RTNI—A symbolic integrator for Haar-random tensor networks

Fukuda, Motohisa; König, Robert; Nechita, Ion

doi:10.1088/1751-8121/ab434b

Cited by 22 publications

(16 citation statements)

References 37 publications

(45 reference statements)

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“…Finally, let us remark that while we employ a sub-space search algorithm, in the presence of barren plateaus all optimization methods will (on average) fail unless the algorithm has a precision (i.e., number of shots) that grows exponentially with n. The latter is due to the fact that an exponentially vanishing gradient implies that on average the cost function landscape will essentially be flat, with the slope of the order of Oð1=2 n Þ. Hence, unless one has a precision that can detect such small changes in the cost value, one will not be able to determine a cost minimization direction with gradient-based, or even with black-box optimizers such as the Nelder-Mead method [58][59][60][61] .…”

Section: Methodsmentioning

confidence: 99%

Cost function dependent barren plateaus in shallow parametrized quantum circuits

et al. 2021

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Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V(θ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming V(θ) is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining C in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when V(θ) is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining C with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of V(θ) is $${\mathcal{O}}(\mathrm{log}\,n)$$ O ( log n ) . Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.

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Section: Methodsmentioning

confidence: 99%

Cost function dependent barren plateaus in shallow parametrized quantum circuits

et al. 2021

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“…These involve the fourth moment of the Haar measure, which results in cumbersome analytical expressions of hundreds of coefficients coming from the Weingarten calculus [48]. We deal with these analytically with the recently introduced RTNI package [60]. In contrast, the calculations with the same model of e.g., [7] only involve second moments, which can be done by hand.…”

Section: Random Matrix Theory Analysismentioning

confidence: 99%

Relaxation of non-integrable systems and correlation functions

Riddell¹,

García-Pintos²,

Alhambra³

2021

Preprint

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We investigate early-time equilibration rates of observables in closed many-body quantum systems and compare them to those of two correlation functions, first introduced by Kubo and Srednicki. We explore whether these different rates coincide at a universal value that sets the timescales of processes at a finite energy density. We find evidence for this coincidence when the initial conditions are sufficiently generic, or typical. We quantify this with the effective dimension of the state and with a state-observable effective dimension, which estimate the number of energy levels that participate in the dynamics. Our findings are confirmed by proving that these different timescales coincide for dynamics generated by Haar-random Hamiltonians. This also allows to quantitatively understand the scope of previous theoretical results on equilibration timescales and on random matrix formalisms. We approach this problem with exact, full spectrum diagonalization. The numerics are carried out in a non-integrable Heisenberg-like Hamiltonian, and the dynamics are investigated for several pairs of observables and states.

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“…Proof. To calculate the integral, we employ the Weingarten formula [26][27][28], which for the relevant case reads:…”

Section: And Arbitrary Operators a ∈ B(hmentioning

confidence: 99%

Quantum and classical dynamical semigroups of superchannels and semicausal channels

Hasenöhrl¹,

Caro²

2021

Preprint

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Quantum devices are subject to natural decay. We propose to study these decay processes as the Markovian evolution of quantum channels, which leads us to dynamical semigroups of superchannels. A superchannel is a linear map that maps quantum channels to quantum channels, while satisfying suitable consistency relations. If the input and output quantum channels act on the same space, then we can consider dynamical semigroups of superchannels. No useful constructive characterization of the generators of such semigroups is known. We characterize these generators in two ways: First, we give an efficiently checkable criterion for whether a given map generates a dynamical semigroup of superchannels. Second, we identify a normal form for the generators of semigroups of quantum superchannels, analogous to the GKLS form in the case of quantum channels. To derive the normal form, we exploit the relation between superchannels and semicausal completely positive maps, reducing the problem to finding a normal form for the generators of semigroups of semicausal completely positive maps. We derive a normal for these generators using a novel technique, which applies also to infinite-dimensional systems. Our work paves the way to a thorough investigation of semigroups of superchannels: Numerical studies become feasible because admissible generators can now be explicitly generated and checked. And analytic properties of the corresponding evolution equations are now accessible via our normal form.

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RTNI—A symbolic integrator for Haar-random tensor networks

Cited by 22 publications

References 37 publications

Cost function dependent barren plateaus in shallow parametrized quantum circuits

Cost function dependent barren plateaus in shallow parametrized quantum circuits

Relaxation of non-integrable systems and correlation functions

Quantum and classical dynamical semigroups of superchannels and semicausal channels

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