Matrix Operations

Introduction to Quantum Computing

Matrix Operations

• Inner Product - vector multiplication that produces a scalar
  • results from multiplying a row vector with a column vector                  
  • <a|b> = [a0 a1 ... an ] x [ b0] = a1b0 + a1b1 + ... + anbn
                                                [ b1 ]
  • its square denotes the probability
  • use the inner product to obtain the probability amplitude of measuring a particular value from a qubit state bn
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• Outer Product - vector multiplication that produces a matrix
  • results from multiplying a column vector multiplied by a row vector
  • |a><b| =
• Identity Matrix
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  • applying the I-gate results in unchanged vector
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• Not (X) Matrix
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  • applying the X-gate flips the probability amplitudes
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• Hadamard (H) Matrix 
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  • applying the Hadamard gate puts the basis state into superposition
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• Tensor Product
  • the tensor product is represented as |a> ⊗ |b>
  • two qubits yield four combinations of the basis states
  • these form a quantum system of their own, so they can be represented by a single equation - |ab> where ab is just a name
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  • used to calculate the amplitudes and the amplitude squared must be 1
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  • apply a H-gate and a X-gate
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