Some Aspects of Affleck–Kennedy–Lieb–Tasaki Models: Tensor Network, Physical Properties, Spectral Gap, Deformation, and Quantum Computation

Wei, Tzu-Chieh; Raussendorf, Robert; Affleck, Ian

doi:10.1007/978-3-031-03998-0_5

Cited by 3 publications

(4 citation statements)

References 73 publications

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“…Here ⃗ S i is the angular momentum vector operator for the spin-1 particle on the i th site. This model was first proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) in 1987 as an exactly solvable model exemplifying a gapped excitation spectrum [57][58][59] and a symmetry-protected topological (SPT) order for odd-integer spin chain [60,61]. The topological order of the 1-D AKLT chain is protected by the Z 2 × Z 2 symmetry group, and can be detected by a string order parameter [62,63] or characterized by the entanglement spectrum [64].…”

Section: Proposal a The Aklt Statementioning

confidence: 99%

“…The topological order of the 1-D AKLT chain is protected by the Z 2 × Z 2 symmetry group, and can be detected by a string order parameter [62,63] or characterized by the entanglement spectrum [64]. The AKLT ground state is short-range entangled, can be efficiently represented via MPS, and cannot be modified into a non-entangled product state without closing the gap of the Hamiltonian or breaking the Z 2 × Z 2 symmetry [59]. Also, the computation capability of an open AKLT chain as a quantum wire in measurementbased quantum computation is shared by all states in the Z 2 ×Z 2 symmetry-protected topological phase [65].…”

Section: Proposal a The Aklt Statementioning

confidence: 99%

“…Here P S=2 i,i+1 projects a pair of neighboring spin-1 particles (the i th and the (i + 1) th ) onto the subspace where their total spin equals 2, thereby adding an energetic cost to the S total = 2 subspace. Since the AKLT Hamiltonian is frustration-free [59], its ground state can be reached by independently driving each adjacent pair of qutrits into the S total ∈ {0, 1} subspace. The AKLT state can then be obtained whenever the projection onto S total = 2 is eliminated for each adjacent pair of sites, which is schematically indicated in Figure 1(a).…”

Section: Proposal a The Aklt Statementioning

confidence: 99%

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Dissipative preparation and stabilization of many-body quantum states in a superconducting qutrit array

et al. 2023

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We present and analyze a protocol for driven-dissipatively preparing and stabilizing a manifold of quantum manybody entangled states with symmetry-protected topological order. Specifically, we consider the experimental platform consisting of superconducting transmon circuits and linear microwave resonators. We perform theoretical modeling of this platform via pulse-level simulations based on physical features of real devices. In our protocol, transmon qutrits are mapped onto spin-1 systems. The qutrits' sharing of nearest-neighbor dispersive coupling to a dissipative microwave resonator enables elimination of state population in the S total = 2 subspace for each adjacent pair, and thus, the stabilization of the manybody system into the Affleck, Kennedy, Lieb, and Tasaki (AKLT) state up to the edge mode configuration. We also analyze the performance of our protocol as the system size scales up to four qutrits, in terms of its fidelity as well as the stabilization time.Our work shows the capacity of driven-dissipative superconducting cQED systems to host robust and self-corrected quantum manybody states that are topologically non-trivial.

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Section: Proposal a The Aklt Statementioning

confidence: 99%

Section: Proposal a The Aklt Statementioning

confidence: 99%

Section: Proposal a The Aklt Statementioning

confidence: 99%

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Dissipative preparation and stabilization of many-body quantum states in a superconducting qutrit array

et al. 2023

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“…As a type of Valence-Bond-Solid (VBS) state [37], it is the exact ground state of the spin-1 AKLT model, which is the paradigm of a strongly correlated symmetry protected topological (SPT) phase with a Haldane gap [38] and fractional excitations at its boundaries [36,39,40]. SPT phases of matter received much attention recently on quantum computers [17,[41][42][43][44][45][46], and the two-dimensional generalization of the AKLT model on a trivalent lattice is proposed to be a universal resource [47,48] for measurement-based quantum computation [49][50][51]. So far, the 1D AKLT state has been experimentally realized and characterized on photonic implementations [52] using cluster states [53] and in trapped ions [54].…”

Section: Introductionmentioning

confidence: 99%

High-fidelity realization of the AKLT state on a NISQ-era quantum processor

Chen,

Shen,

Lee

et al. 2023

SciPost Phys.

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The AKLT state is the ground state of an isotropic quantum Heisenberg spin-1 model. It exhibits an excitation gap and an exponentially decaying correlation function, with fractionalized excitations at its boundaries. So far, the one-dimensional AKLT model has only been experimentally realized with trapped-ions as well as photonic systems. In this work, we successfully prepared the AKLT state on a noisy intermediate-scale quantum (NISQ) era quantum device. In particular, we developed a non-deterministic algorithm on the IBM quantum processor, where the non-unitary operator necessary for the AKLT state preparation is embedded in a unitary operator with an additional ancilla qubit for each pair of auxiliary spin-1/2’s. Such a unitary operator is effectively represented by a parametrized circuit composed of single-qubit and nearest-neighbor CX gates. Compared with the conventional operator decomposition method from Qiskit, our approach results in a much shallower circuit depth with only nearest-neighbor gates, while maintaining a fidelity in excess of 99.99% with the original operator. By simultaneously post-selecting each ancilla qubit such that it belongs to the subspace of spin-up |↑>, an AKLT state can be systematically obtained by evolving from an initial trivial product state of singlets plus ancilla qubits in spin-up on a quantum computer, and it is subsequently recorded by performing measurements on all the other physical qubits. We show how the accuracy of our implementation can be further improved on the IBM quantum processor with readout error mitigation.

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Efficient Verification of Ground States of Frustration-Free Hamiltonians

Zhu,

Li,

Chen

2024

Quantum

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Ground states of local Hamiltonians are of key interest in many-body physics and also in quantum information processing. Efficient verification of these states are crucial to many applications, but very challenging. Here we propose a simple, but powerful recipe for verifying the ground states of general frustration-free Hamiltonians based on local measurements. Moreover, we derive rigorous bounds on the sample complexity by virtue of the quantum detectability lemma (with improvement) and quantum union bound. Notably, the number of samples required does not increase with the system size when the underlying Hamiltonian is local and gapped, which is the case of most interest. As an application, we propose a general approach for verifying Affleck-Kennedy-Lieb-Tasaki (AKLT) states on arbitrary graphs based on local spin measurements, which requires only a constant number of samples for AKLT states defined on various lattices. Our work is of interest not only to many tasks in quantum information processing, but also to the study of many-body physics.

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Some Aspects of Affleck–Kennedy–Lieb–Tasaki Models: Tensor Network, Physical Properties, Spectral Gap, Deformation, and Quantum Computation

Cited by 3 publications

References 73 publications

Dissipative preparation and stabilization of many-body quantum states in a superconducting qutrit array

Dissipative preparation and stabilization of many-body quantum states in a superconducting qutrit array

High-fidelity realization of the AKLT state on a NISQ-era quantum processor

Efficient Verification of Ground States of Frustration-Free Hamiltonians

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