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Basis States, Bitstrings, And Qubit Order

This chapter exists because many beginners lose hours to bit ordering.

The rule to remember

Qiskit uses little-endian ordering for basis states.

That means qubit 0 is the least significant bit in a computational-basis label.

For example, the label |10> means:

  • qubit 1 is 1
  • qubit 0 is 0

This feels backwards at first. You still need to internalize it.

A quick sanity check

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

qc = QuantumCircuit(2)
qc.x(1)
print(Statevector.from_instruction(qc))

This prepares |10>, not |01>.

Why this matters

All of these depend on qubit order:

  • reading statevectors
  • interpreting counts
  • preparing a target basis state
  • writing arithmetic circuits
  • solving QCoder problems with explicit target strings

When a circuit looks almost right, ordering is one of the first things to check.

State labels and measurement strings

The same bit-ordering convention shows up in measurement output.

If you do:

from qiskit import QuantumCircuit
from qiskit.primitives import StatevectorSampler

qc = QuantumCircuit(2)
qc.x(0)
qc.measure_all()

sampler = StatevectorSampler()
result = sampler.run([qc], shots=16).result()
print(result[0].data.meas.get_counts())

the counts show "01", because qubit 1 is the left bit and qubit 0 is the right bit.

A debugging habit that pays off

Whenever you target a basis state, write down both:

  • which qubits you are changing
  • which basis label you expect to see

For example, if you want |101>, say it out loud:

  • qubit 2 should be 1
  • qubit 1 should be 0
  • qubit 0 should be 1

That kind of explicitness prevents a large class of mistakes.

A small table worth memorizing

For two qubits:

  • |00> means q1=0, q0=0
  • |01> means q1=0, q0=1
  • |10> means q1=1, q0=0
  • |11> means q1=1, q0=1

If this table feels natural, later arithmetic and QFT chapters become much easier.

Checkpoint Exercises

  1. Prepare |10> on two qubits.
  2. Prepare |101> on three qubits.
  3. Explain why qc.x(0) on two qubits gives |01>.
  4. Create a small table for all 2-qubit basis states and the qubit values they represent.

Try These On QCoder