Fault-tolerant quantum computation with constant error
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
A new kind of science
Quantum computation and quantum information
Quantum computation and quantum information
Problems of Information Transmission
Adiabatic quantum state generation and statistical zero knowledge
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
The Quantum Communication Complexity of Sampling
SIAM Journal on Computing
Multi-linear formulas for permanent and determinant are of super-polynomial size
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Counting, fanout and the complexity of quantum ACC
Quantum Information & Computation
Quantum Information & Computation
Multi-linear formulas for permanent and determinant are of super-polynomial size
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Quantum Circuit Simplification Using Templates
Proceedings of the conference on Design, Automation and Test in Europe - Volume 2
Learning mixtures of product distributions over discrete domains
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Multi-linear formulas for permanent and determinant are of super-polynomial size
Journal of the ACM (JACM)
A full characterization of quantum advice
Proceedings of the forty-second ACM symposium on Theory of computing
Tensor-rank and lower bounds for arithmetic formulas
Proceedings of the forty-second ACM symposium on Theory of computing
Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors
Journal of Computer and System Sciences
Quantum advantage without entanglement
Quantum Information & Computation
Counterexamples to Kalai's conjecture C
Quantum Information & Computation
Tensor-Rank and Lower Bounds for Arithmetic Formulas
Journal of the ACM (JACM)
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Several researchers, including Leonid Levin, Gerard 't Hooft, and Stephen Wolfram, have argued that quantum mechanics will break down before the factoring of large numbers becomes possible. If this is true, then there should be a natural set of quantum states that can account for all quantum computing experiments performed to date, but not for Shor's factoring algorithm. We investigate as a candidate the set of states expressible by a polynomial number of additions and tensor products. Using a recent lower bound on multilinear formula size due to Raz, we then show that states arising in quantum error-correction require nΩ(log n) additions and tensor products even to approximate, which incidentally yields the first superpolynomial gap between general and multilinear formula size of functions. More broadly, we introduce a complexity classification of pure quantum states, and prove many basic facts about this classification. Our goal is to refine vague ideas about a breakdown of quantum mechanics into specific hypotheses that might be experimentally testable in the near future.