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Analysis of Quantum Entanglement in Quantum Programs using Stabilizer Formalism The University of Tokyo Kentaro Honda Purpose Analyse how separated the output of a program are. q0 := | 0 i ; q1 := | 0 i ; q2 := | 0 i ; q3 := | 1 i ; H q3; H

  1. J · K \ Abstract Semantics q0 := | 1 i ; q1 := | 0 i ; | 100 i q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; q0 Z CX q2, q0; CX q0, q1; q1 Z q2 Z

  2. J · K \ Abstract Semantics q0 := | 1 i ; q1 := | 0 i ; | 10+ i q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; q0 Z CX q2, q0; CX q0, q1; q1 Z q2 X

  3. J · K \ Abstract Semantics q0 := | 1 i ; q1 := | 0 i ; | 10+ i q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; q0 Z CX q2, q0; CX q0, q1; q1 Z q2 X

  4. J · K \ Abstract Semantics q0 := | 1 i ; q1 := | 0 i ; | � 0+ i q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; q0 X CX q2, q0; CX q0, q1; q1 Z q2 X

  5. J · K \ Abstract Semantics q0 := | 1 i ; q1 := | 0 i ; 1 2 | � 0 i ( | 0 i + e i π 4 | 1 i ) p q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; q0 X CX q2, q0; CX q0, q1; q1 Z q2 >

  6. J · K \ Abstract Semantics q0 := | 1 i ; q1 := | 0 i ; 1 2 | � 0 i ( | 0 i + e i π 4 | 1 i ) p q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; q0 X CX q2, q0; CX q0, q1; q1 Z q2 >

  7. J · K \ Abstract Semantics q0 := | 1 i ; q1 := | 0 i ; 1 2 | � 0 i ( | 0 i � e i π 4 | 1 i ) p q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; q0 X CX q2, q0; CX q0, q1; q1 Z q2 >

  8. J · K \ Abstract Semantics q0 := | 1 i ; q1 := | 0 i ; 1 2( | 00 i � | 11 i )( | 0 i � e i π 4 | 1 i ) q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; q0 > CX q2, q0; CX q0, q1; q1 > q2 >

  9. Soundness [Perdrix08] ∀ ρ : quantum state ∀ ( π , b ) ∈ A Q ∀ C : program ( π , b ) ✏ ρ ⇒ J C K \ ( π , b ) ✏ J C K ( ρ )

  10. Soundness [Perdrix08] ∀ ρ : quantum state ∀ ( π , b ) ∈ A Q ∀ C : program ( π , b ) ✏ ρ ⇒ J C K \ ( π , b ) ✏ J C K ( ρ ) valid approximation valid approximation

  11. J · K \ Abstract Semantics q0 := | 1 i ; q1 := | 0 i ; 1 2( | 00 i � | 11 i )( | 0 i � e i π 4 | 1 i ) q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; q0 > CX q2, q0; CX q0, q1; q1 > q2 >

  12. J · K \ Abstract Semantics q0 := | 1 i ; q1 := | 0 i ; 1 2( | 00 i � | 11 i )( | 0 i � e i π 4 | 1 i ) q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; q0 > CX q2, q0; CX q0, q1; q1 > CX q1, q0; q2 > CX q1, q2;

  13. Previous work • Measured qubit is separated from the others [Perdrix07] • Multi-qubit gate may entangle qubits [Perdrix07] • Restriction to CX gate [Perdrix08][Prost&Zerrari09] • C-qubit / preserves separability [Perdrix08][Prost&Zerrari09] | 0 i | 1 i • T-qubit preserves separability [Perdrix08] |± i • Type system, Abstract interpretation, Hoare-like logic [Perdrix07] [Perdrix08] [Prost&Zerrari09]

  14. J · K \ Abstract Semantics q0 := | 1 i ; q1 := | 0 i ; 1 2( | 00 i � | 11 i )( | 0 i � e i π 4 | 1 i ) q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; q0 > CX q2, q0; CX q0, q1; q1 > CX q1, q0; q2 > CX q1, q2;

  15. J · K \ Abstract Semantics q0 := | 1 i ; q1 := | 0 i ; 1 2 | 0 �i ( | 0 i � e i π 4 | 1 i ) p q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; q0 > CX q2, q0; CX q0, q1; q1 > CX q1, q0; q2 > CX q1, q2;

