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Dirac Notation and Spectral decomposition Michele Mosca

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Dirac notation For any vector, we let denote, the complex conjugate of. We denote by the inner product between two vectors and defines a linear function that maps (I.e. … it maps any state to the coefficient of its component )

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More Dirac notation defines a linear operator that maps (Aside: this projection operator also corresponds to the “density matrix” for ) More generally, we can also have operators like (I.e. projects a state to its component)

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More Dirac notation For example, the one qubit NOT gate corresponds to the operator e.g. The NOT gate is a 1-qubit unitary operation.

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Special unitaries: Pauli Matrices The NOT operation, is often called the X or σ X operation.

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Special unitaries: Pauli Matrices

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What is ?? It helps to start with the spectral decomposition theorem.

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Spectral decomposition l Definition: an operator (or matrix) M is “normal” if MM t =M t M l E.g. Unitary matrices U satisfy UU t =U t U=I l E.g. Density matrices (since they satisfy = t ; i.e. “Hermitian”) are also normal

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Spectral decomposition l Theorem: For any normal matrix M, there is a unitary matrix P so that M=P P t where is a diagonal matrix. l The diagonal entries of are the eigenvalues. The columns of P encode the eigenvectors.

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e.g. NOT gate

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Spectral decomposition

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Verifying eigenvectors and eigenvalues

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Why is spectral decomposition useful? Note that So recall Consider e.g.

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Why is spectral decomposition useful?

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Same thing in matrix notation

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“Von Neumann measurement in the computational basis” l Suppose we have a universal set of quantum gates, and the ability to measure each qubit in the basis l If we measure we get with probability

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In section 2.2.5, this is described as follows l We have the projection operators and satisfying l We consider the projection operator or “observable” l Note that 0 and 1 are the eigenvalues l When we measure this observable M, the probability of getting the eigenvalue is and we are in that case left with the state

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