A full characterization of quantum advice

Abstract

We prove the following surprising result: given any quantum state ρ 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 ρ 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 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 ∈ S can be expressed as the pointwise majority of m = O (n) functions f 1 , . . . , f m ∈ S, such that each f i is the unique function in S compatible with O (log |S|) input/output constraints.

Quantum Proofs

“…2 The group non-membership problem is perhaps most naturally described within the black-box group setting [23]. Here, one considers that there is an underlying finite group G n that has been specified for each positive integer n, and elements of G n are encoded as binary strings of length n (so that it must necessarily hold that |G n | ≤ 2 n ).…”

“…This leads to the class QMA (2). While it is an open question whether or not QMA(2) and QMA are equal, some evidence exists to suggest that QMA(2) is larger than QMA.…”

“…The proof provides a method, based on a tool called the majority-certificates lemma [2], to achieve this. Given any state ρ n , the lemma specifies that there exists a polynomial number of tests, each of which can be specified by a polynomial-size quantum circuit, that the verifier V can perform on the untrusted |ψ such that, provided |ψ passes all tests, it is guaranteed that |ψ will also approximately reproduce the same outcome as ρ n with respect to all polynomial-size quantum circuits.…”

“…At the time it was not known if this problem was contained in NP, but the authors later "de-quantized" their result to obtain this fact [10]. The class BQP/qpoly was first considered in [132], and the inclusion BQP/qpoly ⊆ QMA/poly was proved in [2]. The class QMA(t) for t ≥ 2 was introduced in [120].…”

“…It will be possible to avoid the need to handle special cases at the beginning and end of interactions by adopting the convention that "absent" registers, such as Z 0 , X n , X 0 , and W −1 , and possibly Y 0 , are identified with trivial registers comprising zero qubits. 2 Our primary interest is in the situation in which the specification of a verifier is fixed, and it is to be viewed that one's goal is to optimize the actions of the prover so as to cause the verifier to produce the output 1 (representing acceptance in the terminology of interactive proofs). Generalizing the notation and terminology used in the previous chapter, we write ω(V ) to denote the value of a given verifier V in an interactive game, which is defined as the maximum probability with which a prover (compatible with V ) can cause V to output 1.…”