Multi-qubit systems and operations

Two Qubit Systems and Operations

The joint states of two qubit can represent binary numbers, for example, $\vert\psi\rangle \bigotimes\vert\phi\rangle$, where $\bigotimes$ is the tensor product, denotes the joint state of two atoms. If we have \(\vert\psi\rangle = a\vert 0 \rangle + b \vert 1\rangle\) and \(\vert\phi\rangle = ac\vert 0 \rangle + d \vert 1\rangle\), Then

\[\vert\psi\rangle\bigotimes\vert\phi\rangle = ac\vert 00\rangle + ad \vert 01\rangle + bc\vert 10\rangle + bd \vert 11\rangle\]

The tensor product of atoms can be represented by:

\[\left[\begin{matrix} a \\ b \end{matrix}\right]\bigotimes\left[\begin{matrix} c \\ d \end{matrix}\right]=\left[\begin{matrix} ac \\ ad \\ bc \\ bd \end{matrix}\right]\rightarrow \left[\begin{matrix} 00 \\ 01 \\ 10 \\ 11 \end{matrix}\right]\]

Operators can be tensor producted as well:

\[\left[\begin{matrix} a & {\color{red} b} \\ {\color{blue} c} & {\color{green}d} \end{matrix}\right] \bigotimes\left[\begin{matrix} m & n \\ p & q \end{matrix}\right] = \left[\begin{matrix} am & an & \color{red}{bm} & \color{red}{bn} \\ ap & aq & \color{red}{bp} & \color{red}{bq} \\ \color{blue}{cm} & \color{blue}{cn} & \color{green}{dm} & \color{green}{dn} \\ \color{blue}{cp} & \color{blue}{cq} & \color{green}{dp} & \color{green}{dq} \end{matrix}\right]\]

Single qubit gates can be applied on multi-qubit systems:

Two qubit logic gates:

• CNOT: Controlled-Not, flip the target qubit iff the control qubit is 1. example: before the CNOT gate, sate is:

  \[\vert\psi\rangle = (\alpha\vert 0\rangle+\beta\vert 1\rangle)\bigotimes\vert 0\rangle = \alpha\vert 00 \rangle + \beta\vert 10 \rangle\]

  Then after CNOT:

  \[\text{CNOT}\vert\psi\rangle = \vert\psi'\rangle = \alpha\vert 00\rangle + \beta\vert 11\rangle\]

• CZ: Control Phase Gate: the phase of the target flipped iff the control qubit is 1. Since CZ matrix is diagonal, so control/target is interchangeable, in other words, only flip phase when they are all 11.

• SWAP: Swap gate: The two qubit states are swapped.

• Toffoli: Toffoli gate, CNOT gate with two control qubits.

  In quantum computing, every quantum circuit can be expressed as a sequence of CNOT gate and a Single-qubit Rotation, Or sequence of CNOT + Hadamard + T gate.
  Any Two qubit gate can be decomposed into just 3 CNOTs and single qubit rotation.

  Tofolli -> 3 CNOTs: CNOT c1->c2, X c2, CNOT c2->c3

Two qubit measurement: we can measure a single qubit by applying projection operators onto the two qubit system, for example:

To measure a two-qubit state, first apply projector operator into the measured state, then renormalize the state by \(\vert\psi'\rangle=\frac{P_m\vert\psi\rangle}{\sqrt{p(m)}}\)

Entanglement

Quantum states may be separable, \(\vert\Phi\rangle = \vert\psi\rangle\bigotimes\vert\phi\rangle\), and further decomposed by: \(\vert\psi\rangle = ac\vert 00\rangle + ad\vert 01\rangle + bc \vert 10 \rangle + bd \vert 11\rangle\).

Two zero qubits after a Hardamard gate and a CNOT gate, is not separable, and is called a Bell State.

This state, a state that is not separable is called an entangled state.

The entanglement measure: How much entanglement a state has, the Entropy of Entanglement.

\[S=-\sum_i p_i\log p_i\]

Entanglement is a type of correlation between two systems, say A and B. To see how much correlation there is between A and B: We will measure B, throw away the result, and ask how many bits information of information do we need to determine the state of A.

Dense coding

Teleportation

communication between quantum computer.

After the Alice’s measurement, There are four outcome with equal probability: 00, 01, 10, 11. And Bob’s state will collapse into one of the four sates listed on the right. Once Alice know her mesurement result, she can tell Bob what's her result and what corrections should Bob take to reconstruct the Alice's sending state.. This process is a teleport.