Complexity phase diagram for interacting and long-range bosonic Hamiltonians

Maskara, Nishad; Deshpande, Abhinav V.; Ehrenberg, Adam; Trân, Minh; Fefferman, Bill; Gorshkov, Alexey V.

doi:10.48550/arxiv.1906.04178

Cited by 9 publications

(23 citation statements)

References 35 publications

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“…Since these models do not typically conserve the number of bosons, our methods will need to be modified somewhat to remain applicable, although in certain limiting cases Lieb-Robinson bounds have nevertheless been obtained for such models [26]. We anticipate that our theorem will also have broad implications for the simulatability of the Bose-Hubbard model, which is conjectured to be a test of quantum supremacy in a future quantum computer [13,[43][44][45][46]. We hope to return to these interesting extensions of our work in the near future.…”

Section: Discussionmentioning

confidence: 99%

Finite speed of quantum information in models of interacting bosons at finite density

Yin,

Lucas

2021

Preprint

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We prove that quantum information propagates with a finite velocity in any model of interacting bosons whose (possibly time-dependent) Hamiltonian contains spatially local single-boson hopping terms along with arbitrary local density-dependent interactions. More precisely, with density matrix ρ ∝ exp[−µN ] (with N the total boson number), ensemble averaged correlators of the form [A0, Br(t)] , along with out-of-time-ordered correlators, must vanish as the distance r between two local operators grows, unless t ≥ r/v for some finite speed v. In one dimensional models, we give a useful extension of this result that demonstrates the smallness of all matrix elements of the commutator [A0, Br(t)] between finite density states if t/r is sufficiently small. Our bounds are relevant for physically realistic initial conditions in experimentally realized models of interacting bosons. In particular, we prove that v can scale no faster than linear in number density in the Bose-Hubbard model: this scaling matches previous results in the high density limit. The quantum walk formalism underlying our proof provides an alternative method for bounding quantum dynamics in models with unbounded operators and infinite-dimensional Hilbert spaces, where Lieb-Robinson bounds have been notoriously challenging to prove.

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Section: Discussionmentioning

confidence: 99%

Finite speed of quantum information in models of interacting bosons at finite density

Yin,

Lucas

2021

Preprint

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“…Discussion.-The above analysis of 2d circuits suggests that the finite-time teleportation transition may generally correspond to a transition in approximate sampling complexity [22,44]. Specifically, we consider the problem of sampling measurement outcomes from N qubits initialized in a product state and evolved under a finite-time unitary circuit.…”

Section: However S (N)mentioning

confidence: 99%

Finite time teleportation phase transition in random quantum circuits

Bao¹,

Block²,

Altman³

2021

Preprint

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How long does it take to entangle two distant qubits in a quantum circuit evolved by generic unitary dynamics? We show that if the time evolution is followed by measurement of all but the two infinitely separated test qubits, then the entanglement between them can undergo a phase transition and become nonzero at a finite critical time tc. The fidelity of teleporting a quantum state from an input qubit to an infinitely distant output qubit shows the same critical onset. Specifically, these finite time transitions occur in short-range interacting two-dimensional random unitary circuits and in sufficiently long-range interacting one-dimensional circuits. The phase transition is understood by mapping the random continuous-time evolution to a finite temperature thermal state of an effective spin Hamiltonian, where the inverse temperature equals the evolution time in the circuit. In this framework, the entanglement between two distant qubits at times t > tc corresponds to the emergence of long-range ferromagnetic spin correlations below the critical temperature. We verify these predictions using numerical simulation of Clifford circuits and propose potential realizations in existing platforms for quantum simulation.

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“…For example, the Lieb-Robinson light cone limits the correlation between sublattices [21]. Using this, it has been shown that up to time t ∼ (M/N ) 1/d , there exists an efficient classical algorithm to sample from the output distribution of general quadratic bosonic Hamiltonian dynamics [19,20]. In this work, we focus on a circuit model of random 2-local passive transformation and find a depth in which classical simulation of sampling is efficient.…”

Section: Problem Setupmentioning

confidence: 99%

“…On the other hand, when a quantum circuit is deep enough to implement a global Haarrandom unitary circuit that generates a large amount of entanglement, sampling from the probability distributions of the output state becomes classically intractable under plausible assumptions [9,10,17,18]. Indeed, under bosonic Hamiltonian dynamics, phase transition behavior of sampling arising from the evolution time has been diagnosed [19,20]. Whereas time-evolution dynamics under a bosonic Hamitonian using the Lieb-Robinson bound [21] has been studied in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…Whereas time-evolution dynamics under a bosonic Hamitonian using the Lieb-Robinson bound [21] has been studied in Refs. [19,20], we focus on linearoptical circuits composed of Haar-random beam splitter arrays and employ a mapping from random linear-optical circuits to classical random walk [22]. Investigating random circuits enables us to analyze a generic property of linear-optical circuits.…”

Section: Introductionmentioning

confidence: 99%

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Classical simulation of bosonic linear-optical random circuits beyond linear light cone

Oh¹,

Lim²,

Fefferman³

et al. 2021

Preprint

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Sampling from probability distributions of quantum circuits is a fundamentally and practically important task which can be used to demonstrate quantum supremacy using noisy intermediatescale quantum devices. In the present work, we examine classical simulability of sampling from the output photon-number distribution of linear-optical circuits composed of random beam splitters with equally distributed squeezed vacuum states and single-photon states input. We provide efficient classical algorithms to simulate linear-optical random circuits and show that the algorithms' error is exponentially small up to a depth less than quadratic in the distance between sources using a classical random walk behavior of random linear-optical circuits. Notably, the average-case depth allowing an efficient classical simulation is larger than the worst-case depth limit, which is linear in the distance. Besides, our results together with the hardness of boson sampling give a lower-bound on the depth for constituting global Haar-random unitary circuits.

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Complexity phase diagram for interacting and long-range bosonic Hamiltonians

Cited by 9 publications

References 35 publications

Finite speed of quantum information in models of interacting bosons at finite density

Finite speed of quantum information in models of interacting bosons at finite density

Finite time teleportation phase transition in random quantum circuits

Classical simulation of bosonic linear-optical random circuits beyond linear light cone

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