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Advance in quantum error correction

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Quantum computers are largely theoretical devices that could perform some computations exponentially faster than conventional computers can. Crucial to most designs for quantum computers is quantum error correction, which helps preserve the fragile quantum states on which quantum computation depends.

The ideal quantum error correction code would correct any errors in quantum data, and it would require measurement of only a few quantum bits, or qubits, at a time. But until now, codes that could make do with limited measurements could correct only a limited number of errors — one roughly equal to the square root of the total number of qubits. So they could correct eight errors in a 64-qubit quantum computer, for instance, but not 10.

In a paper they’re presenting at the Association for Computing Machinery’s Symposium on Theory of Computing in June, researchers from MIT, Google, the University of Sydney, and Cornell University present a new code that can correct errors afflicting — almost — a specified fraction of a computer’s qubits, not just the square root of their number. And for reasonably sized quantum computers, that fraction can be arbitrarily large — although the larger it is, the more qubits the computer requires.

“There were many, many different proposals, all of which seemed to get stuck at this square-root point,” says Aram Harrow, an assistant professor of physics at MIT, who led the research. “So going above that is one of the reasons we’re excited about this work.”

Like a bit in a conventional computer, a qubit can represent 1 or 0, but it can also inhabit a state known as “quantum superposition,” where it represents 1 and 0 simultaneously. This is the reason for quantum computers’ potential advantages: A string of qubits in superposition could, in some sense, perform a huge number of computations in parallel.

Once you perform a measurement on the qubits, however, the superposition collapses, and the qubits take on definite values. The key to quantum algorithm design is manipulating the quantum state of the qubits so that when the superposition collapses, the result is (with high probability) the solution to a problem.

Baby, bathwater

But the need to preserve superposition makes error correction difficult. “People thought that error correction was impossible in the ’90s,” Harrow explains. “It seemed that to figure out what the error was you had to measure, and measurement destroys your quantum information.”

The first quantum error correction code was invented in 1994 by Peter Shor, now the Morss Professor of Applied Mathematics at MIT, with an office just down the hall from Harrow’s. Shor is also responsible for the theoretical result that put quantum computing on the map, an algorithm that would enable a quantum computer to factor large numbers exponentially faster than a conventional computer can. In fact, his error-correction code was a response to skepticism about the feasibility of implementing his factoring algorithm.

Shor’s insight was that it’s possible to measure relationships between qubits without measuring the values stored by the qubits themselves. A simple error-correcting code could, for instance, instantiate a single qubit of data as three physical qubits. It’s possible to determine whether the first and second qubit have the same value, and whether the second and third qubit have the same value, without determining what that value is. If one of the qubits turns out to disagree with the other two, it can be reset to their value.

In quantum error correction, Harrow explains, “These measurement always have the form ‘Does A disagree with B?’ Except it might be, instead of A and B, A B C D E F G, a whole block of things. Those types of measurements, in a real system, can be very hard to do. That’s why it’s really desirable to reduce the number of qubits you have to measure at once.”

Time embodied

A quantum computation is a succession of states of quantum bits. The bits are in some state; then they’re modified, so that they assume another state; then they’re modified again; and so on. The final state represents the result of the computation.

In their paper, Harrow and his colleagues assign each state of the computation its own bank of qubits; it’s like turning the time dimension of the computation into a spatial dimension. Suppose that the state of qubit 8 at time 5 has implications for the states of both qubit 8 and qubit 11 at time 6. The researchers’ protocol performs one of those agreement measurements on all three qubits, modifying the state of any qubit that’s out of alignment with the other two.

Since the measurement doesn’t reveal the state of any of the qubits, modification of a misaligned qubit could actually introduce an error where none existed previously. But that’s by design: The purpose of the protocol is to ensure that errors spread through the qubits in a lawful way. That way, measurements made on the final state of the qubits are guaranteed to reveal relationships between qubits without revealing their values. If an error is detected, the protocol can trace it back to its origin and correct it.

It may be possible to implement the researchers’ scheme without actually duplicating banks of qubits. But, Harrow says, some redundancy in the hardware will probably be necessary to make the scheme efficient. How much redundancy remains to be seen: Certainly, if each state of a computation required its own bank of qubits, the computer might become so complex as to offset the advantages of good error correction.

But, Harrow says, “Almost all of the sparse schemes started out with not very many logical qubits, and then people figured out how to get a lot more. Usually, it’s been easier to increase the number of logical qubits than to increase the distance — the number of errors you can correct. So we’re hoping that will be the case for ours, too.”

Stephen Bartlett, a physics professor at the University of Sydney who studies quantum computing, doesn’t find the additional qubits required by Harrow and his colleagues’ scheme particularly daunting.

“It looks like a lot,” Bartlett says, “but compared with existing structures, it’s a massive reduction. So one of the highlights of this construction is that they actually got that down a lot.”

“People had all of these examples of codes that were pretty bad, limited by that square root ‘N,’” Bartlett adds. “But people try to put bounds on what may be possible, and those bounds suggested that maybe you could do way better. But we didn’t have constructive examples of getting here. And that’s what’s really got people excited. We know we can get there now, and it’s now a matter of making it a bit more practical.”


Source: http://news.mit.edu/2015/quantum-error-correction-0526

Quantum

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

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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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Quantum

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

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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

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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.

The above citations are from SAO/NASA ADS (last updated successfully 2020-10-28 10:51:27). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2020-10-28 10:51:25: Could not fetch cited-by data for 10.22331/q-2020-10-28-351 from Crossref. This is normal if the DOI was registered recently.

Source: https://quantum-journal.org/papers/q-2020-10-28-351/

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