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# Random Strain Fluctuations as Dominant Disorder Source for High-Quality On-Substrate Graphene Devices

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Graphene is a honeycomb lattice entirely made of carbon only one atom thick; it is the first system discovered to be a perfect two-dimensional crystal. However, even graphene, with all of its ideal properties, is unavoidably deformed when isolated and placed on a supporting substrate. For instance, graphene may not lie perfectly flat but instead form “bumps.” Furthermore, its atoms may be randomly displaced from their ideal positions in the honeycomb lattice because of the force exerted by the atoms in the substrate. Scientists call these types of mechanical deformations “strain.” We show that when graphene exfoliated from graphite is placed on a supporting hexagonal boron nitride substrate, the random strain that appears in the carbon honeycomb lattice reduces the speed with which electrons can move inside the material.

Scientists have puzzled for a long time about the mechanism limiting the speed of electrons in real-world graphene devices. Many possibilities have been proposed, such as missing atoms or chemical impurities, structural defects in the honeycomb lattice, and unintentional “dirt.” All of these mechanisms can play a role in limiting the speed of electrons, which is why the situation has remained unclear for a long time. We use both statistical analyses and experimental verification and provide ample evidence that random strain fluctuations in the material set a limit on the electron speed for temperatures as low as 250 mK. We find that electron scattering is local (i.e., within the same valley) and is due to long-range potentials. The way in which strain affects electrons in graphene is very similar to the effect of a magnetic field. In other words, random strain generates a fluctuating pseudomagnetic field, which randomly deflects electrons and limits their speed. Since the amount of strain can strongly depend on the substrate or on the way in which the graphene layer is mounted on the substrate, the electron speed or mobility can vary significantly. We find that our results can also be applied to graphene mounted on

$SiO2$

and

$SrTiO3$

substrates.

Future electronic applications will necessitate a high electron speed. To optimize applications, therefore, we will need to focus our efforts on selecting substrates and assembly procedures that minimize strain.

# Bell nonlocality with a single shot

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Mateus Araújo1, Flavien Hirsch1, and Marco Túlio Quintino2,1,3

1Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
2Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
3Department of Physics, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan

### Abstract

In order to reject the local hidden variables hypothesis, the usefulness of a Bell inequality can be quantified by how small a \$p\$-value it will give for a physical experiment. Here we show that to obtain a small expected \$p\$-value it is sufficient to have a large gap between the local and Tsirelson bounds of the Bell inequality, when it is formulated as a nonlocal game. We develop an algorithm for transforming an arbitrary Bell inequality into an equivalent nonlocal game with the largest possible gap, and show its results for the CGLMP and \$I_{nn22}\$ inequalities.

We present explicit examples of Bell inequalities with gap arbitrarily close to one, and show that this makes it possible to reject local hidden variables with arbitrarily small \$p\$-value in a single shot, without needing to collect statistics. We also develop an algorithm for calculating local bounds of general Bell inequalities which is significantly faster than the naïve approach, which may be of independent interest.

Nonlocal games are cooperative games between two parties, Alice and Bob, that are not allowed to communicate. The maximal probability with which Alice and Bob can win the game depends on how the world fundamentally works: if it respects classical ideas about locality and determinism, this maximal probability is given by the local bound. On the other hand, if the world works according to quantum mechanics, the maximal probability is given by the Tsirelson bound, which is larger than the local bound. This makes it possible to experimentally falsify the classical worldview: let Alice and Bob play a nonlocal game with quantum devices for many rounds, and if they win more often than the local bound predicts, that’s it.

The number of rounds it takes for a decisive rejection of the classical worldview depends on the statistical power of the nonlocal game: a more powerful game requires fewer rounds to reach a conclusion with the same degree of confidence. We show that in order to get a large statistical power, it is enough to have a large gap between the local bound and the Tsirelson bound of the nonlocal game. Moreover, we show that this gap depends on how precisely a nonlocal game is formulated, so we develop an algorithm to maximise the gap over all possible formulations of a nonlocal game. With this, we derive the most powerful version of several well-known nonlocal games, such as the CHSH game, the CGLMP games, and the Inn22 games.

A natural question to ask is how high can the statistical power of a nonlocal game get. We show that it can get arbitrarily high, by constructing two nonlocal games with gap between their local and Tsirelson bounds arbitrarily close to one. This makes it possible to conclusively falsify the classical worldview with a single round of the nonlocal game, without needing to collect statistics. Unfortunately, neither of these games is experimentally feasible, so the question of whether a single-shot falsification is possible in practice is still open.

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### Cited by

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# Optimization of the surface code design for Majorana-based qubits

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Rui Chao1, Michael E. Beverland2, Nicolas Delfosse2, and Jeongwan Haah2

1University of Southern California, Los Angeles, CA, USA
2Microsoft Quantum and Microsoft Research, Redmond, WA, USA

### Abstract

The surface code is a prominent topological error-correcting code exhibiting high fault-tolerance accuracy thresholds. Conventional schemes for error correction with the surface code place qubits on a planar grid and assume native CNOT gates between the data qubits with nearest-neighbor ancilla qubits.

Here, we present surface code error-correction schemes using \$textit{only}\$ Pauli measurements on single qubits and on pairs of nearest-neighbor qubits. In particular, we provide several qubit layouts that offer favorable trade-offs between qubit overhead, circuit depth and connectivity degree. We also develop minimized measurement sequences for syndrome extraction, enabling reduced logical error rates and improved fault-tolerance thresholds.

Our work applies to topologically protected qubits realized with Majorana zero modes and to similar systems in which multi-qubit Pauli measurements rather than CNOT gates are the native operations.

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# Classical Simulations of Quantum Field Theory in Curved Spacetime I: Fermionic Hawking-Hartle Vacua from a Staggered Lattice Scheme

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Adam G. M. Lewis1 and Guifré Vidal1,2

1Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario, Canada, N2L 2Y5
2X, The Moonshot Factory, Mountain View, CA 94043

### Abstract

We numerically compute renormalized expectation values of quadratic operators in a quantum field theory (QFT) of free Dirac fermions in curved two-dimensional (Lorentzian) spacetime. First, we use a staggered-fermion discretization to generate a sequence of lattice theories yielding the desired QFT in the continuum limit. Numerically-computed lattice correlators are then used to approximate, through extrapolation, those in the continuum. Finally, we use so-called point-splitting regularization and Hadamard renormalization to remove divergences, and thus obtain finite, renormalized expectation values of quadratic operators in the continuum. As illustrative applications, we show how to recover the Unruh effect in flat spacetime and how to compute renormalized expectation values in the Hawking-Hartle vacuum of a Schwarzschild black hole and in the Bunch-Davies vacuum of an expanding universe described by de Sitter spacetime. Although here we address a non-interacting QFT using free fermion techniques, the framework described in this paper lays the groundwork for a series of subsequent studies involving simulation of interacting QFTs in curved spacetime by tensor network techniques.

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### Cited by

[1] Adam G. M. Lewis, “Hadamard renormalization of a two-dimensional Dirac field”, Physical Review D 101 12, 125019 (2020).

[2] Yue-Zhou Li and Junyu Liu, “On Quantum Simulation Of Cosmic Inflation”, arXiv:2009.10921.

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