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arXiv:1208.5334 · New Journal of Physics

The power of noisy fermionic quantum computation

Fernando de Melo, Piotr Ćwikliński, Barbara M. Terhal

Published Jan 9, 2013• 41 citations
arXivHTMLPDF

Abstract

We consider the realization of universal quantum computation through braiding of Majorana fermions supplemented by unprotected preparation of noisy ancillae. It has been shown by Bravyi (2006 Phys. Rev. A 73 042313) that under the assumption of perfect braiding operations, universal quantum computation is possible if the noise rate on a particular four-fermion ancilla is below 40%. We show that beyond a noise rate of 89% on this ancilla the quantum computation can be efficiently simulated classically: we explicitly show that the noisy ancilla is a convex mixture of Gaussian fermionic states in this region, while for noise rates below 53% we prove that the state is not a mixture of Gaussian states. These results are obtained by generalizing concepts in entanglement theory to the setting of Gaussian states and their convex mixtures. In particular, we develop a complete set of criteria, namely the existence of a Gaussian-symmetric extension, which determine whether a state is a convex mixture of Gaussian states.

DOI
10.1088/1367-2630/15/1/013015

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arXiv:quant-ph/0403025•Physical Review A•Feb 22, 2005•1,524 citations
We consider the realization of universal quantum computation through braiding of Majorana fermions supplemented by unprotected preparation of noisy ancillae. It has been shown by Bravyi (2006 Phys. Rev. A 73 042313) that under the assumption of perfect braiding operations, universal quantum computation is possible if the noise rate on a particular four-fermion ancilla is below 40%. We show that beyond a noise rate of 89% on this ancilla the quantum computation can be efficiently simulated classically: we explicitly show that the noisy ancilla is a convex mixture of Gaussian fermionic states in this region, while for noise rates below 53% we prove that the state is not a mixture of Gaussian states. These results are obtained by generalizing concepts in entanglement theory to the setting of Gaussian states and their convex mixtures. In particular, we develop a complete set of criteria, namely the existence of a Gaussian-symmetric extension, which determine whether a state is a convex mixture of Gaussian states.
Braiding Majorana zero modes does not by itself provide a universal set of quantum gates. Additional operations can be supplied through ancillary states, whose preparation need not have the same protection as braiding. Here, we consider purification of the non-Gaussian ancilla used for an entangling gate in Bravyi's scheme. We keep his elementary braid-and-fusion comparison and change how its outputs are combined. An additional layer compares the product of the three stabilizer signs. The first accepted errors giving an incorrect output then require three faulty input blocks rather than two. As a result, the preparation error decreases cubically instead of quadratically, and the asymptotic raw-copy overhead is reduced. We also construct a preliminary sequence in which a processed block is paired with a fresh copy from the same noisy source. For independent inputs supported in the even-parity sector and ideal purification operations, this sequence establishes a sufficient distillability fidelity of 59.95 percent after free Gaussian twirling. At a common source fidelity, it reduces the expected cost of preparing an input for the recurrence by more than seven orders of magnitude relative to the balanced construction studied here. Finally, we examine imperfect purification operations in an effective syndrome-noise model. The additional layer leaves the leading operation-error coefficient unchanged, although a nonzero error floor remains.
Determining the highest amount of noise that quantum circuits can handle is an interesting and crucial task in the development of fault-tolerant quantum computation. Previous work has constrained this upper limit for local depolarizing noise to ≈45 %\approx 45~\%≈45 % for circuits constructed using the universal Clifford with T gate set by finding the noise threshold where the gate set loses universality. In this work, we use a similar method with the matchgate with non-Gaussian resource universal gate set to find a limit of p=(8−26)/5p=(8 - 2 \sqrt{6})/5p=(8−26​)/5 or ≈62 %\approx 62~\%≈62 % for fermionic quantum computing that is independent of circuit depth. To do this, we use Uhlmann-Wootters concurrences for a 4-mode fermionic Choi state representing a resourceful gate combined with local depolarizing noise to determine when the combined channel is convex Gaussian. These bounds are not directly comparable due to differences in noise model making the bound of ≈62 %\approx 62~\%≈62 % the best known for fermions.
