Quantum 101

Table of Contents Introduction What Is Property Testing? Classical vs Quantum Property Testing Quantum Query Models Why Property Testing Is Important in Quantum Computing Key Measures: Query Complexity and Distance Property...

Table of Contents Introduction What Are Lattices in Computational Mathematics? Key Lattice Problems: SVP, CVP, SIVP Relevance to Cryptography and Complexity Lattices and Worst-Case to Average-Case Reductions Ajtai’s Theorem and...

Table of Contents Introduction What Are Relativized Worlds? Classical Context: P vs NP with Oracles Motivation for Studying Relativization Quantum Oracle Machines BQP with Oracles (BQP^O) Relativized Separations: BQP vs BPP BQP...

Table of Contents Introduction Classical PCP Theorem: A Quick Recap Motivation for Quantum PCPs What Are Probabilistically Checkable Quantum Proofs? Quantum PCP Conjecture Quantum Locally Checkable Proofs Complexity Classes: QMA and...

Table of Contents Introduction Classical Hardness vs Randomness: A Brief Overview Quantum Analogs of Pseudorandomness Derandomization and Quantum Algorithms Quantum Pseudorandom Generators (QPRGs) BQP vs BPP and Randomness Elimination Hardness Assumptions...

Table of Contents Introduction What Are Black-Box Models? Motivation for Black-Box Separations Classical Black-Box Separation Examples Quantum Black-Box Access Oracle Constructions in Quantum Complexity Quantum vs Classical Separation Techniques Separating BQP from...

Table of Contents Introduction What Are Reductions in Complexity Theory? Classical vs Quantum Reductions Types of Quantum Reductions Turing Reductions in Quantum Settings Many-One Quantum Reductions Oracle Reductions and Relativized Worlds Reductions...

Table of Contents Introduction Classical Computability: A Brief Review What Is Quantum Computability? Quantum Turing Machines (QTMs) Relationship to Church-Turing Thesis Quantum Extensions of Recursive Functions Quantum Decidability and Semidecidability Halting Problem...

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...