A Full Characterization of Quantum Advice
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Author(s)
Aaronson, Scott
Drucker, Andrew Donald
Date Issued
June 2010
Journal
Proceedings of the 42nd ACM Symposium on Theory of Computing
Publisher
Association for Computing Machinery
Citation
Aaronson, Scott, and Andrew Drucker. “A full characterization of quantum advice.” Proceedings of the 42nd ACM symposium on Theory of computing. Cambridge, Massachusetts, USA: ACM, 2010. 131-140.
Version
Original manuscript
Abstract
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.
Subjects
quantum computation
nonuniform computation
local hamiltonians
learning
karp-lipton theorem
compression
boosting
advice
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Attribution-Noncommercial-Share Alike 3.0 Unported
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DOI of Published Version
https://doi.org/10.1145/1806689.1806710
