Local Pauli stabilizers of symmetric hypergraph states

Lyons, David W.; Gibbons, Nathaniel P.; Peters, Mark; Upchurch, Daniel J; Walck, Scott N.; Wertz, Ezekiel W.

doi:10.1088/1751-8121/aa70b4

Cited by 10 publications

(15 citation statements)

References 32 publications

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“…The first term in the above Hamiltonian is a Isinglike system on the H with many-body interactions corresponding to each hyperedge of the H and the second term is a transverse field. Interesting point is that the role of the magnetic field h and coupling constant J of the original Hamiltonian (7) have been switched in the new Hamiltonian (9). In this way, a regime of the strong magnetic field in the original model, where the ratio h J is bigger than 1, is mapped to a regime of weak magnetic field in the dual Hamiltonian where the ratio J h is smaller than 1.…”

Section: Mapping Quantum Css Codes To Ising-like Systems; a Strong-we...mentioning

confidence: 99%

“…In Fig. (9) we show such a hypergraph. The number of vertices and hyperedges for this hypergraph are the same and since each vertex is a member of two neighboring hyperedges, it is simple to check that such a hypergraph is self dual, see Fig.…”

Section: Self-dual Modelsmentioning

confidence: 99%

“…The number of vertices and hyperedges for this hypergraph are the same and since each vertex is a member of two neighboring hyperedges, it is simple to check that such a hypergraph is self dual, see Fig. (9). In this way, according to the duality mapping a Hamiltonian corresponding to the above X-type stabilizers in a uniform magnetic field is self-dual and the 9: (Color online) a) A one-dimensional lattice where qubits live in edges of the lattice and an X-type stabilizer is defined corresponding to each vertex.…”

Section: Self-dual Modelsmentioning

confidence: 99%

“…In particular, quantum hypergraph states have been introduced and many applications have been considered [6]. Mapping quantum states to a hypergraphs helps one to use some relations in graph theory to study different properties of quantum entangled states [7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

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Strong-weak coupling duality between two perturbed quantum many-body systems: Calderbank-Shor-Steane codes and Ising-like systems

Zarei

2017

Phys. Rev. B

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Graphs and recently hypergraphs have been known as an important tool for considering different properties of quantum many-body systems. In this paper, we study a mapping between an important class of quantum systems namely quantum Calderbank-Shor-Steane (CSS) codes and Ising-like systems by using hypergraphs. We show that the Hamiltonian corresponding to a CSS code on a hypergraph H which is perturbed by a uniform magnetic field is mapped to Hamiltonian of a Isinglike system on dual hypergraphH in a transverse field. Interestingly, we show that a strong regime of couplings in one of the systems is mapped to a weak regime of couplings in another one. We also give some applications for such a mapping where we study robustness of different topological CSS codes against a uniform magnetic field including Kitaev's toric codes defined on graphs and color codes in different dimensions. We show that a perturbed Kitaev's toric code on an arbitrary graph is mapped to an Ising model in a transverse field on the same graph and a perturbed color code on a D-colex is mapped to a Ising-like model on a D-simplicial lattice in a transverse field. In particular, we use these results to explicitly compare the robustness of TC on different graphs in different dimensions. Interestingly, our results show that the robustness of such topological codes defined on graphs decreases with increasing dimension. Furthermore, we also use the duality mapping for some self-dual models where we exactly derive the point of phase transition.

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Section: Mapping Quantum Css Codes To Ising-like Systems; a Strong-we...mentioning

confidence: 99%

Section: Self-dual Modelsmentioning

confidence: 99%

Section: Self-dual Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Strong-weak coupling duality between two perturbed quantum many-body systems: Calderbank-Shor-Steane codes and Ising-like systems

Zarei

2017

Phys. Rev. B

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“…Transversality means that, in order to perform gates on the encoded logical qubits, one can apply corresponding gates to the physical qubits in a parallel fashion, and this convenience has sparked recent interest on this gate set [9][10][11][12][13][14]. Third, the manybody entangled states generated by the CCZ gates are known as hypergraph states in entanglement theory [15][16][17][18][19]. They found applications in quantum algorithms [20] and Bell inequalities [21].…”

Section: Introductionmentioning

confidence: 99%

Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states

2019

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While the circuit model of quantum computation defines its logical depth or "computational time" in terms of temporal gate sequences, the measurement-based model could allow totally different temporal ordering and parallelization of logical gates. By developing techniques to analyze Pauli measurements on multi-qubit hypergraph states generated by the Controlled-Controlled-Z (CCZ) gates, we introduce a deterministic scheme of universal measurement-based computation. In contrast to the cluster-state scheme where the Clifford gates are parallelizable, our scheme enjoys massive parallelization of CCZ and SWAP gates, so that the computational depth grows with the number of global applications of Hadamard gates, or, in other words, with the number of changing computational bases. A logarithmic-depth implementation of an N -times Controlled-Z gate illustrates a novel trade-off between space and time complexity.

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