Inspecting the Statevector (Simulation Only)

Overview

This page demonstrates a powerful learning tool: inspecting the statevector.

It answers: What amplitudes did my circuit create, and how do they map to measurement probabilities?

Important: statevectors are available in simulation, not from real hardware.

Conceptual Mapping

In Foundations:

• a pure quantum state can be written as a vector of amplitudes over basis states
• probabilities come from amplitudes (Born rule)

In code:

• a simulator can keep the full state internally
• you can ask the simulator to return it

But on real hardware:

• you cannot “peek” at amplitudes directly
• the only standard access is through repeated measurements (shots)

This is why statevector inspection is a learning and debugging tool, not an experimental shortcut.

Code Walkthrough

We’ll prepare a one-qubit state and request the simulator’s statevector.

from qiskit import QuantumCircuit
from qiskit_aer import AerSimulator
 
qc = QuantumCircuit(1)
qc.h(0)
qc.save_statevector()
 
sim = AerSimulator(method="statevector")
result = sim.run(qc).result()
state = result.get_statevector()
 
print(state)

Line by line:

• QuantumCircuit(1) creates one qubit (no classical bits needed here).
• qc.h(0) prepares a superposition.
• qc.save_statevector() tells the simulator to record the statevector.
• AerSimulator(method="statevector") uses a simulation mode that tracks amplitudes.
• get_statevector() returns the final statevector.

Now connect this to measurement:

from qiskit import QuantumCircuit
from qiskit_aer import AerSimulator
 
qc = QuantumCircuit(1, 1)
qc.h(0)
qc.measure(0, 0)
 
sim = AerSimulator()
counts = sim.run(qc, shots=500).result().get_counts()
print(counts)

Same state preparation, but now we sample measurement outcomes.

Results & Interpretation

A statevector printout may look like “two complex numbers.” Conceptually:

• entry 0 corresponds to the amplitude of basis state |0⟩
• entry 1 corresponds to the amplitude of basis state |1⟩

Mapping amplitudes to probabilities:

• probability of 0 is the squared magnitude of the |0⟩ amplitude
• probability of 1 is the squared magnitude of the |1⟩ amplitude

Then counts approximate those probabilities with sampling noise.

Bloch sphere connection (conceptual):

• for one qubit, the statevector represents a point on the Bloch sphere (up to a global phase)
• gates like H rotate that point
• measurement in the computational basis samples outcomes based on where that point lies relative to the measurement axis

Why this is not possible on real hardware:

• hardware does not hand you its internal amplitudes
• measurement collapses the state
• you learn the state only indirectly through many measurements and careful experimental design

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

Now that you can connect amplitudes to probabilities, we’ll build a two-qubit Bell state and measure correlations. That’s where entanglement becomes visible in real measurement results.