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My research interest is in quantum computation, in particular computational models, quantum fault-tolerance and foundational aspects. I have invented the one-way quantum computer (QCc) jointly with Hans Briegel (UK patent GB 2382892, US patent 7,277,872). The QCc is a scheme of universal quantum computation by local measurements on a multi-particle entangled quantum state, the so-called cluster state. Quantum information is written into the cluster state, processed and read out by one-qubit measurements only. As the computation proceeds, the entanglement in the resource cluster state is progressively destroyed. Measurements replace unitary evolution as the elementary process driving a quantum computation.
The one-way quantum computer (QCc): A universal resource for the QCc is the cluster state, a highly entangled mult-qubit quantum state that can be easily generated unitarily by the Ising interaction on a square lattice. In the figure to the left, the qubits forming the cluster state are represented by dots and arrows. The symbol used indicates the basis of local measurement. Dots represent cluster qubits measured in the eigenbasis of the Pauli operator Z, arrows denote measurement in a basis in the equator of the Bloch sphere. The pattern of measurement bases can be regarded as representing a quantum circuit, i.e., the "vertical" direction on the cluster specifies the location of a logical qubit in a quantum register, and the "horizontal" direction on the cluster represents circuit time. However, this simple picture should be taken with a grain of salt: The optimal temporal order of measurements has very little to do with the temporal sequence of gates in the corresponding circuit.
I also work in the field of fault-tolerant quantum computation. Error-correction is what a large-scale quantum computer spends most of its computation time with, and it is important to devise error-correction methods which allow for a high error threshold at a moderate operational overhead. My research interest is in fault-tolerance for quantum systems with a geometrical constraint, e.g. low-dimensional lattice systems, and in topological methods.
With my collaborators Jim Harrington and Kovid Goyal, I have presented a fault-tolerant one-way quantum computer [arXiv:quant-ph/0510135], and have described a method for fault-tolerant quantum computation in a two-dimensional lattice of qubits requiring local and translation-invariant nearest-neighbor interaction only [arXiv:quant-ph/0610082], [arXiv:quant-ph/0703143]. For our method, we have obtained by far the highest known threshold for a two-dimensional architecture with nearest-neighbor interaction, namely 0.75 percent. A high value of the error threshold is important for realization of fault-tolerant quantum computation because it relaxes the accuracy requirements of the experiment. The imposed constraint of nearest-neighbor interaction in a two-dimensional qubit array is suggested by experimental reality: Many physical systems envisioned for the realization of a quantum computer are confined to two dimensions and prefer short-range interaction, for example optical lattices, arrays of superconducting qubits and quantum dots.
Fault-tolerant topological CNOT-gate: Holes puncture a the surface of a Kitaev's surface code, creating pieces of boundary. Each pair of holes gives rise to an encoded qubit. There are two types of holes and hence qubits, primal and dual. The CNOT-gate is implemented by moving two holes around another, one being primal and the other dual. Also shown is the string corresponding to an encoded Pauli operator X on the control qubit and its evolution from the initial to the final codes surface. As expected for conjugation under the CNOT, X_c evolves into X_c X_t. The CNOT in the opposite direction - the primal qubit being the target and the dual qubit being the control - is also possible. It requires pairwise insertion and removal of holes from the code surface, i.e., the topology of the code surface for that gate changes with time.
Tzu-Chieh Wei, Ian Affleck, Robert Raussendorf, The 2D AKLT state is a universal quantum computational resource, Phys. Rev. Lett. 106, 070501 (2011).
R. Raussendorf and H.-J. Briegel, Computational model underlying the one-way quantum computer, arXiv:quant-ph/0108067, Quant. Inf. Comp. 6, 443 (2002).
R. Raussendorf and H.-J. Briegel, A one-way quantum computer, Phys. Rev. Lett. 86, 5188 (2001).
A computationally universal phase of quantum matter [Published March 4, 2019] We provide the first example of a symmetry protected quantum phase that has universal computational power. This two-dimensional phase is protected by one-dimensional line-like symmetries that can be understood in terms of local symmetries of a tensor network. These local symmetries imply that every ground state in the phase is a universal resource for measurement based quantum computation.
In the presence of symmetry, quantum phases of matter can have computational power. The important property is that the computational power is uniform. It does not depend on the precise choice of the state within the phase, and is thus a property of the phase itself. In this way, phases of quantum matter acquire a computational characterization and computational value.
Quantum computational power of physical phases is utilized by measurement based quantum computation (MBQC), where the process of computation is driven by local measurements on an initial entangled state. Here, we consider initial states that originate from symmetry protected topological phases. Work on the usefulness of SPT phases of matter for MBQC have to date been focussed on spatial dimension 1. Computationally, physical phases in dimension 2 and higher are more interesting than in dimension 1. The reason is that, in MBQC, one spatial dimension plays the role of circuit model time. Therefore, MBQC in dimension D corresponds to the circuit model in dimension D-1, and universal MBQC is possible only in D >=2.
Here, we prove the existence of a computationally universal phase of quantum matter in spatial dimension two. The phase we consider is protected by one-dimensional line-like symmetries, generalizing the conventional notion of symmetry protected topological order defined by global on-site symmetries. As in the case of global symmetries, these line symmetries can be built from the local symmetries of a tensor network which persist throughout the phase. Using this, we establish that computational universality persists throughout the entire phase.
Symmetry Lego. Ground states throughout the whole cluster states have an enhanced symmetry, and the functioning of MBQC is based on it. This symmetry is best under stood in terms of tensor networks. (a) Tensor network representing an MBQC resource state in the 2D cluster phase. Every local tensor represents one physical spin 1/2 particle. (b) Throughout the cluster phase, each local tensor is invariant under a number of symmetries. These symmetries form the backbone of measurement-based computation in the cluster phase, as is illustrated in Figs. (c) and (d). (c) The symmetries shown in (b) are composed into a larger pattern propagating a logical Z-operator forward in time. The area shown represents the ``clock cycle'' of the compution. If all physical qubits shown are measured in the X-eigenbasis, the resulting operation is the logical identity (computational wire). (d) If the qubit marked by the double-headed arrow is measured off the X-basis, the result is an entangling gate. Again, the functioning of the gate can be understood in terms of the tensor symmetries shown in (b).
R. Raussendorf, C. Okay, D.S. Wang, D.T. Stephen, H. Poulsen Nautrup, A computationally universal quantum phase of matter, Phys. Rev. Lett. 122, 090501 (2019).
Further computationally universal SPT phases, and the connection to quantum cellular automata: D.T. Stephen, H.P Nautrup, J. Bermejo-Vega, J. Eisert, R. Raussendorf, Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter, arXiv:1806.08780.
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I obtained my PhD from the Ludwig Maximilians University in Munich, Germany in 2003. My PhD thesis [Int. J. of Quantum Information 7, 1053 - 1203 (2009).] is on measurement-based quantum computation. I was postdoc at Caltech (2003-06) and at the Perimeter Institute for Theoretical Physics (2006-07), and Sloan Research Fellow 2009 - 2011. I am Associate Professor at the Department of Physics and Astronomy of the University of British Columbia, and scholar of the Cifar Quantum Information program.
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