QUIE2T Quantum Information Entanglement-Enabled Technologies

Topological quantum information processing and computation

Topological quantum computation (TQC) is an approach to quantum information processing that eliminates decoherence at the hardware level by encoding quantum states and gates in global, delocalized properties of the hardware medium.

Most of the current quantum computing schemes assume nearly perfect shielding from the environment. Decoherence makes quantum computing prone to error and nonscalable, allowing only for very small "proof of principle" devices. Error correction software can in principle solve this problem, but progress along this path will take a long time. While much of the current research on other approaches to quantum computation is focused on improving control over well-understood physical systems, TQC research promises fundamental breakthroughs.

Delocalized, or topological degrees of freedom are intrinsically immune to all forms of noise which do not impact the entire medium at once and coherently. For media which exhibit an energy gap, kept at low enough temperatures, this is in fact all conceivable noise. If such materials can be constructed or found in nature, they will allow a much cleaner and faster realization of scalable quantum computation than other schemes.

TQC can be realized in effectively planar (2D) systems whose quasiparticles are anyons, that is they have nontrivial exchange behavior, different from that of bosons or fermions. If, in a system of three or more anyons, the result of sequential exchanges depends on the order in which they are performed, they are called non-Abelian anyons. Systems with non-abelian anyons allow for scalable quantum computation: many-anyon systems have an exponentially large set of topologically protected low-energy states which can be manipulated and distinguished from one another by experimental techniques, such as anyon interferometry recently realized in fractional quantum Hall systems.

A physical system which harbours anyons is said to be topologically ordered, or in a topological phase. One of the most important goals is to study such phases and their non-Abelian anyonic quasiparticles. The most advanced experiments in this direction are done in the context of the fractional quantum Hall effect (FQHE), where phases with fractionally charged Abelian anyons have already been seen and strong experimental evidence for the existence of non-Abelian anyons is emerging. In addition, very promising results have recently been obtained on engineered topologically ordered phases in Josephson junction arrays.

In addition to its natural fault-tolerance, topological quantum computation - though computationally equivalent to the conventional quantum circuit model - is a unique operational model of computation, which represents an original path to new quantum algorithms. New algorithms for approximation of certain hard #P hard computational problems have already been developed and this is opening up new areas of quantum algorithmic research.

The research objectives cover all aspects of topological quantum computation and include:

• Produce clear experimental evidence of topological phases suitable for TQC;
• design, simulate and build devices for fully scalable topological memory and gates;
• develop theoretical and algorithmic aspects of topological quantum computation as a new quantum computing paradigm;
• characterize topological phases and topological phase transitions, and link this scaling to properties of the topological entanglement entropy;
• propose engineered experimental realizations of topological phases;
• develop analytical and numerical computing skills for the FQHE and other topological systems;

Key references
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Quantum error correction and purification

The ability to carry out coherent quantum operation even in the presence of inevitable noise is a key requirement for quantum information processing. To cope with this decoherence problem, active strategies (quantum error correcting codes) as well as passive ones (error avoiding codes) have been developed.

Error correcting codes allow one to reduce errors by suitable encoding of logical qubits into larger systems. It has been shown that, with operations of accuracy above some threshold, the ideal quantum algorithms can be implemented. Recent ideas involving error correcting teleportation have made the threshold estimate more favorable by several orders of magnitude. This path has to be continued and adapted to realistic error models and to alternative models of quantum computation like the adiabatic model or the cluster model (see section 4.3.3).

In error avoiding codes, no active monitoring/intervention on the system is in principle necessary, since errors are simply circumvented. Error avoiding is based on the symmetry structure of the system-environment interaction that in some circumstances allows for the existence of decoherence-free subspaces (DFS), i.e. subspaces of the system Hilbert state-space over which the dynamics is still unitary. The prototype noise model for which this situation occurs is provided by the so-called collective decoherence, where all the qubits are affected by the environment in the same way. For encoding a single logical noiseless qubit for general collective decoherence (dephasing), four (two) physical qubits are needed. DFSs have been experimentally demonstrated in a host of physical systems, and their scope extended by generalizing the idea of symmetry-aided protection to noiseless subsystems.

