Constraints on Measurement-Based Quantum Computation in Effective Cluster States

Abstract

The aim of this work is to study the physical properties of a one-way quantum computer in an effective low-energy cluster state. We calculate the optimal working conditions as a function of the temperature and of the system parameters. The central result of our work is that any effective cluster state implemented in a perturbative framework is fragile against special kinds of external perturbations. Qualitative aspects of our work are important for any implementation of effective low-energy models containing strong multisite interactions.

Symmetry protection of measurement-based quantum computation in ground states

“…Let us suppose that G k results from the measurement sequence corresponding to a unitary gate U k . Then, after tracing out the physical site, equation (48) corresponds to an evolution on the ancilla qubit described by the CPTP map…”

“…Furthermore, E k acts non-trivially only on the same qubits that were acted on by the corresponding gate in the original noiseless quantum circuit. The strength of the noise, as given by E k − I ♦ , is determined by the deviation (in the trace norm) of the reduced density operator on Q k from its unperturbed value (see equation (48) in section 2.6), which should be small for small perturbations.…”

“…Now we will return to the model we are actually interested in: the 2D cluster model on a square lattice. Investigations of the effect on this model of perturbations [13,[47][48][49] have demonstrated a variety of results depending on the perturbation. Here, we will focus on perturbations respecting an appropriate symmetry.…”

Competing topological orders in three dimensions

“…We computed several high-order series expansions using the Löwdin method [53]. This method has been applied successfully in related contexts like the robustness of effective cluster states in measurement-based quantum computation [54] as well as topological string-net phases [55]. Furthermore, the application of the Löwdin method to perturbed topological models is well described in Ref.…”

Measurement-Based Quantum Computation on Two-Body Interacting Qubits with Adiabatic Evolution

“…Each logical qubit j is encoded in four spin-1=2 particles and, there exists an exact ground state after controlled-phase transformation (CZ) on every bond [22,23], i.e.,…”