Toric codes and quantum doubles from two-body Hamiltonians

Brell, Courtney G.; Flammia, Steven T.; Bartlett, Stephen D.; Doherty, Andrew C.

doi:10.1088/1367-2630/13/5/053039

Cited by 31 publications

(53 citation statements)

References 66 publications

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“…Perturbation gadgets formalism allows one to construct a simpler highenergy simulator Hamiltonian with only two-body interactions whose low-energy properties (such as the ground state energy) approximate the ones of H target . For example, various perturbation gadgets have been constructed for target Hamiltonians with the topological quantum order [21,22,23]. The SW method provides a natural framework for constructing and analyzing perturbation gadgets [24].…”

Section: Applications Of the Sw Methodsmentioning

confidence: 99%

Schrieffer–Wolff transformation for quantum many-body systems

2011

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The Schrieffer-Wolff (SW) method is a version of degenerate perturbation theory in which the low-energy effective Hamiltonian H eff is obtained from the exact Hamiltonian by a unitary transformation decoupling the low-energy and high-energy subspaces. We give a self-contained summary of the SW method with a focus on rigorous results. We begin with an exact definition of the SW transformation in terms of the so-called direct rotation between linear subspaces. From this we obtain elementary proofs of several important properties of H eff such as the linked cluster theorem. We then study the perturbative version of the SW transformation obtained from a Taylor series representation of the direct rotation. Our perturbative approach provides a systematic diagram technique for computing high-order corrections to H eff . We then specialize the SW method to quantum spin lattices with short-range interactions. We establish unitary equivalence between effective low-energy Hamiltonians obtained using two different versions of the SW method studied in the literature. Finally, we derive an upper bound on the precision up to which the ground state energy of the n-th order effective Hamiltonian approximates the exact ground state energy.

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Section: Applications Of the Sw Methodsmentioning

confidence: 99%

Schrieffer–Wolff transformation for quantum many-body systems

2011

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“…Indeed, the simplest example of a Clifford circuit -a wire of identity gates -maps directly onto the quantum XY model, which is gapless when the coupling strengths are equal [32], but somewhat encouragingly is otherwise gapped and maps onto Kitaev's proposal for a quantum wire [27]. Other models of subsystem code Hamiltonians exist; some are gapped [13,8,12] and some are not [4,21]. Addressing the lack of a satisfying general theory of gauge Hamiltonians is perhaps a natural first step in trying to understand the power of our construction in the quest for a self-correcting memory.…”

Section: Discussionmentioning

confidence: 99%

“…The discovery of subsystem codes [37,30] led to the study of sparse subsystem codes, first in the context of topological subsystem codes, of which there are now many examples [4,5,19,13,42,39,8,6,12]. However, these codes are all concerned with the case k = O(1).…”

Section: Sparse Quantum Codes and Related Workmentioning

confidence: 99%

Sparse Quantum Codes from Quantum Circuits

Bacon

Flammia

Harrow

et al. 2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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Sparse quantum codes are analogous to LDPC codes in that their check operators require examining only a constant number of qubits. In contrast to LDPC codes, good sparse quantum codes are not known, and even to encode a single qubit, the best known distance is O( n log(n)), due to Freedman, Meyer and Luo.We construct a new family of sparse quantum subsystem codes with minimum distance n 1− for = O(1/ √ log n). A variant of these codes exists in D spatial dimensions and has d = n 1− −1/D , nearly saturating a bound due to Bravyi and Terhal.Our construction is based on a new general method for turning quantum circuits into sparse quantum subsystem codes. Using this prescription, we can map an arbitrary stabilizer code into a new subsystem code with the same distance and number of encoded qubits but where all the generators have constant weight, at the cost of adding some ancilla qubits. With an additional overhead of ancilla qubits, the new code can also be made spatially local.

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“…Indeed, the 4-local toric code Hamiltonian can be recovered effectively in the right parameter regime of a nearest-neighbor 2-local, yet frustrated, Hamiltonian on a honeycomb lattice. More generally, there is a procedure to turn a 4-local quantum double Hamiltonian for arbitrary group into a frustrated 2-local Hamiltonian thanks to a so-called "gadget construction" [14].…”

Section: Locality Of the Hamiltonianmentioning

confidence: 99%

Anyons are not energy eigenspaces of quantum double Hamiltonians

Kómár

Landon-Cardinal

2017

Phys. Rev. B

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Kitaev's quantum double models, including the toric code, are canonical examples of quantum topological models on a two-dimensional spin lattice. Their Hamiltonian defines the ground space by imposing an energy penalty to any nontrivial flux or charge, but does not distinguish among those. We generalize this construction by introducing a family of Hamiltonians made of commuting four-body projectors that provide an intricate splitting of the Hilbert space by discriminating among nontrivial charges and fluxes. Our construction highlights that anyons are not in one-to-one correspondence with energy eigenspaces, a feature already present in Kitaev's construction. This discrepancy is due to the presence of local degrees of freedom in addition to topological ones on a lattice.

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Toric codes and quantum doubles from two-body Hamiltonians

Cited by 31 publications

References 66 publications

Schrieffer–Wolff transformation for quantum many-body systems

Schrieffer–Wolff transformation for quantum many-body systems

Sparse Quantum Codes from Quantum Circuits

Anyons are not energy eigenspaces of quantum double Hamiltonians

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