  16. J · K \ Abstract Semantics q0 := | 1 i ; 1 q1 := | 0 i ; 2 | 0 i ( | 00 i � e i π 4 | 01 i q2 := | 0 i ; + e i π 4 | 10 i � | 11 i ) H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; q0 > CX q2, q0; CX q0, q1; q1 > CX q1, q0; q2 > CX q1, q2;

  17. Previous work • Measured qubit is separated from the others [Perdrix07] • Multi-qubit gate may entangle qubits [Perdrix07] • Restriction to CX gate [Perdrix08][Prost&Zerrari09] • C-qubit / preserves separability [Perdrix08][Prost&Zerrari09] | 0 i | 1 i • T-qubit preserves separability [Perdrix08] |± i • Type system, Abstract interpretation, Hoare-like logic [Perdrix07] [Perdrix08] [Prost&Zerrari09]

  18. Previous work • Measured qubit is separated from the others [Perdrix07] • Multi-qubit gate may entangle qubits [Perdrix07] disentangle • Restriction to CX gate [Perdrix08][Prost&Zerrari09] • C-qubit / preserves separability [Perdrix08][Prost&Zerrari09] | 0 i | 1 i • T-qubit preserves separability [Perdrix08] |± i • Type system, Abstract interpretation, Hoare-like logic [Perdrix07] [Perdrix08] [Prost&Zerrari09]

  19. J · K \ Abstract Semantics q0 := | + i ; ρ q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; q0 q1 q2 CX q1, q2; q0 − q1 − q2 −

  20. J · K \ Abstract Semantics q0 := | + i ; | +00 i q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; q0 q1 q2 CX q1, q2; q0 X q1 Z q2 Z

  21. J · K \ Abstract Semantics 1 q0 := | + i ; p 2( | 00 i + | 11 i ) | 0 i q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; q0 q1 q2 CX q1, q2; q0 > q1 > q2 Z

  22. J · K \ Abstract Semantics 1 q0 := | + i ; p 2( | 000 i + | 111 i ) q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; q0 q1 q2 CX q1, q2; q0 > q1 > q2 >

  23. J · K \ Abstract Semantics 1 q0 := | + i ; p 2( | 000 i + | 111 i ) q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; q0 q1 q2 CX q1, q2; meas q0; q0 > q1 > q2 >

  24. Previous work • Measured qubit is separated from the others [Perdrix07] • Multi-qubit gate may entangle qubits [Perdrix07] • Restriction to CX gate [Perdrix08][Prost&Zerrari09] • C-qubit / preserves separability [Perdrix08][Prost&Zerrari09] | 0 i | 1 i • T-qubit preserves separability [Perdrix08] |± i • Type system, Abstract interpretation, Hoare-like logic [Perdrix07] [Perdrix08] [Prost&Zerrari09]

  25. J · K \ Abstract Semantics 1 q0 := | + i ; p 2( | 000 i + | 111 i ) q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; q0 q1 q2 CX q1, q2; meas q0; q0 > q1 > q2 >

  26. J · K \ Abstract Semantics 1 q0 := | + i ; 2( | 000 i h 000 | + | 111 i h 111 | ) q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; q0 q1 q2 CX q1, q2; meas q0; q0 Z q1 > q2 >

  27. Previous work • Measured qubit is separated from the others [Perdrix07] • Multi-qubit gate may entangle qubits [Perdrix07] • Restriction to CX gate [Perdrix08][Prost&Zerrari09] • C-qubit / preserves separability [Perdrix08][Prost&Zerrari09] | 0 i | 1 i • T-qubit preserves separability [Perdrix08] |± i • Type system, Abstract interpretation, Hoare-like logic [Perdrix07] [Perdrix08] [Prost&Zerrari09]

  28. Previous work • Measured qubit is separated from the others [Perdrix07] Measurement may affect unmeasured qubits • Multi-qubit gate may entangle qubits [Perdrix07] • Restriction to CX gate [Perdrix08][Prost&Zerrari09] • C-qubit / preserves separability [Perdrix08][Prost&Zerrari09] | 0 i | 1 i • T-qubit preserves separability [Perdrix08] |± i • Type system, Abstract interpretation, Hoare-like logic [Perdrix07] [Perdrix08] [Prost&Zerrari09]

  29. Observation • Multi-qubit gate may undo entangled qubits • Measurement may destroy multipartite entanglement

  30. Observation • Multi-qubit gate may undo entangled qubits • Measurement may destroy multipartite entanglement • Information about entangled qubits is needed • Semantics should be easily computable

  31. Observation • Multi-qubit gate may undo entangled qubits • Measurement may destroy multipartite entanglement • Information about entangled qubits is needed • Semantics should be easily computable Stabilizer Formalism

