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A Full Characterization of Quantum Advice Scott Aaronson #

Summary: A Full Characterization of Quantum Advice
Scott Aaronson #
Andrew Drucker +
We prove the following surprising result: given any quantum state # on n qubits, there exists
a local Hamiltonian H on poly (n) qubits (e.g., a sum of two­qubit interactions), such that any
ground state of H can be used to simulate # on all quantum circuits of fixed polynomial size.
In terms of complexity classes, this implies that BQP/qpoly # QMA/poly, which supersedes the
previous result of Aaronson that BQP/qpoly # 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 real­valued concept classes, a result of Aaronson on the learn­
ability of quantum states, and a result of Aharonov and Regev on `QMA+ super­verifiers'---and
also creating some new ones. The main new tool is a so­called majority­certificates lemma,
which is closely related to boosting in machine learning, and which seems likely to find inde­
pendent applications. In its simplest version, this lemma says the following. Given any set
S of Boolean functions on n variables, any function f # S can be expressed as the pointwise


Source: Aaronson, Scott - Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (MIT)


Collections: Physics; Computer Technologies and Information Sciences