Sabee Grewal

We investigate two resources whose effects on quantum interactive proofs remain poorly understood: the promise of unentanglement, and the verifier’s ability to condition on an intermediate measurement, which we call post-measurement branching. We first show that unentanglement can dramatically increase computational power: three-round unentangled quantum interactive proofs equal NEXP, even if only the first message is quantum. By contrast, we prove that if the verifier uses no post-measurement branching, then the same type of unentangled proof system has at most the power of QAM. Finally, we investigate post-measurement branching in two-round quantum-classical proof systems. Unlike the equivalence between public-coin and private-coin classical interactive proofs, we give evidence of a separation in the quantum setting that arises from post-measurement branching.

We prove several new results concerning the pure quantum polynomial hierarchy pureQPH. First, we show that QMA(2) ⊆ pureQΣ_2, i.e., two unentangled existential provers can be simulated by competing existential and universal provers. We further prove that pureQΣ_2 ⊆ QΣ_3 ⊆ NEXP. Second, we give an error reduction result for pureQPH, and, as a consequence, prove that pureQPH = QPH. A key ingredient in this result is an improved dimension-independent disentangler. Finally, we initiate the study of quantified Hamiltonian complexity, the quantum analogue of quantified Boolean formulae. We prove that the quantified pure sparse Hamiltonian problem is pureQΣ_i-complete. By contrast, other natural variants (pure/local, mixed/local, and mixed/sparse) admit nontrivial containments but fail to be complete under known techniques. For example, we show that the ∃∀-mixed local Hamiltonian problem lies in NP^QMA ∩ coNP^QMA.

We show that any pseudoentangled state ensemble with a gap of t bits of entropy requires Ω(t) non-Clifford gates to prepare. This bound is tight up to polylogarithmic factors if linear-time quantum-secure pseudorandom functions exist. Our result follows from a polynomial-time algorithm to estimate the entanglement entropy of a quantum state across any cut of qubits. When run on an n-qubit state that is stabilized by at least 2^n−t Pauli operators, our algorithm produces an estimate that is within an additive factor of t/2 bits of the true entanglement entropy.