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Symmetry, Phases of Matter, and Resources in Quantum Computing (with Robert Raussendorf, Physics). Areas of interest include magic state distilliation, extensions of the Pauli stabilizer formalism, measurement-based quantum computation and symmetry-proteced topological order.
This position is part of the collaborative quantum projects initiative supported by the Canada First Research Excellence Fund (CFREF). It is a joint project between the Universty of Sherbrooke, UBC and the University of Waterloo (IQC). Interaction with the other sites is strongly enocouraged.
Applications and inquiries should be sent to Robert Raussendorf.
Algebraic methods in quantum computing (supervised by Ian Affleck and Robert Raussendorf). Development of novel methods for mapping Fermionic systems to bosons, and/or Classification of measurement-based quantum computation within the framework of symmetry protected topological order.
Candidates should have a background in both quantum information and condensed matter physics; the former covering computational models such as circuit, measurement-based, adiabatic and topological, plus quantum error correction and the stabilizer formalism. The latter should cover fermionic systems and symmetry-protectedtopological order, and topological order.
This position is located at the Stuart Blusson Quantum Matter Institute at UBC, and is part of the QMI Grand Challenge "Pushing the Boundaries of NISQ-era Quantum Computing by Quantum Materials Problems". Visit this site for more information.
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).
Cohomological framework for contextual quantum computations [Published December 2019]. We describe a cohomological framework for measurement based quantum computation, in which symmetry plays a central role. Therein, the essential information about the computational output is contained in topological invariants, namely elements of two cohomology groups. One of those invariants applies to the deterministic case, and the other to the general probabilistic case. The same invariants also witness quantumness in the form of contextuality. In result, they give rise to fundamental algebraic structures underlying quantum computation.
The Boolean algebra is at the foundation of all digital classical computation. From a quantum perspective it is thus pertinent to ask what its counterpart in quantum computation is. Which fundamental algebraic structures can quantum computation be based on? This is the question we are concerned with here. We address it for the model of measurement-based quantum computation (MBQC), in which the process of computation is driven by measurements rather than unitary evolution.
To address the above question, we introduce a generalized notion of MBQC based on symmetry. Each such H-MBQC transforms under a characteristic symmetry group H, whose action preserves the outputted function o. We establish the following properties.
The input structure to each H-MBQC is a group Q related to H. Denote by N the normal subgroup N of H, such that each element in it maps the measurable observables of the computation element-wise to themselves, up to sign. Then, it holds that Q=H/N.
The output function o is preserved under the action of H.
In the deterministic case, the output function o is determined by a 2-cocycle in the cohomology of chain complexes, and the action of Q on it. In the probabilistic case, the output function o is determined up to an additive constant by the phase function, a 1-cocycle in group cohomology that derives from the action of H on the set of measurable observables.
We impose as minimal requirements on our description of H-MBQCs that it contains (i) a witness of quantumness, and (ii) the computed function. It turns out that this information is topological. Namely, the witness of quantumness arises as a contextuality witness, provided by the above cohomological invariants. The same invariants also characterize the outputted function.
R. Raussendorf, Cohomological framework for contextual quantum computations, arXiv:1602:04155, Quant. Inf. Comp. 19, 1141 - 1170 (2019).
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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.