A Full Characterization of Quantum Advice

TL;DR

This paper characterizes quantum advice by showing it can be simulated with local Hamiltonians and ground states, leading to a new understanding of quantum complexity classes and introducing a novel majority-certificates lemma.

Contribution

It proves quantum advice is equivalent to untrusted quantum plus trusted classical advice and introduces the majority-certificates lemma for boosting in learning theory.

Findings

Quantum states can be simulated by ground states of local Hamiltonians.

BQP/qpoly is contained in QMA/poly, expanding previous complexity class relations.

Introduces the majority-certificates lemma, a new tool with potential independent applications.

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.