Representation of Qubits

Detailed Description

A qubit is the fundamental unit of quantum information, analogous to a classical bit but capable of existing in superpositions. In Hilbert space, a qubit is represented as a two-dimensional complex vector in the computational basis {|0⟩, |1⟩}, where the state is ψ = α|0⟩ + β|1⟩ with |α|^2 + |β|^2 = 1. This allows for quantum parallelism and entanglement.

History

The concept of qubits was formalized by David Deutsch in 1985, building on Richard Feynman's 1982 proposal for quantum computers. Roots trace to quantum mechanics by Dirac in 1930 with bra-ket notation.

Derivation of Formula

From linearity of quantum mechanics: The state is a unit vector in 2D Hilbert space H^2. Basis vectors |0⟩ = [1,0]^T, |1⟩ = [0,1]^T. General state ψ = α|0⟩ + β|1⟩, normalized ⟨ψ|ψ⟩ = 1.

Formula: \( |\psi\rangle = \alpha |0\rangle + \beta |1\rangle \)

Where: α, β ∈ ℂ, |α|^2 + |β|^2 = 1; |0⟩, |1⟩ are basis states.

Solved Questions

Question 1: Normalize the state |ψ⟩ = 3|0⟩ + 4|1⟩.

Step 1: Norm = √(9 + 16) = 5.

Step 2: |ψ⟩ = (3/5)|0⟩ + (4/5)|1⟩.

Question 2: Find probability of measuring 1 for |ψ⟩ = (1/√2)(|0⟩ + |1⟩).

Step 1: |β|^2 = (1/√2)^2 = 1/2.

Bloch Sphere

Detailed Description

The Bloch sphere is a geometrical representation of a qubit's state as a point on a unit sphere. Pure states lie on the surface, with poles as |0⟩ and |1⟩, equator as superpositions. Parameters θ, φ define the state.

History

Introduced by Felix Bloch in 1946 for nuclear magnetic resonance, adapted to qubits in quantum optics in the 1970s.

Derivation of Formula

From |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩, mapping to spherical coordinates: x = sinθ cosφ, y = sinθ sinφ, z = cosθ.

Formula: \( |\psi\rangle = \cos\left(\frac{\theta}{2}\right) |0\rangle + e^{i \phi} \sin\left(\frac{\theta}{2}\right) |1\rangle \)

Where: θ ∈ [0, π] polar angle, φ ∈ [0, 2π) azimuthal angle.

Solved Questions

Question 1: Find Bloch vector for |+⟩ = (1/√2)(|0⟩ + |1⟩).

Step 1: θ = π/2, φ = 0.

Step 2: (1, 0, 0).

Question 2: State for vector (0,0,1).

Step 1: θ=0.

Step 2: |0⟩.

Multi-qubit Systems

Detailed Description

Multi-qubit systems are described in tensor product Hilbert spaces H = H_1 ⊗ H_2 ⊗ ... ⊗ H_n, dimension 2^n. Allows for entanglement beyond classical correlations.

History

Tensor products in QM by von Neumann 1932; applied to quantum info in 1990s with Shor's algorithm.

Derivation of Formula

For two qubits: |ψ⟩ = ∑_{ij} c_{ij} |i⟩ ⊗ |j⟩, with ∑ |c_{ij}|^2 = 1.

Formula: \( |\psi\rangle = \sum_{i,j=0}^{1} c_{ij} |i j\rangle \)

Where: c_{ij} amplitudes, |ij⟩ = |i⟩ ⊗ |j⟩.

Solved Questions

Question 1: Dimension of 3-qubit Hilbert space.

Step 1: 2^3 = 8.

Question 2: Write |00⟩ in vector form.

Step 1: [1,0] ⊗ [1,0] = [1,0,0,0]^T.

Single Qubit Gates (X, Y, Z, H, S, T, Rφ)

Detailed Description

Single-qubit gates are unitary operations on one qubit. Pauli X flips bit, Y combines flip and phase, Z phase flip; H creates superposition; S, T are phase gates; Rφ general phase.

History

Pauli matrices 1927; Hadamard from Fourier transform; Clifford gates in fault-tolerance 1990s.

Derivation of Formula

X from σ_x = [[0,1],[1,0]], rotation by π around x-axis.

Formulas: X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, etc.

Where: Matrices act on state vectors.

Solved Questions

Question 1: Apply X to |0⟩.

Step 1: X [1,0]^T = [0,1]^T = |1⟩.

Question 2: Apply H to |0⟩.

Step 1: (1/√2) [1,1]^T = |+⟩.

