2 := val(λx.val(0)) val(λx.val(1)). [Deng and Zhang, TCS, 2015] 6

7 An example f 1 := val(λx.val(0) val(1)) f 2 := val(λx.val(0)) val(λx.val(1)). f 1 f 2 C := bind f = [ ] in bind x = f(0) in bind y = f(0) in val(x = y). 7

8 Linear context? f 1 := val(λx.val(0) val(1)) f 2 := val(λx.val(0)) val(λx.val(1)). Equivalence under linear contexts. 8

9 A Quantum λ-calculus 9

10 Types A,B,C ::= qubit A B!(A B) 1 A B A B A l 10

11 Terms M, N, P ::= x Variables λx A.M M N Abstractions / applications skip M;N Skip / seq. compositions M N let x A y B = M in N Tensor products / proj. in l M in r M Sums match P with (x A : M y B : N) Matches split A Split letrec f A B x = M in N Recursions new meas U Quantum operators 11

12 Values V,W ::= x c λx A.M V W in l V in r W where c {skip,split A,meas,new,U}. As syntactic sugar bit = 1 1, tt = in r skip, and ff = in l skip. 12

13 Typing rules A linear!,x : A x : A,x : A M : B λx A.M : A B!, M : A!,x :!(A B) x : A B!, M : A B!, N : A!,, MN : B!, M : B!, in l M : A B!, in r M : A B!, P : A B!,,x : A M : C!,,y : B N : C!,, match P with (x A : M y B : N) : C!,f :!(A B),x : A M : B!,,f :!(A B) N : C!, letrec f A B x = M in N : C U of arity n! new : bit qubit! meas : qubit bit! U : qubit n qubit n 13

14 Quantum closure Def. A quantum closure is a triple [q,l,m] where q is a normalized vector of C 2n, for some integer n 0. It is called the quantum state; M is a term, not necessarily closed; l is a linking function that is an injective map from fqv(m) to the set {1,...,n}. A closure [q,l,m] is total if l is surjective. In that case we write l as x 1,...,x n if dom(l) = {x 1,...,x n } and l(x i ) = i for all i {1...n}. Non-total closures are allowed. E.g. [ ,{x 1},x] 14

15 Small-step reduction axioms [q, l, (λx A.M)V] 1 [q, l, M{V/x}] [q, l, let x A y B = V W in N] [q, l, skip;n] 1 [q, l, N] [q, l, match in l V with (x A : M y B : N)] [q, l, match in r V with (x A : M y B : N)] [q, l, letrec f A B x = M in N] [q,, newff] [q,, newtt] 1 [q 0, {x n+1}, x] 1 [q 1, {x n+1}, x] [αq 0 +βq 1, {x i}, measx] [αq 0 +βq 1, {x i}, measx] [q, l, U(x 1 x k )] 1 [q, l, N{V/x,W/y}] 1 [q, l, M{V/x}] 1 [q, l, N{V/y}] 1 [q, l, N{(λx A.letrec f A B x = M in M)/f}] α 2 [r 0,, ff] β 2 [r 1,, tt] 1 [r, l, (x 1 x k )] 15

16 Structural rule [q, l, M] [q, j l, E[M]] p [r, i, N] p [r, j i, E[N]] where E is any evaluation context generated by the grammar E ::= [] E M V E E;M E M V E in l E in r E let x A y B = E in M match E with (x A : M y B : N). 16

17 Extreme derivative Def. Suppose we have subdistributions µ, µ k, µ k following properties: for k 0 with the µ = µ 0 +µ 0 µ 0 µ 1 +µ 1 µ 1 µ 2 +µ 2. and each µ k is stable in the sense that C, for all C µ k. Then we call µ := k=0 µ k an extreme derivative of µ, and write µ µ. NB: µ could be a proper subdistribution. 17

18 Example Consider a Markov chain with three states {s 1,s 2,s 3 } and two transitions s s s 3 and s 3 s 3. Then s s 2. Let C be a quantum closure in the Markov chain (Cl, ). Then C [C] for a unique subdistribution [C]. 18

19 Big-step reduction C ε [q,l,v] [q,l,v] [q,l,m] k Kp k [r k,i k,v k ] {[r k,i k,n] µ k } k K [q,l,m N] k Kp k (V k µ k ) [q,l,m] k Kp k [r k,i k,v k W k ] {[r k,i k,(n{v k /x,w k /y})] µ k } k K [q,l,let x A y B = M in N] k Kp k µ k Lem. [C] = sup{µ C µ} 19

20 Linear contextual equivalence Def. A linear context is a term with a hole, written C( ;A), such that C[M] is a closed program when the hole is filled in by a term M, where M : A, and the hole lies in linear position. Def. Linear contextual equivalence is the typed relation given by M N : A if for every linear context C, quantum state q and linking function l such that C( ;A) : B, and both [q,l,c[m]] and [q,l,c[n]] are total quantum closures, [[q,l,c[m]]] = [[q,l,c[n]]] 20

