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Third Workshop on "Algebraic Structures in Quantum Computation"
May 21 - 25 , 2018
University of British Columbia, Vancouver, Canada
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 (Los Alamos National Laboratory) and Kovid Goyal (Caltech), 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 complete list of my publications can be found here.
Computational power of symmetry protected topological phases [Published July 5, 2017] We consider ground states of quantum spin chains with symmetry-protected topological (SPT) order as resources for measurement-based quantum computation (MBQC). We show that, for a wide range of SPT phases, the computational power of ground states is uniform throughout each phase. This computational power, defined by the Lie group of executable gates in MBQC, is determined by the same algebraic information that labels the SPT phase itself. We prove that these Lie groups always contain a full set of single-qubit gates, thereby affirming the long-standing conjecture that general SPT phases can serve as computationally useful phases of matter.
Changeover from unitary evolution to measurement. For any (unique) ground state in a given SPT phase, the same set of logical unitaries and measurements can be implemented via MBQC on that state. Now, when is a logical unitary implemented, and when a measurement? -- This is decided by the interplay between (i) the basis for physical measurement implementing the logical operation and (ii) the number of repitions of that physical measurement along the spin chain. There is a preferred measurement basis, the so-called wire basis. If the spins of the chain are measured in this basis, the logical identity is implemented. For all other operations, one needs to deviate from the wire basis (Herein actually lies the difficulty of computing in SPT phases: away from the wire basis, the logical processing is a priori prone to decoherence, which needs to be circumvented.).
Parametrizing the deviation of the actual measurement basis from the wire basis by an angle alpha, and denoting the number of repitions of the measurement by N, the angle of the unitary rotation is Phi ~ alpha * N, and the deviations from unitarity are ~ alpha^2 * N. Therefore, the strategy to implement a unitary with rotation angle Phi is to choose alpha ~ Phi/N, with N large, and then repeat N times. The optimal strategy for a measurement is to choose alpha = Pi/4. However, for sufficiently large number N of repitions, all angles alpha <> 0 will result in a logical measurement. As per the above relation, the changeover occurs around alpha = 1/sqrt(N).
Now to the figure itself, which shows a numerical simulation of MBQC gate implementation. Here, the number of repitions of the physical measurement is N=1600. On the horizontal axis, the angle alpha is plotted. The vertical axis shows the fraction of outcomes "0" among the outcomes "0" or "1". Each point in the plot represents a sequence of 1600 repitions of the same physical measurement. We find that, for every angle alpha, the frequencies of obtaining "0" cluster around certain values. If alpha is small, there is -- up to noise -- only one value. Thus, the physical measurement does not learn anything about the processed logical state. The logical evolution is unitary. On the other hand, if alpha is large, there are several values around which the frequency N_0/N can cluster, corresponding to measurement of the logical state. As expected, the changeover occurs around alpha=1/sqrt(N).
For a detailed description of the computational scheme, demonstrating how to perform logical unitary gates and measurements, see Robert Raussendorf, Dong-Sheng Wang, Abhishodh Prakash, Tzu-Chieh Wei, and David T. Stephen, Symmetry-protected topological phases with uniform computational power in one dimension, Phys. Rev. A 96, 012302 (2017).
For the algebraic aspects and their implications, see David T. Stephen, Dong-Sheng Wang, Abhishodh Prakash, Tzu-Chieh Wei, and Robert Raussendorf, Computational Power of Symmetry-Protected Topological Phases, Phys. Rev. Lett. 119, 010504 (2017).
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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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