scholarly journals The boundaries and twist defects of the color code and their applications to topological quantum computation

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 101 ◽  
Author(s):  
Markus S. Kesselring ◽  
Fernando Pastawski ◽  
Jens Eisert ◽  
Benjamin J. Brown
Keyword(s):  
Quantum Computation ◽  
Domain Walls ◽  
Fault Tolerant ◽  
Error Correcting Codes ◽  
Two Dimensions ◽  
Abstract Theory ◽  
Color Code ◽  
Topological Quantum ◽  
Multiple Copies ◽  
Surface Code

The color code is both an interesting example of an exactly solved topologically ordered phase of matter and also among the most promising candidate models to realize fault-tolerant quantum computation with minimal resource overhead. The contributions of this work are threefold. First of all, we build upon the abstract theory of boundaries and domain walls of topological phases of matter to comprehensively catalog the objects realizable in color codes. Together with our classification we also provide lattice representations of these objects which include three new types of boundaries as well as a generating set for all 72 color code twist defects. Our work thus provides an explicit toy model that will help to better understand the abstract theory of domain walls. Secondly, we discover a number of interesting new applications of the cataloged objects for quantum information protocols. These include improved methods for performing quantum computations by code deformation, a new four-qubit error-detecting code, as well as families of new quantum error-correcting codes we call stellated color codes, which encode logical qubits at the same distance as the next best color code, but using approximately half the number of physical qubits. To the best of our knowledge, our new topological codes have the highest encoding rate of local stabilizer codes with bounded-weight stabilizers in two dimensions. Finally, we show how the boundaries and twist defects of the color code are represented by multiple copies of other phases. Indeed, in addition to the well studied comparison between the color code and two copies of the surface code, we also compare the color code to two copies of the three-fermion model. In particular, we find that this analogy offers a very clear lens through which we can view the symmetries of the color code which gives rise to its multitude of domain walls.

scholarly journals The XZZX surface code

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
J. Pablo Bonilla Ataides ◽  
David K. Tuckett ◽  
Stephen D. Bartlett ◽  
Steven T. Flammia ◽  
Benjamin J. Brown
Keyword(s):  
Quantum Computation ◽  
High Performance ◽  
Quantum Computer ◽  
Fault Tolerant ◽  
Error Correcting Codes ◽  
The Common ◽  
Surface Code ◽  
Random Codes ◽  
Quantum Error ◽  
Relevant Range

AbstractPerforming large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code—the XZZX code—offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation.

Graphical algorithms and threshold error rates for the 2d color code

2010 ◽  
Vol 10 (9&10) ◽  
pp. 780-802
Author(s):  
David S. Wang ◽  
Austin G. Fowler ◽  
Charles D. Hill ◽  
Lloyd C.L. Hollenberg
Keyword(s):  
Quantum Computation ◽  
Error Rate ◽  
Graph Matching ◽  
Fault Tolerant ◽  
Error Rates ◽  
Color Code ◽  
Clifford Group ◽  
Universal Quantum ◽  
Surface Code ◽  
Topological Error

Recent work on fault-tolerant quantum computation making use of topological error correction shows great potential, with the 2d surface code possessing a threshold error rate approaching 1\%. However, the 2d surface code requires the use of a complex state distillation procedure to achieve universal quantum computation. The color code of is a related scheme partially solving the problem, providing a means to perform all Clifford group gates transversally. We review the color code and its error correcting methodology, discussing one approximate technique based on graph matching. We derive an analytic lower bound to the threshold error rate of 6.25\% under error-free syndrome extraction, while numerical simulations indicate it may be as high as 13.3\%. Inclusion of faulty syndrome extraction circuits drops the threshold to approximately 0.10 \pm 0.01\%.

scholarly journals Competing ν = 5/2 fractional quantum Hall states in confined geometry

2016 ◽  
Vol 113 (44) ◽  
pp. 12386-12390 ◽  
Author(s):  
Hailong Fu ◽  
Pengjie Wang ◽  
Pujia Shan ◽  
Lin Xiong ◽  
Loren N. Pfeiffer ◽  
...  
Keyword(s):  
Quantum Computation ◽  
Filling Factor ◽  
Fault Tolerant ◽  
Quantum Hall ◽  
Point Contact ◽  
Fractional Quantum ◽  
Fractional Quantum Hall ◽  
Topological Quantum ◽  
Quantum Point ◽  
Topological Quantum Computation