  32. Stabilizer Formalism standard stabilizer { P | P : Pauli matrix | ψ i = α 0 ··· 00 | 0 · · · 00 i s.t. P | ψ i = | ψ i } + · · · + α 1 ··· 11 | 1 · · · 11 i e.g. { III , XXX , − ZZI , − IZZ , 1 p 2( | 010 i + | 101 i ) ZIZ , YYX , − YXY , XYY }

  33. Stabilizer Formalism standard stabilizer { P | P : Pauli matrix | ψ i = α 0 ··· 00 | 0 · · · 00 i s.t. P | ψ i = | ψ i } + · · · + α 1 ··· 11 | 1 · · · 11 i e.g. { III , XXX , − ZZI , − IZZ , 1 p 2( | 010 i + | 101 i ) ZIZ , YYX , − YXY , XYY }   X X X + Z Z I −   Z I Z +

  34. Stabilizer Formalism standard stabilizer { P | P : Pauli matrix | ψ i = α 0 ··· 00 | 0 · · · 00 i s.t. P | ψ i = | ψ i } + · · · + α 1 ··· 11 | 1 · · · 11 i e.g. { III , XXX , − ZZI , − IZZ , 1 p 2( | 010 i + | 101 i ) ZIZ , YYX , − YXY , XYY }   X X X + Z Z I −   Z I Z +

  35. Stabilizer Formalism standard stabilizer { P | P : Pauli matrix | ψ i = α 0 ··· 00 | 0 · · · 00 i s.t. P | ψ i = | ψ i } + · · · + α 1 ··· 11 | 1 · · · 11 i e.g. { III , XXX , − ZZI , − IZZ , 1 p 2( | 010 i + | 101 i ) ZIZ , YYX , − YXY , XYY }   X X X + X | ψ a i h ψ a | ⌦ ρ a ρ = Z Z I −   a Z I Z +

  36. Stabilizer Formalism standard stabilizer { P | P : Pauli matrix | ψ i = α 0 ··· 00 | 0 · · · 00 i s.t. P | ψ i = | ψ i } + · · · + α 1 ··· 11 | 1 · · · 11 i e.g. { III , XXX , − ZZI , − IZZ , 1 p 2( | 010 i + | 101 i ) ZIZ , YYX , − YXY , XYY }   h GHZ | σ 0 ρ = p 0 | GHZ i X X X + Z Z I h GHZ | (IIX) σ 1 + p 1 (IIX) | GHZ i −   Z I Z + + · · ·

  37. Our Abstract Domain C Q Partition with stabilizers

  38. Our Abstract Domain C Q Partition with stabilizers q0 q1 q2

  39. Our Abstract Domain C Q Partition with stabilizers q0 q1 q2  X � [Z] X Z Z

  40. Our Abstract Domain C Q Partition with stabilizers q0 q1 q2  X � X ⌅ Z Z

  41. Our Abstract Domain C Q Partition with stabilizers q0 q1 q2  X � X ⌅ Z Z may be a non-stabilizer state

  42. Our Abstract Domain C Q Partition with stabilizers q0 q1 q2  X � X ⌅ Z Z

  43. Our Abstract Domain C Q Partition with stabilizers q0 q1 q2 ⌅ ⌅

  44. Our Abstract Domain C Q Partition with stabilizers q0 q1 q2 ⌅ ⌅ J C K C : C Q → C Q

  45. Abstract Semantics J · K C q0 := | 1 i ; q1 := | 0 i ; | 100 i q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; CX q2, q0; [Z] [Z] [Z] CX q0, q1; CX q1, q0; CX q1, q2;

  46. Abstract Semantics J · K C q0 := | 1 i ; q1 := | 0 i ; | 10+ i q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; CX q2, q0; [Z] [Z] [X] CX q0, q1; CX q1, q0; CX q1, q2;

  47. Abstract Semantics J · K C q0 := | 1 i ; q1 := | 0 i ; | 10+ i q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; CX q2, q0; [Z] [Z] [X] CX q0, q1; CX q1, q0; CX q1, q2;

  48. Abstract Semantics J · K C q0 := | 1 i ; q1 := | 0 i ; | � 0+ i q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; CX q2, q0; [X] [Z] [X] CX q0, q1; CX q1, q0; CX q1, q2;

  49. Abstract Semantics J · K C q0 := | 1 i ; q1 := | 0 i ; 1 2 | � 0 i ( | 0 i + e i π 4 | 1 i ) p q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; CX q2, q0; [X] [Z] ⌅ CX q0, q1; CX q1, q0; CX q1, q2;

