Registers of n Qubits

A quantum register is a group of qubits treated as a single unit — the quantum analogue of a classical integer register. Understanding how to allocate registers and address individual qubits is the first step toward writing multi-qubit programs.

Allocating a register

Calling circuit(n) creates a fresh $n$ -qubit register, with every qubit initialised to $|0\rangle$ . The full starting state is the tensor product

$|\psi_0\rangle = \underbrace{|0\rangle \otimes |0\rangle \otimes \cdots \otimes |0\rangle}_{n} = |0\cdots 0\rangle.$

For three qubits the state vector has $2^3 = 8$ amplitudes, one for each basis state $|000\rangle, |001\rangle, \ldots, |111\rangle$ . Only the amplitude of $|000\rangle$ is non-zero at the start.

Qubit indexing

Qubits are indexed $0, 1, \ldots, n-1$ . In this platform qubit 0 is the most-significant bit (the leftmost position in the ket label). So for a 3-qubit register:

| Qubit | Position in ket | Bit value | |-------|-----------------|-----------| | 0 | leftmost | $2^2 = 4$ | | 1 | middle | $2^1 = 2$ | | 2 | rightmost | $2^0 = 1$ |

Applying $X$ to qubit 2 flips the least-significant bit, so $|000\rangle \to |001\rangle$ . Applying $X$ to qubit 0 instead would give $|100\rangle$ (decimal 4).

State vector after a single flip

After c.x(2) the register is in $|001\rangle$ . In the length-8 state vector only amplitude index 1 is non-zero (decimal value of the bit string $001$ ):

$|\psi\rangle = |001\rangle, \qquad \text{amplitude vector } = (0,\,1,\,0,\,0,\,0,\,0,\,0,\,0)^T.$

More generally, $c.x(k)$ flips the bit at position $k$ and shifts the non-zero amplitude to the corresponding new basis state.

Try it

Allocate a 3-qubit register and flip qubit 2. After running, inspect the state vector to confirm the amplitude is on $|001\rangle$ (index 1).

Run your code to see the quantum state.

Look at the simulator output: the single non-zero amplitude sits at index 1, confirming the state is $|001\rangle$ . Changing the x argument to 0 or 1 would move that amplitude to index 4 ( $|100\rangle$ ) or index 2 ( $|010\rangle$ ) respectively.