Hardness of approximation for quantum problems

The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact hard to approximate. Using these techniques, we also obtain hardness of approximation for the class QCMA. Our approach is based on the use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy [Umans, FOCS 1999]. The problems for which we prove hardness of approximation for include, among others, a quantum version of the Succinct Set Cover problem, and a variant of the local Hamiltonian problem with hybrid classical-quantum ground states.Comment: 21 pages, 1 figure, extended abstract appeared in Proceedings of the 39th International Colloquium on Automata, Languages and Programming (ICALP), pages 387-398, Springer, 201

Combined Error Correction Techniques for Quantum Computing Architectures

Proposals for quantum computing devices are many and varied. They each have unique noise processes that make none of them fully reliable at this time. There are several error correction/avoidance techniques which are valuable for reducing or eliminating errors, but not one, alone, will serve as a panacea. One must therefore take advantage of the strength of each of these techniques so that we may extend the coherence times of the quantum systems and create more reliable computing devices. To this end we give a general strategy for using dynamical decoupling operations on encoded subspaces. These encodings may be of any form; of particular importance are decoherence-free subspaces and quantum error correction codes. We then give means for empirically determining an appropriate set of dynamical decoupling operations for a given experiment. Using these techniques, we then propose a comprehensive encoding solution to many of the problems of quantum computing proposals which use exchange-type interactions. This uses a decoherence-free subspace and an efficient set of dynamical decoupling operations. It also addresses the problems of controllability in solid state quantum dot devices.Comment: Contribution to Proceedings of the 2002 Physics of Quantum Electronics Conference", to be published in J. Mod. Optics. This paper provides a summary and review of quant-ph/0205156 and quant-ph/0112054, and some new result

Riverine transfer of heavy metals from Patagonia to the southwestern Atlantic Ocean

The occurrence and geochemical behaviour of Fe, Mn, Pb, Cu, Ni, Cr, Zn and Co are studied in riverine detrital materials transported by Patagonian rivers. Their riverine inputs have been estimated and the nature of these inputs to the Atlantic Ocean is discussed. Most of the metals are transported to the ocean via the suspended load; there is evidence that Fe oxides and organic matter are important phases controlling their distribution in the detrital non-residual fraction. Most heavy metal concentrations found in bed sediments, in suspended matter, and in the dissolved load of Patagonian rivers were comparable to those reported for non-polluted rivers. There is indication that human activity is altering riverine metal inputs to the ocean. In the northern basins – and indicatinganthropogenic effects – heavy metals distribution in the suspended load is very different from that found in bed sediments. The use of pesticides in the Negro River valley seems correlated with increased riverine input of Cu, mostly bound to the suspended load. The Deseado and Chico Rivers exhibit increased specific yield of metals as a consequence of extended erosion within their respective basins. The Santa Cruz is the drainage basin least affected by human activity and its metal-exporting capacity should be taken as an example of a relatively unaffected large hydrological system. In contrast, coal mining modifies the transport pattern of heavy metals in the Gallegos River, inasmuch as they are exported to the coastal zone mainly as dissolved load

Universal 2-local Hamiltonian Quantum Computing

We present a Hamiltonian quantum computation scheme universal for quantum computation (BQP). Our Hamiltonian is a sum of a polynomial number (in the number of gates L in the quantum circuit) of time-independent, constant-norm, 2-local qubit-qubit interaction terms. Furthermore, each qubit in the system interacts only with a constant number of other qubits. The computer runs in three steps - starts in a simple initial product-state, evolves it for time of order L^2 (up to logarithmic factors) and wraps up with a two-qubit measurement. Our model differs from the previous universal 2-local Hamiltonian constructions in that it does not use perturbation gadgets, does not need large energy penalties in the Hamiltonian and does not need to run slowly to ensure adiabatic evolution.Comment: recomputed the necessary number of interactions, new geometric layout, added reference

Decoherence and Quantum Walks: anomalous diffusion and ballistic tails

The common perception is that strong coupling to the environment will always render the evolution of the system density matrix quasi-classical (in fact, diffusive) in the long time limit. We present here a counter-example, in which a particle makes quantum transitions between the sites of a d-dimensional hypercubic lattice whilst strongly coupled to a bath of two-level systems which 'record' the transitions. The long-time evolution of an initial wave packet is found to be most unusual: the mean square displacement of the particle density matrix shows long-range ballitic behaviour, but simultaneously a kind of weakly-localised behaviour near the origin. This result may have important implications for the design of quantum computing algorithms, since it describes a class of quantum walks.Comment: 4 pages, 1 figur