The Strength of Weak Learnability
Machine Learning
Local expansion of vertex-transitive graphs and random generation in finite groups
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Oracles and queries that are sufficient for exact learning
Journal of Computer and System Sciences
A decision-theoretic generalization of on-line learning and an application to boosting
Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
Scale-sensitive dimensions, uniform convergence, and learnability
Journal of the ACM (JACM)
Prediction, learning, uniform convergence, and scale-sensitive dimensions
Journal of Computer and System Sciences - Special issue on the eighth annual workshop on computational learning theory, July 5–8, 1995
Dense quantum coding and quantum finite automata
Journal of the ACM (JACM)
Hard-core distributions for somewhat hard problems
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Succinct quantum proofs for properties of finite groups
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
A Lattice Problem in Quantum NP
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Polynomial time quantum computation with advice
Information Processing Letters
Multilinear formulas and skepticism of quantum computing
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Journal of the ACM (JACM)
Quantum Information and the PCP Theorem
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
The Complexity of the Local Hamiltonian Problem
SIAM Journal on Computing
QMA/qpoly \subseteq PSPACE/poly: De-Merlinizing Quantum Protocols
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
Oracles Are Subtle But Not Malicious
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
Computational Complexity
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
ACM Transactions on Computation Theory (TOCT) - Special issue on innovations in theoretical computer science 2012
Hi-index | 0.00 |
We prove the following surprising result: given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly is contained in PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice. Proving our main result requires combining a large number of previous tools -- including a result of Alon et al. on learning of real-valued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new ones. The main new tool is a so-called majority-certificates lemma, which is closely related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set S of Boolean functions on n variables, any function f in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm in S, such that each fi is the unique function in S compatible with O(log|S|) input/output constraints.