5. Unitary transformations and gates

5.1. The constraints on types of operations allowed in quantum physics.

• It is possible to perform operation on qubits.
• Constraints: not any arbitrary operations allowed. Some of them are physically impossible.

5.2. Unitary transformation (幺正变换)

• Definition: The operation allowed by q. physics is called Unitary transformation $U$

  $U:|\psi_{in}\rang\rightarrow|\psi_{out}\rang=U(|\psi_{in}\rang)$

• Necessary conditions (properties) for $U$ to be a valid quantum operation:

  • Linear property (for any $U$):

    $U(\alpha|\psi_1\rang+\beta|\psi_2\rang)=\alpha U(|\psi_1\rang)+\beta U(|\psi_2\rang)$

    • For d-dimensional qudit: $U$: matrix $dd$ $|\psi\rang$: vector 1d

  • Length preservation (For all possible state $|\phi_{in}\rang$):

    $\lang \phi_{out}|\phi_{out}\rang=\lang\phi_{in}|U^\dagger\cdot U|\phi_{in}\rang=\lang\phi_{in}|\phi_{in}\rang=1$

    • Length condition: $U^\dagger U=UU^\dagger=II$ —Identity operator matrix (单位矩阵/算符)
    • $U^{-1}=U^\dagger$
    • Ps: $U^\dagger=(U^*)^T$ → $(U|\phi\rang)^\dagger=\lang\phi|U^\dagger$
    • Ps: The identity matrix leaves all quantum states invariant: $\forall|\psi\rang, II|\psi\rang=|\psi\rang$

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Example:

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$H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}$ is the Hadamard gate (unitary matrix)

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Example:

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• Lemma of the unitary transformation:
  • Let $U$ be a linear map on $C^d$ represented by a $d\times d$ matrix.
  • $U$ is unitary if and only if the columns of U form an orthnonormal basis of $C^d$.
  • $U$ is unitary if and only if it sends the canonical basis $\{|e_0\rang, \ldots,|e_{d-1}\rang\}$ to $\{ |u_0\rang=U|e_0\rang,\ldots,|u_{d-1}\rang=U|e_{d-1}\rang\}$ such that $\{|u_0\rang,\ldots,|u_{d-1}\rang\}$ is also an orthonormal basis of $C^d$
  • Generally, $U$ is unitary if and only if it transforms any orthonormal basis of $C^d$ into an orthonormal basis.

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Proof:

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5.3. Pauli matrixs (unitary)

• $X=\begin{pmatrix} 0&1\\1&0\end{pmatrix}$ “Not gate” “bit-flip in standard basis” “phase flip in hadmard basis”
  • $X|0\rang=\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}0\\1\end{pmatrix}=|1\rang$
  • $X|1\rang=\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}0\\1\end{pmatrix}=\begin{pmatrix}1\\0\end{pmatrix}=|0\rang$