Completeness of qubit ZX calculus via elementary operations Quanlong Wang Department of Computer Science, University of Oxford Third Workshop on String Diagrams in Computation, Logic and Physics 5 September, 2019
Outline Background Complete axiomatisation of ZX-calculus with total linearity Proof of completeness
What is ZX-calculus ◮ ZX-calculus is a graphical language for quantum computing proposed by Coecke and Duncan [ICALP’08, New J. Phys., 2011]. ◮ It gives all the details of interacting processes in quantum computation using qubits. ◮ ZX-calculus can be formalised in the framework of PROPs, which are strict symmetric monoidal categories having the natural numbers as objects, with the tensor product of objects given by addition. ◮ As a PROP , ZX-calculus can be presented by generators and relations (rewriting rules), just like the presentation of a group.
How useful is completeness ◮ Completeness of ZX-calculus means quantum computing can be done pure diagrammatically. ◮ Completeness offers a complete set of rules based on which one could develop an efficient rule set for particular application purpose. ◮ The key idea of applying ZX-calculus is first encoding matrices into diagrams then choosing suitable rules to rewrite diagrams into a form as simple as you can.
Original generators of ZX-calculus n n ... ... R ( n , m ) R ( n , m ) : n → m : n → m α α Z ,α X ,α ... ... m m H : 1 → 1 σ : 2 → 2 H · · · · · · · I : 1 → 1 e : 0 → 0 · · · · · · · · · C a : 0 → 2 C u : 2 → 0 where m , n ∈ N , α ∈ [ 0 , 2 π ) , a ∈ C , and e represents an empty diagram.
Standard interpretation of ZX-calculus � � � � n n � � � � � � � � � � � � � � � � � ... � � ... � � � � � � � � � � � � � � � � � = | 0 � ⊗ m � 0 | ⊗ n + e i α | 1 � ⊗ m � 1 | ⊗ n , = | + � ⊗ m � + | ⊗ n + e i α |−� ⊗ m �−| ⊗ n , � � � � � α � � α � � � � � � � � � � � � � � � � � ... ... � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � m m · · · · · � · � � � � � � � � � 1 1 1 · 1 0 � � � � � � � = , � · · � = 1 , � � = , � H � � √ � � � � � � � � � � � � 1 − 1 · · � � 0 1 � � 2 � � · · · · · 1 0 0 0 1 � � � � � � 0 0 1 0 0 � � � � � � � = , � � = � , = 1 0 0 1 , � � � � � � � � � � 0 1 0 0 0 � � � � 0 0 0 1 1 � D 1 ⊗ D 2 � = � D 1 � ⊗ � D 2 � , � D 1 ◦ D 2 � = � D 1 � ◦ � D 2 � , where � � � � 1 0 � � � � | 0 � = � 0 | = | 1 � = � 1 | = , 1 0 , , 0 1 , 0 1 � � � � 1 1 1 1 1 1 � � � � | + � = � + | = |−� = �−| = √ , √ 1 1 , √ , √ 1 − 1 . 1 − 1 2 2 2 2
Typical rewriting rules of ZX-calculus ... ... ... α ... = ( S 1 ) = ( S 2 ) α + β β ... ... ... H = = ( S 3 ) H = ( H 2 ) ... ... H H ( B 2 ′ ) = α = α ( H ) H H ... ... = ( B 1 ) = ( B 2 ) π / 2 - π / 2 α = α π H = ( EU ) ( K 2 ) π π - α π / 2 · · · · · · · = = · · ( IV ) ( Hopf ) · · · · · · ·
Three properties of the ZX-calculus ◮ The ZX-calculus is sound: for any two diagrams D 1 and D 2 , ZX ⊢ D 1 = D 2 must imply that � D 1 � = � D 2 � . [Coecke, Duncan, New J. Phys., 2011] ◮ The ZX-calculus is universal: for any linear map L , there must exist a diagram D in the ZX-calculus such that � D � = L . [Coecke, Duncan, New J. Phys., 2011] ◮ The ZX-calculus is complete: for any two diagrams D 1 and D 2 , � D 1 � = � D 2 � must imply that ZX ⊢ D 1 = D 2 . [Hadzihasanovic, Ng, Wang, LICS’18; Jeandel, Perdrix, Vilmart, LICS’18]
Why another complete axiomatisation for qubit ZX-calculus ◮ The following non-linear axiom was presented in [Jeandel, Perdrix, and Vilmart, LICS’18] and [Jeandel, Perdrix, and Vilmart, LICS’19]:
Why another complete axiomatisation for qubit ZX-calculus ◮ The following non-linear axiom was presented in [Vilmart, LICS’19]:
Why another complete axiomatisation for qubit ZX-calculus ◮ The following non-linear axiom was presented in [Hadzihasanovic, Ng, Wang, LICS’18]: α λ 1 γ = β λ 2 λ where λ e i γ = λ 1 e i β + λ 2 e i α . ◮ Except for [Jeandel, Perdrix, and Vilmart, LICS’19], all the other completeness proofs need the translation from the ZW-calculus. ◮ All these proofs are not easy to generalise to qudit cases.
