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This mini lecture series provides an introduction to measurement based quantum computation (MBQC). We discuss the following subjects: the overall computational scheme, cluster states, quantum computational universality, MBQC with probabilistic heralded entangling gates, and fault tolerance with 3D cluster states.
My research interest is in quantum computation, in particular computational models, quantum fault-tolerance and foundational aspects. See the Featured Publications for recent examples of my and my group's work.
A hidden variable model for universal quantum computation with magic states on qubits [Published December 2020]. We show that every quantum computation can be described by probabilistic update of a probability distribution on a finite phase space. Negativity in a quasiprobability function is not required, neither in states nor the operations. Our result is consistent with Gleason's Theorem and the Pusey-Barrett-Rudolph theorem.
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HVM state space. The image on the left shows a cross-section of the state space of the hidden variable model (HVM) constructed, for the case of 2 qubits. The egg-shaped region represents the states allowed by quantum mechanics, i.e., proper density matrices. The kite-shaped subset of the proper quantum states are stabilizer mixtures. The trapezoid represents the set of states allowed by the HVM constructed. Note that it contains the proper quantum states as a subset. This is not a feature of the particular cross section shown. Rather, it holds for the entire state polytope, and does so for any number of qubits. |
The central result of our paper is that a probability distribution over a finite state space--and its probabilistic update under the necessary quantum operations--suffices to describe universal quantum computation, hence all quantum mechanics in finite Hilbert space dimension. Previously it was thought that such descriptions require quasi-probabilities (like probabilities but alowed to go negative), either in the states or in the operations. An example of this are the Wigner functions, originally developed in the field of quantum optics, and recently applied to quantum computation. We show that negativity in quasiprobability functions is not required; rather, probability functions suffice.
Our result builds on the one showcased below [PRA 101,012350 (2020)], and at a technical level remains rather closely related. But the conclusion is near opposite--how can that be? What [PRA 101] shows is that a result that was previously know in odd Hilbert space dimension carries over to qubits: Namely that a given quantum computation can have a quantum speedup only if a certain quasiprobability function does indeed become negative in the computation. What the new result shows is that this negativity is an artifact of the special quasiprobability function chosen. Negativity can always be avoided with a different choice of fuction.
What does all this mean for quantum speedup and the hardness of classical simulation of quantum computation? -- We do not have an answer to that question yet, but one word of caution is in order: The fact that in our model every quantum computation can be described by iterated sampling does not a priori mean that the classical simulation is efficient. Indeed, all intuition gathered on the matter so far would suggest that it is inefficient. However, presently we can neither prove efficiency nor inefficiency.
Journal Reference:
Michael Zurel, Cihan Okay and Robert Raussendorf, A hidden variable model for universal quantum computation with magic states on qubits, Phys. Rev. Lett. 125, 260404 (2020).
Phase space simulation method for quantum computation with magic states on qubits [Published January 2020]. We propose a method for classical simulation of finite-dimensional quantum systems, based on sampling from a quasiprobability distribution, i.e., a generalized Wigner function. Our construction applies to all finite dimensions, with the most interesting case being that of qubits. For multiple qubits, we find that quantum computation by Clifford gates and Pauli measurements on magic states can be efficiently classically simulated if the quasiprobability distribution of the magic states is non-negative. This provides the so far missing qubit counterpart of the corresponding result [V. Veitch et al., New J. Phys. 14, 113011 (2012)] applying only to odd dimension. Our approach is more general than previous ones based on mixtures of stabilizer states. Namely, all mixtures of stabilizer states can be efficiently simulated, but for any number of qubits there also exist efficiently simulable states outside the stabilizer polytope. Further, our simulation method extends to negative quasiprobability distributions, where it provides probability estimation. The simulation cost is then proportional to a robustness measure squared. For all quantum states, this robustness is smaller than or equal to robustness of magic.
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Update under measurement. Unlike in the odd-dimensional scenario, the multi-qubit phase space is a complicated place. It is splintered into many segments each of which can be travelled by translation, but cannot be passed between by translation. Some of these components are connected by symplectic transformations, but not all. The image shows two points in phase space, for the setting of two rebits. The description of each point in phase space consists of a set of 2-rebit Pauli observables, embedded into a Mermin square and displayed in green. These sets have to satisfy certian consistency contraints.The red dot indicates a measured Pauli observable, and the arrow how a point in phase space is updated under the Pauli measurement (left (right) of the arrow = before (after) the measurement). |
The technical problem solved in this paper is the construction of a quasiprobability function for multi-qubit systems that (a) is covariant under all Clifford transformations, and (b) represents all Pauli measurements positively. Those two criteria are imposed by the computational scheme discussed, quantum computation with magic states. Different from the odd-dimensional case, these criteria have proved difficulty to satisfy, for reasons related to certain proofs of the Kochen-Specker theorem such as Mermin's square.
With the present paper, we have overcome these difficulties. The key ingredient is to admit non-unique quasiprobability distributions. That is, a given quantum state can be represented by many such quasiprobability functions.
Journal Reference:
Robert Raussendorf, Juani Bermejo-Vega, Emily Tyhurst, Cihan Okay, and Michael Zurel, Phase space simulation method for quantum computation with magic states on qubits, Phys. Rev. A 101, 012350 (2020).
Click here for previous posts.
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)].
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 J. Harrington, Fault-tolerant quantum computation with high threshold in two dimensions, [Phys. Rev. Lett. 98, 150504 (2007)].
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)].
The one way quantum computer (QCc), aka. measurement based quantum computation. I have invented the QCc jointly with Hans Briegel (US patent 7,277,872). It 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.
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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. |
Fault tolerant quantum computation in low spatial dimension. With my collaborators Jim Harrington and Kovid Goyal, I have presented a fault-tolerant one-way quantum computer [arXiv:quant-ph/0510135] based on 3D cluster states, 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 [Phys. Rev. Lett. 98, 190504 (2007)]. For our method, we have obtained an error threshold of 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.
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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. |
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 Professor at the Department of Physics and Astronomy of the University of British Columbia.