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Diagrammatic Quantum Reasoning: Completeness and Incompleteness

Simon Perdrix

CNRS, Loria, Nancy, France

Workshop on Topology and Languages, Toulouse, June 2016

Diagrammatic Language for Reasoning in Quantum Computing ZX-Calculus1

• 1B. Coecke, R. Duncan. Interacting quantum observables. ICALP’08.

Diagrammatic Language for Reasoning in Quantum Computing ZX-Calculus1

π

π/2 π/4

• 1B. Coecke, R. Duncan. Interacting quantum observables. ICALP’08.

Diagrammatic Language for Reasoning in Quantum Computing ZX-Calculus1 Categorical Quantum Mechanics2

• Proving properties: protocols, algorithms, models of quantum

computing. Proof assistant software: Quantomatic.

• 1B. Coecke, R. Duncan. Interacting quantum observables. ICALP’08.
• 2S. Abramsky, B. Coecke. A categorical semantics for quantum protocols. LiCS’04.

Diagrammatic Language for Reasoning in Quantum Computing ZX-Calculus1 Categorical Quantum Mechanics2

• Proving properties: protocols, algorithms, models of quantum

computing. Proof assistant software: Quantomatic.

Diagrammatic Language for Reasoning in Quantum Computing ZX-Calculus1 Categorical Quantum Mechanics2

• Proving properties: protocols, algorithms, models of quantum

computing. Proof assistant software: Quantomatic.

• Foundations:

entanglement, causality aximatisation of quantum mechanics.

• Pedagogical.
• 1B. Coecke, R. Duncan. Interacting quantum observables. ICALP’08.
• 2S. Abramsky, B. Coecke. A categorical semantics for quantum protocols. LiCS’04.

Motivating Example: Post-Selected Teleportation

Motivating Example: Post-Selected Teleportation

c Aleks Kissinger

Frobenius Algebras

• (special commutative) Frobenius algebra (

, , , ) =

3B.Coecke, D.Pavlovic, J. Vicary. A new description of orthogonal bases. MSCS 23, pp 555-567. 2013.]

Frobenius Algebras

• (special commutative) Frobenius algebra (

, , , ) = =:

3B.Coecke, D.Pavlovic, J. Vicary. A new description of orthogonal bases. MSCS 23, pp 555-567. 2013.]

Frobenius Algebras

• (special commutative) Frobenius algebra (

, , , ) , in bijection with orthonormal basis in FdHilb [Coecke,Pavlovic,Vicary’133] = =:

3B.Coecke, D.Pavlovic, J. Vicary. A new description of orthogonal bases. MSCS 23, pp 555-567. 2013.

Frobenius Algebras

• (special commutative) Frobenius algebra (

, , , ) , in bijection with orthonormal basis in FdHilb [Coecke,Pavlovic,Vicary’133]

• Frobenius Algebra with Phases

Phase: α = α = α α β γ = α+β+γ

3B.Coecke, D.Pavlovic, J. Vicary. A new description of orthogonal bases. MSCS 23, pp 555-567. 2013.

Complementary basis

Frobenius algebra Frobenius algebra

3Duncan, Dunne. Interacting Frobenius Algebras are Hopf. LiCS’16. 3Bonchi, Sobocinski, Zanasi. Interacting Hopf Algebras, Journal of Pure and Applied Algebra, 2016

Complementary basis

Frobenius algebra Frobenius algebra = =

α π = π

• α

3Duncan, Dunne. Interacting Frobenius Algebras are Hopf. LiCS’16. 3Bonchi, Sobocinski, Zanasi. Interacting Hopf Algebras, Journal of Pure and Applied Algebra, 2016

Complementary basis

Hopf algebra Frobenius algebra Frobenius algebra = =

α π = π

• α

3Duncan, Dunne. Interacting Frobenius Algebras are Hopf. LiCS’16. 3Bonchi, Sobocinski, Zanasi. Interacting Hopf Algebras, Journal of Pure and Applied Algebra, 2016

Complementary basis

Hopf algebra Hopf algebra Frobenius algebra Frobenius algebra = =

α π = π

• α

3Duncan, Dunne. Interacting Frobenius Algebras are Hopf. LiCS’16. 3Bonchi, Sobocinski, Zanasi. Interacting Hopf Algebras, Journal of Pure and Applied Algebra, 2016

Hadamard

... ... ...

