Classical Coding Problem from Transversal T Gates Narayanan - PowerPoint PPT Presentation

Classical Coding Problem from Transversal T Gates Narayanan Rengaswamy Rhodes Information Initiative at Duke (iiD), Duke University Joint Work: Michael Newman, Robert Calderbank and Henry Pfister 2020 International Symposium on Information Theory (ISIT ’20) arXiv:2001.04887, 1910.09333, 1902.04022, 1907.00310 June 21-26, 2020 Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 1 / 14

Overview Motivation and Related Work 1 Essential Algebraic Setup 2 Quadratic Form Diagonal (QFD) Gates 3 Stabilizer Codes Matched to QFD Gates 4 Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 1 / 14

Goal: Logical Operations from Physical Gates Information | x � L Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 1 / 14

Goal: Logical Operations from Physical Gates logical operation Information | x � L | ˜ x � L Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 1 / 14

Goal: Logical Operations from Physical Gates QECC: Quantum Error Correcting Code logical operation Information | x � L | ˜ x � L [ [ n , k , d ] ] QECC encode | ψ x � Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 1 / 14

Goal: Logical Operations from Physical Gates QECC: Quantum Error Correcting Code logical operation Information | x � L | ˜ x � L [ [ n , k , d ] ] QECC encode relevant physical operation | ψ x � | ψ ˜ x � Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 1 / 14

Goal: Logical Operations from Physical Gates QECC: Quantum Error Correcting Code logical operation Information | x � L | ˜ x � L [ [ n , k , d ] ] [ [ n , k , d ] ] QECC QECC encode decode relevant physical operation | ψ x � | ψ ˜ x � Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 1 / 14

Goal: Logical Operations from Physical Gates QECC: Quantum Error Correcting Code logical operation Information | x � L | ˜ x � L Need to translate [ [ n , k , d ] ] [ [ n , k , d ] ] for the QECC QECC [ [ n , k , d ] ] encode decode QECC relevant physical operation | ψ x � | ψ ˜ x � Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 1 / 14

Goal: Logical Operations from Physical Gates QECC: Quantum Error Correcting Code logical operation Information | x � L | ˜ x � L Need to translate [ [ n , k , d ] ] [ [ n , k , d ] ] for the QECC QECC [ [ n , k , d ] ] encode decode QECC relevant physical operation | ψ x � | ψ ˜ x � What QECC structure is required so that the physical application of certain gates preserves the code subspace? Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 1 / 14

Line of Thought What QECC structure is required so that the physical application of certain gates preserves the code subspace? Key Idea Pauli operators form an orthonormal basis for all operators! Understand action of those certain gates on Pauli operators Use the action to study effect on QECC subspaces Finally, restrict to gates that are reliable in the lab Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 2 / 14

Literature related to Magic State Distillation (MSD) [GC99]: Universal computation via quantum teleportation [BK05]: Ideal Clifford gates and noisy ancillas – MSD [BH12]: Distillation with low overhead, triorthogonal codes [KB15]: Transversal gates on color codes [CH17]: Quasitransversality [HH17]: Generalized triorthogonality [Haa+17]: Distillation with optimal asymptotic input count [KT18]: Punctured polar codes from decreasing monomial codes [VB19]: Quantum Pin Codes . . . (see arXiv:1910.09333 for explicit connections) Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 3 / 14

Main Distinction of Our Work Prior works (“Schr¨ odinger Perspective”): Focus on Calderbank-Shor-Steane (CSS) type stabilizer codes Examine action of the (physical) gates on the basis quantum states in the CSS code subspace Our strategy (“Heisenberg Perspective”): Work with arbitrary stabilizer codes; results can be specialized to CSS Examine action of the (physical) gates on the Pauli operators defining the code subspace Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 4 / 14

In this talk . . . Motivation and Related Work 1 Essential Algebraic Setup 2 Quadratic Form Diagonal (QFD) Gates 3 Stabilizer Codes Matched to QFD Gates 4 Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 4 / 14

Overview Motivation and Related Work 1 Essential Algebraic Setup 2 Quadratic Form Diagonal (QFD) Gates 3 Stabilizer Codes Matched to QFD Gates 4 Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 4 / 14

Pauli Group, Clifford Group and Symplectic Matrices 2 , κ ∈ Z 4 } ( ı = √− 1) Heisenberg-Weyl Group HW N := { ı κ E ( a , b ): a , b ∈ F n a = 1 0 1 E ( a , b ) , a , b ∈ F n 2 : X ⊗ Z ⊗ Y = E ( 1 0 1 , 0 1 1 ) b = 0 1 1 ���� ���� � �� � a b n =3 qubits E ( a , b ) = X 1 Z 2 Y 3 � 0 � I n Symplectic Inner Product: � [ a , b ] , [ c , d ] � s := [ a , b ] Ω [ c , d ] T , Ω := 0 I n Clifford Group: All unitaries that map Paulis to Paulis under conjugation Symplectic Matrices: If g ∈ Cliff N (Cliffords on n = log 2 N qubits) then g E ( a , b ) g † = ± E ([ a , b ] F g ) , where F g Ω F T g = Ω F g ∈ F 2 n × 2 n is symplectic: preserves the symplectic inner product 2 Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 5 / 14

