Abstract:
We show how to bound the accuracy of a family of semi-definite programming relaxations for the problem of polynomial optimization on the hypersphere. Our method is inspired by a set of results from quantum information known as quantum de Finetti theorems. In particular, we prove a de Finetti theorem for a special class of real symmetric matrices to establish the existence of approximate representing measures for moment matrix relaxations.

Abstract:
We investigate the anisotropic quantum orbital compass model on an infinite square lattice by means of the infinite projected entangled-pair state algorithm. For varying values of the $J_x$ and $J_z$ coupling constants of the model, we approximate the ground state and evaluate quantities such as its expected energy and local order parameters. We also compute adiabatic time evolutions of the ground state, and show that several ground states with different local properties coexist at $J_x = J_z$. All our calculations are fully consistent with a first order quantum phase transition at this point, thus corroborating previous numerical evidence. Our results also suggest that tensor network algorithms are particularly fitted to characterize first order quantum phase transitions.

Abstract:
We consider separating the problem of designing Hamiltonian quantum feedback control algorithms into a measurement (estimation) strategy and a feedback (control) strategy, and consider optimizing desirable properties of each under the minimal constraint that the available strength of both is limited. This motivates concepts of information extraction and disturbance which are distinct from those usually considered in quantum information theory. Using these concepts we identify an information trade-off in quantum feedback control.

Abstract:
Quantum computation can proceed solely through single-qubit measurements on an appropriate quantum state, such as the ground state of an interacting many-body system. We investigate a simple spin-lattice system based on the cluster-state model, and by using nonlocal correlation functions that quantify the fidelity of quantum gates performed between distant qubits, we demonstrate that it possesses a quantum (zero-temperature) phase transition between a disordered phase and an ordered "cluster phase" in which it is possible to perform a universal set of quantum gates.

Abstract:
We explicitly compute the optimal cost for a class of example problems in geometric quantum control. These problems are defined by a Cartan decomposition of $su(2^n)$ into orthogonal subspaces $\mathfrak{l}$ and $\mathfrak{p}$ such that $[\mathfrak{l},\mathfrak{l}] \subseteq \mathfrak{p}, [\mathfrak{p},\mathfrak{l}] = \mathfrak{p}, [\mathfrak{p},\mathfrak{p}] \subseteq \mathfrak{l}$. Motion in the $\mathfrak{l}$ direction are assumed to have negligible cost, where motion in the $\mathfrak{p}$ direction do not. In the special case of two qubits, our results correspond to the minimal interaction cost of a given unitary.

Abstract:
We imagine an experiment on an unknown quantum mechanical system in which the system is prepared in various ways and a range of measurements are performed. For each measurement M and preparation rho the experimenter can determine, given enough time, the probability of a given outcome a: p(a|M,rho). How large does the Hilbert space of the quantum system have to be in order to allow us to find density matrices and measurement operators that will reproduce the given probability distribution? In this note, we prove a simple lower bound for the dimension of the Hilbert space. The main insight is to relate this problem to the construction of quantum random access codes, for which interesting bounds on Hilbert space dimension already exist. We discuss several applications of our result to hidden variable, or ontological models, to Bell inequalities and to properties of the smooth min-entropy.

Abstract:
When a logical qubit is protected using a quantum error-correcting code, the net effect of coding, decoherence (a physical channel acting on qubits in the codeword) and recovery can be represented exactly by an effective channel acting directly on the logical qubit. In this paper we describe a procedure for deriving the map between physical and effective channels that results from a given coding and recovery procedure. We show that the map for a concatenation of codes is given by the composition of the maps for the constituent codes. This perspective leads to an efficient means for calculating the exact performance of quantum codes with arbitrary levels of concatenation. We present explicit results for single-bit Pauli channels. For certain codes under the symmetric depolarizing channel, we use the coding maps to compute exact threshold error probabilities for achievability of perfect fidelity in the infinite concatenation limit.

Abstract:
We analyse a simple exchange-based two-qubit gate for singlet-triplet qubits in gate-defined semiconductor quantum dots that can be implemented in a single exchange pulse. Excitations from the logical subspace are suppressed by a magnetic field gradient that causes spin-flip transitions to be non-energy-conserving. We show that the use of adiabatic pulses greatly reduces leakage processes compared to square pulses. We also characterise the effect of charge noise on the entanglement fidelity of the gate both analytically and in simulations; demonstrating high entanglement fidelities for physically realistic experimental parameters. Specifically we find that it is possible to achieve fidelities and gate times that are comparable to single-qubit states using realistic magnetic field gradients.

Abstract:
The standard Bell inequality experiments test for violation of local realism by repeatedly making local measurements on individual copies of an entangled quantum state. Here we investigate the possibility of increasing the violation of a Bell inequality by making collective measurements. We show that nonlocality of bipartite pure entangled states, quantified by their maximal violation of the Bell-Clauser-Horne inequality, can always be enhanced by collective measurements, even without communication between the parties. For mixed states we also show that collective measurements can increase the violation of Bell inequalities, although numerical evidence suggests that the phenomenon is not common as it is for pure states.

Abstract:
We derive the effective channel for a logical qubit protected by an arbitrary quantum error-correcting code, and derive the map between channels induced by concatenation. For certain codes in the presence of single-bit Pauli errors, we calculate the exact threshold error probability for perfect fidelity in the infinite concatenation limit. We then use the control theory technique of balanced truncation to find low-order non-asymptotic approximations for the effective channel dynamics.