α

... =

α

= =

π/2 π/2 π/2

Universality, Soundness, and Completeness

Universality

• =
• |0

→

|0+|1 √ 2

=: |+ |1 →

|0−|1 √ 2

=: |−

• α

· · · · · ·

• =

|0 . . . 0 → |0 . . . 0 |1 . . . 1 → eiα |1 . . . 1

• α

· · · · · ·

• =

|+ . . . + → |+ . . . + |− . . . − → eiα |− . . . −

• Universality: for any n-qubit linear map U, ∃D s.t. D = U.
• π/4-fragment is approximately universal: ∀ǫ > 0 and any n-qubit linear

map U, ∃D with angles multiple of π/4 s.t. || D − U|| < ǫ.

• π/2-fragment is not (approximately) universal.

... β ... ... α ... ...

=

α+β ... ...

... ... ...

α

... =

α

= =

π/2 π/2 π/2

= =

α π = π

• α
• Soundness: (ZX ⊢ D1 = D2) ⇒ (D1 ≃ D2)

where D1 ≃ D2 if it exists a non zero s ∈ C s.t. D1 = s D2

... β ... ... α ... ...

=

α+β ... ...

... ... ...

α

... =

α

= =

π/2 π/2 π/2

= =

α π = π

• α
• Soundness: (ZX ⊢ D1 = D2) ⇒ (D1 ≃ D2)

where D1 ≃ D2 if it exists a non zero s ∈ C s.t. D1 = s D2

• Completeness: (D1 ≃ D2) =

⇒ ? (ZX ⊢ D1 = D2) “The most fundamental open problem related to the zx-calculus is establishing its completeness properties for some of the calculus’ variants”

CQM wiki

Completeness of the π/2-fragment

Theorem [Backens’124] Completeness of the π/2 fragment of the zx-calculus. ∀D1, D2 involving angles multiple of π/2 only, D1 ≃ D2 ⇔ (ZX ⊢ D1 = D2)

• 4M. Backens. The ZX-calculus is complete for stabilizer quantum mechanics. New J. Phys. 16 (2014) 093021

Incompleteness of zx-calculus

Theorem [Schr¨

• der, Zamdzhiev’145]. zx-calculus is incomplete for

Qubit Quantum Mechanics. Proof.

• 5C. Schr¨
• der de Witt, V. Zamdzhiev. The ZX-calculus is incomplete for quantum mechanics. EPTCS 172,

2014

Incompleteness of zx-calculus

Theorem [Schr¨

• der, Zamdzhiev’145]. zx-calculus is incomplete for

Qubit Quantum Mechanics. Proof.

• π/3

π/3 2π/3 π/3 π/3

• ≃
• α0

β0 γ0

• α0 = − arccos
• 5

2 √ 13

• , β0 = −2 arcsin

√

3 4

• , γ0 = arcsin

√

3 4

• −α0
• 5C. Schr¨
• der de Witt, V. Zamdzhiev. The ZX-calculus is incomplete for quantum mechanics. EPTCS 172,

2014

Incompleteness of zx-calculus

Theorem [Schr¨

• der, Zamdzhiev’145]. zx-calculus is incomplete for

Qubit Quantum Mechanics. Proof.

• π/3

π/3 2π/3 π/3 π/3

• ≃
• α0

β0 γ0

• α0 = − arccos
• 5

2 √ 13

• , β0 = −2 arcsin

√

3 4

• , γ0 = arcsin

√

3 4

• −α0
• α
• 3

:=

• 3α
• If ZX ⊢ D1 = D2 then D13 ≃ D23.
• 5C. Schr¨
• der de Witt, V. Zamdzhiev. The ZX-calculus is incomplete for quantum mechanics. EPTCS 172,

2014

Incompleteness of zx-calculus

Theorem [Schr¨

• der, Zamdzhiev’145]. zx-calculus is incomplete for

Qubit Quantum Mechanics. Proof.

