Freek Witteveen

A long-standing open problem in quantum complexity theory is whether QMA has perfect completeness, i.e. whether any QMA verifier can be made to have completeness $c=1$. Previous constructions have yielded a completeness parameter exponentially close to 1. We improve this to doubly-exponentially close to 1. Additionally, we show that QMA has perfect completeness if one allows the verifier an infinite-dimensional (witness) space. We show that this can be achieved using a gate set which is such that the ability to use an infinite-dimensional space does not increase the computational power of QMA. We also show that when using a finite-dimensional space of polynomially many qubits, a completeness doubly-exponentially close to 1 is optimal among black-box constructions. We show that the soundness can at most be made exponentially small using black-box reductions.

Gaussian states are widely regarded as the most important class of continuous-variable (CV) quantum states, as they naturally arise in physical systems and play a key role in quantum technologies. This motivates a fundamental question: given copies of an unknown CV state, how can we efficiently test whether it is Gaussian? We address this problem from the perspective of representation theory and quantum learning theory, characterizing the sample complexity of Gaussianity testing as a function of the number of modes. For pure states, we prove that just a constant number of copies is sufficient to decide whether the state is exactly Gaussian. We then extend this to the tolerant setting, showing that a polynomial number of copies suffices to distinguish states that are close to Gaussian from those that are far. In contrast, we establish that testing Gaussianity of general mixed states necessarily requires exponentially many copies, thereby identifying a fundamental limitation in testing CV systems. Our approach relies on rotation-invariant symmetries of Gaussian states together with the recently introduced toolbox of CV trace-distance bounds.

Abstract Unitary dynamics with a strict causal cone (or "light cone") have been studied extensively, under the name of locality preserving unitaries (LPUs) or quantum cellular automata. In particular, LPUs in one dimension have been completely classified by an index theory. Physical systems often exhibit only approximate causal cones; Hamiltonian evolutions on the lattice satisfy Lieb-Robinson bounds rather than strict locality. This motivates us to study approximately locality preserving unitaries (ALPUs). We show that the index theory is robust and completely extends to one-dimensional ALPUs. As a consequence, we achieve a converse to the Lieb-Robinson bounds: any ALPU of index zero can be exactly generated by some time-dependent, quasi-local Hamiltonian in constant time. For the special case of finite chains with open boundaries, any unitary satisfying the Lieb-Robinson bound may be generated by such a Hamiltonian. We also discuss some results on the stability of operator algebras which may be of independent interest. Session 1C Stage C