Clifford group from scratch Maris Ozols University of Waterloo - PowerPoint PPT Presentation

Introduction Definition Single qubit case Order Generators Applications References Clifford group from scratch Maris Ozols University of Waterloo July 28, 2008

Introduction Definition Single qubit case Order Generators Applications References Outline 1 Introduction 2 Definition of the Clifford group C n on n qubits 3 Clifford group C 1 of a single qubit 4 Number of elements in C n 5 Generators of C n 6 Applications

Introduction Definition Single qubit case Order Generators Applications References Motivation Everybody knows what the Clifford group is

Introduction Definition Single qubit case Order Generators Applications References Motivation Everybody knows what the Clifford group is, only Maris doesn’t know. . .

Introduction Definition Single qubit case Order Generators Applications References Motivation Everybody knows what the Clifford group is, only Maris doesn’t know. . . I’m obsessed with symmetric structures in the Hilbert space

Introduction Definition Single qubit case Order Generators Applications References Motivation Everybody knows what the Clifford group is, only Maris doesn’t know. . . I’m obsessed with symmetric structures in the Hilbert space Clifford group has lots of applications

Introduction Definition Single qubit case Order Generators Applications References Motivation Everybody knows what the Clifford group is, only Maris doesn’t know. . . I’m obsessed with symmetric structures in the Hilbert space Clifford group has lots of applications I know the results, but I haven’t seen the proofs

Introduction Definition Single qubit case Order Generators Applications References Motivation Everybody knows what the Clifford group is, only Maris doesn’t know. . . I’m obsessed with symmetric structures in the Hilbert space Clifford group has lots of applications I know the results, but I haven’t seen the proofs Some folklore results with no proofs available

Introduction Definition Single qubit case Order Generators Applications References Pauli matrices Single qubit The set of Pauli matrices is P = { I, X, Y, Z } , where � 1 0 � 0 1 � 0 − i � 1 0 � � � � I = , X = , Y = , Z = . 0 − 1 0 1 1 0 i 0

Introduction Definition Single qubit case Order Generators Applications References Pauli matrices Single qubit The set of Pauli matrices is P = { I, X, Y, Z } , where � 1 0 � 0 1 � 0 − i � 1 0 � � � � I = , X = , Y = , Z = . 0 − 1 0 1 1 0 i 0 For n qubits P n = { σ 1 ⊗ σ 2 ⊗ · · · ⊗ σ n | σ i ∈ P } .

Introduction Definition Single qubit case Order Generators Applications References Pauli matrices Single qubit The set of Pauli matrices is P = { I, X, Y, Z } , where � 1 0 � 0 1 � 0 − i � 1 0 � � � � I = , X = , Y = , Z = . 0 − 1 0 1 1 0 i 0 For n qubits P n = { σ 1 ⊗ σ 2 ⊗ · · · ⊗ σ n | σ i ∈ P } . Vector space structure The group P n /U (1) is isomorphic to a vector space over F 2 with dimension 2 n via identification Z Y (0 , 1) (1 , 1) | | | | ⇐ ⇒ I X (0 , 0) (1 , 0) multiply add

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation.

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 .

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } .

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } . All matrices in P ∗ n have eigenvalues ± 1 with equal multiplicity.

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } . All matrices in ± P ∗ n have eigenvalues ± 1 with equal multiplicity.

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } . All matrices in ± P ∗ n have eigenvalues ± 1 with equal multiplicity. You can

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } . All matrices in ± P ∗ n have eigenvalues ± 1 with equal multiplicity. You can X �→ − X ,

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } . All matrices in ± P ∗ n have eigenvalues ± 1 with equal multiplicity. You can X �→ − X , e.g., ZXZ = − X ,

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } . All matrices in ± P ∗ n have eigenvalues ± 1 with equal multiplicity. You can X �→ − X , e.g., ZXZ = − X , X ⊗ I �→ X ⊗ X ,

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } . All matrices in ± P ∗ n have eigenvalues ± 1 with equal multiplicity. You can X �→ − X , e.g., ZXZ = − X , X ⊗ I �→ X ⊗ X , e.g., CNOT ( X ⊗ I ) CNOT † = X ⊗ X .

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } . All matrices in ± P ∗ n have eigenvalues ± 1 with equal multiplicity. You can You cannot X �→ − X , e.g., ZXZ = − X , X ⊗ I �→ X ⊗ X , e.g., CNOT ( X ⊗ I ) CNOT † = X ⊗ X .

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } . All matrices in ± P ∗ n have eigenvalues ± 1 with equal multiplicity. You can You cannot X �→ − X , e.g., ZXZ = − X , X �→ I , X ⊗ I �→ X ⊗ X , e.g., CNOT ( X ⊗ I ) CNOT † = X ⊗ X .

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } . All matrices in ± P ∗ n have eigenvalues ± 1 with equal multiplicity. You can You cannot X �→ − X , e.g., ZXZ = − X , X �→ I , X ⊗ I �→ X ⊗ X , e.g., X �→ iX . CNOT ( X ⊗ I ) CNOT † = X ⊗ X .

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take Paulis to Paulis via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } . All matrices in ± P ∗ n have eigenvalues ± 1 with equal multiplicity. You can You cannot X �→ − X , e.g., ZXZ = − X , X �→ I , X ⊗ I �→ X ⊗ X , e.g., X �→ iX . CNOT ( X ⊗ I ) CNOT † = X ⊗ X .

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take ± P ∗ n to ± P ∗ n via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } . All matrices in ± P ∗ n have eigenvalues ± 1 with equal multiplicity. You can You cannot X �→ − X , e.g., ZXZ = − X , X �→ I , X ⊗ I �→ X ⊗ X , e.g., X �→ iX . CNOT ( X ⊗ I ) CNOT † = X ⊗ X .

Introduction Definition Single qubit case Order Generators Applications References Clifford group Definition (sloppy) Unitaries that take ± P ∗ n to ± P ∗ n via conjugation. Eigenvalues The eigenvalues of X , Y , Z are ± 1 . Let P ∗ n = P n \ { I ⊗ n } . All matrices in ± P ∗ n have eigenvalues ± 1 with equal multiplicity. You can You cannot X �→ − X , e.g., ZXZ = − X , X �→ I , X ⊗ I �→ X ⊗ X , e.g., X �→ iX . CNOT ( X ⊗ I ) CNOT † = X ⊗ X . Global phase U and e iϕ U act identically, i.e., UMU † = ( e iϕ U ) M ( e iϕ U ) † .