Tag: Quantum Complexity

Table of Contents Introduction What Are Restricted Quantum Models? Motivation: Why Study Non-Universal Models? The IQP Model: Instantaneous Quantum Polynomial-Time Structure of IQP Circuits IQP Complexity Assumptions Hardness of Simulating IQP...

Table of Contents Introduction What Are Sampling Problems? Importance of Sampling in Complexity Theory Classical vs Quantum Sampling Models Boson Sampling: Basic Concept Theoretical Foundation by Aaronson and Arkhipov Matrix Permanents...

Table of Contents Introduction What Is Quantum Supremacy? Early Theoretical Proposals Google's Sycamore Experiment Random Circuit Sampling Explained Sycamore’s Hardware and Architecture Verification Methods and Cross-Entropy Benchmarking IBM’s Response and Classical Simulation...

Table of Contents Introduction What Is Quantum Advantage? Historical Milestones in Demonstrating Quantum Advantage Circuit-Based Quantum Supremacy Sampling Problems and Quantum Speedups Query Complexity Separations Communication Complexity Gaps Oracle Separations and Limitations Fine-Grained...

Table of Contents Introduction Why Study Complexity in Quantum ML? Overview of Quantum Machine Learning (QML) Models Circuit Depth and Width in QML Architectures Complexity Classes Related to QML VC...

Table of Contents Introduction Classical vs Quantum Search Structures Motivation for Quantum Search Trees Binary Search and Quantum Speedups Grover's Algorithm and Unstructured Trees Tree-Like Data in Quantum Walks Quantum Tree...

Table of Contents Introduction Why Graph Problems Matter in Quantum Computing Graph Representation in Quantum Systems Quantum Walks and Their Graph Applications Element Distinctness and Collision Finding Triangle Finding in...

Table of Contents Introduction Why Quantum Data Structures Matter Differences Between Classical and Quantum Data Models Superposition and Data Representation Quantum RAM (QRAM) Quantum Access Patterns and Query Models Quantum Hashing...

Table of Contents Introduction Noise in Quantum Computation Fault-Tolerance vs Noise-Tolerance Importance of Noise Tolerance in Complexity Theory Noise Models: Depolarizing, Dephasing, and Stochastic Errors Robustness of BQP Error Threshold Theorems Fault-Tolerant...

Table of Contents Introduction Circuit Complexity in Quantum Computation The Class QNC: Quantum Nick’s Class Depth and Width Constraints in QNC QNC^0 and Constant-Depth Circuits QTC: Quantum Threshold Circuits Comparing QNC...

Table of Contents Introduction What Is Circuit Complexity? Classical vs Quantum Circuit Complexity Quantum Circuits: Structure and Basics Size, Depth, and Width Metrics Quantum Gate Sets and Universality Measuring Quantum Circuit...

Table of Contents Introduction Motivation: Fault-Tolerant Universal Quantum Computation Clifford Gates and the Gottesman-Knill Theorem The Limitation of Clifford-Only Computation Magic States: Definition and Examples The T Gate and Its...