Completeness of qubit ZX calculus via elementary operations - - PowerPoint PPT Presentation

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

• ne 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

R(n,m)

Z,α

: n → m

m n

... α ... R(n,m)

X,α

: n → m

m n

α ... ... H : 1 → 1

H

σ : 2 → 2 I : 1 → 1 e : 0 → 0 · · · · · · · · · · · · · · · · Ca : 0 → 2 Cu : 2 → 0 where m, n ∈ N, α ∈ [0, 2π), a ∈ C, and e represents an empty diagram.

Standard interpretation of ZX-calculus

• m

n

... α ...

• = |0⊗m 0|⊗n+eiα |1⊗m 1|⊗n ,
• m

n

α ... ...

• = |+⊗m +|⊗n+eiα |−⊗m −|⊗n ,
• H
• =

1 √ 2

• 1

1 1 −1

• ,
• ·

· · · · · · · · · · · · · · ·

• = 1,
• =
• 1

1

• ,
• =

               1 1 1 1                ,

• =

               1 1                ,

• =
• 1

1

• ,

D1 ⊗ D2 = D1 ⊗ D2, D1 ◦ D2 = D1 ◦ D2, where |0 =

• 1
• ,

0| =

• 1
• ,

|1 =

• 1
• ,

1| =

• 1
• ,

|+ = 1 √ 2

• 1

1

• ,

+| = 1 √ 2

• 1

1

• ,

|− = 1 √ 2

• 1

−1

• ,

−| = 1 √ 2

• 1

−1

• .

Typical rewriting rules of ZX-calculus

... β ... ... α ... ...

=

α+β ... ...

(S1) = (S2) = = (S3)

H H =

(H2) = (B2′) α ... ... = ... α

H H H H

... (H) = (B1) = (B2)

H = π/2 π/2

• π/2

(EU) π α =

• α

π α π (K2) · · · · · · · · · · · · · · · · = (IV) = (Hopf)

Three properties of the ZX-calculus

◮ The ZX-calculus is sound: for any two diagrams D1 and D2,

ZX ⊢ D1 = D2 must imply that D1 = D2. [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 D1 and D2,

D1 = D2 must imply that ZX ⊢ D1 = D2. [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

λ1

β

λ2

=

λ

γ

where λeiγ = λ1eiβ + λ2eiα.

◮ Except for [Jeandel, Perdrix, and Vilmart, LICS’19], all the

• ther 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 ΛR : C → ZX[1, 0] as: ◮ Theorem [Jeandel, Perdrix, and Vilmart, LICS’19] Any ZX-diagram

can be put into a normal form with respect toSR, and the ZX-calculus is complete for the full pure quibt QM.

Generators for pure linear complete axiomatisation of qubit ZX-calculus

R(n,m)

Z,α

: n → m

m n

a ... ... R(n,m)

X,α

: n → m

m n

... ... a H : 1 → 1

H

σ : 2 → 2 I : 1 → 1 e : 0 → 0 · · · · · · · · · · · · · · · · Ca : 0 → 2 Cu : 2 → 0 T : 1 → 1 T−1 : 1 → 1

• 1

Table: Generators of qubit ZX-calculus

where m, n ∈ N, α ∈ [0, 2π), a ∈ C, and e represents an empty diagram.

Standard interpretation of new generators

• m

n

a ... ...

• = |0⊗m 0|⊗n + a |1⊗m 1|⊗n ,
• m

n

... ... a

• = |+⊗m +|⊗n + a |−⊗m −|⊗n ,
• =
• 1

1 1

• ,
• 1
• =
• 1

−1

1

• .

where a is an arbitrary complex number.

Rules for pure linear complete axiomatisation of qubit ZX-calculus

... ... a ... . . . ... b ab ... ... = (S1) = (S2) = = (S3) α = eiα (S4) · · · · · · · · · · · · · · · · = (Inv) =

H

... ... ...

H ... H

a a

H

(H) = (B1) = (B2)

H

= π =

π/2

• π/2

π/2

(EU)

Figure: Rules I, where α, β ∈ [0, 2π), a, b ∈ C.

