Quantum Physics

In this talk I address the problem of simultaneously inferring unknown quantum states and unknown quantum measurements from empirical data. This task goes beyond state tomography because we are not assuming anything about the measurement devices. I am going to talk about the time and sample complexity of the inference of states and measurements, and I am going to talk about the robustness of the minimal Hilbert space dimension. Moreover, I will describe a simple heuristic algorithm (alternating optimization) to fit states and measurements to empirical data. For this algorithm the dataset does not need to be quantum. Hence, the proposed algorithm enables us to interpret general datasets from a quantum perspective. By analyzing movie ratings, we demonstrate the power of quantum models in the context of item recommendation which is a key discipline in machine learning. We observe that quantum models can compete with state-of-the-art algorithms for item recommendation. Based on joint work with Aram Harrow. Relevant preprints: arXiv:1412.7437 and arXiv:1510.02800.

In this talk we will discuss how C*-algebras can be identified and characterised in terms of certain cosheaves with self-action. The reason we are interested in such a study is it to try and give a rigorous mathematical derivation of the axioms of quantum theory. In particular, many of the standard axioms for C*-algebras have unclear physical and operational meaning, but by defining an equivalence of categories between C*-algebras and cosheaves with self-action, we believe that these axioms can acquire a clear operational meaning. So far, however, we have only managed to show that every C*-algebra becomes a cosheaf with self-action, but we don't know whether the converse holds as well. This work is done in collaboration with Tobias Fritz.

The accumulated intuition from the last decades of research on quantum entanglement is that this phenomenon is highly non-robust, and very hard to maintain in the presence of de-cohering noise at non-zero temperatures. In recent years however, and motivated, in part, by a quest for a quantum analog of the PCP theorem researches have tried to establish, at least in theory, whether or not we can preserve quantum entanglement at "constant" temperatures that are independent of system size. This would imply that any quantum state with energy at most, say 0.05 of the total available energy of the Hamiltonian, would be highly-entangled.

A conjecture formalizing this notion was defined by Freedman and Hastings : called NLTS - it stipulates the existence of locally-defined quantum systems that retain long-range entanglement even at high temperatures. Such a conjecture does not only present a necessary condition for quantum PCP, but also poses a fundamental question on the nature of entanglement itself. To this date, no such systems were found, and moreover, it became evident that even embedding local Hamiltonians on robust, albeit "non-physical" topologies, namely expanders, does not guarantee entanglement robustness.

In this study, refute the intuition that entanglement is inherently fragile: we show that locally-defined quantum systems can, in fact, retain long-range entanglement at high temperatures.  To do this, we construct an explicit family of 7-local Hamiltonians, and prove that for such local Hamiltonians ANY low-energy state is hard to even approximately simulate by low-depth quantum circuits of depth o(log(n)). In particular, this resolves the NLTS conjecture in the affirmative, and suggests the existence of quantum systems whose low-energy states are not only highly-entangled but also "usefully"-entangled, in the computational-theoretic sense. 

I will introduce the unitarity, a parameter quantifying the coherence of a channel and show that it is useful for two reasons. First, it can be efficiently estimated via a variant of randomized benchmarking. Second, it captures useful information about the channel, such as the optimal fidelity achievable with unitary corrections and an improved bound on the diamond distance.

I’ll present new approaches to the problems of quantum control and quantum tomography wherein no classical simulation is required. The experiment itself performs the simulation (in situ) and, in a sense, guides itself to the correct solution. The algorithm is iterative and makes use of ideas from stochastic optimization theory.