Multi-qubit Gates (CNOT, Toffoli, Fredkin, Swap)

Detailed Description

Multi-qubit gates operate on multiple qubits. CNOT flips target if control 1; Toffoli CCNOT for three; Fredkin controlled swap; Swap exchanges states.

History

CNOT proposed by Feynman 1985; Toffoli 1980 for reversible computing.

Derivation of Formula

CNOT: Projector on control |0><0| ⊗ I + |1><1| ⊗ X.

Formula: CNOT = |0\rangle\langle0| \otimes I + |1\rangle\langle1| \otimes X

Where: I identity, X Pauli X.

Solved Questions

Question 1: CNOT on |10⟩.

Step 1: Control 1, flip target: |11⟩.

Question 2: Swap on |01⟩.

Step 1: Exchanges to |10⟩.

Universal Gate Sets

Detailed Description

Universal gate sets can approximate any unitary operation. Examples: {H, S, CNOT, T} via Solovay-Kitaev theorem.

History

Solovay-Kitaev 1995 proved efficient approximation.

Derivation of Formula

Any U approximated by sequence of gates to epsilon, length O(log^c (1/epsilon)).

Solved Questions

Question 1: Is {H, T, CNOT} universal?

Step 1: Yes, generates Clifford + T for universal approximation.

Question 2: Approximate Rz(π/8).

Step 1: T gate is Rz(π/4), sequences for smaller angles.

Projective Measurements

Detailed Description

Projective measurements collapse the wavefunction to an eigenvector with probability |⟨a|ψ⟩|^2, outcome a.

History

Postulated by von Neumann 1932 in measurement problem.

Derivation of Formula

From Born rule: P(m) = ⟨ψ| P_m |ψ⟩, post-state P_m |ψ⟩ / √P(m), where P_m projector.

Formula: \( P(m) = \langle \psi | P_m | \psi \rangle \)

Where: P_m = |m⟩⟨m|, sum P_m = I.

Solved Questions

Question 1: Measure Z on |+⟩.

Step 1: P(0) = P(1) = 1/2.

Question 2: Post-state for 0.

Step 1: |0⟩.

POVMs (Positive Operator Valued Measures)

Detailed Description

POVMs generalize measurements with operators E_m >0, sum E_m = I, P(m) = ⟨ψ| E_m |ψ⟩, no collapse specified.

History

Introduced by Helstrom 1976 for quantum detection.

Derivation of Formula

From Naimark's theorem: POVM as projective on larger space.

Formula: \( P(m) = \langle \psi | E_m | \psi \rangle \)

Where: E_m positive, sum E_m = I.

Solved Questions

Question 1: Check if E1= |0><0|/2, E2= |+><+|/2, E3= I - E1 - E2 is POVM.

Step 1: Check positive and sum to I.

Question 2: P(1) for |ψ⟩=|0⟩.

Step 1: Tr(E1 |0><0|) = 1/2.

EPR Pairs

Detailed Description

EPR pairs are maximally entangled two-qubit states, highlighting non-locality in QM as per Einstein, Podolsky, Rosen paradox.

History

Proposed in 1935 EPR paper questioning QM completeness.

Derivation of Formula

From singlet state, rotationally invariant.

Formula: \( |\Phi^+\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle) \)

Where: One of Bell states.

Solved Questions

Question 1: Measure first qubit 0, second?

Step 1: Collapses to |00⟩, second 0.

Question 2: Correlation.

Step 1: Perfect.

Bell States

Detailed Description

Bell states are four maximally entangled two-qubit states forming a basis for entanglement.

History

John Bell 1964 used them for inequality testing locality.

Derivation of Formula

From applying Pauli on EPR.

Formulas: |\Phi^\pm\rangle = \frac{1}{\sqrt{2}} (|00\rangle \pm |11\rangle), |\Psi^\pm\rangle = \frac{1}{\sqrt{2}} (|01\rangle \pm |10\rangle)

Solved Questions

Question 1: Circuit for |\Phi^+\rangle.

Step 1: H on first, CNOT.

Question 2: Distinguish them.

Step 1: Bell measurement.

GHZ States

Detailed Description

GHZ states are multi-particle entangled states like \frac{1}{\sqrt{2}} (|000\rangle + |111\rangle) for three qubits, used in quantum tests.

History

Greenberger, Horne, Zeilinger 1989 for stronger non-locality argument.

Derivation of Formula

Generalize Bell: \frac{1}{\sqrt{2}} (|0...0\rangle + |1...1\rangle).

Formula: \( |GHZ\rangle = \frac{1}{\sqrt{2}} (|000\rangle + |111\rangle) \)

Solved Questions

Question 1: Measure two 0, third?

Step 1: 0.

Question 2: Parity.

Step 1: Even.