Skip to main contentIBM Quantum Documentation

Bit-ordering in Qiskit

If you have a set of nn bits (or qubits), you'll usually label each bit 0n10 \rightarrow n-1. Different softwares and resources must choose how they order these bits both in computer memory and when displayed on-screen.


Qiskit conventions

Here's how Qiskit® orders bits in different scenarios.

Quantum circuits

The QuantumCircuit class stores its qubits in a list (QuantumCircuit.qubits). The index of a qubit in this list defines the qubit's label.

from qiskit import QuantumCircuit
qc = QuantumCircuit(2)
qc.qubits[0]  # qubit "0"
Qubit(QuantumRegister(2, 'q'), 0)

Circuit diagrams

On a circuit diagram, qubit 00 is the topmost qubit, and qubit nn the bottommost qubit. You can change this with the reverse_bits argument of QuantumCircuit.draw (see Change ordering in Qiskit).

qc.x(1)
qc.draw()
q_0: ─────
     ┌───┐
q_1: ┤ X ├
     └───┘

Integers

When interpreting bits as a number, bit 00 is the least significant bit, and bit nn the most significant. This is helpful when coding because each bit has the value 2label2^\text{label} (label being the qubit's index in QuantumCircuit.qubits). For example, the following circuit execution ends with bit 00 being 0, and bit 11 being 1. This is interpreted as the decimal integer 2 (measured with probability 1.0).

from qiskit.primitives import Sampler
qc.measure_all()
Sampler().run(qc).result().quasi_dists[0]
{2: 1.0}

Strings

When displaying or interpreting a list of bits (or qubits) as a string, bit nn is the leftmost bit, and bit 00 is the rightmost bit. This is because we usually write numbers with the most significant digit on the left, and in Qiskit, bit nn is interpreted as the most significant bit.

For example, the following cell defines a Statevector from a string of single-qubit states. In this case, qubit 00 is in state +|+\rangle, and qubit 11 in state 0|0\rangle.

from qiskit.quantum_info import Statevector
sv = Statevector.from_label("0+")
sv.probabilities_dict()
{'00': 0.4999999999999999, '01': 0.4999999999999999}

This occasionally causes confusion when interpreting a string of bits, as you might expect the leftmost bit to be bit 00, whereas it usually represents bit nn.

Statevector matrices

When representing a statevector as a list of complex numbers (amplitudes), Qiskit orders these amplitudes such that the amplitude at index xx represents the computational basis state x|x\rangle.

print(sv[1])  # amplitude of state |01>
print(sv[2])  # amplitude of state |10>
(0.7071067811865475+0j)
0j

Gates

Each gate in Qiskit can interpret a list of qubits in its own way, but controlled gates usually follow the convention (control, target).

For example, the following cell adds a controlled-X gate where qubit 00 is the control and qubit 11 is the target.

from qiskit import QuantumCircuit
qc = QuantumCircuit(2)
qc.cx(0, 1)
qc.draw()
q_0: ──■──
     ┌─┴─┐
q_1: ┤ X ├
     └───┘

Following all the previously mentioned conventions in Qiskit, this CX-gate performs the transformation 0111|01\rangle \leftrightarrow |11\rangle, so has the following matrix.

(1000000100100100)\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ \end{pmatrix}

Change ordering in Qiskit

To draw a circuit with qubits in reversed order (that is, qubit 00 at the bottom), use the reverse_bits argument. This only affects the generated diagram and does not affect the circuit; the X-gate still acts on qubit 00.

from qiskit import QuantumCircuit
qc = QuantumCircuit(2)
qc.x(0)
qc.draw(reverse_bits=True)
q_1: ─────
     ┌───┐
q_0: ┤ X ├
     └───┘

You can use the reverse_bits method to return a new circuit with the qubits' labels reversed (this does not mutate the original circuit).

qc.reverse_bits().draw()
q_0: ─────
     ┌───┐
q_1: ┤ X ├
     └───┘

Note that in this new circuit, the X-gate acts on qubit 11.


Next steps

Recommendations
Was this page helpful?