On entanglement distillation and quantum error correction for unknown states and channels

Open Physics ◽  
2003 ◽  
Vol 1 (4) ◽  
Author(s):  
Pawel Horodecki
Keyword(s):  
Quantum State ◽  
Transmission Rate ◽  
Single Copy ◽  
Quantum Error Correction ◽  
Quantum Channels ◽  
Level System ◽  
Open Problems ◽  
Single Qubit ◽  
N Level ◽  
Quantum Error

AbstractWe consider the problem of invariance of distillable entanglement D and quantum capacities Q under erasure of information about single copy of quantum state or channel respectively. We argue that any 2 ⊗N two-way distillable state is still two-way distillable after erasure of single copy information. For some known distillation protocols the obtained two-way distillation rate is the same as if Alice and Bob knew the state from the very beginning. The isomorphism between quantum states and quantum channels is also investigated. In particular it is pointed out that any transmission rate down the channel is equal to distillation rate with formal LOCC-like superoperator that uses in general nonphysical Alice actions. This allows to we prove that if given channel Λ has nonzero capacity (Q → or Q ⟺) then the corresponding quantum state ϱ(Λ) has nonzero distillable entanglement (D → or D ⟺). Follwoing the latter arguments are provided that any channel mapping single qubit into N level system allows for reliable two-way transmission after erasure of information about single copy. Some open problems are discussed.

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 On Families of Wigner Functions for N-Level Quantum Systems

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1013
Author(s):  
Vahagn Abgaryan ◽  
Arsen Khvedelidze
Keyword(s):  
Flag Manifold ◽  
Quantum Systems ◽  
Master Equations ◽  
Wigner Functions ◽  
Level System ◽  
Single Qubit ◽  
N Level ◽  
Weyl Correspondence ◽  
Quasiprobability Distributions ◽  
Complex Flag

A method for constructing all admissible unitary non-equivalent Wigner quasiprobability distributions providing the Stratonovic-h-Weyl correspondence for an arbitrary N-level quantum system is proposed. The method is based on the reformulation of the Stratonovich–Weyl correspondence in the form of algebraic “master equations” for the spectrum of the Stratonovich–Weyl kernel. The later implements a map between the operators in the Hilbert space and the functions in the phase space identified by the complex flag manifold. The non-uniqueness of the solutions to the master equations leads to diversity among the Wigner quasiprobability distributions. It is shown that among all possible Stratonovich–Weyl kernels for a N=(2j+1)-level system, one can always identify the representative that realizes the so-called SU(2)-symmetric spin-j symbol correspondence. The method is exemplified by considering the Wigner functions of a single qubit and a single qutrit.

scholarly journals A Software Method For Mitigating Single Qubit Errors On Superconducting Quantum Devices

2020 ◽  
Author(s):  
Macauley Coggins ◽  
Devanshi Arora
Keyword(s):  
Error Correction ◽  
Quantum Error Correction ◽  
Superconducting Qubits ◽  
Quantum Decoherence ◽  
Small Scale ◽  
Quantum Devices ◽  
Single Qubit ◽  
Error Mitigation ◽  
Quantum Error

Quantum error correction schemes have gained a lot of attention in recent years. This is due to the emergence of small scale quantum devices that make use of superconducting qubits. However these devices are noisy and prone to quantum decoherence and thus errors. Along with quantum error correction there has been a push for new schemes in quantum error mitigation that take a more passive approach in eliminating readout errors. In this research we introduce a software method for quantum error mitigation that maps virtual qubits in a circuit to physical qubits with the least error. The method developed was tested on 9 IBM quantum devices. Results in the study have shown the method can reduce readout errors by up to 35.52%.

Stabilizing qubit coherence via tracking-control

2005 ◽  
Vol 5 (4&5) ◽  
pp. 350-363
Author(s):  
D.A. Lidar ◽  
S. Schneider
Keyword(s):  
Error Correction ◽  
Finite Time ◽  
Tracking Control ◽  
Quantum Error Correction ◽  
The State ◽  
Control Field ◽  
Dynamical Decoupling ◽  
Single Qubit ◽  
Quantum Error

We consider the problem of stabilizing the coherence of a single qubit subject to Markovian decoherence, via the application of a control Hamiltonian, without any additional resources. In this case neither quantum error correction/avoidance, nor dynamical decoupling applies. We show that using tracking-control, i.e., the conditioning of the control field on the state of the qubit, it is possible to maintain coherence for finite time durations, until the control field diverges.

scholarly journals Jordan products of quantum channels and their compatibility

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Mark Girard ◽  
Martin Plávala ◽  
Jamie Sikora
Keyword(s):  
Quantum State ◽  
Sufficient Conditions ◽  
The Other ◽  
Independent Interest ◽  
Quantum Channels ◽  
Semidefinite Programs ◽  
Jordan Product ◽  
Marginal Problem ◽  
Jordan Products ◽  
Product Of Matrices

AbstractGiven two quantum channels, we examine the task of determining whether they are compatible—meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolarizing channels are compatible.

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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