Normal form by [Jeandel, Perdrix, and Vilmart, LICS’19] ◮ The normal form used in [Jeandel, Perdrix, and Vilmart, LICS’19] is defined recursively. ◮ (Controlled scalars) . A ZX-diagram D : 1 → 0 is a controlled scalar if � D � | 0 � = 1. ◮ (Controlled Normal Form) . Given a set S of controlled scalars, the diagrams in normal controlled form with respect to S (S-CNF) are inductively defined as follows:
Normal form by [Jeandel, Perdrix, and Vilmart, LICS’19] ◮ (Normal Form) . Given a set S of controlled scalars, for any n , m ∈ N , and any D : 1 → n + m in S-CNF , the following diagram is called a normal form with respect to S (S-NF): ◮ Define Λ R : C → ZX [ 1 , 0 ] as: ◮ Theorem [Jeandel, Perdrix, and Vilmart, LICS’19] Any ZX-diagram can be put into a normal form with respect to S R , and the ZX-calculus is complete for the full pure quibt QM.
Generators for pure linear complete axiomatisation of qubit ZX-calculus n n ... ... R ( n , m ) R ( n , m ) : n → m a : n → m a Z ,α X ,α ... ... m m H : 1 → 1 σ : 2 → 2 H · · · · · · · I : 1 → 1 e : 0 → 0 · · · · · · · · · C a : 0 → 2 C u : 2 → 0 - 1 T − 1 : 1 → 1 T : 1 → 1 Table: Generators of qubit ZX-calculus where m , n ∈ N , α ∈ [ 0 , 2 π ) , a ∈ C , and e represents an empty diagram.
Standard interpretation of new generators � � n � � � � � � � � � ... � � � � � � � � � = | 0 � ⊗ m � 0 | ⊗ n + a | 1 � ⊗ m � 1 | ⊗ n , � � � a � � � � � � � � � ... � � � � � � � � � � � � � � � � � � m � � n � � � � � � � � � ... � � � � � � � � � = | + � ⊗ m � + | ⊗ n + a |−� ⊗ m �−| ⊗ n , � � � a � � � � � � � � � ... � � � � � � � � � � � � � � � � � � m � � � � � � � � 1 1 - 1 1 − 1 � � = � � � � = , . � � � � � � � � � � 0 1 � � 0 1 � � where a is an arbitrary complex number.
Rules for pure linear complete axiomatisation of qubit ZX-calculus ... ... ... a . . = ( S 1 ) = ( S 2 ) . ab b ... ... ... = = ( S 3 ) α = e i α ( S 4 ) ... ... · · · · · H H · · = = · · ( Inv ) a a ( H ) · · · · · · · H ... ... H = ( B 1 ) = ( B 2 ) π / 2 - π / 2 = = ( EU ) π H π / 2 Figure: Rules I, where α, β ∈ [ 0 , 2 π ) , a , b ∈ C .
Rules for pure linear complete axiomatisation of qubit ZX-calculus π = ( TR 2 ) = ( TR 3 ) π = π ( TR 7 ) = ( TR 8 ) H - 1 - 1 a b ( TR 8 ′ ) ( AD ′ ) = = a + b a = = ( TR 9 ) ( Asso ) a a - 1 = ( SYM ) - 1 = = ( InvTR ) Figure: Ruels II, a , b ∈ C
Rules for pure linear complete axiomatisation of qubit ZX-calculus a π - 1 π = ( TR 10 ) a = ( TR 11 ) - 1 a π π a a - 1 a a b = - 1 ( TR 12 ) = ( TR 13 ) - 1 - 1 b a π π π π π a a π π ab - 1 = ( TR 14 ) π = ( TR 15 ) b π - 1 a a π b b π a π a b = ( TR 16 ) = ( TR 17 ) π - 1 - 1 a + b π a b b a a = = ( TR 18 ) = ( TR 19 ) - 1 - 1 - 1 - 1 b b Figure: Ruels III, a , b ∈ C
Derivable rules π H H = ◮ = π π ... ... Proved in [Backens, Perdrix, Wang, QPL ’16] π = - 1 (TR3’) ◮ Directly obtained by plugging a triangle on both sides of (TR3).
Derivable rules π = (TR1) Proof: ◮ π TR 8 ′ = = ⇒ = π π π = π TR 3 TR 3 = = π π π = = ⇒ ⇒ π
Derivable rules = (TR2’) Proof: ◮ π TR 1 TR 1 π = = = π = Proof: ◮ SYM Asso B 1 SYM = = = =
Derivable rules a = a + b b (AD) Proof: ◮ - 1 π a b a a b b a + b ⇒ TR 3 ′ AD ′ TR 3 = = = a b a = = b a + b
Derivable rules = (TR4) Proof: ◮ - 1 π TR 8 ′ TR 3 ′ = TR 3 = = =
Derivable rules π - 1 = (IVT) Proof: ◮ π π π π π π π π H TR 2 ′ , S 1 π TR 2 , B 1 TR 7 B 1 , S 1 , H S 2 = = = = =
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