Symplectic Geometry of Quantum Noise

Polterovich, Leonid

doi:10.1007/s00220-014-1937-9

Cited by 32 publications

(80 citation statements)

References 36 publications

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“…We expect that time-smoothing will suppress high terms for some Fourier-like expansion, thus relating Predictions 6 and 5. We also note that Prediction 7 resembles the picture of the "unsharpness principle" from symplectic geometry and quantization [49].…”

Section: 3supporting

confidence: 52%

The Quantum Computer Puzzle

Kalai¹

2016

Notices Amer. Math. Soc.

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Q uantum computers are hypothetical devices, based on quantum physics, which would enable us to perform certain computations hundreds of orders of magnitude faster than digital computers. This feature is coined "quantum supremacy", and one aspect or another of such quantum computational supremacy might be seen by experiments in the near future: by implementing quantum error-correction or by systems of noninteracting bosons or by exotic new phases of matter called anyons or by quantum annealing, or in various other ways. We concentrate in this paper on the model of a universal quantum computer that allows the full computational potential for quantum systems, and on the restricted model, called "BosonSampling", based on noninteracting bosons.A main reason for concern regarding the feasibility of quantum computers is that quantum systems are inherently noisy. We will describe an optimistic hypothesis regarding quantum noise that will allow quantum computing and a pessimistic hypothesis that won't. The quantum computer puzzle is to decide between these two hypotheses. We list some remarkable consequences of the optimistic hypothesis, giving strong reasons for the intensive efforts to build quantum computers, as well as good reasons for suspecting that this might not be possible. For systems of noninteracting bosons, we explain how quantum supremacy achieved without noise is replaced, in the presence of noise, by a very low yet fascinating computational power. 1 Finally, we describe eight predictions about quantum physics and computation from the pessimistic hypothesis. 2 Are quantum computers feasible? Is quantum supremacy possible? My expectation is that the pessimistic hypothesis will prevail, leading to a negative answer. Rather than regarding this possibility as an unfortunate failure that impedes the progress of humanity, I believe that the failure of quantum supremacy itself leads to important consequences for quantum physics, the theory of computing, and mathematics. Some of these will be explored here. Gil Kalai is Henry and Manya Noskwith Professor of Mathematics at the Hebrew University of Jerusalem and is also affiliated with A Brief SummaryHere is a brief summary of the author's pessimistic point of view as explained in the paper: understanding quantum computers in the presence of noise requires consideration of behavior at different scales. In the small scale, standard models of noise from the mid-90s are suitable, and quantum evolutions and states described by them manifest a very low-level computational power. This small-scale behavior has far-reaching consequences for the behavior of noisy quantum systems at larger scales. On the one hand, it does not allow reaching the starting points for quantum fault tolerance and quantum supremacy, making them both impossible at all scales. On the other hand, it leads to novel implicit ways for modeling noise at larger scales and to various predictions on the behavior of noisy quantum systems. 508

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Section: 3supporting

confidence: 52%

The Quantum Computer Puzzle

Kalai¹

2016

Notices Amer. Math. Soc.

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“…An important consequence of Principle 2 is that there are inherent lower bounds on the quality of qubits, or, more concretely, a universal upper bound on the number of non-commuting Pauli operators you can apply to a qubit before it get destroyed. We note that lower bounds on the rate of noise in terms of a measure of non-commutativity arise also in Polterovich (2014) in the study of quantizations in symplectic geometry.…”

Section: Reaching Ground Statesmentioning

confidence: 85%

The Argument Against Quantum Computers

Kalai

2020

Jerusalem Studies in Philosophy and History of Science

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Dedicated to the memory of Itamar PitowskyMy purpose in this paper is to examine, in a general way, the relations between abstract ideas about computation and the performance of actual physical computers.Itamar Pitowsky -The physical Church thesis and physical computational complexity 1990 AbstractWe give a computational complexity argument against the feasibility of quantum computers. We identify a very low complexity class of probability distributions described by noisy intermediate-scale quantum computers, and explain why it will allow neither good-quality quantum error-correction nor a demonstration of "quantum supremacy." Some general principles governing the behavior of noisy quantum systems are derived. Our work supports the "physical Church thesis" studied by Pitowsky (1990) and follows his vision of using abstract ideas about computation to study the performance of actual physical computers.

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“…We refer to works by the author [34], Seyfaddini [39] and Ishikawa [22] for more information about this constant. An interpretation of this result comes from the phase localization problem in quantum mechanics.…”

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confidence: 99%

“…The noise and the symplectic size turn out to be related by the following noise-localization uncertainty relation [34]:…”

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confidence: 99%

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Quantum Footprints of Symplectic Rigidity

Polterovich¹

2016

EMS Newsl.

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In this note, we discuss an interaction between symplectic topology and quantum mechanics. The interaction goes in both directions. On one hand, ideas from quantum mechanics give rise to new notions and structures on the symplectic side and, furthermore, quantum mechanical insights lead to useful symplectic predictions when topological intuition fails. On the other hand, some phenomena discovered within symplectic topology admit a meaningful translation into the language of quantum mechanics, thus revealing quantum footprints of symplectic rigidity. What is . . . symplectic?A symplectic manifold is an even-dimensional manifold M 2n equipped with a closed differential 2-form ω that can be written as n i=1 dp i ∧ dq i in appropriate local coordinates (p, q). For an oriented surface Σ ⊂ M, the integral Σ ω plays the role of a generalised area, which, in contrast to the Riemannian area, can be negative or vanish.To have some interesting examples in mind, think of surfaces with an area form and their products, as well as complex projective spaces equipped with the Fubini-Study form, and their complex submanifolds.Symplectic manifolds model the phase spaces of systems of classical mechanics. Observables (i.e. physical quantities such as energy, momentum, etc.) are represented by functions on M. The states of the system are encoded by Borel probability measures µ on M. The simplest states are given by the Dirac measure δ z concentrated at a point z ∈ M.The laws of motion are governed by the Poisson bracket, a canonical operation on smooth functions on M, given by. The evolution of the system is determined by its energy, a time-dependent function f t : M → R called its Hamiltonian. Hamilton's famous equation describing the motion of a system is given, in the Heisenberg picture, byġ t = { f t , g t }, where g t = g • φ −1 t stands for the time evolution of an observable function g on M under the Hamiltonian flow φ t . The maps φ t are called Hamiltonian diffeomorphisms. They preserve the symplectic form ω and constitute a group with respect to composition.In the 1980s, new methods, such as Gromov's theory of pseudo-holomorphic curves and the Floer-Morse theory on loop spaces, gave birth to "hard" symplectic topology. It detected surprising symplectic rigidity phenomena involving symplectic manifolds, their subsets and diffeomorphisms. A number of recent advances show that there is yet another manifestation of symplectic rigidity taking place in function spaces associated to a symplectic manifold. Its study forms the subject of function theory on symplectic manifolds, a rapidly evolving area whose development has led to the interactions with quantum mechanics described below. 2The non-displaceable fiber theoremIn 1990, Hofer [21] introduced an intrinsic "small scale" on a symplectic manifold: a subset X ⊂ M is called displaceable if there exists a Hamiltonian diffeomorphism φ such that φ(X) ∩ X = ∅.

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Symplectic Geometry of Quantum Noise

Cited by 32 publications

References 36 publications

The Quantum Computer Puzzle

The Quantum Computer Puzzle

The Argument Against Quantum Computers

Quantum Footprints of Symplectic Rigidity

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