Qubits - the basis of Quantum Computing
Ranjani Seshadri
October 30, 2025
1 Introduction to Qubits
A qubit (quantum bit) is the basic unit of quantum information. Unlike a classical bit, a qubit can exist in a
superposition of the computational basis states |0⟩ and |1⟩ simultaneously:
|ψ⟩ = α |0⟩ + β |1⟩
where α and β are complex probability amplitudes. The normalization condition ensures that the total
probability is 1:
|α|2 + |β|2 = 1
1 0
In matrix form, the basis states are |0⟩ = and |1⟩ = .
0 1
3
Problem 1.1: Normalization: A qubit is in the state |ψ⟩ = 5 |0⟩ + β |1⟩. Find the possible values for the
complex number β.
—
2 Classical Bits vs. Quantum Bits (Qubits)
The fundamental differences between a classical bit (c-bit) and a quantum bit (qubit) are summarized below,
highlighting the unique properties that enable quantum computation.
S. No Property Classical Bit (c-bit) Quantum Bit (Qubit)
1 State Representation Must be in a single, defi- Exists in a superposition of |0⟩ and |1⟩:
nite state: 0 or 1. α |0⟩ + β |1⟩.
2 Information Storage Deterministic value based Probabilistic value defined by continuous
on a voltage or magnetic complex amplitudes (α, β).
state.
3 Underlying Basis Binary or Boolean alge- Quantum mechanical basis vectors in a
bra. Hilbert space.
4 Logic Gates AND, OR, NOT. Gates Hadamard, CNOT, Pauli, Phase, T gates.
are typically irre- Gates are required to be Unitary (re-
versible. versible).
5 Inter-bit Correlation States are independent. Qubits can be entangled, leading to non-
No correlation beyond local correlations.
classical dependence.
6 Information Density Strictly 1 bit of informa- Effectively infinite (continuous ampli-
tion. tudes) until measurement. N qubits en-
code 2N amplitudes.
7 Computation Principle Sequential processing of Parallel computation (quantum paral-
single values. lelism) by acting on all superposed states
simultaneously.
8 Error Sensitivity Low sensitivity; protected High sensitivity to decoherence; requires
by redundancy. specialized quantum error correction
codes.
Problem 1.2: State Space: How many complex amplitudes are required to fully describe the state of a
5-qubit system? Compare this to the number of classical states a 5-bit system can represent.
—
1
, Figure 1: Note that θ is the angle between the vector and the z-axis (mislabelled in the figure)
3 Single Qubit States and the Bloch Sphere
Any pure single-qubit state |ψ⟩ can be represented by two angles (θ and ϕ) on the surface of the Bloch Sphere:
θ iϕ θ
|ψ⟩ = cos |0⟩ + e sin |1⟩
2 2
• |0⟩ is the North Pole (θ = 0).
• |1⟩ is the South Pole (θ = π).
• States like |+⟩ = √1 (|0⟩
2
+ |1⟩) and |−⟩ = √1 (|0⟩
2
− |1⟩) lie on the equator.
Problem 2.1: Bloch Sphere Coordinates: What are the angles (θ, ϕ) for the state |y+⟩ = √1 (|0⟩ + i |1⟩)?
2
On which axis of the Bloch sphere does this state lie?
—
4 Two-Qubit Systems
A system of two qubits, |ψA ⟩ and |ψB ⟩, lives in a 4-dimensional Hilbert space, spanned by the basis states |00⟩,
|01⟩, |10⟩, and |11⟩. A general two-qubit state is:
|Ψ⟩ = c00 |00⟩ + c01 |01⟩ + c10 |10⟩ + c11 |11⟩
|cij |2 = 1.
P
where i,j
—
5 Quantum Entanglement and Bell States
5.1 Quantum Entanglement
Entanglement is a non-classical correlation where the quantum states of two or more qubits are linked such
that they cannot be described independently.
A state |Ψ⟩ is separable (not entangled) if it can be written as a tensor product: |Ψ⟩ = |ψA ⟩ ⊗ |ψB ⟩. If it
cannot be factored, it is entangled.
