Parafermions in a Kagome Lattice of Qubits for Topological Quantum Computation

Hutter, Adrian; Wootton, James R.; Loss, Daniel

doi:10.1103/physrevx.5.041040

Cited by 19 publications

(26 citation statements)

References 85 publications

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“…Exchange and transverse-field clock-model couplings take on a particular simple form in this language: (35) and then introduce spinful fermions d a,α via a Bogoliubov transformation:…”

Section: E Spin-1/2 Representation and Alternative Fermionization Scmentioning

confidence: 99%

“…Finally, parafermion chains are related to bosonic clock models (for any Z N ) [15,32]-a relation that we will frequently exploit. In the Z 4 limit, one can decompose clock spins into two sets of Pauli matrices [33][34][35] that can be fermionized by standard methods [36]. We will later draw further connections to all of these works, particularly the results for quantum-spin-Hall systems.…”

Section: Introductionmentioning

confidence: 99%

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Fermionized parafermions and symmetry-enriched Majorana modes

2018

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Parafermion zero modes are generalizations of Majorana modes that underlie comparatively rich non-Abeliananyon properties. We introduce exact mappings that connect parafermion chains, which can emerge in twodimensional fractionalized media, to strictly one-dimensional fermionic systems. In particular, we show that parafermion zero modes in the former setting translate into symmetry-enriched Majorana modes that intertwine with a bulk order parameter-yielding braiding and fusion properties that are impossible in standard Majorana platforms. Fusion characteristics of symmetry-enriched Majorana modes are directly inherited from the associated parafermion setup and can be probed via two kinds of anomalous pumping cycles that we construct. Most notably, our mappings relate Z 4 parafermions to conventional electrons with time-reversal symmetry. In this case, one of our pumping protocols entails fairly minimal experimental requirements: Cycling a weakly correlated wire between a trivial phase and time-reversal-invariant topological superconducting state produces an edge magnetization with quadrupled periodicity. Our work highlights new avenues for exploring beyond-Majorana physics in experimentally relevant one-dimensional electronic platforms, including proximitized ferromagnetic chains.

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“…Exchange and transverse-field clock-model couplings take on a particular simple form in this language: (35) and then introduce spinful fermions d a,α via a Bogoliubov transformation:…”

Section: E Spin-1/2 Representation and Alternative Fermionization Scmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fermionized parafermions and symmetry-enriched Majorana modes

2018

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“…Unfortunately, many proposals to physically realize parafermions concern the case where d is even [7,[10][11][12]23,25]. On the other hand, Z 3 parafermions can, for example, emerge in interacting nanowires [17,19,20]; see also Ref.…”

Section: Discussionmentioning

confidence: 99%

“…This includes the case d = 2, i.e., Majorana fermions in a qubit toric code. Reference [25] describes in detail how this can be achieved for d = 4, and the generalization to arbitrary d is straightforward.…”

Section: Discussionmentioning

confidence: 99%

Quantum computing with parafermions

Hutter

Loss

2016

Phys. Rev. B

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Z d parafermions are exotic non-Abelian quasiparticles generalizing Majorana fermions, which correspond to the case d = 2. In contrast to Majorana fermions, braiding of parafermions with d > 2 allows one to perform an entangling gate. This has spurred interest in parafermions, and a variety of condensed matter systems have been proposed as potential hosts for them. In this work, we study the computational power of braiding parafermions more systematically. We make no assumptions on the underlying physical model but derive all our results from the algebraical relations that define parafermions. We find a family of 2d representations of the braid group that are compatible with these relations. The braiding operators derived this way reproduce those derived previously from physical grounds as special cases. We show that if a d-level qudit is encoded in the fusion space of four parafermions, braiding of these four parafermions allows one to generate the entire single-qudit Clifford group (up to phases), for any d. If d is odd, then we show that in fact the entire many-qudit Clifford group can be generated.

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“…Alternatively, braiding can be done using either interaction-based proposals [31,32] or a measurement-only approach [33][34][35] without physically moving anyons. These two approaches are shown to be equivalent [36] and they can avoid diabatic errors associated with moving anyons.…”

Section: Introductionmentioning

confidence: 99%

Measurement-only topological quantum computation without forced measurements

2017

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We investigate the measurement-only topological quantum computation (MOTQC) approach proposed by Bonderson et al (2008 Phys. Rev. Lett. 101 010501) where the braiding operation is shown to be equivalent to a series of topological charge 'forced measurements' of anyons. In a forced measurement, the charge measurement is forced to yield the desired outcome (e.g. charge 0) via repeatedly measuring charges in different bases. This is a probabilistic process with a certain success probability for each trial. In practice, the number of measurements needed will vary from run to run. We show that such an uncertainty associated with forced measurements can be removed by simulating the braiding operation using a fixed number of three measurements supplemented by a correction operator. Furthermore, we demonstrate that in practice we can avoid applying the correction operator in hardware by implementing it in software. Our findings greatly simplify the MOTQC proposal and only require the capability of performing charge measurements to implement topologically protected transformations generated by braiding exchanges without physically moving anyons.

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Parafermions in a Kagome Lattice of Qubits for Topological Quantum Computation

Cited by 19 publications

References 85 publications

Fermionized parafermions and symmetry-enriched Majorana modes

Fermionized parafermions and symmetry-enriched Majorana modes

Quantum computing with parafermions

Measurement-only topological quantum computation without forced measurements

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