scholarly journals Surface code quantum error correction incorporating accurate error propagation

2011 ◽  
Vol 11 (1&2) ◽  
pp. 8-18
Author(s):  
Austin G. Fowler ◽  
David S. Wang ◽  
Lloyd C. L. Hollenberg
Keyword(s):  
Nearest Neighbor ◽  
Minimum Weight ◽  
Error Correcting Code ◽  
Square Lattice ◽  
Checkerboard Pattern ◽  
Space And Time ◽  
Logical Error ◽  
Surface Code ◽  
Pattern Information ◽  
Quantum Error

The surface code is a powerful quantum error correcting code that can be defined on a 2-D square lattice of qubits with only nearest neighbor interactions. Syndrome and data qubits form a checkerboard pattern. Information about errors is obtained by repeatedly measuring each syndrome qubit after appropriate interaction with its four nearest neighbor data qubits. Changes in the measurement value indicate the presence of chains of errors in space and time. The standard method of determining operations likely to return the code to its error-free state is to use the minimum weight matching algorithm to connect pairs of measurement changes with chains of corrections such that the minimum total number of corrections is used. Prior work has not taken into account the propagation of errors in space and time by the two-qubit interactions. We show that taking this into account leads to a quadratic improvement of the logical error rate.

scholarly journals Optimization of the surface code design for Majorana-based qubits

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 352
Author(s):  
Rui Chao ◽  
Michael E. Beverland ◽  
Nicolas Delfosse ◽  
Jeongwan Haah
Keyword(s):  
Fault Tolerance ◽  
Error Correction ◽  
Nearest Neighbor ◽  
Error Correcting Code ◽  
Error Rates ◽  
Code Design ◽  
Trade Offs ◽  
Planar Grid ◽  
Connectivity Degree ◽  
Surface Code

The surface code is a prominent topological error-correcting code exhibiting high fault-tolerance accuracy thresholds. Conventional schemes for error correction with the surface code place qubits on a planar grid and assume native CNOT gates between the data qubits with nearest-neighbor ancilla qubits.Here, we present surface code error-correction schemes using only Pauli measurements on single qubits and on pairs of nearest-neighbor qubits. In particular, we provide several qubit layouts that offer favorable trade-offs between qubit overhead, circuit depth and connectivity degree. We also develop minimized measurement sequences for syndrome extraction, enabling reduced logical error rates and improved fault-tolerance thresholds.Our work applies to topologically protected qubits realized with Majorana zero modes and to similar systems in which multi-qubit Pauli measurements rather than CNOT gates are the native operations.

Polyestimate: a library for near-instantaneous surface code analysis

2015 ◽  
Vol 15 (1&2) ◽  
pp. 1034-1444
Author(s):  
Austin G. Fowler
Keyword(s):  
Quantum Computation ◽  
Error Rate ◽  
Nearest Neighbor ◽  
Careful Analysis ◽  
Error Rates ◽  
Code Analysis ◽  
Logical Error ◽  
Surface Code ◽  
User Friendly ◽  
Library User

The surface code is highly practical, enabling arbitrarily reliable quantum computation given a 2-D nearest-neighbor coupled array of qubits with gate error rates below approximately 1\%. We describe an open source library, Polyestimate, enabling a user with no knowledge of the surface code to specify realistic physical quantum gate error models and obtain logical error rate estimates. Functions allowing the user to specify simple depolarizing error rates for each gate have also been included. Every effort has been made to make this library user-friendly. Polyestimate provides data essentially instantaneously that previously required hundreds to thousands of hours of simulation, statements which we discuss and make precise. This advance has been made possible through careful analysis of the error structure of the surface code and extensive pre-simulation.

scholarly journals Rectangular surface code under biased noise

2021 ◽  
Vol 20 (7) ◽  
Author(s):  
Jonghyun Lee ◽  
Jooyoun Park ◽  
Jun Heo
Keyword(s):  
Error Rate ◽  
Minimum Weight ◽  
Estimation Method ◽  
Error Correcting Codes ◽  
Error Rates ◽  
Noise Model ◽  
High Threshold ◽  
Lattice Size ◽  
Logical Error ◽  
Surface Code

