Almost all known superpolynomial quantum speedups over classical algorithms have used the quantum Fourier transform (QFT). Most known applications of the QFT make use of the QFT over abelian groups, including Shor’s well known factoring algorithm [1]. However, the QFT can be generalised to act on non-abelian groups allowing different applications. For example, Kuperberg solves the dihedral hidden subgroup problem in subexponential time using the QFT on the dihedral group. The aim of this research is to construct an efficient QFT on SU(2). Most of the progress in constructing QFTs has come from applying ideas from classical algorithms such as subgroup adapted bases. For example, Moore et al. have applied classical ideas from e.g. to build non-abelian QFTs. Applying these ideas to infinite groups such as SU(2) requires new tools. The function must be sampled or discretised in a way so as to minimise the error. I will present some ideas based on classical algorithms which may lead to a QFT over the group SU(2) for band limited functions. There are problems with making these algorithms unitary that must be addressed and efficient methods for calculating coefficients (cf. the controlled phase gates in the abelian case) must be found.
An approximate quantum encryption scheme uses a private key to encrypt a quantum state while leaking only a very small (though non-zero) amount of information to the adversary. Previous work has shown that while we need 2n bits of key to encrypt n qubits exactly, we can get away with only n bits in the approximate case, provided that we know that the state to be encrypted is not entangled with something that the adversary already has in his possession. In this talk I will show a generalization of this result: approximate quantum encryption requires roughly n-t bits of key, where t is a lower bound on the conditional min-entropy of the state to be encrypted given the adversary's prior knowledge. Along the way, I will introduce a quantum version of entropic security and show how the approximate quantum encryption scheme fits within this framework. This is joint work with Simon-Pierre Desrosiers.
Imperfections in devices are inevitable in practice. In this talk, we focus on the imperfection of QKD systems in the detectors, namely that the efficiencies of the detectors are not completely identical. We show some practical attacks that specifically exploit this efficiency mismatch and demonstrate how Eve may obtain some information on the final key if Alice and Bob are unaware of the attack. Also, we discuss the upper and lower bounds on the secret key rates both with and without the assumption of the efficiency mismatch
I will introduce Kitaev's suface codes as a block quantum error-correcting code. Recovery procedures will be described in the case of imperfect syndrome measurements. More might be covered if time permits.
Imagine that Alice and Bob share a quantum state, from which they want to distill something useful like entanglement or secret key. For this they need to communicate classically and they want to do this by one way communication from Alice to Bob. For some states, it might happen that the state is a part of a tripartite state shared with Charlie, which is invariant if Bob's and Charlie's systems are switched. Such a state is called a symmetric extension, and if it exists Alice and Bob have no chance of distilling key or entanglement by one way communication. I will present some results characterizing which quantum states have symmetric extension.