Quantum Information

Quantum field theory provides the framework for the Standard Model of particle physics and plays a key role in many areas of physics. However, calculations are generally computationally complex and limited to weak interaction strengths. I shall describe a polynomial-time algorithm for computing, on a quantum computer, relativistic scattering amplitudes in massive scalar quantum field theories. The quantum algorithm applies at both weak and strong coupling, achieving exponential speedup over known classical methods at high precision or strong coupling. The study of such quantum algorithms may also help us learn more about the nature and foundations of quantum field theory itself.

Many interesting physical systems in condensed matter physics may be described in the language of error-correcting codes. In this talk, we illustrate the applications of coding theoretical techniques to problems in many-body physics by reviewing our recent works. In particular, we discuss (1) the classification of quantum phases via local quantum codes, (2) thermal stability of topological order and its relation to feasibility of self-correcting quantum memory, and (3) information storage capacity of discrete spin systems.

One of the major obstacles in quantum information processing is to prevent a quantum bit from decoherence. One powerful approach to protect quantum coherence is dynamical decoupling. I will present some recent progress of diamond-based quantum information processing using dynamical decoupling. The other promising approach is to use topological quantum systems, which are intrinsically insensitive to local perturbations. I will discuss some ideas to create and probe topological quantum systems. Furthermore, I will propose a hybrid platform between topological and conventional quantum systems, which can combine the advantages from both.

Relativistic quantum information theory uses well-known tools coming from quantum information and quantum optics to study quantum effects provoked by gravity and to learn information about the spacetime. One can take advantage of our knowledge about quantum optics and quantum information theory to analyse from a new perspective the effects produced by the gravitational interaction. I will present some results and new ideas in this topic: from experimental proposals for detection of the Unruh and Hawking effects and cosmological implications of gravitationally generated entanglement to quantum simulation of general relativistic settings.

We generalize the result of Bravyi et al. on the stability of the spectral gap for frustration-free, commuting Hamiltonians, by removing the assumption of commutativity and weakening the assumptions needed for stability.

The model of an arbitrarily varying quantum channel will be introduced in strict analogy to the classical definition by Blackwell, Breiman and Thomasian. We will then consider the task of entanglement transmission over such a channel and take a look at the methods, both from classical and quantum information theory, that enter the direct part of our proof of a quantum version of Ahlswede's dichotomy for the capacity of classical arbitrarily varying channels. Differences to the classical setting will be pointed out.

A symmetric informationally complete positive-operator-valued measure (SIC POVM) is a special POVM that is composed of d^2 subnormalized pure projectors with equal pairwise fidelity. It may be considered a fiducial POVM for reasons of its high symmetry and high tomographic efficiency. Most known SIC POVMs are covariant with respect to the Heisenberg-Weyl (HW) group. We show that in prime dimensions the HW group is the unique group that may generate a SIC POVM. In particular, in prime dimensions not equal to three, each group covariant SIC POVM is covariant with respect to a unique HW group. In addition, the symmetry group of the SIC POVM is a subgroup of the Clifford group, which is the normalizer of the HW group. Hence, two SIC POVMs covariant with respect to the HW group are unitarily equivalent if and only if they are on the same orbit of the Clifford group. In dimension three, each group covariant SIC POVM may be covariant with respect to three or nine HW groups, and the symmetry group of the SIC POVM is a subgroup of at least one of the Clifford groups of these HW groups respectively. There may exist two or three orbits of equivalent SIC POVMs for each group covariant SIC POVM, depending on the order of its symmetry group.

We review situations under which a standard quantum adiabatic condition fails. We reformulate the problem of adiabatic evolution as the problem of Hamiltonian eigenpath traversal, and give convergence conditions in terms of the length of the eigenpath and the minimum energy gap of the Hamiltonians. We introduce a randomized evolution method that can be used to traverse the eigenpath and prove its convergence and cost. We then describe more efficient methods for the same task and show that their implementation complexity is close to optimal.

Quantum key distribution (QKD) is an application of quantum theory as its security relies on quantum foundations, at the same time there is development in the information-theoretic point of view to quantum theory. The security is related to impossible quantum performance, for instance, neither perfect quantum cloning nor perfect quantum state discrimination are possible. In this talk, I would like to discuss issues relevant to practical and fundamental sides of QKD: i) toward characterization of quantum correlations from which a secret key can be distilled, ii) determination whether quantum states shared by two honest parties in distance are entangled or separable, and iii) limitations on quantum performance by fundamental principles in quantum theory.

We find analytic models that can perfectly transfer, without state initialization or remote collaboration, arbitrary functions in two- and three-dimensional interacting bosonic and fermionic networks. This provides for the possible experimental implementation of state transfer through bosonic or fermionic atoms trapped in optical lattices. A significant finding is that the state of a spin qubit can be perfectly transferred through a fermionic system. Families of Hamiltonians are described that are related to the linear models and that enable the perfect transfer of arbitrary functions. Furthermore, we propose methods for eliminating certain types of errors

Topological order is a new kind of collective order which appears in two-dimensional quantum systems such as the fractional quantum Hall effect and brings about rather unusual particles: unlike bosons or fermions these anyons obey exotic statistics and can be exploited to perform quantum computation. Topological order also implies that quantum states at low energies exhibit a very subtle, yet intricate inner structure. Remarkably, both phenomena can be studied in relatively simple spin systems (like Kitaev's quantum double models and the ubiquitous toric code) which in fact capture the essential properties of entire topological phases of matter in many important cases. What is the relationship between these topological phases? Reviewing recent work I will explain how they arrange themselves in a landscape of dualities and hierarchies. In particular, I will focus on two aspects: first, a duality between electric and magnetic quasiparticles in generalized quantum double models and second, a hierarchy construction of quantum states which are related by the condensation of topological charges.