AbstractTo date, the surface code has become a promising candidate for quantum error correcting codes because it achieves a high threshold and is composed of only the nearest gate operations and low-weight stabilizers. Here, we have exhibited that the logical failure rate can be enhanced by manipulating the lattice size of surface codes that they can show an enormous improvement in the number of physical qubits for a noise model where dephasing errors dominate over relaxation errors. We estimated the logical error rate in terms of the lattice size and physical error rate. When the physical error rate was high, the parameter estimation method was applied, and when it was low, the most frequently occurring logical error cases were considered. By using the minimum weight perfect matching decoding algorithm, we obtained the optimal lattice size by minimizing the number of qubits to achieve the required failure rates when physical error rates and bias are provided .

scholarly journals Optimal local unitary encoding circuits for the surface code

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 517
Author(s):  
Oscar Higgott ◽  
Matthew Wilson ◽  
James Hefford ◽  
James Dborin ◽  
Farhan Hanif ◽  
...  
Keyword(s):  
Error Correcting Code ◽  
High Threshold ◽  
Planar Surface ◽  
Compact Mapping ◽  
Lattice Size ◽  
Unknown State ◽  
Non Local ◽  
Surface Code ◽  
Quantum Error ◽  
Toric Code

The surface code is a leading candidate quantum error correcting code, owing to its high threshold, and compatibility with existing experimental architectures. Bravyi et al. (2006) showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size L, however the most efficient known method for encoding an unknown state, introduced by Dennis et al. (2002), has O(L2) time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly 2L time steps to encode an unknown state in a distance L planar code. We further show how an O(L) complexity local unitary encoder for the toric code can be found by enforcing locality in the O(log⁡L)-depth non-local renormalisation encoder. We relate these techniques by providing an O(L) local unitary circuit to convert between a toric code and a planar code, and also provide optimal encoders for the rectangular, rotated and 3D surface codes. Furthermore, we show how our encoding circuit for the planar code can be used to prepare fermionic states in the compact mapping, a recently introduced fermion to qubit mapping that has a stabiliser structure similar to that of the surface code and is particularly efficient for simulating the Fermi-Hubbard model.

scholarly journals Scalable Neural Network Decoders for Higher Dimensional Quantum Codes

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 68 ◽  
Author(s):  
Nikolas P. Breuckmann ◽  
Xiaotong Ni
Keyword(s):  
Neural Network ◽  
Machine Learning ◽  
Convolutional Neural Network ◽  
Nearest Neighbor ◽  
System Size ◽  
Higher Dimensional ◽  
Power Machine ◽  
Theoretical And Numerical Analysis ◽  
Surface Code ◽  
Quantum Error

Machine learning has the potential to become an important tool in quantum error correction as it allows the decoder to adapt to the error distribution of a quantum chip. An additional motivation for using neural networks is the fact that they can be evaluated by dedicated hardware which is very fast and consumes little power. Machine learning has been previously applied to decode the surface code. However, these approaches are not scalable as the training has to be redone for every system size which becomes increasingly difficult. In this work the existence of local decoders for higher dimensional codes leads us to use a low-depth convolutional neural network to locally assign a likelihood of error on each qubit. For noiseless syndrome measurements, numerical simulations show that the decoder has a threshold of around 7.1% when applied to the 4D toric code. When the syndrome measurements are noisy, the decoder performs better for larger code sizes when the error probability is low. We also give theoretical and numerical analysis to show how a convolutional neural network is different from the 1-nearest neighbor algorithm, which is a baseline machine learning method.

scholarly journals Machine-learning-assisted correction of correlated qubit errors in a topological code

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 48 ◽  
Author(s):  
Paul Baireuther ◽  
Thomas E. O'Brien ◽  
Brian Tarasinski ◽  
Carlo W. J. Beenakker
Keyword(s):  
Neural Network ◽  
Machine Learning ◽  
Error Correction ◽  
Recurrent Neural Network ◽  
Learning Algorithm ◽  
Minimum Weight ◽  
Quantum Error Correction ◽  
Noise Model ◽  
Surface Code ◽  
Quantum Error

