 
Summary: A Full Characterization of Quantum Advice
Scott Aaronson
MIT
Andrew Drucker
MIT
Abstract
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 toolsincluding 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
