A Full Characterization of Quantum Advice

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.

💡 Research Summary

The paper presents a striking new characterization of quantum advice, showing that any n‑qubit quantum state ρ can be replaced by the ground state of a polynomial‑size local Hamiltonian H (e.g., a sum of two‑qubit interactions) without loss of computational power. The authors prove that for any fixed polynomial‑size quantum circuit family, a ground state of H can simulate the behavior of ρ on all such circuits. This structural result yields the complexity‑theoretic inclusion BQP/qpoly ⊆ QMA/poly, improving on Aaronson’s earlier containment BQP/qpoly ⊆ PP/poly. Consequently, quantum advice is exactly as powerful as untrusted quantum advice supplemented with trusted classical advice.

The proof combines several existing tools with a novel “majority‑certificates lemma.” First, the authors rely on Aaronson’s result that an n‑qubit quantum state can be learned to ε‑precision from a polynomial number of classical samples, providing a compact classical description. However, this description alone does not suffice to verify an untrusted quantum witness. To bridge this gap, they invoke the QMA+ framework of Aharonov and Regev, which allows a verifier to check that a quantum proof satisfies certain expected‑value constraints, thereby strengthening the verification power of QMA.

The new majority‑certificates lemma is a combinatorial statement about Boolean function families. Given a set S of Boolean functions on n variables, any f ∈ S can be expressed as the pointwise majority of m = O(n) functions f₁,…,f_m ∈ S, where each f_i is uniquely determined within S by only O(log|S|) input‑output constraints. This is reminiscent of boosting in machine learning: a small number of “certificates” (constraints) suffice to isolate each component function, and a modest collection of such components collectively reproduces any target function.

Applying this lemma to quantum advice proceeds as follows. The space of possible n‑qubit states is discretized into a finite set S of “candidate” states that are distinguishable by polynomial‑size circuits. By the learning results, each candidate can be identified by a short list of measurement outcomes (the constraints). The majority‑certificates lemma guarantees that any target state ρ can be written as the majority of O(poly(n)) candidates, each uniquely fixed by O(log|S|) constraints. These constraints are then encoded as local terms of a Hamiltonian H: each constraint corresponds to a two‑qubit interaction whose ground‑state energy penalizes violations. Consequently, the unique ground state of H satisfies all constraints simultaneously and therefore implements the majority‑certificate representation of ρ.

From a complexity‑theoretic perspective, a BQP/qpoly algorithm that receives ρ as advice can be simulated by a QMA verifier that receives as classical advice the description of H (poly(n) bits) and as quantum proof the ground state of H. The verifier checks the ground‑state property using standard QMA techniques (e.g., phase estimation) and then runs the original circuit on this state. Because the ground state reproduces the behavior of ρ on all relevant circuits, the simulation is exact. Hence every language in BQP/qpoly also lies in QMA/poly.

The paper concludes with several implications. First, it shows that quantum advice does not provide any advantage beyond what can be achieved with a trusted classical description of a local Hamiltonian plus an untrusted quantum witness. Second, the majority‑certificates lemma may find independent applications in quantum learning theory, quantum cryptography, and the design of approximate Hamiltonians, as it provides a systematic way to compress a large function class into a small set of certifiable components. Finally, the work deepens the connection between quantum complexity classes and physical models, suggesting that the study of ground‑state physics can illuminate the limits of quantum information processing.