A fault-tolerant quantum computation requires an efficient means to detect and correct errors that accumulate in encoded quantum information. In the context of machine learning, neural networks are a promising new approach to quantum error correction. Here we show that a recurrent neural network can be trained, using only experimentally accessible data, to detect errors in a widely used topological code, the surface code, with a performance above that of the established minimum-weight perfect matching (or blossom) decoder. The performance gain is achieved because the neural network decoder can detect correlations between bit-flip (X) and phase-flip (Z) errors. The machine learning algorithm adapts to the physical system, hence no noise model is needed. The long short-term memory layers of the recurrent neural network maintain their performance over a large number of quantum error correction cycles, making it a practical decoder for forthcoming experimental realizations of the surface code.

scholarly journals Leakage detection for a transmon-based surface code

2020 ◽  
Vol 6 (1) ◽  
Author(s):  
Boris Mihailov Varbanov ◽  
Francesco Battistel ◽  
Brian Michael Tarasinski ◽  
Viacheslav Petrovych Ostroukh ◽  
Thomas Eugene O’Brien ◽  
...  
Keyword(s):  
Markov Models ◽  
Leakage Detection ◽  
Detection Scheme ◽  
Defect Probability ◽  
Logical Error ◽  
Near Term ◽  
Surface Code ◽  
Break Even Point ◽  
Quantum Error ◽  
The Given

AbstractLeakage outside of the qubit computational subspace, present in many leading experimental platforms, constitutes a threatening error for quantum error correction (QEC) for qubits. We develop a leakage-detection scheme via Hidden Markov models (HMMs) for transmon-based implementations of the surface code. By performing realistic density-matrix simulations of the distance-3 surface code (Surface-17), we observe that leakage is sharply projected and leads to an increase in the surface-code defect probability of neighboring stabilizers. Together with the analog readout of the ancilla qubits, this increase enables the accurate detection of the time and location of leakage. We restore the logical error rate below the memory break-even point by post-selecting out leakage, discarding less than half of the data for the given noise parameters. Leakage detection via HMMs opens the prospect for near-term QEC demonstrations, targeted leakage reduction and leakage-aware decoding and is applicable to other experimental platforms.

scholarly journals Exponential suppression of bit or phase errors with cyclic error correction

Nature ◽  
2021 ◽  
Vol 595 (7867) ◽  
pp. 383-387
Author(s):  
◽  
Zijun Chen ◽  
Kevin J. Satzinger ◽  
Juan Atalaya ◽  
Alexander N. Korotkov ◽  
...  
Keyword(s):  
Error Correction ◽  
Error Detection ◽  
Fault Tolerant ◽  
Error Rates ◽  
Quantum Error Correction ◽  
Superconducting Qubits ◽  
Two Dimensional ◽  
Logical Error ◽  
Quantum Error ◽  
Logical Qubit

AbstractRealizing the potential of quantum computing requires sufficiently low logical error rates1. Many applications call for error rates as low as 10−15 (refs. 2–9), but state-of-the-art quantum platforms typically have physical error rates near 10−3 (refs. 10–14). Quantum error correction15–17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.

THE CRITICAL BEHAVIOR OF THE 2D SPIN-1 ISING MODEL WITH THE BILINEAR AND POSITIVE BIQUADRATIC NEAREST NEIGHBOR INTERACTIONS ON A CELLULAR AUTOMATON

2004 ◽  
Vol 15 (10) ◽  
pp. 1425-1438 ◽  
Author(s):  
A. SOLAK ◽  
B. KUTLU
Keyword(s):  
Cellular Automaton ◽  
Phase Boundary ◽  
Critical Behavior ◽  
Nearest Neighbor ◽  
Finite Size ◽  
Square Lattice ◽  
Two Dimensional ◽  
Finite Size Scaling ◽  
Creutz Cellular Automaton ◽  
Beg Model

The two-dimensional BEG model with nearest neighbor bilinear and positive biquadratic interaction is simulated on a cellular automaton, which is based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transitions of the model are presented for comparison with those obtained from other calculations. We confirm the existence of the tricritical points over the phase boundary for D/K>0. The values of static critical exponents (α, β, γ and ν) are estimated within the framework of the finite size scaling theory along D/K=-1 and 1 lines. The results are compatible with the universal Ising critical behavior except the points over phase boundary.

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