Quantum State Space
Number of basis states an n-qubit register can be in superposition of simultaneously.
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Example
n = 50 qubits → 2⁵⁰ ≈ 10¹⁵ states at once
How it works
An n-qubit register has 2 possible basis states for each qubit, because each qubit can be measured as 0 or 1. When you combine independent qubits, the total number of possible basis states multiplies, so the full state space has 2 × 2 × ... × 2 = 2^n states. This does not mean you can read out all 2^n values at once; it means the quantum state can be a superposition across that many basis states.
When to use it
Use this formula when you want to know how large the Hilbert space is for a quantum register, or when comparing how quantum systems scale as more qubits are added. It is especially useful in introductory quantum computing, quantum algorithms, and estimating why even small numbers of qubits can represent very large state spaces.
Worked example, step by step
Suppose a quantum circuit uses n = 8 qubits. States = 2^n States = 2^8 States = 256 So an 8-qubit register can be in a superposition of 256 basis states.
Common mistakes to avoid
- Thinking 2^n means you can directly observe all states at once in a measurement
- Confusing the number of qubits with the number of basis states, which grows exponentially
- Forgetting that the formula counts basis states in the state space, not separate physical copies of the qubits
Did You Know?
A 300-qubit register could represent more states than there are atoms in the observable universe.
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More in Quantum Computing
View allQubit Superposition State
IntermediateGeneral state of a single qubit as a superposition of the computational basis states.
Normalisation Constraint
BasicThe squared amplitudes of a qubit must sum to 1, ensuring total probability is 100 %.
Hadamard Gate
IntermediatePuts a qubit into an equal superposition — the most common single-qubit gate.
Pauli-X Gate (Quantum NOT)
BasicFlips |0⟩ to |1⟩ and vice versa — the quantum equivalent of a classical NOT gate.
CNOT Gate
IntermediateTwo-qubit gate that flips the target qubit only when the control qubit is |1⟩.
Bell State (EPR Pair)
IntermediateMaximally entangled two-qubit state — measuring one qubit instantly determines the other.