Rules for pure linear complete axiomatisation of qubit ZX-calculus

= (TR2) = π (TR3) = π π

H

(TR7) = (TR8)

• 1

= (TR8′) = b a

• 1

a + b (AD′) = a a a (TR9) = (Asso) = (SYM)

• 1 =
• 1

= (InvTR) Figure: Ruels II, a, b ∈ C

Rules for pure linear complete axiomatisation of qubit ZX-calculus

=

• 1
• 1

π π

• 1

a a a (TR10) = π a a π a (TR11) =

• 1
• 1
• 1

(TR12) b a = b a (TR13) π b

• 1

π a = b π

• 1

a π (TR14) π a = π π π b a π π ab (TR15) = a b π π a b π π (TR16) b

• 1

a =

• 1

a + b (TR17)

• 1
• 1

= = (TR18) b

• 1

a = a

• 1

b (TR19) Figure: Ruels III, a, b ∈ C

Derivable rules

◮

π π π ... ... =

H H =

Proved in [Backens, Perdrix, Wang, QPL ’16]

◮

= π

• 1 (TR3’)

Directly obtained by plugging a triangle on both sides of (TR3).

Derivable rules

◮

π

=

π

(TR1) Proof:

TR8′

= = = ⇒

TR3

= π π = π

TR3

= π ⇒ π = π ⇒ = π π

Derivable rules

◮

= (TR2’) Proof:

TR1

= π π

TR1

= π =

◮

= Proof:

SYM

=

Asso

=

B1

=

SYM

=

Derivable rules

◮

= a + b a b (AD) Proof:

TR3′

=

• 1

b a

TR3

= a b π b a

AD′

= a + b ⇒ = a b a b a + b =

Derivable rules

◮

= (TR4) Proof: =

TR8′

=

• 1

TR3′

=

TR3

= π

Derivable rules

◮

• 1

π

=

π

(IVT) Proof:

TR7

= π π

S2

= π π

B1,S1,H

= π

H

TR2′,S1

= π π

TR2,B1

= π

Derivable rules

◮

π α =

• α

π α π (K2) Proof: a π a a a π π

TR3

= = π a a

TR9

= a π = a π π

TR3

= π a π

Derivable rules

◮

= π π (TR5) Proof:

• 1

IVT

= π π π π = π π π

TR8′

= π = π ⇒ π π π = π ⇒ = π π π π = π π π π π = π π π π =

Normal form

Any complex vector (a0, a1, · · · , a2m−1)T can be uniquely represented by

• 1

π

a2m−1 ai

π π · · · . . . . . . · · · · · · · · · . . . · · · π · · ·

where ai connects to wires by red nodes depending on i, and all possible connections are included in the normal form.

Where does this normal form come from

                 . . .

1

                

row

− − − − − − →

addition

                

a0

. . .

1

                

row

− − − − − − →

addition

                      

a0 a1

. . .

a2m−1 1

                      

row

− − − − − − − − − − →

multiplication

                      

a0 a1

. . .

a2m−2 a2m−1

                      

How to prove completeness

◮ the juxtaposition of any two diagrams in normal form can be

rewritten into a normal form.

◮ a self-plugging on a diagram in normal form can be rewritten

into a normal form.

◮ all generators can be rewritten into normal forms.

One simple application of the linear version of ZX

◮ Translate arbitrary H-box in ZH to ZX:

a a − 1

→

a a − 1

→ = a

a a − 1

→

a − 1

→

a

· · · · · · · · · · · ·

where the general H-box corresponds to the matrix:

           

1

· · ·

1

. . . · · · . . .

1

· · ·

a

            .

◮ With this translation, we can say that ZH is “SLOCC

equivalent” to ZX.

Further work

◮ Generalise the completeness result of the ZX-calculus from

qubit to qudit for arbitrary dimension d. Normal form for

                      

a0 a1

. . .

adm−2 adm−1

                       :

K1

• 1

. . . − →

a dm−1

· · · − →

a i

. . . . . . · · · · · · · · ·

K1

· · · · · ·

K1 K1

... ...

. where K1 = |d − 1 , −

→

a i = (0, · · · , 0, ai), −

→

a dm−1 = (1, · · · , 1, adm−1).

Further work

◮ Achieve a complete axiomatization of the ZX-calculus with

mixed dimensions.

◮ Find useful ZX rules for optimisation of Benchmark quantum

circuits.

◮ Apply to linguistics.

Thank you!