• π/3

π/3 2π/3 π/3 π/3

• 3

=

• π

π π π

• =

1 1

• ≃
• 3α0

3β0 3γ0

• =
• α0

β0 γ0

• 3

α0 = − arccos

• 5

2 √ 13

• , β0 = −2 arcsin

√

3 4

• , γ0 = arcsin

√

3 4

• −α0
• α
• 3

:=

• 3α
• If ZX ⊢ D1 = D2 then D13 ≃ D23.
• 5C. Schr¨
• der de Witt, V. Zamdzhiev. The ZX-calculus is incomplete for quantum mechanics. EPTCS 172,

2014

(In)-completeness

• Completeness of the π/2-fragment [Backens’12]
• Incompleteness for Qubit QM [Schr¨
• der,Zamdzhiev’14]

No obvious way to extend the zx-calculus

(In)-completeness

• Completeness of the π/2-fragment [Backens’12]
• Incompleteness for Qubit QM [Schr¨
• der,Zamdzhiev’14]

No obvious way to extend the zx-calculus

• Completeness of the 1-qubit π/4-fragment (path diagrams)

[Backens’146]

• 6M. Backens. The ZX-calculus is complete for the single-qubit Clifford+T group. EPTCS 172, 2014.

(In)-completeness

• Completeness of the π/2-fragment [Backens’12]
• Incompleteness for Qubit QM [Schr¨
• der,Zamdzhiev’14]

No obvious way to extend the zx-calculus

• Completeness of the 1-qubit π/4-fragment (path diagrams)

[Backens’146]

• Incompleteness of the π/4-fragment [Perdrix, Wang’167]
• 6M. Backens. The ZX-calculus is complete for the single-qubit Clifford+T group. EPTCS 172, 2014.
• 7S. Perdrix, Q. Wang. Supplementarity is necessary for quantum diagram reasoning. MFCS’16

Supplementarity, a candidate for incompleteness

• α+π

α

• =
• 2α

+π

• Inspired by [Coecke,Edwards’10]: supplementarity.
• Can be proven in ZX when α = ± π

2 .

Theorem:    ZX ⊢

α+π = α 2α

+π

    ⇔ α = 0 mod π

2

Alternative interpretation

• ♯

=

• ♯

=

• ♯

=

• α
• ♯

=

α α α 2α

Soundness: (ZX ⊢ D1 = D2) ⇒ D1♯ ≃ D2♯ Counterexample: ∀α = 0 mod π

2 ,

• α+π

α

• ♯

≃

• 2α

+π

• ♯

Sound interpretation (1)

• π
• ♯

= π π π = π π π

Sound interpretation (1)

• π
• ♯

= π π π = π π π

• π

2

• ♯

=

π 2 π 2

π

π 2

=

• π

2

• π

2

• π

2

=

• π

2

• π

2

• π

2

Sound interpretation (1)

• π
• ♯

= π π π = π π π

• π

2

• ♯

=

π 2 π 2

π

π 2

=

• π

2

• π

2

• π

2

=

• π

2

• π

2

• π

2

• ♯

= =

π 2 π 2 π 2 π 2 π 2 π 2 π 2 π 2 π 2

=

• π

2

• π

2

• π

2

• π

2

• π

2

• π

2

• π

2

• π

2

• π

2

=

• π

2 π 2 π 2

• ♯

Sound interpretation (2)

• α

π

• ♯

= 2α π π π α α α = α π 2α α α 3π π π =

• 2α

π

• α
• α
• α

π π =

• π
• α
• ♯

Sound interpretation (3)

• α

β

• ♯

=

α+β α+β 2α α+β 2β

=

α+β α+β 2β 2α α+β

=

α+β α+β α+β 2α 2β

= α+β

α+β α+β 2α 2β

=

2(α+β) α+β α+β α+β

=

• α+β
• ♯

Incompleteness

• α+π

α

• ♯

≃

• 2α

+π

• ♯

⇒ α = 0 mod π 2 Corollary:

π 4 -fragment of ZX-calculus is not complete as the following

equation cannot be derived:

π 4

=

−3π 4 −π 2

Graphical interpretation

• Theorem. In ZX-calculus, antiphase twins can be merged if and only if

∀α,

α+π = α 2α

+π

where two dots are antiphase twins if: – they have the same colour; – the difference between their angles is π; – they have the same neighbourhood.

α α+π γ β

→

β 2α

+π

γ α+π β α γ

→

2α

+π

γ β

Conclusion

• π

4 -fragment of zx-calculus is completeness [Backens’12]

• Incompleteness in general [Schr¨
• der,Zamdzhiev’14]

No obvious way to extend the zx-calculus

• π

4 -fragment is incompleteness [Perdrix,Wang’16]

Supplementarity as an axiom: zx-calculus := zx-calculus +‘Supplementarity’ ∀α,

α+π = α 2α

+π

Open question. Is π

4 -fragment of zx-calculus +‘Supplementarity’ complete?