Quantum Complexity

Table of Contents 1. Introduction Quantum complexity theory (QCT) is evolving rapidly, with breakthroughs in interactive proofs, quantum learning, and cryptography. This review summarizes major recent results that define and refine the structure and power of quantum computations. 2. MIP* = RE and Its Impact This landmark result shows that multiprover interactive … [Read more...] about Research Review: Recent Results in Quantum Complexity Theory (QCT)

Table of Contents 1. Introduction Quantum computation theory explores the power, limits, and structures of quantum algorithms and machines. As quantum devices scale, theory must evolve to answer foundational and applied questions in complexity, learning, and verification. 2. The Landscape of Quantum Complexity Classes Quantum classes like BQP, QMA, QIP, and MIP* … [Read more...] about Frontiers of Quantum Computation Theory: Open Questions and Emerging Paradigms

Table of Contents 1. Introduction Quantum meta-complexity studies the complexity of problems about quantum complexity itself. This includes reasoning about the minimal description of quantum circuits, the verification of circuit hardness, and introspective models of computation. 2. What Is Meta-Complexity? In classical complexity, meta-complexity addresses … [Read more...] about Quantum Meta-Complexity: Self-Referential Questions in Quantum Computation

Table of Contents 1. Introduction Quantum advice extends the concept of non-uniform computation into the quantum world. It involves providing a quantum state—rather than a classical string—as auxiliary input to a quantum machine, enabling potentially more powerful computation. 2. Classical Advice and Non-uniform Computation In classical complexity, P/poly and … [Read more...] about Quantum Advice and Non-uniform Models in Quantum Complexity Theory

Table of Contents 1. Introduction Quantum property testing is the study of algorithms that determine whether a function or object possesses a certain global property or is far from having it, using only a few quantum queries. It offers exponential speedups over classical methods in certain settings. 2. What Is Property Testing? Property testing involves designing … [Read more...] about Quantum Property Testing: Verifying Global Properties with Few Quantum Queries

Table of Contents 1. Introduction Lattice-based problems play a central role in post-quantum cryptography and quantum complexity. Quantum reductions allow us to connect the hardness of cryptographic schemes to worst-case assumptions, ensuring security even in the face of quantum adversaries. 2. What Are Lattices in Computational Mathematics? A lattice is a … [Read more...] about Lattice Problems and Quantum Reductions: Foundations for Post-Quantum Security

Table of Contents 1. Introduction Relativized worlds are hypothetical universes where all computations have access to the same oracle function. In quantum complexity theory, these models allow researchers to explore the power of quantum algorithms under controlled assumptions. 2. What Are Relativized Worlds? A relativized world gives all Turing or quantum machines … [Read more...] about Relativized Worlds in Quantum Complexity Theory

Table of Contents 1. Introduction Probabilistically Checkable Quantum Proofs (PCQPs) extend the classical PCP framework into the quantum domain. They aim to define proof systems where the correctness of a quantum statement can be verified with high confidence using only limited quantum queries. 2. Classical PCP Theorem: A Quick Recap The classical PCP theorem … [Read more...] about Probabilistically Checkable Quantum Proofs (PCQPs): Extending the PCP Paradigm

Table of Contents 1. Introduction The "hardness vs randomness" paradigm explores how computational hardness can substitute for randomness, and vice versa. In quantum computing, this relationship is reshaped by entanglement, quantum advice, and non-classical correlations. 2. Classical Hardness vs Randomness: A Brief Overview In classical complexity, the idea is … [Read more...] about Hardness vs Randomness in Quantum Computation

Table of Contents 1. Introduction Black-box separations (also called oracle separations) are used to demonstrate that two computational models or complexity classes behave differently when given access to the same abstract subroutine or "oracle." In quantum computing, they highlight fundamental differences between quantum and classical complexity. 2. What Are … [Read more...] about Black-Box Separations in Quantum Complexity Theory