Welcome to Combinatorics and Optimization

Abstract:The satisfiability threshold of a random constraint satisfaction problem (CSP) is the density of constraints at which a random CSP instance transitions from being satisfiable to unsatisfiable with high probability. Much of the research on well known CSPs, including the $k$-SAT problem, $k$-XORSAT problem, hypergraph colouring, and systems of linear equations, has focused on determining satisfiability thresholds.

In this talk we consider systems of linear equations over finite commutative rings as CSPs, and build on the work of Ayre, Coja-Oghlan, Gao, and Müller, who determined the satisfiability threshold for random linear equations over a finite field. We determine when the satisfiability threshold is linear in the number of variables, and show that any linear threshold over a principal ideal ring coincides with the (unique) linear threshold over fields. We also determine the satisfiability threshold for some examples of non-principal ideal rings.

This is joint work with Jane Gao.

There will be a pre-seminar presenting relevant background at beginning graduate level starting at 1:30pm in MC 5417.

Abstract:  Given copies of a quantum state, a shadow tomography protocol aims to learn all expectation values from a fixed set of observables, to within a given precision. We say that such a protocol is triply efficient if it is sample efficient, time efficient, and uses measurements that entangle a constant number of copies of the state at a time.   A natural family of shadow tomography protocols based on random single-copy Clifford measurements can be understood as arising from fractional colorings of a graph G that encodes the commutation structure of the set of observables. Here we describe a framework for two-copy shadow tomography that uses an initial round of Bell measurements to reduce to a fractional coloring problem in an induced subgraph of G with bounded clique number. This coloring problem can be addressed using techniques from graph theory known as chi-boundedness. Using this framework we give the first triply efficient  shadow tomography scheme for the set of local fermionic observables, which arise in a broad class of interacting fermionic systems in physics and chemistry. We also give a triply efficient scheme for the set of all -qubit Pauli observables. Our protocols for these tasks use two-copy measurements, which is necessary: sample-efficient schemes are provably impossible using only single-copy measurements. This is joint work with Robbie King, Robin Kothari, and Ryan Babbush.