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The quantum PCP conjecture is one of the major open problems in quantum complexity theory. It has resisted attack in part because many primitives used in the proof of the classical PCP theorem, such as locality-preserving gap amplification and alphabet reduction, have no obvious quantum analogues due to quantum no-cloning. Locality-preserving gap amplification is a procedure that takes as input a local Hamiltonian problem instance and produces a new instance with a larger promise gap, without increasing the locality of the Hamiltonian, and instead moderately increasing its local qudit dimension. Obtaining this kind of control over the locality during gap amplification is critical to the success of every known strategy for proving the classical PCP theorem. Here, we put forth the first known viable template for quantum locality-preserving gap amplification, and we prove that our procedure amplifies combinatorial gap. Our work introduces a new framework for reasoning about quantum gap amplification in terms of fault-tolerant computation, and illuminates a route toward importing one of the central structural ingredients in classical PCPs into the quantum setting.
The early days of fault-tolerant quantum computing (FTQC) are rapidly approaching. Many contemporary architecture proposals are built on quantum low-density parity-check (QLDPC) codes — a subject that was largely an asymptotic theory five years ago. How did this happen? What do these computers look like and how do they operate? In this talk, I will present an overview of modern QLDPC architectures, highlight the key methods and design choices, and contemplate where we should go from here.
We consider a family of general Heisenberg models studied by Suzuki and Fisher in 1971. This family includes the Heisenberg antiferromagnet on any bipartite graph as well as certain models with sign problems. The ground and Gibbs states of these models are Lee-Yang tensors, meaning that their generating polynomials do not vanish in the unit polydisk in the complex plane. We show that each Hamiltonian in this family has spectral gap proportional to the magnetic field strength in the Z direction. Using this result, we obtain an efficient quantum adiabatic algorithm for the ground energy of any model in this family, hence providing candidates for quantum advantage. The proof is based on a new inequality that bounds the relative spectral gap of positive semidefinite operators which are Lee-Yang tensors.
Modern classical processors are built at the scale of 100B+ transistors in a single device while only possessing a small number of connections to the outside world. In this talk, I will present a potential direction to build fault-tolerant quantum computers at similar scales.