Diagrammatic Quantum Reasoning: Completeness and Incompleteness - PowerPoint PPT Presentation

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

Diagrammatic Language for Reasoning in Quantum Computing ZX-Calculus 1 π π / 2 π / 4 1B. Coecke, R. Duncan. Interacting quantum observables. ICALP’08.

Diagrammatic Language for Reasoning in Quantum Computing ZX-Calculus 1 Categorical Quantum Mechanics 2 • 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-Calculus 1 Categorical Quantum Mechanics 2 • Proving properties: protocols, algorithms, models of quantum computing. Proof assistant software: Quantomatic.

Diagrammatic Language for Reasoning in Quantum Computing ZX-Calculus 1 Categorical Quantum Mechanics 2 • 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 � Aleks Kissinger c

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’13 3 ] = =: 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’13 3 ] • Frobenius Algebra with Phases α α α Phase: = = α β = α + β + γ 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 π = - α = = α π 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 Hopf algebra π = - α = = α π 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 Hopf algebra Hopf algebra π = - α = = α π 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 � + | 1 � | 0 � �→ =: | + � √ 2 = | 0 �−| 1 � | 1 � �→ =: |−� √ 2 � � · · · � | 0 . . . 0 � �→ | 0 . . . 0 � � � α � � = e iα | 1 . . . 1 � | 1 . . . 1 � �→ · · · � � · · · � | + . . . + � �→ | + . . . + � � � α � � = e iα |− . . . −� |− . . . −� �→ · · · • 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 .

... ... ... ... ... π / 2 α ... = = α + β α α = = π / 2 β ... ... ... π / 2 ... ... π = - α = = α π • Soundness : ( ZX ⊢ D 1 = D 2 ) ⇒ ( � D 1 � ≃ � D 2 � ) where � D 1 � ≃ � D 2 � if it exists a non zero s ∈ C s.t. � D 1 � = s � D 2 �

... ... ... ... ... π / 2 α ... = = α + β α α = = π / 2 β ... ... ... π / 2 ... ... π = - α = = α π • Soundness : ( ZX ⊢ D 1 = D 2 ) ⇒ ( � D 1 � ≃ � D 2 � ) where � D 1 � ≃ � D 2 � if it exists a non zero s ∈ C s.t. � D 1 � = s � D 2 � • Completeness: ( � D 1 � ≃ � D 2 � ) = ? ( ZX ⊢ D 1 = D 2 ) ⇒ “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’12 4 ] Completeness of the π/ 2 fragment of the zx -calculus. ∀ D 1 , D 2 involving angles multiple of π/ 2 only, � D 1 � ≃ � D 2 � ⇔ ( ZX ⊢ D 1 = D 2 ) 4M. Backens. The ZX-calculus is complete for stabilizer quantum mechanics. New J. Phys. 16 (2014) 093021

Incompleteness of zx -calculus oder, Zamdzhiev’14 5 ] . zx -calculus is incomplete for Theorem [Schr¨ Qubit Quantum Mechanics. Proof. 5C. Schr¨ oder de Witt, V. Zamdzhiev. The ZX-calculus is incomplete for quantum mechanics. EPTCS 172, 2014

Incompleteness of zx -calculus oder, Zamdzhiev’14 5 ] . zx -calculus is incomplete for Theorem [Schr¨ Qubit Quantum Mechanics. Proof. � � � � � � � π / 3 � � � � � � � � � α 0 π / 3 � � � � � � � � ≃ 2 π / 3 β 0 � � � � � � � � � � γ 0 π / 3 � � � � � � π / 3 � � � √ � � √ � 5 3 3 α 0 = − arccos , β 0 = − 2 arcsin , γ 0 = arcsin − α 0 √ 4 4 2 13 5C. Schr¨ oder de Witt, V. Zamdzhiev. The ZX-calculus is incomplete for quantum mechanics. EPTCS 172, 2014

Incompleteness of zx -calculus oder, Zamdzhiev’14 5 ] . zx -calculus is incomplete for Theorem [Schr¨ Qubit Quantum Mechanics. Proof. � � � � � � � π / 3 � � � � � � � � � α 0 π / 3 � � � � � � � � ≃ 2 π / 3 β 0 � � � � � � � � � � γ 0 π / 3 � � � � � � π / 3 � � � √ � � √ � 5 3 3 α 0 = − arccos , β 0 = − 2 arcsin , γ 0 = arcsin − α 0 √ 4 4 2 13 � � � � α := If ZX ⊢ D 1 = D 2 then � D 1 � 3 ≃ � D 2 � 3 . 3 α 3 5C. Schr¨ oder de Witt, V. Zamdzhiev. The ZX-calculus is incomplete for quantum mechanics. EPTCS 172, 2014

Incompleteness of zx -calculus oder, Zamdzhiev’14 5 ] . zx -calculus is incomplete for Theorem [Schr¨ Qubit Quantum Mechanics. Proof. � � � � � � � � � � � � π � π / 3 � � � � � � � � � � � � � � � � � � � � 1 α 0 π / 3 π 3 α 0 � � � � � � � � � 0 � � � � � � � � = = �≃ = 2 π / 3 0 3 β 0 β 0 � � � � � � � � 0 1 � � � � � � � � � � � � γ 0 π / 3 π � 3 γ 0 � � � � � � � � � � � π / 3 π 3 3 � � � √ � � √ � 5 3 3 α 0 = − arccos , β 0 = − 2 arcsin , γ 0 = arcsin − α 0 √ 4 4 2 13 � � � � α := 3 α If ZX ⊢ D 1 = D 2 then � D 1 � 3 ≃ � D 2 � 3 . 3 5C. Schr¨ oder 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¨ oder,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¨ oder,Zamdzhiev’14] No obvious way to extend the zx -calculus • Completeness of the 1 -qubit π/ 4 -fragment (path diagrams) [Backens’14 6 ] 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¨ oder,Zamdzhiev’14] No obvious way to extend the zx -calculus • Completeness of the 1 -qubit π/ 4 -fragment (path diagrams) [Backens’14 6 ] • Incompleteness of the π/ 4 -fragment [Perdrix, Wang’16 7 ] 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