Combining Topological Hardware and Topological Software: Color-Code Quantum Computing with Topological Superconductor Networks

Litinski, Daniel; Kesselring, Markus S.; Eisert, Jens; Oppen, Felix von

doi:10.1103/physrevx.7.031048

Cited by 79 publications

(89 citation statements)

References 88 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Each segment hosts two Majorana bound states which form a qubit. (This is in contrast to the definite-parity Majorana qubits which are composed of four Majoranas [32,33].) The Hamiltonian arXiv:1906.01658v2 [cond-mat.mes-hall] 6 Aug 2019 2 for the device iswhere Λ decreases as e −L/ξ and which we assume to be homogeneous to each segment, L is the length of each segment, ξ is the Majorana decay length, and g i is the coupling between Majoranas at the junction between segments i and i + 1.…”

mentioning

confidence: 84%

See 1 more Smart Citation

One-dimensional error-correcting code for Majorana qubits

Stenger

Mong

2020

Phys. Rev. A

View full text Add to dashboard Cite

Although Majorana platforms are promising avenues to realizing topological quantum computing, they are still susceptible to errors from thermal noise and other sources. We show that the error rate of Majorana qubits can be drastically reduced using a 1D repetition code. The success of the code is due the imbalance between the phase error rate and the flip error rate. We demonstrate how a repetition code can be naturally constructed from segments of Majorana nanowires. We find the optimal lifetime may be extended from a millisecond to over one second.The main road block in achieving quantum computation [1] is dealing with quantum error. Isolating a bit of quantum information from its environment is challenging enough, however, in order to realize a useful quantum computation machine it is necessary to maintain coherence for thousands of entangled qubits. Topological qubits are useful in that they have built-in fault tolerance due to the spatial separations between the anyons and the boundary modes [2]. Majorana zero modes [3][4][5], which appear as end modes of p-wave superconducting nanowires, are one the most promising directions in topological quantum computing [4,[6][7][8][9][10][11][12][13][14]. These Majorana end modes can store information non-locally and can be braided to perform topologically protected logic gates [15][16][17][18][19][20][21][22].Although topological qubits have some level of protection from error they will still require error correction in order to be fully implemented as computational qubits. A perfect Majorana qubit would be infinitely long and held at zero temperature. Nonzero temperatures lead to a finite quasiparticle density, which will cause errors in the qubit. There exist error correction codes such as the toric code [2], surface codes [23][24][25][26], and color codes [27][28][29], which can be implemented on Majorana qubits [30][31][32][33][34][35][36][37] or in other schemes such as planar codes [38,39]. However, these error correction schemes require a great deal of overhead, having a large number of redundant qubits in order to catch and correct error. As Kitaev pointed out [2], any topological phase of matter can be identified as an error correcting code. In this vain, we ask if the 1D fermionic topological phase [40,41] built from a chain of Majorana nanowires can be identified with a "fermion-parity protected error correcting code". Such a chain would require only a line of physical qubits instead of a surface.In this paper, we show how a chain of Majorana nanowires can be used to significantly improve the qubit lifetime, because of a hierarchy of different error types in Majorana qubits. Due to an unexpectedly high observed density of quasiparticles [42][43][44][45][46][47], we argue that phase errors in Majorana qubits are orders of magnitude greater than bit flip errors. This phase error can be corrected at the expense of the much smaller bit flip error using the repetition code [48][49][50]. We describe the repetition code in the language of Majorana qubits. The code...

show abstract

mentioning

confidence: 84%

mentioning

confidence: 99%

One-dimensional error-correcting code for Majorana qubits

Stenger

Mong

2020

Phys. Rev. A

View full text Add to dashboard Cite

show abstract

“…While this is not yet computationally universal, it can be rendered universal using gate teleportation [18] and magic state distillation [19]. Moreover, color codes are particularly suitable for topological quantum computation with Majorana qubits, since high-fidelity Clifford gates are accessible by braiding [20,21].…”

Section: Introductionmentioning

confidence: 99%

Neural network decoder for topological color codes with circuit level noise

et al. 2019

View full text Add to dashboard Cite

A quantum computer needs the assistance of a classical algorithm to detect and identify errors that affect encoded quantum information. At this interface of classical and quantum computing the technique of machine learning has appeared as a way to tailor such an algorithm to the specific error processes of an experiment-without the need for a priori knowledge of the error model. Here, we apply this technique to topological color codes. We demonstrate that a recurrent neural network with long short-term memory cells can be trained to reduce the error rate ò L of the encoded logical qubit to values much below the error rate ò phys of the physical qubits-fitting the expected power law scaling   µ + ( ) d L phys 1 2 , with d the code distance. The neural network incorporates the information from 'flag qubits' to avoid reduction in the effective code distance caused by the circuit. As a test, we apply the neural network decoder to a density-matrix based simulation of a superconducting quantum computer, demonstrating that the logical qubit has a longer life-time than the constituting physical qubits with near-term experimental parameters.

show abstract

“…3 In this work, we concentrate on a different implementation of the latter approach involving a lattice of mesoscopic superconducting islands with Majorana zero modes (MZM-s) on each island. [7][8][9][10][11][12][13][14][15][16][17][18] Note that any physical implementation of either approaches eventually also involves active measurements which evacuate the entropy due to thermal fluctuations since the topological ordering of the toric code disappears at any nonzero temperature. A toric code built out of MZM-based physical qubits holds the promise of better error correction properties compared to its superconducting transmon qubit-based counterpart.…”

Section: Introductionmentioning

confidence: 99%

Topological ordering in the Majorana toric code

2019

View full text Add to dashboard Cite

At zero temperature, a two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands exhibits a topologically-ordered toric code phase. Recently, a Landau field theory was used to describe the different phases of the aforementioned system and the phase-transitions separating them. While the field theory provides details on the properties of the system close to the phase-transitions, signatures of topological ordering in the different phases have not been computed. This is the primary goal of the current work. We describe a lattice gauge theory of the Majorana toric code in terms of U(1) matter fields coupled to an emergent Z2 gauge field. Subsequently, we use a generalized Wilson-loop order-parameter, the equal-time Fredenhagen-Marcu order parameter, to characterize the topological ordering in the different phases. Our computation provides evidence of the toric code phase both in the Mott insulator and the charge-2e superconductor phases, while showing that the toric code phase disappears in the charge-e superconductor phase. In addition, we perturbatively analyze the influence of Cooper pair tunneling on the topological gap of the toric code in the limit of strong charging energy and show that the toric code phase is, in fact, stabilized by the Cooper pair tunneling. Our results are relevant for experimental realizations of the Majorana toric code.

show abstract

Combining Topological Hardware and Topological Software: Color-Code Quantum Computing with Topological Superconductor Networks

Cited by 79 publications

References 88 publications

One-dimensional error-correcting code for Majorana qubits

One-dimensional error-correcting code for Majorana qubits

Neural network decoder for topological color codes with circuit level noise

Topological ordering in the Majorana toric code

Contact Info

Product

Resources

About