scholarly journals Quantum error correction and large $N$

2021 ◽  
Vol 11 (5) ◽  
Author(s):  
Alexey Milekhin
Keyword(s):  
Error Correction ◽  
Error Correcting Code ◽  
Quantum Error Correction ◽  
Gauge Groups ◽  
Rank Tensor ◽  
Fermionic Sector ◽  
Large N ◽  
Singlet States ◽  
Quantum Error ◽  
Matrix Quantum

In recent years quantum error correction (QEC) has become an important part of AdS/CFT. Unfortunately, there are no field-theoretic arguments about why QEC holds in known holographic systems. The purpose of this paper is to fill this gap by studying the error correcting properties of the fermionic sector of various large NN theories. Specifically we examine SU(N)SU(N) matrix quantum mechanics and 3-rank tensor O(N)^3O(N)3 theories. Both of these theories contain large gauge groups. We argue that gauge singlet states indeed form a quantum error correcting code. Our considerations are based purely on large NN analysis and do not appeal to a particular form of Hamiltonian or holography.

Quantum Error Correction

2020 ◽  
Author(s):  
Todd A. Brun
Keyword(s):  
Error Correction ◽  
Quantum Computation ◽  
Single Photon ◽  
Error Correcting Code ◽  
Quantum Systems ◽  
Quantum Error Correction ◽  
The State ◽  
Quantum States ◽  
Quantum Code ◽  
Quantum Error

Quantum error correction is a set of methods to protect quantum information—that is, quantum states—from unwanted environmental interactions (decoherence) and other forms of noise. The information is stored in a quantum error-correcting code, which is a subspace in a larger Hilbert space. This code is designed so that the most common errors move the state into an error space orthogonal to the original code space while preserving the information in the state. It is possible to determine whether an error has occurred by a suitable measurement and to apply a unitary correction that returns the state to the code space without measuring (and hence disturbing) the protected state itself. In general, codewords of a quantum code are entangled states. No code that stores information can protect against all possible errors; instead, codes are designed to correct a specific error set, which should be chosen to match the most likely types of noise. An error set is represented by a set of operators that can multiply the codeword state. Most work on quantum error correction has focused on systems of quantum bits, or qubits, which are two-level quantum systems. These can be physically realized by the states of a spin-1/2 particle, the polarization of a single photon, two distinguished levels of a trapped atom or ion, the current states of a microscopic superconducting loop, or many other physical systems. The most widely used codes are the stabilizer codes, which are closely related to classical linear codes. The code space is the joint +1 eigenspace of a set of commuting Pauli operators on n qubits, called stabilizer generators; the error syndrome is determined by measuring these operators, which allows errors to be diagnosed and corrected. A stabilizer code is characterized by three parameters [[n,k,d]], where n is the number of physical qubits, k is the number of encoded logical qubits, and d is the minimum distance of the code (the smallest number of simultaneous qubit errors that can transform one valid codeword into another). Every useful code has n>k; this physical redundancy is necessary to detect and correct errors without disturbing the logical state. Quantum error correction is used to protect information in quantum communication (where quantum states pass through noisy channels) and quantum computation (where quantum states are transformed through a sequence of imperfect computational steps in the presence of environmental decoherence to solve a computational problem). In quantum computation, error correction is just one component of fault-tolerant design. Other approaches to error mitigation in quantum systems include decoherence-free subspaces, noiseless subsystems, and dynamical decoupling.

Improvement of quantum correlations by repetitive quantum error correction

2019 ◽  
Vol 17 (05) ◽  
pp. 1950044
Author(s):  
A. El Allati ◽  
H. Amellal ◽  
A. Meslouhi
Keyword(s):  
Error Correction ◽  
Quantum Correlation ◽  
Coherent States ◽  
Quantum Discord ◽  
Error Correcting Code ◽  
Quantum Correlations ◽  
Quantum Error Correction ◽  
Optical Field ◽  
Quantum Error ◽  
Entangled Coherent States

A quantum error-correcting code is established in entangled coherent states (CSs) with Markovian and non-Markovian environments. However, the dynamic behavior of these optical states is discussed in terms of quantum correlation measurements, entanglement and discord. By using the correcting codes, these correlations can be as robust as possible against environmental effects. As the number of redundant CSs increases due to the repetitive error correction, the probabilities of success also increase significantly. Based on different optical field parameters, the discord can withstand more than an entanglement. Furthermore, the behavior of quantum discord under decoherence may exhibit sudden death and sudden birth phenomena as functions of dimensionless parameters.

scholarly journals Experimental exploration of five-qubit quantum error correcting code with superconducting qubits

2021 ◽  
Author(s):  
Ming Gong ◽  
Xiao Yuan ◽  
Shiyu Wang ◽  
Yulin Wu ◽  
Youwei Zhao ◽  
...  
Keyword(s):  
Error Correction ◽  
Error Correcting Code ◽  
Error Correcting Codes ◽  
Quantum Error Correction ◽  
Superconducting Qubits ◽  
Perfect Code ◽  
Experimental Realization ◽  
Single Qubit ◽  
Universal Quantum ◽  
Quantum Error

Abstract Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error correcting code, with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state, and state decoding. To address this challenge, we experimentally realise the [[5, 1, 3]] code, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the [[5, 1, 3]] code for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of $57.1(3)\%$ while with a high fidelity of $98.6(1)\%$ in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilizers. We further implement logical Pauli operations with a fidelity of $97.2(2)\%$ within the code space. Finally, we realise the decoding circuit and recover the input state with an overall fidelity of $74.5(6)\%$, in total with 92 gates. Our work demonstrates each key aspect of the [[5, 1, 3]] code and verifies the viability of experimental realization of quantum error correcting codes with superconducting qubits.

scholarly journals A Holographic Quantum Code

2019 ◽  
Vol 20 (2) ◽  
Author(s):  
Grant Elliot
Keyword(s):  
Error Correction ◽  
Error Correcting Code ◽  
Quantum Error Correction ◽  
Quantum Code ◽  
Tensor Networks ◽  
Logical Qubits ◽  
Quantum Error ◽  
The Way

Abstract: It was shown by [2] how bulk operators in the AdS/CFT correspondence can be represented on the boundary analogously to the way logical qubits are represented in an encoded subspace in quantum error correction. Then in [1]  holographic tensor networks that serve as toy models of the bulk boundary. This paper reviews some of the developments of [1] and [2]. Then it is demonstrated explicitly how to construct perfect tensors, which are essential to the tensor networks mentioned in [2]. Lastly a new example of a holographic quantum error-correcting code based on an eight index perfect tensor is presented.

Quantum Error Correction Code Scheme used for Homomorphic Encryption like Quantum Computation

2019 ◽  
Vol 19 (3) ◽  
pp. 61-70
Author(s):  
Il Kwon Sohn ◽  
◽  
Jonghyun Lee ◽  
Wonhyuk Lee ◽  
Woojin Seok ◽  
...  
Keyword(s):  
Error Correction ◽  
Quantum Computation ◽  
Homomorphic Encryption ◽  
Quantum Error Correction ◽  
Code Scheme ◽  
Error Correction Code ◽  
Quantum Error

scholarly journals 2-designs and redundant syndrome extraction for quantum error correction

2021 ◽  
Vol 20 (3) ◽  
Author(s):  
Vickram N. Premakumar ◽  
Hele Sha ◽  
Daniel Crow ◽  
Eric Bach ◽  
Robert Joynt
Keyword(s):  
Error Correction ◽  
Quantum Error Correction ◽  
Quantum Error

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.

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