Pauli Group, Clifford Group and Symplectic Matrices 2 , κ ∈ Z 4 } ( ı = √− 1) Heisenberg-Weyl Group HW N := { ı κ E ( a , b ): a , b ∈ F n a = 1 0 1 E ( a , b ) , a , b ∈ F n 2 : X ⊗ Z ⊗ Y = E ( 1 0 1 , 0 1 1 ) b = 0 1 1 ���� ���� � �� � a b n =3 qubits E ( a , b ) = X 1 Z 2 Y 3 � 0 � I n Symplectic Inner Product: � [ a , b ] , [ c , d ] � s := [ a , b ] Ω [ c , d ] T , Ω := 0 I n Clifford Group: All unitaries that map Paulis to Paulis under conjugation Symplectic Matrices: If g ∈ Cliff N (Cliffords on n = log 2 N qubits) then g E ( a , b ) g † = ± E ([ a , b ] F g ) , where F g Ω F T g = Ω F g ∈ F 2 n × 2 n is symplectic: preserves the symplectic inner product 2 Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 5 / 14

Pauli Group, Clifford Group and Symplectic Matrices 2 , κ ∈ Z 4 } ( ı = √− 1) Heisenberg-Weyl Group HW N := { ı κ E ( a , b ): a , b ∈ F n a = 1 0 1 E ( a , b ) , a , b ∈ F n 2 : X ⊗ Z ⊗ Y = E ( 1 0 1 , 0 1 1 ) b = 0 1 1 ���� ���� � �� � a b n =3 qubits E ( a , b ) = X 1 Z 2 Y 3 � 0 � I n Symplectic Inner Product: � [ a , b ] , [ c , d ] � s := [ a , b ] Ω [ c , d ] T , Ω := 0 I n Clifford Group: All unitaries that map Paulis to Paulis under conjugation Symplectic Matrices: If g ∈ Cliff N (Cliffords on n = log 2 N qubits) then g E ( a , b ) g † = ± E ([ a , b ] F g ) , where F g Ω F T g = Ω F g ∈ F 2 n × 2 n is symplectic: preserves the symplectic inner product 2 Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 5 / 14

Pauli Group, Clifford Group and Symplectic Matrices 2 , κ ∈ Z 4 } ( ı = √− 1) Heisenberg-Weyl Group HW N := { ı κ E ( a , b ): a , b ∈ F n a = 1 0 1 E ( a , b ) , a , b ∈ F n 2 : X ⊗ Z ⊗ Y = E ( 1 0 1 , 0 1 1 ) b = 0 1 1 ���� ���� � �� � a b n =3 qubits E ( a , b ) = X 1 Z 2 Y 3 � 0 � I n Symplectic Inner Product: � [ a , b ] , [ c , d ] � s := [ a , b ] Ω [ c , d ] T , Ω := 0 I n Clifford Group: All unitaries that map Paulis to Paulis under conjugation Symplectic Matrices: If g ∈ Cliff N (Cliffords on n = log 2 N qubits) then g E ( a , b ) g † = ± E ([ a , b ] F g ) , where F g Ω F T g = Ω F g ∈ F 2 n × 2 n is symplectic: preserves the symplectic inner product 2 Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 5 / 14

Stabilizer Codes ( N = 2 n ) r -dimensional Stabilizer: Generated by r commuting Pauli operators: S = � ǫ i E ( a i , b i ) ; i = 1 , . . . , r � , ǫ i ∈ {± 1 } , − I N / ∈ S ] Stabilizer Code: The 2 k dimensional subspace, V ( S ), [ [ n , k = n − r , d ] jointly fixed by all elements of S � � | ψ � ∈ C N : g | ψ � = | ψ � for all g ∈ S V ( S ) := Example: ] CSS Code: S := � X ⊗ 6 = E ( a , 0) , Z ⊗ 6 = E (0 , a ) � , a := [ 1 1 1 1 1 1 ] [ [6 , 4 , 2] � 0 � 0 0 0 0 0 1 1 1 1 1 1 Generator Matrix: G S = 1 1 1 1 1 1 0 0 0 0 0 0 Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 6 / 14

Overview Motivation and Related Work 1 Essential Algebraic Setup 2 Quadratic Form Diagonal (QFD) Gates 3 Stabilizer Codes Matched to QFD Gates 4 Narayanan Rengaswamy (Duke) Classical Coding Problem from Transversal T ISIT 2020 6 / 14