On Coinduction and Quantum Lambda Calculi Yuxin Deng East China - PowerPoint PPT Presentation

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

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

Motivation 3

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

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

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

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

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

A Quantum λ -Calculus 9

Types A, B, C ::= qubit | A ⊸ B | !( A ⊸ B ) | 1 | A ⊗ B | A ⊕ B | A l 10

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

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

Typing rules A linear !∆ , x : !( A ⊸ B ) ⊢ x : A ⊸ B !∆ , x : A ⊢ x : A !∆ , ∆ ′ ⊢ M : A ⊸ B !∆ , ∆ ′′ ⊢ N : A ∆ , x : A ⊢ M : B !∆ , ∆ ′ , ∆ ′′ ⊢ MN : B ∆ ⊢ λx A .M : A ⊸ B !∆ , ∆ ′ ⊢ M : A !∆ , ∆ ′ ⊢ 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 ) ⊢ N : C !∆ , f : !( A ⊸ B ) , x : A ⊢ M : B !∆ , ∆ ′ ⊢ 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

Quantum closure Def. A quantum closure is a triple [ q, l, M ] where • q is a normalized vector of C 2 n , 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. [ | 00 � + | 11 � , { x �→ 1 } , x ] √ 2 14

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

Structural rule p [ q, l, M ] � [ r, i, N ] p [ q, j ⊎ l, E [ M ]] � [ 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

Extreme derivative k , µ × Def. Suppose we have subdistributions µ , µ → k for k ≥ 0 with the following properties: 0 + µ × µ → µ = 0 1 + µ × µ → µ → → 0 1 2 + µ × µ → µ → → 1 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

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

Big-step reduction C ⇓ ε [ q, l, V ] ⇓ [ q, l, V ] � [ q, l, M ] ⇓ p k · [ r k , i k , V k ] { [ r k , i k , N ] ⇓ µ k } k ∈ K k ∈ K � [ q, l, M ⊗ N ] ⇓ p k ( V k ⊗ µ k ) k ∈ K � [ q, l, M ] ⇓ p k · [ r k , i k , V k ⊗ W k ] { [ r k , i k , ( N { V k /x, W k /y } )] ⇓ µ k } k ∈ K k ∈ K [ q, l, let x A ⊗ y B = M in N ] ⇓ � p k µ k k ∈ K Lem. [ [ C ] ] = sup { µ | C ⇓ µ } 19

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

Coinductive proof techniques 21

A Probabilistic Labelled Transition System i U i meas [ q, l, x 1 ⊗ · · · ⊗ x n ] − − → [ q, l, U ( x 1 ⊗ · · · ⊗ x n )] − − − − → [ q, l, meas x ] [ q, l, x ] ∅ ⊲ V : A ⊸ B ∅ ⊲ W : A skip [ q, ∅ , skip ] − − − → [ q, ∅ , Ω Ω Ω] @[ r,W ] [ q, l, V ] − − − − − → [ q, l ⊎ r, V W ] ∅ ⊲ in l V : A ⊕ B x : A ⊲ M : C l [ r,M ] − − − − − → [ q, l ⊎ r, M { V/x } ] [ q, l, in l V ] ∅ ⊲ V ⊗ W : A ⊗ B x : A, y : B ⊲ M : C eval − − − → [ ⊗ [ r,M ] C [ C ] ] [ q, l, V ⊗ W ] − − − − − → [ l ⊎ r, M { V/x, W/y } ] 22

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

State-based bisimilarity Def. C ∼ s D iff • env ( C ) = env ( D ); ] ∼ s † [ • [ [ C ] [ D ] ]; → ν with µ ∼ s † ν , and a a • if C, D are values then C − → µ implies D − 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. i p i · tr fqv ( M ) q i q † env ( µ ) = � i for any µ = � i p i · [ q i , l i , M i ]. 24

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

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

Similar behaviour by quantum closures [ | 00 � + | 11 � [ ∅ , ∅ , ( λxy. meas ( H ( new ff ))) x ] , � x 1 x 2 � , ( λxy. meas x 1 )( meas x 2 )] √ 2 eval eval [ ∅ , ∅ , λy. meas ( H ( new ff ))] 1 1 @[ ∅ , V ] 2 2 [ ∅ , ∅ , meas ( H ( new ff ))] [ | 0 � , � x 1 � , λy. meas x 1 ] [ | 1 � , � x 1 � , λy. meas x 1 ] @[ ∅ , V ] @[ ∅ , V ] eval [ | 0 � , � x 1 � , meas x 1 ] [ | 1 � , � x 1 � , meas x 1 ] 1 1 eval eval 2 2 [ ∅ , ∅ , ff ] [ ∅ , ∅ , tt ] [ ∅ , ∅ , ff ] [ ∅ , ∅ , tt ] ff tt ff tt [ ∅ , ∅ , Ω Ω [ ∅ , ∅ , Ω Ω Ω] Ω] 27

Soundness 28