21 Coinductive proof techniques 21

22 A Probabilistic Labelled Transition System [q,l,x 1 x n ] iu [q,l,u(x 1 x n )] [q,l,x] imeas [q,l,measx] [q,,skip] skip [q,,ω] V : A B W : A [q, l, [q, l r, VW] in l V : A B x : A M : C [q,l,in l V] l[r,m] [q,l r,m{v/x}] V W : A B [q,l,v W] x : A,y : B M : C [r,m] [l r,m{v/x,w/y}] C eval [C] 22

23 Lifting relations Def. Let S,T be two countable sets and R S T be a binary relation. The lifted relation R D(S) D(T) is defined by letting µ R ν iff µ(x) ν(r(x)) for all X S. Here R(X) = {t T s X. s R t} and µ(x) = s X µ(s). There are alternative formulations; related to the Kantorovich metric and the maximum network flow problem. See e.g. 23

24 State-based bisimilarity Def. C s D iff env(c) = env(d); [C] s [D]; if C,D are values then C a µ implies D a ν with µ s ν, and vice-versa. Write M s N : A if [q,l,m] s [q,l,n] for any q and l such that [q,l,m] and [q,l,n] are both typable quantum closures. env(µ) = i p i tr fqv(m) q i q i for any µ = i p i [q i,l i,m i ]. 24

25 Distribution-based bisimilarity Def. µ a ρ if ρ = s µ µ(s) µ s, where µ s is determined as follows: either s a µ s or there is no ν with s a ν, and in this case we set µ s = ε. Def. µ d ν iff env(µ) = env(ν); [µ] d [ν]; if µ and ν are value distributions and µ a ρ, then ν a ξ for some ξ with ρ d ξ, and vice-versa. Write M d N : A if [[q,l,m]] d [[q,l,n]] for any q and l such that [q,l,m] and [q,l,n] are quantum closures. 25

26 s is finer than d s t a a s 1 b t 1 t b b s 2 s 3 t 3 t 4 c d c d s 4 t 5 s s t 26

27 Similar behaviour by quantum closures [,,(λxy.meas(h(new ff)))x] eval [ , x 1 x 2,(λxy.measx 1 )(measx 2 )] eval [,, λy.meas(h(new [,, meas(h(new ff))] [ 0, x 1,λy.measx 1 ] [ 1, x 1,λy.measx 1 [ 0, x 1,measx 1 ] [ 1, x 1,measx 1 ] eval eval [,,ff] [,,tt] [,,ff] [,,tt] ff tt ff tt [,,Ω] [,,Ω] 27

28 Soundness 28

29 Congruence Basic idea: Given a relation R, construct a congruence candidate R H, and then show R=R H. 29

30 Howe s construction,x : A [q,l,m] R H [r,j,n] [r,j,λx A.N] R [p,i,l] [q,l,λx A.M] R H [p,i,l]!, [q,l,m] R H [r,j,n]!, [q,i,l] R H [r,m,p]!,, [r,j m,np] R [s,n,q]!,, [q,l i,ml] R H [s,n,q]!, [q,l,m] R H [r,j,n]!, [q,i,l] R H [r,m,p]!,, [r,j m,n P] R [s,n,q]!,, [q,l i,m L] R H [s,n,q] 30

31 Congruence Lem. If [q,l,m] s H [r,j,n] then [[q,l,m]] ( s H ) [[r,j,n]]. Lem. If [q,l,v] H s [r,j,w] then we have that [q,l,v] a [r,j,w] ν and µ ( H s ) ν. a µ implies Consequently, s = s H. Similar arguments apply to d. 31

32 Soundness Thm. Both s and d are included in. 32

33 Completeness 33

34 A simple testing language The tests: t ::= ω a t Apply a test to a distribution in a reactive plts Pr(µ,ω) = µ Pr(µ,a t) = Pr(ρ,t) where µ a ρ µ = T ν iff t T : Pr(µ,t) = Pr(ν,t). 34

35 Characterisation of d by tests Thm. Let µ and ν be two distributions in a reactive plts. Then µ d ν if and only if µ = T ν. 35

36 Converting a test into a context Lem. Let A be a type and t a test. There is a context C A t such that C A t ( ;A) : bit and for every M with M : A, we have Pr([q,l,M],t) = [[q,l,c A t [M]]] where [q,l,m] and [q,l,c A t [M]] are quantum closures for any q and l. 36

37 Full abstraction Thm. coincides with d. 37

38 Summary 38

39 Conclusion Two notions of bisimilarity for reasoning about higher-order quantum programs Both bisimilarities are sound with respect to the linear contextual equivalence The distribution-based one is complete. 39

40 Future work A denotational model fully abstract with respect to the linear contextual equivalence. 40

41 Thank you! 41