[Paper Review] Limits on Efficient Computation in the Physical World

Scott Aaronson|ArXiv.org|Dec 20, 2004

Quantum Computing Algorithms and Architecture213 references22 citations

TL;DR

This paper establishes fundamental limits on quantum computing by proving strong lower bounds on query complexity for key problems like the collision problem, local search, and graph isomorphism, showing quantum algorithms cannot solve NP-complete problems efficiently even with quantum advice. It also introduces a complexity-theoretic classification of quantum states and demonstrates that postselection or history-sampling models do not yield super-quantum computational power beyond PP.

ABSTRACT

More than a speculative technology, quantum computing seems to challenge our most basic intuitions about how the physical world should behave. In this thesis I show that, while some intuitions from classical computer science must be jettisoned in the light of modern physics, many others emerge nearly unscathed; and I use powerful tools from computational complexity theory to help determine which are which.

Motivation & Objective

• To rigorously establish the limits of quantum computation by proving strong lower bounds on query complexity for fundamental problems such as the collision problem and local search.
• To investigate whether quantum computers can solve NP-complete problems efficiently, even with nonuniform quantum advice states.
• To develop a complexity-theoretic classification of quantum states to address skepticism about the physical realizability of quantum computing.
• To analyze whether models extending beyond standard quantum computing—such as those allowing postselection or sampling of hidden-variable histories—yield computational advantages beyond known complexity classes.
• To determine whether the principles of quantum mechanics, including finite speed of light and locality, preserve the quadratic speedup of Grover’s algorithm in spatial search.

Proposed method

• Proved a lower bound of Ω(n^{1/5}) queries for the collision problem using a relational adversary method and reduction to bivariate polynomial degree bounds.
• Applied the trace distance method and direct product theorems to analyze quantum search with quantum advice and simulate quantum messages using classical oracles.
• Introduced the concept of a 'Sure/Shor separator' to distinguish physically realizable quantum states from those in Shor’s algorithm, based on complexity classification.
• Used the quantum black-box model with oracles to separate quantum complexity classes, including showing PostBQP = PP via postselection.
• Modeled spatial search on graphs and grids using amplitude amplification and locality criteria, proving that Grover’s speedup holds even in 2D spatial regions.
• Defined and analyzed hidden-variable theories (e.g., flow and Schrödinger theories) using axioms of indifference and robustness, and studied their computational complexity.

Experimental results

Research questions

• RQ1 What is the minimum number of queries a quantum algorithm needs to solve the collision problem, and can this be improved beyond constant lower bounds?
• RQ2 Can quantum computers solve NP-complete problems in polynomial time with the help of nonuniform quantum advice states?
• RQ3 Is there a complexity-theoretic criterion—such as a 'Sure/Shor separator'—that distinguishes physically realizable quantum states from those in large-scale quantum algorithms?
• RQ4 Does the finite speed of light or spatial locality affect the quadratic speedup of Grover’s algorithm in searching spatial databases?
• RQ5 Can models that allow sampling of entire hidden-variable histories compute more than standard quantum computers, and if so, how do they compare to known complexity classes?

Key findings

• Any quantum algorithm solving the collision problem requires Ω(n^{1/5}) queries, resolving a long-open question and implying no black-box quantum speedup for breaking cryptographic hash functions.
• Relative to an oracle, quantum computers cannot solve NP-complete problems in polynomial time, even with quantum advice, showing that NP is not in BQP/qpoly.
• A quantum algorithm needs Ω(2^{n/4}/n) queries to find a local minimum on the n-dimensional hypercube, a result that also yields new classical lower bounds for local search.
• The quantum query complexity of total Boolean functions is bounded by O(Q^6) where Q is the certificate complexity, establishing a tight relation between quantum and classical complexity.
• Postselection in quantum computing yields the complexity class PostBQP, which equals PP, providing a new quantum-based proof that PP is closed under intersection.
• A model allowing sampling of the entire history of a hidden variable is only slightly more powerful than standard quantum computing, and does not exceed PP in power, suggesting fundamental limits on such models.