Some theories predict that the filling factor 5/2 fractional quantum Hall state can exhibit non-Abelian statistics, which makes it a candidate for fault-tolerant topological quantum computation. Although the non-Abelian Pfaffian state and its particle-hole conjugate, the anti-Pfaffian state, are the most plausible wave functions for the 5/2 state, there are a number of alternatives with either Abelian or non-Abelian statistics. Recent experiments suggest that the tunneling exponents are more consistent with an Abelian state rather than a non-Abelian state. Here, we present edge-current–tunneling experiments in geometrically confined quantum point contacts, which indicate that Abelian and non-Abelian states compete at filling factor 5/2. Our results are consistent with a transition from an Abelian state to a non-Abelian state in a single quantum point contact when the confinement is tuned. Our observation suggests that there is an intrinsic non-Abelian 5/2 ground state but that the appropriate confinement is necessary to maintain it. This observation is important not only for understanding the physics of the 5/2 state but also for the design of future topological quantum computation devices.

scholarly journals GEOMETRIC PHASES AND TOPOLOGICAL QUANTUM COMPUTATION

2003 ◽  
Vol 01 (01) ◽  
pp. 1-23 ◽  
Author(s):  
VLATKO VEDRAL
Keyword(s):  
Quantum Computation ◽  
Large Scale ◽  
Quantum Computer ◽  
Fault Tolerant ◽  
Geometric Phases ◽  
Quantum Phases ◽  
Topological Quantum ◽  
Universal Quantum ◽  
The Subject ◽  
Topological Quantum Computation

In the first part of this review we introduce the basics theory behind geometric phases and emphasize their importance in quantum theory. The subject is presented in a general way so as to illustrate its wide applicability, but we also introduce a number of examples that will help the reader understand the basic issues involved. In the second part we show how to perform a universal quantum computation using only geometric effects appearing in quantum phases. It is then finally discussed how this geometric way of performing quantum gates can lead to a stable, large scale, intrinsically fault-tolerant quantum computer.

scholarly journals Accuracy threshold for postselected quantum computation

2008 ◽  
Vol 8 (3&4) ◽  
pp. 181-244 ◽  
Author(s):  
P. Aliferis ◽  
D. Gottesman ◽  
J. Preskill
Keyword(s):  
Lower Bound ◽  
Quantum Computation ◽  
Error Detection ◽  
Fault Tolerant ◽  
Error Correcting Codes ◽  
Stochastic Noise ◽  
Strongly Correlated ◽  
Error Detecting ◽  
Error Detecting Code

We prove an accuracy threshold theorem for fault-tolerant quantum computation based on error detection and postselection. Our proof provides a rigorous foundation for the scheme suggested by Knill, in which preparation circuits for ancilla states are protected by a concatenated error-detecting code and the preparation is aborted if an error is detected. The proof applies to independent stochastic noise but (in contrast to proofs of the quantum accuracy threshold theorem based on concatenated error-correcting codes) not to strongly-correlated adversarial noise. Our rigorously established lower bound on the accuracy threshold, $1.04\times 10^{-3}$, is well below Knill's numerical estimates.

scholarly journals Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 62 ◽  
Author(s):  
Daniel Litinski ◽  
Felix von Oppen
Keyword(s):  
Quantum Computation ◽  
Long Range ◽  
Fault Tolerant ◽  
Planar Surface ◽  
Single Qubit ◽  
Control Target ◽  
Surface Code ◽  
Time Overhead

We present a planar surface-code-based scheme for fault-tolerant quantum computation which eliminates the time overhead of single-qubit Clifford gates, and implements long-range multi-target CNOT gates with a time overhead that scales only logarithmically with the control-target separation. This is done by replacing hardware operations for single-qubit Clifford gates with a classical tracking protocol. Inter-qubit communication is added via a modified lattice surgery protocol that employs twist defects of the surface code. The long-range multi-target CNOT gates facilitate magic state distillation, which renders our scheme fault-tolerant and universal.

scholarly journals Non-Pauli topological stabilizer codes from twisted quantum doubles

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 398
Author(s):  
Julio Carlos Magdalena de la Fuente ◽  
Nicolas Tarantino ◽  
Jens Eisert
Keyword(s):  
Quantum Information ◽  
Error Correction ◽  
Lattice Models ◽  
Full Rank ◽  
Error Correcting Codes ◽  
Quantum Error Correction ◽  
Topological Order ◽  
Topological Quantum ◽  
Surface Code ◽  
Quantum Error

It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by non-Pauli commuting stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models.

Sign in / Sign up

Export Citation Format

Share Document