A qubit is the basic unit of quantum information which can exist in a superposition of |0⟩ and |1⟩ states simultaneously.

What is superposition in quantum computing?

Superposition refers to a quantum system’s ability to be in multiple states simultaneously until measured.

What is quantum entanglement?

Quantum entanglement is a phenomenon where qubits become correlated such that the state of one instantaneously influences the state of another, regardless of distance.

What are quantum gates?

Quantum gates are unitary operations applied to qubits to change their state and form the building blocks of quantum algorithms.

What role does measurement play in quantum computing?

Measurement collapses a qubit’s superposition into one of its basis states, providing a classical outcome.

Quantum Computing 1Mark Questions

What is the probability interpretation of |α|² + |β|² = 1 for a qubit?
For a qubit state \( |\psi\rangle = \alpha|0\rangle + \beta|1\rangle \), \( |\alpha|^2 \) represents the probability of measuring the qubit in state \( |0\rangle \), and \( |\beta|^2 \) represents the probability of measuring it in state \( |1\rangle \). The condition \( |\alpha|^2 + |\beta|^2 = 1 \) ensures that the total probability of all possible outcomes is unity.
State and briefly explain one advantage of the Bloch sphere representation of a qubit.
The Bloch sphere provides a clear geometric visualization of a qubit state, where states are represented as points on a sphere and single-qubit gates appear as rotations, simplifying understanding and analysis.
Write the tensor product representation of \( |0\rangle \otimes |1\rangle \).
The matrix representations are:

\( |0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \)

The tensor product is:

\( |0\rangle \otimes |1\rangle = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} = |01\rangle \)
Use the Bloch sphere representation to show how a single qubit state can be visualized.
In the Bloch sphere representation, a single qubit state is visualized as a point on the surface of a unit sphere. The north and south poles represent \( |0\rangle \) and \( |1\rangle \), while superposition states lie elsewhere on the sphere. Quantum gates act as rotations of this state vector.
What happens to a qubit state after measurement?
After measurement, the qubit collapses from a superposition into one of the basis states \( |0\rangle \) or \( |1\rangle \), with probabilities determined by the squared magnitudes of the probability amplitudes.
State the type of problem solved by the Deutsch–Jozsa algorithm and why it shows quantum advantage.
The Deutsch–Jozsa algorithm determines whether a function is constant or balanced. It demonstrates quantum advantage by solving the problem with a single function evaluation, whereas classical methods require multiple evaluations.
Define Hilbert space and explain its importance in describing quantum states.
A Hilbert space is a mathematical vector space containing all possible states of a quantum system. Quantum states, operators, and observables are defined within this space, making it fundamental to quantum mechanics.
Give one major advantage of quantum computing over classical computing.
Quantum computing can solve certain problems exponentially faster than classical computing by exploiting superposition and entanglement.
Differentiate between classical bit states and qubit states with one example each.
A classical bit exists only in one of two states, 0 or 1, whereas a qubit can exist in a superposition of states.
Example: Classical bit: 0 or 1; Qubit: \( \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \).
Explain what a multi-qubit system is and how multiple qubits combine.
A multi-qubit system consists of two or more qubits combined using the tensor product. A two-qubit system can exist in a superposition of four basis states:

\( |\psi\rangle = \alpha_{00}|00\rangle + \alpha_{01}|01\rangle + \alpha_{10}|10\rangle + \alpha_{11}|11\rangle \)
What makes the Toffoli gate important for universal quantum computation?
The Toffoli gate is a universal reversible gate, meaning any classical computation can be implemented using Toffoli gates, making it essential for universal quantum computation.
Explain what quantum measurement is and how it affects the state of a quantum system.
Quantum measurement is the act of observing a quantum system, causing the system to collapse from a superposition into one of its eigenstates corresponding to the measured observable.
Why is Shor’s algorithm considered a threat to RSA cryptography?
Shor’s algorithm can efficiently factor large integers using a quantum computer, threatening RSA cryptography, which relies on the difficulty of integer factorization.
Compare classical search and Grover’s search.
Classical search requires \( O(N) \) steps to search an unsorted database, while Grover’s algorithm requires only \( O(\sqrt{N}) \) steps, providing a quadratic speedup.