The Plus/Minus Basis and Basis Changes

This chapter ties together everything you’ve learned about single-qubit gates and introduces the powerful concept of basis changes—a mental model that underlies many quantum algorithms.

The Four Cardinal Bases

Every single-qubit state can be expressed in any of these bases:

Understanding how to switch between bases is crucial for quantum algorithms.

The Computational Basis (Z Basis)

The default basis:

• Eigenstates of the Z operator
• \(|0\rangle\) and \(|1\rangle\)
• Created by: \(I\) (for \(|0\rangle\)), \(X\) (for \(|1\rangle\))

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

# |0⟩ in Z basis
qc = QuantumCircuit(1)
print("|0⟩:", Statevector(qc))

# |1⟩ in Z basis
qc.x(0)
print("|1⟩:", Statevector(qc))

The Plus/Minus Basis (X Basis)

Eigenstates of the X operator:

• \(|+\rangle = \frac{|0\rangle+|1\rangle}{\sqrt{2}}\)
• \(|-\rangle = \frac{|0\rangle-|1\rangle}{\sqrt{2}}\)
• Created by: \(H\) (for \(|+\rangle\)), \(XH\) or \(HX\) (for \(|-\rangle\))

# |+⟩ in X basis
qc = QuantumCircuit(1)
qc.h(0)
print("|+⟩:", Statevector(qc))

# |−⟩ in X basis
qc.x(0)
qc.h(0)
print("|−⟩:", Statevector(qc))

The Key Identity: HZH = X

This identity reveals the connection between bases:

\[HZH = X\]

What does this mean?

• \(H\) changes from Z basis to X basis
• \(Z\) acts (phase flip)
• \(H\) changes back from X basis to Z basis
• Net effect: bit flip!

from qiskit import QuantumCircuit
from qiskit.quantum_info import Operator

# Build HZH
qc_hzh = QuantumCircuit(1)
qc_hzh.h(0)
qc_hzh.z(0)
qc_hzh.h(0)

# Build X
qc_x = QuantumCircuit(1)
qc_x.x(0)

# Are they equal?
print("HZH == X:", Operator(qc_hzh) == Operator(qc_x))

General Basis Change Formula

In general, to transform an operation from one basis to another:

\[B \cdot \text{Operation} \cdot B^\dagger = \text{New Operation}\]

Where \(B\) is the basis change operation.

For the X-Z connection: \(HZH = X\)

Basis Changes in Practice

Problem: Apply X in the X basis

In the X basis, the eigenstates are \(|+\rangle\) and \(|-\rangle\). But we want to “flip” \(|+\rangle\) to \(|-\rangle\).

Solution: Change to Z basis, flip, change back.

# Flip |+⟩ to |−⟩
qc = QuantumCircuit(1)

# Start with |+⟩
qc.h(0)

# Change to Z basis, apply X, change back
qc.h(0)  # |+⟩ → |0⟩
qc.x(0)  # |0⟩ → |1⟩
qc.h(0)  # |1⟩ → |−⟩

print(Statevector(qc))

This is equivalent to applying Z in the original basis!

The Complete Picture: All Pauli Conjugates

The Hadamard gate transforms all Pauli gates:

from qiskit.quantum_info import Operator

# Verify all three
for op, name in [('x', 'X'), ('y', 'Y'), ('z', 'Z')]:
    qc = QuantumCircuit(1)
    qc.h(0)
    getattr(qc, op)(0)
    qc.h(0)
    
    x_qc = QuantumCircuit(1)
    if name == 'Y':
        x_qc.y(0)
        x_qc.x(0)
    else:
        x_qc.x(0)
    
    # We expect X → Z, Z → X, Y → Y
    print(f"H {name} H =", Operator(qc).simplify())

Why Basis Changes Matter

Basis changes are central to many quantum algorithms:

1. Phase kickback: Use basis changes to move phase information
2. Measurement basis: Measure in different bases to extract different information
3. Algorithm design: QFT, Grover, and others use basis changes extensively

Measuring in Different Bases

You can measure in any basis by changing to that basis first:

from qiskit import QuantumCircuit
from qiskit.primitives import StatevectorSampler

# Measure |+⟩ in Z basis: 50-50
qc = QuantumCircuit(1)
qc.h(0)  # Create |+⟩
qc.measure_all()

sampler = StatevectorSampler()
result = sampler.run([qc], shots=100).result()
print("|+⟩ in Z basis:", result[0].data.meas.get_counts())

# Measure |+⟩ in X basis: always 0 (because |+⟩ is |0⟩ in X basis)
qc2 = QuantumCircuit(1)
qc2.h(0)  # Create |+⟩
qc2.h(0)  # Change to X basis for measurement
qc2.measure_all()

result2 = sampler.run([qc2], shots=100).result()
print("|+⟩ in X basis:", result2[0].data.meas.get_counts())

Summary of Single-Qubit Mastery

You now know all the single-qubit tools:

Checkpoint Exercises

Exercise 1

Create \(|-\rangle\) starting from \(|0\rangle\) using only \(H\) and \(Z\).

Exercise 2

Verify that \(HZH = X\) on both \(|0\rangle\) and \(|1\rangle\).

Exercise 3

Measure \(|-\rangle\) in the Z basis. What do you expect? Verify with code.

Exercise 4

Design a circuit that flips \(|+\rangle\) to \(|-\rangle\).

Exercise 5

Explain in your own words why \(HZH = X\).

Summary

Key concepts:

• Four main bases: Z (computational), X (plus/minus), Y (circular)
• \(H\) connects Z and X bases
• \(HZH = X\) shows the relationship between operations in different bases
• You can measure in any basis by changing bases first
• Basis changes are fundamental to quantum algorithms

With single-qubit gates mastered, you’re ready for the multi-qubit world.

Try These On QCoder

• QPC001 A3: Generate Minus state
• QPC002 B1: Generate State e^(i theta)|0>
• QPC003 B3: Generate State tensor_i (cos T_i |0> + sin T_i |1>)

Next: Controlled Operations: CNOT