Talks and presentations

The power of a single Haar random state: constructing and separating quantum pseudorandomness

September 03, 2024

Talk, QCrypt 2024, Vigo, Spain

In this work, we focus on the following question: what are the cryptographic implications of having access to an oracle that provides a single Haar random quantum state? We show, perhaps surprisingly, that such an oracle is sufficient to construct quantum pseudorandomness. Pseudorandom states (PRS) are a family of states for which it is hard to distinguish between polynomially many copies of either a state sampled uniformly from the family or a Haar random state. A weaker notion, called single-copy pseudorandom states (1PRS), satisfies this property with respect to a single copy. We obtain the following results:

1. First, we show, perhaps surprisingly, that 1PRS (as well as bit-commitments) exist relative to an oracle that provides a single Haar random state.
2. Second, we build on this result to show the existence of a unitary oracle relative to which 1PRS exist, but PRS do not. Taken together, our contributions yield one of the first black-box separations between central notions of quantum pseudorandomness, and introduce a new framework to study black-box separations between various inherently quantum primitives.

A Convergence Theory for Over-parameterized Variational Quantum Eigensolvers

February 06, 2023

Talk, QIP 2023, Ghent, Belgium

The Variational Quantum Eigensolver (VQE) is a promising candidate for quantum applications on near-term Noisy Intermediate-Scale Quantum (NISQ) computers. Despite a lot of empirical studies and recent progress in theoretical understanding of VQE’s optimization landscape, the convergence for optimizing VQE is far less understood. We provide the first rigorous analysis of the convergence of VQEs in the over-parameterization regime. By connecting the training dynamics with the Riemannian Gradient Flow on the unit-sphere, we establish a threshold on the sufficient number of parameters for efficient convergence, which depends polynomially on the system dimension and the spectral ratio, a property of the problem Hamiltonian, and could be resilient to gradient noise to some extent. We further illustrate that this over-parameterization threshold could be vastly reduced for specific VQE instances by establishing an ansatz-dependent threshold paralleling our main result. We showcase that our ansatz-dependent threshold could serve as a proxy of the trainability of different VQE ansatzes without performing empirical experiments, which hence leads to a principled way of evaluating ansatz design. Finally, we conclude with a comprehensive empirical study that supports our theoretical findings.