Piotr Ćwikliński

Convergence to equilibrium under a random Hamiltonian.

We analyze equilibration times of subsystems of a larger system under a random total Hamiltonian, in which the basis of the Hamiltonian is drawn from the Haar measure. We obtain that the time of equilibration is of the order of the inverse of the arithmetic average of the Bohr frequencies. To compute the average over a random basis, we compute the inverse of a matrix of overlaps of operators which permute four systems. We first obtain results on such a matrix for a representation of an arbitrary finite group and then apply it to the particular representation of the permutation group under consideration.

The power of noisy fermionic quantum computation

We consider the realization of universal quantum computation through braiding of Majorana fermions supplemented by unprotected preparation of noisy ancillae. It has been shown by Bravyi (2006 Phys. Rev. A 73 042313) that under the assumption of perfect braiding operations, universal quantum computation is possible if the noise rate on a particular four-fermion ancilla is below 40%. We show that beyond a noise rate of 89% on this ancilla the quantum computation can be efficiently simulated classically: we explicitly show that the noisy ancilla is a convex mixture of Gaussian fermionic states in this region, while for noise rates below 53% we prove that the state is not a mixture of Gaussian states. These results are obtained by generalizing concepts in entanglement theory to the setting of Gaussian states and their convex mixtures. In particular, we develop a complete set of criteria, namely the existence of a Gaussian-symmetric extension, which determine whether a state is a convex mixture of Gaussian states.

Local random quantum circuits are approximate polynomial-designs: numerical results

We numerically investigate the statement that local random quantum circuits acting on n qubits composed of polynomially many nearest-neighbor two-qubit gates form an approximate unitary poly(n)-design (Brandao et al 2012 arXiv:1208.0692). Using a group theory formalism, spectral gaps that give a ratio of convergence to a given t-design are evaluated for a different number of qubits n (up to 20) and degrees t (t = 2, 3, 4 and 5), improving previously known results for n = 2 in the case of t = 2 and 3. Their values lead to the conclusion that the previously used lower bound that bounds spectral gaps values may give very little information about the real situation and in most cases, only tells us that a gap is closed. We compare our results to another lower bounding technique, again showing that its results may not be tight.

Group-representation approach to1→Nuniversal quantum cloning machines

In this work, we revisit the problem of finding an admissible region of fidelities obtained after an application of an arbitrary

Region of fidelities for a 1→N universal qubit quantum cloner

1 \rightarrow N

Construction and properties of a class of private states in arbitrary dimensions

universal quantum cloner which has been recently solved in [A. Kay et al., Quant. Inf. Comput 13, 880 (2013)] from the side of cloning machines. Using group-theory formalism, we show that the allowed region for fidelities can be alternatively expressed in terms of overlaps of pure states with recently found irreducible representations of the commutant

A sufficient set of experimentally implementable thermal operations

U \otimes U \otimes \ldots \otimes U \otimes U^*

Generic appearance of objective results in quantum measurements

, which gives the characterization of the allowed region where states being cloned are figure of merit. Additionally, it is sufficient to take pure states with real coefficients only, which makes calculations simpler. To obtain the allowed region, we make a convex hull of possible ranges of fidelities related to a given irrep. Subsequently, two cases:

On distilling secure key from reducible private states and (non)existence of entangled key-undistillable states

1 \rightarrow 2

Irreducible private states

and