  50. Abstract Semantics J · K C q0 := | 1 i ; q1 := | 0 i ; 1 2 | � 0 i ( | 0 i + e i π 4 | 1 i ) p q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; CX q2, q0; [X] [Z] ⌅ CX q0, q1; CX q1, q0; CX q1, q2;

  51. Abstract Semantics J · K C q0 := | 1 i ; q1 := | 0 i ; 1 2 | � 0 i ( | 0 i � e i π 4 | 1 i ) p q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; CX q2, q0; [X] [Z] ⌅ CX q0, q1; CX q1, q0; CX q1, q2;

  52. Abstract Semantics J · K C q0 := | 1 i ; q1 := | 0 i ; 1 2( | 00 i � | 11 i )( | 0 i � e i π 4 | 1 i ) q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2;  X CX q2, q0; � X ⌅ CX q0, q1; Z Z CX q1, q0; CX q1, q2;

  53. Abstract Semantics J · K C q0 := | 1 i ; q1 := | 0 i ; 1 2 | 0 �i ( | 0 i � e i π 4 | 1 i ) p q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2;  I CX q2, q0; � X ⌅ CX q0, q1; Z I CX q1, q0; CX q1, q2;

  54. Abstract Semantics J · K C q0 := | 1 i ; q1 := | 0 i ; 1 2 | 0 �i ( | 0 i � e i π 4 | 1 i ) p q2 := | 0 i ; H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; CX q2, q0; [Z] [X] ⌅ CX q0, q1; CX q1, q0; CX q1, q2;

  55. Abstract Semantics J · K C q0 := | 1 i ; 1 q1 := | 0 i ; 2 | 0 i ( | 00 i � e i π 4 | 01 i q2 := | 0 i ; + e i π 4 | 10 i � | 11 i ) H q2; T q1; H q0; q0 q1 q2 T q2; CX q1, q2; CX q2, q0; [Z] ⌅ CX q0, q1; CX q1, q0; CX q1, q2;

  56. J · K C Abstract Semantics q0 := | + i ; | +00 i q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; q0 q1 q2 CX q1, q2; meas q0; [X] [Z] [Z]

  57. J · K C Abstract Semantics 1 q0 := | + i ; p 2( | 00 i + | 11 i ) | 0 i q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; q0 q1 q2 CX q1, q2; meas q0;  X � [Z] X Z Z

  58. J · K C Abstract Semantics 1 q0 := | + i ; p 2( | 000 i + | 111 i ) q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; q0 q1 q2 CX q1, q2; meas q0;   X X X Z Z I   I Z Z

  59. J · K C Abstract Semantics 1 q0 := | + i ; 2( | 000 i h 000 | + | 111 i h 111 | ) q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; q0 q1 q2 CX q1, q2; meas q0;   Z I I Z Z I   I Z Z

  60. J · K C Abstract Semantics 1 q0 := | + i ; 2( | 000 i h 000 | + | 111 i h 111 | ) q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; q0 q1 q2 CX q1, q2; meas q0;   Z I I I Z I   I I Z

  61. J · K C Abstract Semantics 1 q0 := | + i ; 2( | 000 i h 000 | + | 111 i h 111 | ) q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; q0 q1 q2 CX q1, q2; meas q0; [Z] [Z] [Z]

  62. Soundness ∀ ρ : quantum state ∀ α ∈ C Q ∀ C : program α ✏ ρ ⇒ J C K C ( α ) ✏ J C K ( ρ )

  63. For better approximation

  64. Abstract Semantics J · K C 1 q0 := | + i ; p 2( | 000 i + | 111 i ) q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; CX q1, q2; q0 q1 q2 T q1; meas q0;   X X X Z Z I   I Z Z

  65. Abstract Semantics J · K C 1 q0 := | + i ; 2( | 000 i + e i π 4 | 111 i ) p q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; CX q1, q2; q0 q1 q2 T q1; meas q0; ⌅

  66. Abstract Semantics J · K C 1 q0 := | + i ; 2( | 000 i h 000 | + | 111 i h 111 | ) q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; CX q1, q2; q0 q1 q2 T q1; meas q0; ⌅ [Z]

  67. 0 J · K Abstract Semantics C 1 q0 := | + i ; p 2( | 000 i + | 111 i ) q1 := | 0 i ; q2 := | 0 i ; CX q0, q1; CX q1, q2; q0 q1 q2 T q1; meas q0;   X X X Z Z I   I Z Z

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