This series consists of weekly discussion sessions on foundations of quantum Theory and quantum information theory. The sessions start with an informal exposition of an interesting topic, research result or important question in the field. Everyone is strongly encouraged to participate with questions and comments.
[joint work with: Victor Albert, John Preskill (Caltech), Sepehr Nezami, Grant Salton, Patrick Hayden (Stanford University), and Fernando Pastawski (Freie Universität Berlin)]
Suppose the eigenvalue distributions of two matrices $M_1$ and $M_2$ are known. What is the eigenvalue distribution of the sum $M_1+M_2$? This problem has a rich pure mathematics history dating back to H. Weyl (1912) with many applications in various fields. Free probability theory (FPT) answers this question under certain conditions, which often involves some degree of randomness (disorder). We will describe FPT and show examples of its powers for the qualitative understanding (often approximations) of physical quantities such as density of states, and gapped vs.
When a system interacts with an environment with which it is initially uncorrelated, its evolution is described by a completely positive map. The common wisdom in the field of quantum information theory, however, is that when the system is initially correlated with the environment, the map describing its evolution may fail to be completely positive.
Port-based teleportation (PBT) is a variant of the well-known task of quantum teleportation in which Alice and Bob share multiple entangled states called "ports". While in the standard teleportation protocol using a single entangled state the receiver Bob has to apply a non-trivial correction unitary, in PBT he merely has to pick up the right quantum system at a port specified by the classical message he received from Alice. PBT has applications in instantaneous non-local computation and can be used to attack position-based quantum cryptography.
Several equivalent formulations of quantum error correction condition will be introduced. Subtleties arise
when the error correction conditions hold only approximately. We will discuss an equivalent formulation that is
robust to the approximation error. One can leverage this tool to derive the existence of approximate quantum
error correcting code at low energy subspace of CFT that reproduces aspects of the holographic quantum error
correcting code. Using the same tool, we observe that two operators with greatly differing complexity approximately
Randomness is an essential tool in many disciplines of modern sciences, such as cryptography, black hole physics, random matrix theory, and Monte Carlo sampling. In quantum systems, random operations can be obtained via random circuits thanks to so-called q-designs and play a central role in condensed-matter physics and in the fast scrambling conjecture for black holes. Here, we consider a more physically motivated way of generating random evolutions by exploiting the many-body dynamics of a quantum system driven with stochastic external pulses.
I'll ask whether the knowledge of a single eigenstate of a local lattice Hamiltonian is sufficient to uniquely determine the Hamiltonian. I’ll present evidence that the answer is yes for generic local Hamiltonians, given either the ground state or an excited state. In fact, knowing only the correlation functions of local observables with respect to the eigenstate appears generically sufficient to exactly recover both the eigenstate and the Hamiltonian, with efficient numerical algorithms.
Quantum computers can only offer a computational advantage when they have sufficiently many qubits operating with sufficiently small error rates. In this talk, I will show how both these requirements can be practically characterized by variants of randomized benchmarking protocols. I will first show that a simple modification to protocols based on randomized benchmarking allows multiplicative-precision estimates of error rates. I will then outline a new protocol for estimating the fidelity of arbitrarily large quantum systems using only single-qubit randomizing gates.
Stabilizer states are a rich class of quantum states which can be efficiently classically represented and manipulated. In this talk I will describe some ways in which they can help us to represent and manipulate more general quantum states. I will discuss classical simulation algorithms for quantum circuits which are based on expressing a quantum state as a superposition of (as few as possible) stabilizer states.
Based on arXiv:1601.07601 (with Sergey Bravyi) and work in progress with Sergey Bravyi, Dan Browne, Padraic Calpin, Earl Campbell and Mark Howard.
Adiabatic quantum computation (AQC) is a method for performing universal quantum computation in the ground state of a slowly evolving local Hamiltonian, and in an idea