Tree size complexity of states, and the power of quantum computation
Speaker: Valerio Scarani (Centre for Quantum Technologies and Department of Physics, National University of Singapore)
Tuesday, October 13, 5PM - 6PM, Zoom
Abstract: In some of the early debates on quantum computing, the challenge was put forward that it may be impossible to generate many-qubit states that are "complex enough". To escape vagueness, Aaronson proposed a specific measure of complexity for pure states that he called minimal tree size (TS) [1]. This measure has two important features: it is computable (at worst by brute force), and non-trivial lower bounds have been obtained (that is, examples of states have been found that are indeed complex). A few years ago, my group contributed several studies in this direction [2-4]. One of the main results is the proof that states with polynomial TS cannot give any quantum advantage in measurement-based quantum computation; relatedly, the TS of the family of cluster states was proved super-polynomial by direct calculation. The systematic study of TS, however, was possible up to only 4 qubits. These results were duly published but went virtually unnoticed, either because they were not timely, or because we didn't advertise well enough. With this talk, I hope to revive some interest, both for studying TS for NISQ-size number of qubits, and for assessing its role in the "magic" of quantum computation. [1] S. Aaronson, Multilinear formulas and skepticism of quantum computing, Proc. ACM symposium on Theory of computing, 2004 (ACM, 2004), pp. 118-127. [2] H.N. Le, Y. Cai, X. Wu, V. Scarani, Tree-size complexity of multiqubit states, Phys. Rev. A 88, 012321 (2013) [3] H.N. Le, Y. Cai, X. Wu, R. Rabelo, V. Scarani, Maximal tree size of few-qubit states, Phys. Rev. A 89, 062333 (2014) [4] Y. Cai, H.N. Le, V. Scarani, State complexity and quantum computation, Ann. Phys. 527, 684 (2015)
Recording: click here!
Geometric properties of the stabilizer polytope in qubit and qudit systems
Speaker: Arne Heimendahl (University of Cologne, Germany, Department of Mathematics and Computer Science)
Wednesday, October 28, 10AM - 11AM, Zoom
Abstract: The stabilizer polytope, which is the convex hull of all stabilizer states, plays a central role in the magic state model of fault tolerant quantum computation. In this talk, I will discuss some geometric properties of the stabilizer polytope, such as its dual polytope, defining inequalities and its symmetry group. The focus will be mainly on qudits, but also differences between the qudit and the qubit stabilizer polytope will be highlighted. Besides, I will link some geometric aspects to applications in phase space simulation and features of completely stabilizer preserving channels.
Recording: click here!
Generative training of quantum Boltzmann machines with hidden units
Speaker: Nathan Wiebe (University of Washington)
Friday, September 13, 11AM - 12noon, Brimacombe 311
Abstract: We provide the first method for fully quantum generative training of quantum Boltzmann machines with both visible and hidden units while using quantum relative entropy as an objective. This is significant because prior methods were not able to do so due to mathematical challenges posed by the gradient evaluation. We present two novel methods for solving this problem. The first proposal addresses it, for a class of restricted quantum Boltzmann machines with mutually commuting Hamiltonians on the hidden units, by using a variational upper bound on the quantum relative entropy. The second one uses high-order divided difference methods and linear-combinations of unitaries to approximate the exact gradient of the relative entropy for a generic quantum Boltzmann machine. Both methods are efficient under the assumption that Gibbs state preparation is efficient and that the Hamiltonian are given by a sparse row-computable matrix.
A fault-tolerant non-Clifford gate for the surface code in two dimensions
Speaker: Benjamin Brown (University of Sydney)
Wednesday, October 16, 2 - 3PM, Brimacombe 311
Abstract: Performing non-Clifford gates with magic state distillation will consume a considerable proportion of the resources of a two-dimensional fault-tolerant quantum computing architecture. Here we show how to perform a fault-tolerant non-Clifford gate with the surface code; a quantum error-correcting code now under intensive experimental development. This alleviates the need for distillation or higher-dimensional components to complete a universal set of quantum logical gates. The operation uses local transversal gates and code deformations on a two-dimensional architecture over a time that scales with the size of the qubit array. An important component of the gate is a just-in-time decoder. Such decoding algorithms allow us to draw upon the advantages of a three-dimensional model using only a two-dimensional array of live qubits. Remarkably, our gate is completed using parity checks of weight no greater than four. As such, we expect it to be experimentally amenable with technology that is available in the near-term future. As this gate circumvents the need for magic-state distillation, it may reduce the resource overhead of surface-code quantum computation significantly.
Quantum advantage with noisy shallow circuits in 3D
Speaker: Sergey Bravyi (IBM)
Wednesday, November 20, 2PM, Brimacombe (QMI) 311
Abstract: Quantum effects can significantly enhance information-processing capabilities and speed up solution of certain problems. Whether a computational quantum advantage can be rigorously proved in some settings or demonstrated experimentally is the subject of active debate. Here we show that parallel quantum algorithms running in a constant time are strictly more powerful than their classical counterparts. To this end we define a computational problem associated with the well-known Magic Square Game. We show that the problem can be solved by a constant depth quantum circuit, whereas any classical circuit solving the problem must have logarithmic depth. Moreover, this advantage persists even if the quantum circuit is corrupted by noise, provided that the noise rate is below a certain constant threshold value. Based on arXiv:1904.01502. Joint work with David Gosset, Robert Koenig and Marco Tomamichel
A rigorous version of quantization of transported charge in many-body systems
Speaker: Wojciech De Roeck (Katolieke Universiteit Leuven)
Tuesday, February 4th, 4 - 5 PM, Hennings 309B
Abstract: We propose a many-body index that extends Fredholm index theory to many-body systems. The index is defined for any charge-conserving system with a topologically ordered p-dimensional groundstate sector. The index is fractional with the denominator given by p. In particular, this yields a new short proof of the quantization of the Hall conductance and of Lieb-Schulz-Mattis theorem. In the case that the index is non-integer, the argument provides an explicit construction of Wilson-loop operators exhibiting a non-trivial braiding and that can be used to create fractionally charged Abelian anyons.
Scalable Majorana vortex modes in iron-based superconductors
Speaker: Ching-Kai Chiu (Kavli Institute for Theoretical Sciences, Beijing, China)
Thursday, March 12, 1PM in Brim 311
Abstract: The iron-based superconductor FeTe_xSe_[1-x] is one of the material candidates hosting Majorana vortex modes residing in the vortex cores. It has been observed by recent scanning tunneling spectroscopy measurement that the fraction of vortex cores having zero-bias peaks decreases with increasing magnetic field on the surface of FeTexSe1−x. The hybridization of two Majorana vortex modes cannot simply explain this phenomenon. We construct a three-dimensional tight-binding model simulating the physics of over a hundred Majorana vortex modes in FeTe_xSe_[1-x]. Our simulation shows that the Majorana hybridization and disordered vortex distribution can explain the decreasing fraction of the zero-bias peaks observed in the experiment; the statistics of the energy peaks off zero energy in our Majorana simulation are in agreement with the experiment. These agreements lead to an important indication of scalable Majorana vortex modes in FeTe_xSe_[1-x]. Thus, FeTe_xSe_[1-x] can be one promising platform having scalable Majorana qubits for quantum computing.