Universal quantum computation (UQC) using Majorana fermions on a two-dimensional topological superconducting (TS) medium remains an outstanding open problem. This is because the quantum gate set that can be generated by braiding of the Majorana fermions does not include any two-qubit gate and also no single-qubit $\ensuremath{\pi}/8$ phase gate. In principle, it is possible to create these crucial extra gates using quantum interference of Majorana fermion currents. However, it is not clear if the motion of the various order parameter defects (vortices, domain walls, etc.), to which the Majorana fermions are bound in a TS medium, can be quantum coherent. We show that these obstacles can be overcome using a semiconductor quantum wire network in the vicinity of an sss-wave superconductor, by constructing topologically protected two-qubit gates and any arbitrary single-qubit phase gate in a topologically unprotected manner, which can be error corrected using magic-state distillation. Thus our strategy, using a judicious combination of topologically protected and unprotected gate operations, realizes UQC on a quantum wire network with a remarkably high error threshold of 0.140.140.14 as compared to ${10}^{\ensuremath{-}3}$ to ${10}^{\ensuremath{-}4}$ in ordinary unprotected quantum computation.
We study how much noise can be tolerated by a universal gate set before it loses its quantum-computational power. Specifically we look at circuits with perfect stabilizer operations in addition to imperfect nonstabilizer gates. We prove that for all unitary single-qubit gates there exists a tight depolarizing noise threshold that determines whether the gate enables universal quantum computation or if the gate can be simulated by a mixture of Clifford gates. This exact threshold is determined by the Clifford polytope spanned by the 24 single-qubit Clifford gates. The result is in contrast to the situation wherein nonstabilizer qubit states are used; the thresholds in that case are not currently known to be tight.
Fermionic Gaussian states describe free fermions and underlie the mean-field picture of matter, from metals to superconductors; they are also efficiently simulable on classical computers. Departures from Gaussianity -- the correlations produced by interactions -- are therefore what make a fermionic system hard to simulate classically and useful for quantum computation, analogous to the role of magic in stabilizer-based quantum computation. Yet detecting and quantifying such non-Gaussianity at scale has remained challenging. Here we introduce practical tests and witnesses of fermionic non-Gaussianity built on fermionic antiflatness, a measure derived from the two-point covariance matrix. We estimate it with two protocols -- a two-copy Bell measurement and a single-copy scheme using commuting Majorana bilinears -- that determine whether a state is Gaussian or far from it at lower measurement cost than existing approaches, using only operations native to fault-tolerant hardware. For mixed states, a purity-corrected witness certifies non-Gaussianity and remains robust under strong noise; running it on the IQM quantum processor, we find that noise can both reduce and enhance non-Gaussianity. Finally, we show that preparing pseudorandom fermionic states requires extensive non-Gaussianity. Together, these tools enable the study and certification of non-Gaussian fermionic resources on present-day quantum devices.
We consider a model of quantum computation in which the set of elementary operations is limited to Clifford unitaries, the creation of the state $\ensuremath{\mid}0⟩$, and qubit measurement in the computational basis. In addition, we allow the creation of a one-qubit ancilla in a mixed state $\ensuremath{\rho}$, which should be regarded as a parameter of the model. Our goal is to determine for which $\ensuremath{\rho}$ universal quantum computation (UQC) can be efficiently simulated. To answer this question, we construct purification protocols that consume several copies of $\ensuremath{\rho}$ and produce a single output qubit with higher polarization. The protocols allow one to increase the polarization only along certain ``magic'' directions. If the polarization of $\ensuremath{\rho}$ along a magic direction exceeds a threshold value (about 65%), the purification asymptotically yields a pure state, which we call a magic state. We show that the Clifford group operations combined with magic states preparation are sufficient for UQC. The connection of our results with the Gottesman-Knill theorem is discussed.