A fruitful connection with the theory of entanglement purification, which has been developed primarily in the context of quantum communication, and has been used in protocols such as the quantum repeater, is also emerging. Entanglement purification or distillation is a method to “distill” from a large ensemble of impure and noisy (low-fidelity) entangled states a smaller ensemble of pure (high-fidelity) entangled states. It seems that appropriately generalized procedures can be employed also in general quantum computation (e.g. for quantum gate purification, or for the generation of high fidelity resource states) while benefiting from the relaxed thresholds that exist for entanglement purification.

Key references
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Geometric methods for fault-tolerant quantum computing

An alternative approach to achieve fault-tolerant quantum computation is by geometric means. In this approach, quantum information is encoded in a set of energy degenerate states, depending on dynamically controllable parameters. Quantum gates are then enacted by driving the control parameters along suitable loops. These transformations, termed holonomies, are suitable to realize a set of universal quantum gates. Implementation schemes of geometrical computation have been proposed for several different physical systems, most notably for trapped ions. The existing protocols for fault tolerant quantum computation have been specifically designed for phenomenological uncorrelated noise, while few results are known for a scenario with memory effects, i.e. non-Markovian noise, arising from the Hamiltonian interaction with the environment. In particular this raises the question of fault tolerant schemes for phenomenological noise with memory.

Key references
[1] J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation using nuclear magnetic resonance”, Nature 403, 869 (2000)
[2] P. Zanardi and M. Rasetti, “Holonomic quantum computation”, Phys. Lett. A 264, 94 (1999)
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[4] R. Alicki, M. Horodecki, P. Horodecki, and R. Horodecki, “Dynamical description of quantum computing: Generic non-locality of quantum noise”, Phys. Rev. A 65, 062101 (2002)
[5] M. Terhal and G. Burkard, “Fault-tolerant quantum computation for local non-Markovian noise”, Phys. Rev. A 71, 012336 (2005)

Quantum control theory for quantum information devices

Quantum error correction enables fault-tolerant quantum computation to be performed, provided that each elementary operation meets a certain fidelity threshold, but unfortunately, this puts extremely demanding constraints on the allowable errors. Threshold estimates vary between 0.01% to fractions of a percent, but none of the candidate physical implementations available to date has met such requirements yet. Therefore the main open challenge is a practical one: Will the necessary fidelity ever be reached in practice for elementary operations, and maintained while scaling up qubit number and system complexity? This will ultimately determine the winning hardware platform for future quantum information devices, analog to what has happened with silicon for conventional computing.

One feature is common to all candidate QIP implementations: the need for an extremely accurate control of the quantum dynamics at the individual level, with much better precision than has been achieved before. Optimal control theory is a very powerful set of methods developed over the last decades to optimize the time evolution of a broad variety of complex systems, from aeronautics to economics. The basic underlying idea is to pick a specific path in parameter space to perform a specific task. This is expressed mathematically by a cost functional that depends on the state of the system and is minimized with respect to some control parameters. More recently, this approach is being successfully applied to quantum systems, e.g., in the context of ultra-fast laser pulses and light-assisted molecular reactions. A big advantage is that, in a quantum-mechanical situation, the goal can be reached via interference of many different paths in parameter space, rather than just one. This allows, for instance, to exploit faster non-adiabatic processes, allowing to perform more gate operations within the decoherence time, which is crucial to apply fault-tolerant error correction. In future work, these ideas will also be more closely tied to methods of quantum systems identification.

Over the last few years, quantum optimal control theory (QOCT) has been applied to different aspects of quantum information processing, in particular to the implementation of scalable quantum gates with real physical systems. The figure of merit to be optimized in this case is the fidelity, defined as the projection of the physical state obtained by actually manipulating the chosen system onto the logical state that the gate aims at obtaining. Several examples, from atoms in optical lattices and atom chips to trapped ions and superconducting charge qubits, have indicated systematic improvements in fidelity beyond the fault-tolerance threshold, taking into account experimentally available configurations and known sources of imperfection.

Key references
[1] N. Khaneja, R. Brockett, and S. J. Glaser, “Time optimal control in spin systems”, Phys. Rev. A 63, 032308 (2001)
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[3] C. Brif, R. Chakrabarti, and H. Rabitz, "Control of quantum phenomena: Past, present, and future", arXiv:0912.5121 [quant-ph]
[4] K. Singer, U. Poschinger, M. Murphy, P. Ivanov, F. Ziesel, T. Calarco, and F. Schmidt-Kaler, "Experiments with atomic quantum bits - essential numerical tools", arXiv:0912.0196 [quant-ph]