David Kult

Robustness of the adiabatic quantum search

The effects of open system evolution on single particle interferometry, quantum computation, and the adiabatic approximation are investigated.Single particle interferometry: Three concepts concerning completely positive maps (CPMs) and trace preserving CPMs (channels), named subspace preserving (SP) CPMs, subspace local channels, and gluing of CPMs, are introduced. SP channels preserve probability weights on given orthogonal sum decompositions of the Hilbert space of a quantum system. Subspace locality determines what channels act locally with respect to such decompositions. Gluings are the possible total channels obtainable if two evolution devices, characterized by channels, act jointly on a superposition of a particle in their inputs. It is shown that gluings are not uniquely determined by the two channels. We determine all possible interference patterns in single particle interferometry for given channels acting in the interferometer paths. It is shown that the standard interferometric setup cannot distinguish all gluings, but a generalized setup can.Quantum computing: The robustness of local and global adiabatic quantum search subject to decoherence in the instantaneous eigenbasis of the search Hamiltonian, is examined. In both the global and local search case the asymptotic time-complexity of the ideal closed case is preserved, as long as the Hamiltonian dynamics is present. In the case of pure decoherence, where the environment monitors the search Hamiltonian, it is shown that the local adiabatic quantum search performs as the classical search with scaling N, and that the global search scales like N3/2 , where N is the list length. We consider success probabilities p<1 and prove bounds on the run-time with the same scaling as in the conditions for the p → 1 limit.Adiabatic evolution: We generalize the adiabatic approximation to the case of open quantum systems in the joint limit of slow change and weak open system disturbances.

Noncyclic geometric changes of quantum states

Non-Abelian quantum holonomies, i.e., unitary state changes solely induced by geometric properties of a quantum system, have been much under focus in the physics community as generalizations of the Abelian Berry phase. Apart from being a general phenomenon displayed in various subfields of quantum physics, the use of holonomies has lately been suggested as a robust technique to obtain quantum gates; the building blocks of quantum computers. Non-Abelian holonomies are usually associated with cyclic changes of quantum systems, but here we consider a generalization to noncyclic evolutions. We argue that this open-path holonomy can be used to construct quantum gates. We also show that a structure of partially defined holonomies emerges from the open-path holonomy. This structure has no counterpart in the Abelian setting. We illustrate the general ideas using an example that may be accessible to tests in various physical systems.

Quantum adiabatic search with decoherence in the instantaneous energy eigenbasis

In Phys. Rev. A 71, 060312(R) (2005), the robustness of the local adiabatic quantum search to decoherence in the instantaneous eigenbasis of the search Hamiltonian was examined. We expand this analysis to include the case of the global adiabatic quantum search. As in the case of the local search the asymptotic time complexity for the global search is the same as for the ideal closed case, as long as the Hamiltonian dynamics is present. In the case of pure decoherence, where the environment monitors the search Hamiltonian, we find that the time complexity of the global quantum adiabatic search scales like N{sup 3/2}, where N is the list length. We moreover extend the analysis to include success probabilities p<1 and prove bounds on the run time with the same scaling as in the conditions for the p{yields}1 limit. We supplement the analytical results by numerical simulations of the global and local search.

Operational approach to the Uhlmann holonomy

We suggest a physical interpretation of the Uhlmann amplitude of a density operator. Given this interpretation we propose an operational approach to obtain the Uhlmann condition for parallelity. This allows us to realize parallel transport along a sequence of density operators by an iterative preparation procedure. At the final step the resulting Uhlmann holonomy can be determined via interferometric measurements.

Non-Abelian generalization of off-diagonal geometric phases

If a quantum system evolves in a noncyclic fashion the corresponding geometric phase or holonomy may not be fully defined. Off-diagonal geometric phases have been developed to deal with such cases. Here, we generalize these phases to the non-Abelian case, by introducing off-diagonal holonomies that involve evolution of more than one subspace of the underlying Hilbert space. Physical realizations of the off-diagonal holonomies in adiabatic evolution and interferometry are put forward.

Nodal free geometric phases: Concept and application to geometric quantum computation

Nodal free geometric phases are the eigenvalues of the final member of a parallel transporting family of unitary operators. These phases are gauge invariant, always well defined, and can be measured interferometrically. Nodal free geometric phases can be used to construct various types of quantum phase gates.

Geometry of decomposition dependent evolutions of mixed states

We examine evolutions where each component of a given decomposition of a mixed quantal state evolves independently in a unitary fashion. The geometric phase and parallel transport conditions for th ...

Holonomy for quantum channels

Quantum holonomies are investigated in different contexts.A geometric phase is proposed for decomposition dependent evolution, where each component of a given decomposition of a mixed state evolves independently. It is shown that this geometric phase only depends on the path traversed in the space of decompositions.A holonomy is associated to general paths of subspaces of a Hilbert space, both discrete and continuous. This opens up the possibility of constructing quantum holonomic gates in the open path setting. In the discrete case it is shown that it is possible to associate two distinct holonomies to a given path. Interferometric setups for measuring both holonomies areprovided. It is further shown that there are cases when the holonomy is only partially defined. This has no counterpart in the Abelian setting.An operational interpretation of amplitudes of density operators is provided. This allows for a direct interferometric realization of Uhlmanns parallelity condition, and the possibility of measuring the Uhlmann holonomy for sequences of density operators.Off-diagonal geometric phases are generalized to the non-Abelian case. These off-diagonal holonomies are undefined for cyclic evolution, but must contain members of non-zero rank if all standard holonomies are undefined. Experimental setups for measuring the off-diagonal holonomies are proposed.The concept of nodal free geometric phases is introduced. These are constructed from gauge invariant quantities, but do not share the nodal point structure of geometric phases and off-diagonal geometric phases. An interferometric setup for measuring nodal free geometric phases is provided, and it is shown that these phases could be useful in geometric quantum computation.A holonomy associated to a sequence of quantum maps is introduced. It is shown that this holonomy is related to the Uhlmann holonomy. Explicit examples are provided to illustrate the general idea.