Multi Qubit Circuits - RPIQuantumComputing/QuantumCircuits GitHub Wiki

Multi Qubit Circuits

Multi-qubit circuits: A multi-qubit circuit is a circuit with more than one qubit. Each qubit can have any number of gates on it

Types of Gates:

• Single qubit gates: These are all the gates we are familiar with - X, Z, H, Rx, Rz. They all act on a single qubit at a time. However, any 2x2 Unitary Matrix is a valid single qubit gate composable with this gate set, i.e. the gate set given is computationally complete.
• Control gates: These gates act on multiple qubits and only apply their operation if their control qubits are in the one state. Note, controls can be in a superposition leading to a situation where the gate is both applied and not applied, pending on which basis state one wishes to consider.

Controlled Gates: The control qubit is not affected by the gate - its state remains unchanged. The target qubit is dependent on the control qubit. If the control qubit is in state |0>, no gate is applied to the target qubit. If the control qubit is in state |1>, a gate is applied to the target qubit. The specific gate that gets applied to the target qubit determines the name of the gate. If the gate that gets applied is an X gate, we call it a controlled X (CX) gate. If it is a Z gate, we call it a controlled Z (CZ) gate.

Representing Multi Qubits

Kets for multi-qubit circuits: We can express multi-qubit states using the ket notation by combining the kets for the single qubits together. Ex… a qubit in |1 > and another in |0> would be |10>

Vector notation for multi-qubit states: For a single qubit, the vector had two numbers, corresponding to the contributions of |0> and |1> to the qubit’s state. For 2 qubits, the vector will have 4 numbers, corresponding to the contributions of |00>, |01>, |10>, and |11>.

Finding final state of multi-qubit circuits:

• Find the ket representation of the final state by applying the gates in the circuit. Here, we follow the usual rules for applying gates to kets that we are familiar with.
• Fill in the vector based on the ket.

Measurement on Multi Qubit States

Measurement on multi-qubit circuits: With two qubits, we have four possible results of measurement in the Z basis. The two qubits could be measured to be in the state |00>, |01>, |10>, or |11>.

Born rule - The probability of measuring one of these possible states (|00>, |01>, |10>, and |11>) is equal to the square of the contribution of that state to the overall state vector of the qubits being measured, officially found by taking an inner product of the state with the measurement result and squaring its magnitude.

Predicting the outcome of measurement in multi-qubit circuits: Find the final state of the circuit, using the rules of single-qubit and controlled gates that we have studied over the last few weeks Using the final state and the Born rule, predict the probability of measuring the circuit in each of the possible states.