This post is part of my 21-day quantum computing learning journey, focusing on the three fundamental paradigms of quantum computation: Circuit Model, Adiabatic Quantum Computing, and Measurement-Based Quantum Computing. Today we’re exploring how different approaches to quantum computation shape both algorithm design and hardware development.

Progress: 13/21 days completed. Circuit Model: ✓. Adiabatic QC: ✓. MBQC: mastered.

Circuit Model: The Foundation of Gate-Based Quantum Computing

Universal Gate Sets

The beauty of the circuit model lies in its universality - any quantum computation can be approximated using a small set of quantum gates:

Single-Qubit Gates:

Pauli-X (NOT): X = [[0,1],[1,0]] Pauli-Y: Y = [[0,-i],[i,0]] Pauli-Z: Z = [[1,0],[0,-1]] Hadamard: H = [[1,1],[1,-1]]/√2

Two-Qubit Gates:

CNOT: Controlled-NOT operation CZ: Controlled-Z operation

Mathematical Foundation Any unitary operation U on n qubits can be decomposed into:

U = U₁ ⊗ U₂ ⊗ ... ⊗ Uₙ · CNOT · U₁' ⊗ U₂' ⊗ ... ⊗ Uₙ' · ...

Where each Uᵢ is a single-qubit rotation and CNOT provides entanglement.

Quantum Circuit Representation
|ψ⟩ ——[H]——●——————[Ry(θ)]——————●——[M]
           │                   │
|0⟩ ————————⊕——[X]————————————⊕——[M]
           
     Preparation  Entanglement  Measurement

Adiabatic Quantum Computing: Optimization Through Evolution

Adiabatic Quantum Computing (AQC) leverages the adiabatic theorem to solve optimization problems by slowly evolving a quantum system from an easy-to-prepare ground state to a final state encoding the solution.

The Adiabatic Theorem For a time-dependent Hamiltonian H(t), if the system starts in the ground state and evolves slowly enough, it remains in the instantaneous ground state: H(t) = (1-s(t))H₀ + s(t)H₁ Where:

H₀: Initial “driver” Hamiltonian (easy ground state) H₁: Final “problem” Hamiltonian (encodes optimization problem) s(t): Annealing schedule, s(0) = 0, s(T) = 1

Quantum Annealing Process

Problem Encoding: Map optimization problem to energy landscape Initialization: Prepare ground state of H₀ Adiabatic Evolution: Slowly change from H₀ to H₁ Measurement: Final state encodes optimal solution

Mathematical Formulation

The gap condition ensures adiabatic evolution:

T » ħ

⟨ψ₁(t)

∂H/∂t

ψ₀(t)⟩

² / Δ(t)³

Where Δ(t) is the energy gap between ground and first excited states.

Measurement-Based Quantum Computing: Computation Through Destruction MBQC performs computation entirely through measurements on a pre-prepared entangled resource state, typically a cluster state. Cluster State Preparation

A cluster state is a highly entangled state where qubits are arranged in a graph pattern:

|+⟩ ——●——●——●—— ... ——●——●——●
      │  │  │         │  │  │
|+⟩ ——●——●——●—— ... ——●——●——●
      │  │  │         │  │  │
|+⟩ ——●——●——●—— ... ——●——●——●

Mathematical representation:

\[| \text{Cluster} \rangle = \prod_{i,j} \mathrm{CZ}_{ij} \bigotimes_i |+\rangle_i\]

Adaptive Measurement Strategy

• Single-qubit measurements in bases determined by computation
• Measurement outcomes influence future measurement choices
• Quantum teleportation transfers information through the cluster
• Classical feedforward adapts the computation path

Measurement Basis Selection For qubit at position (i,j), measurement basis depends on:

Desired quantum gate Previous measurement outcomes Byproduct correction requirements

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Reference:

1. Nielsen, M. A., & Chuang, I. L. (2024). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press