Quantum Algorithms

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Dirac Notation and Spectral decomposition
Michele Mosca
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review 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
This is a scalar so I can move it to front
(I.e. projects a state to its component
Recall this projection operator also corresponds
to the density matrix for )
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More Dirac notation
More generally, we can also have operators like
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Example of this Dirac notation
For example, the one qubit NOT gate corresponds
to the operator
e.g.
(sum of matrices applied to ket vector)
This is one more notation to calculate state from
state and operator
The NOT gate is a 1-qubit unitary operation.
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Special unitaries Pauli Matrices in new notation
The NOT operation, is often called the X or sX
operation.
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Recall Special unitaries Pauli Matrices in new
representation
Representation of unitary operator
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What is ??
It helps to start with the spectral decomposition
theorem.
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Spectral decomposition

• Definition an operator (or matrix) M is normal
  if MMtMtM
• E.g. Unitary matrices U satisfy UUtUtUI
• E.g. Density matrices (since they satisfy ??t
  i.e. Hermitian) are also normal

Remember Unitary matrix operators and density
matrices are normal so can be decomposed
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Spectral decomposition Theorem

• Theorem For any normal matrix M, there is a
  unitary matrix P so that
• MP?Pt where ? is a diagonal matrix.
• The diagonal entries of ? are the eigenvalues.
• The columns of P encode the eigenvectors.

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Example Spectral decomposition of the NOT gate
This is the middle matrix in above decomposition
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Spectral decomposition matrix from column vectors
Column vectors
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Spectral decomposition eigenvalues on diagonal
Eigenvalues on the diagonal
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Spectral decomposition matrix as row vectors
Adjont matrix row vectors
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Spectral decomposition using row and column
vectors
From theorem
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Verifying eigenvectors and eigenvalues
Multiply on right by state vector Psi-2
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Verifying eigenvectors and eigenvalues
useful
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Why is spectral decomposition useful? Because we
can calculate f(A)
m-th power
Note that
recall
So
Consider
e.g.
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Why is spectral decomposition useful? Continue
last slide
M
f(? i)
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Now f(M) will be in matrix notation
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Same thing in matrix notation
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Same thing in matrix notation
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Important formula in matrix notation
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Von Neumann measurement in the computational
basis

• Suppose we have a universal set of quantum
  gates, and the ability to measure each qubit in
  the basis
• If we measure we get with probability

We knew it from beginning but now we can
generalize
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Using new notation this can be described like
this

• We have the projection operators
• and satisfying

• We consider the projection operator or
  observable
• Note that 0 and 1 are the eigenvalues

• When we measure this observable M, the
  probability of getting the eigenvalue b is
  and we are in that case left with the state

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Polar Decomposition
Left polar decomposition
Right polar decomposition
This is for square matrices
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Gram-Schmidt Orthogonalization
Hilbert Space
Orthogonality
Norm
Orthonormal basis