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.Comment: We fixed two significant issues: 1. The definition of YQP machines
needed to be changed to preserve our results. The revised definition is more
natural and has the same intuitive interpretation. 2. We needed properties of
Local Hamiltonian reductions going beyond those proved in previous works
(whose results we'd misstated). We now prove the needed properties. See p. 6
for more on both point