2
Ranjani Seshadri
October 30, 2025
1 Introduction to Qubits
A qubit (quantum bit) is the basic unit of quantum information. Unlike a classical bit, a qubit can exist in a
superposition of the computational basis states |0⟩ and |1⟩ simultaneously:
|ψ⟩ = α |0⟩ + β |1⟩
where α and β are complex probability amplitudes. The normalization condition ensures that the total
probability is 1:
|α|2 + |β|2 = 1
1 0
In matrix form, the basis states are |0⟩ = and |1⟩ = .
0 1
3
Problem 1.1: Normalization: A qubit is in the state |ψ⟩ = 5 |0⟩ + β |1⟩. Find the possible values for the
complex number β.
—
2 Classical Bits vs. Quantum Bits (Qubits)
The fundamental differences between a classical bit (c-bit) and a quantum bit (qubit) are summarized below,
highlighting the unique properties that enable quantum computation.
S. No Property Classical Bit (c-bit) Quantum Bit (Qubit)
1 State Representation Must be in a single, defi- Exists in a superposition of |0⟩ and |1⟩:
nite state: 0 or 1. α |0⟩ + β |1⟩.
2 Information Storage Deterministic value based Probabilistic value defined by continuous
on a voltage or magnetic complex amplitudes (α, β).
state.
3 Underlying Basis Binary or Boolean alge- Quantum mechanical basis vectors in a
bra. Hilbert space.
4 Logic Gates AND, OR, NOT. Gates Hadamard, CNOT, Pauli, Phase, T gates.
are typically irre- Gates are required to be Unitary (re-
versible. versible).
5 Inter-bit Correlation States are independent. Qubits can be entangled, leading to non-
No correlation beyond local correlations.
classical dependence.
6 Information Density Strictly 1 bit of informa- Effectively infinite (continuous ampli-
tion. tudes) until measurement. N qubits en-
code 2N amplitudes.
7 Computation Principle Sequential processing of Parallel computation (quantum paral-
single values. lelism) by acting on all superposed states
simultaneously.
8 Error Sensitivity Low sensitivity; protected High sensitivity to decoherence; requires
by redundancy. specialized quantum error correction
codes.
Problem 1.2: State Space: How many complex amplitudes are required to fully describe the state of a
5-qubit system? Compare this to the number of classical states a 5-bit system can represent.
—
1
, Figure 1: Note that θ is the angle between the vector and the z-axis (mislabelled in the figure)
3 Single Qubit States and the Bloch Sphere
Any pure single-qubit state |ψ⟩ can be represented by two angles (θ and ϕ) on the surface of the Bloch Sphere:
θ iϕ θ
|ψ⟩ = cos |0⟩ + e sin |1⟩
2 2
• |0⟩ is the North Pole (θ = 0).
• |1⟩ is the South Pole (θ = π).
• States like |+⟩ = √1 (|0⟩
2
+ |1⟩) and |−⟩ = √1 (|0⟩
2
− |1⟩) lie on the equator.
Problem 2.1: Bloch Sphere Coordinates: What are the angles (θ, ϕ) for the state |y+⟩ = √1 (|0⟩ + i |1⟩)?
2
On which axis of the Bloch sphere does this state lie?
—
4 Two-Qubit Systems
A system of two qubits, |ψA ⟩ and |ψB ⟩, lives in a 4-dimensional Hilbert space, spanned by the basis states |00⟩,
|01⟩, |10⟩, and |11⟩. A general two-qubit state is:
|Ψ⟩ = c00 |00⟩ + c01 |01⟩ + c10 |10⟩ + c11 |11⟩
|cij |2 = 1.
P
where i,j
—
5 Quantum Entanglement and Bell States
5.1 Quantum Entanglement
Entanglement is a non-classical correlation where the quantum states of two or more qubits are linked such
that they cannot be described independently.
A state |Ψ⟩ is separable (not entangled) if it can be written as a tensor product: |Ψ⟩ = |ψA ⟩ ⊗ |ψB ⟩. If it
cannot be factored, it is entangled.
2
