Quantum error correction is essential for realizing fault-tolerant quantum computing,where both the efficiency and accuracy of the decoding algorithms play critical roles.In this work,we introduce the implementation o...Quantum error correction is essential for realizing fault-tolerant quantum computing,where both the efficiency and accuracy of the decoding algorithms play critical roles.In this work,we introduce the implementation of the PLANAR algorithm,a software framework designed for fast and exact decoding of quantum codes with a planar structure.The algorithm first converts the optimal decoding of quantum codes into a partition function computation problem of an Ising spin glass model.Then it utilizes the exact Kac–Ward formula to solve it.In this way,PLANAR offers the exact maximum likelihood decoding in polynomial complexity for quantum codes with a planar structure,including the surface code with independent code-capacity noise and the quantum repetition code with circuit-level noise.Unlike traditional minimumweight decoders such as minimum-weight perfect matching(MWPM),PLANAR achieves theoretically optimal performance while maintaining polynomial-time efficiency.In addition,to demonstrate its capabilities,we exemplify the implementation using the rotated surface code,a commonly used quantum error correction code with a planar structure,and show that PLANAR achieves a threshold of approximately p_(uc)≈0.109 under the depolarizing error model,with a time complexity scaling of O(N^(0.69)),where N is the number of spins in the Ising model.展开更多
For applying the perfect code to transmit quantum information over a noise channel,the standard protocol contains four steps:the encoding,the noise channel,the error-correction operation,and the decoding.In present wo...For applying the perfect code to transmit quantum information over a noise channel,the standard protocol contains four steps:the encoding,the noise channel,the error-correction operation,and the decoding.In present work,we show that this protocol can be simplified.The error-correction operation is not necessary if the decoding is realized by the so-called complete unitary transformation.We also offer a quantum circuit,which can correct the arbitrary single-qubit errors.展开更多
Quantum metrology provides a fundamental limit on the precision of multi-parameter estimation,called the Heisenberg limit,which has been achieved in noiseless quantum systems.However,for systems subject to noises,it i...Quantum metrology provides a fundamental limit on the precision of multi-parameter estimation,called the Heisenberg limit,which has been achieved in noiseless quantum systems.However,for systems subject to noises,it is hard to achieve this limit since noises are inclined to destroy quantum coherence and entanglement.In this paper,a combined control scheme with feedback and quantum error correction(QEC)is proposed to achieve the Heisenberg limit in the presence of spontaneous emission,where the feedback control is used to protect a stabilizer code space containing an optimal probe state and an additional control is applied to eliminate the measurement incompatibility among three parameters.Although an ancilla system is necessary for the preparation of the optimal probe state,our scheme does not require the ancilla system to be noiseless.In addition,the control scheme in this paper has a low-dimensional code space.For the three components of a magnetic field,it can achieve the highest estimation precision with only a 2-dimensional code space,while at least a4-dimensional code space is required in the common optimal error correction protocols.展开更多
The loss of a quantum channel leads to an irretrievable particle loss as well as information. In this paper, the loss of quantum channel is analysed and a method is put forward to recover the particle and information ...The loss of a quantum channel leads to an irretrievable particle loss as well as information. In this paper, the loss of quantum channel is analysed and a method is put forward to recover the particle and information loss effectively using universal quantum error correction. Then a secure direct communication scheme is proposed, such that in a loss channel the information that an eavesdropper can obtain would be limited to arbitrarily small when the code is properly chosen and the correction operation is properly arranged.展开更多
Quantum computing has the potential to solve complex problems that are inefficiently handled by classical computation.However,the high sensitivity of qubits to environmental interference and the high error rates in cu...Quantum computing has the potential to solve complex problems that are inefficiently handled by classical computation.However,the high sensitivity of qubits to environmental interference and the high error rates in current quantum devices exceed the error correction thresholds required for effective algorithm execution.Therefore,quantum error correction technology is crucial to achieving reliable quantum computing.In this work,we study a topological surface code with a two-dimensional lattice structure that protects quantum information by introducing redundancy across multiple qubits and using syndrome qubits to detect and correct errors.However,errors can occur not only in data qubits but also in syndrome qubits,and different types of errors may generate the same syndromes,complicating the decoding task and creating a need for more efficient decoding methods.To address this challenge,we used a transformer decoder based on an attention mechanism.By mapping the surface code lattice,the decoder performs a self-attention process on all input syndromes,thereby obtaining a global receptive field.The performance of the decoder was evaluated under a phenomenological error model.Numerical results demonstrate that the decoder achieved a decoding accuracy of 93.8%.Additionally,we obtained decoding thresholds of 5%and 6.05%at maximum code distances of 7 and 9,respectively.These results indicate that the decoder used demonstrates a certain capability in correcting noise errors in surface codes.展开更多
The present study delves into the application of investigating quantum state behaviour,particularly focusing on coherent and superposition states.These states,characterized by their remarkable stability and precision,...The present study delves into the application of investigating quantum state behaviour,particularly focusing on coherent and superposition states.These states,characterized by their remarkable stability and precision,have found extensive utility in various domains of quantum mechanics and quantum information processing.Coherent states are valuable for manipulating quantum systems with accuracy.Superposition states allow quantum systems to exist in numerous configurations at the same time,which paves the way for quantum computing’s capacity for parallel processing.The research accentuates the crucial role of quantum error correction(QEC)in ensuring the stability and reliability of quantum information processing systems.Quantum systems are prone to errors from decoherence and environmental noise,making QEC essential for ensuring accurate results by employing the Shor code,an error-correcting code devised by Peter Shor,it becomes feasible to detect and rectify errors that may arise during quantum computations.The Shor code detects and corrects both bit-flip and phase-flip errors,greatly enhancing the robustness of quantum information systems.This research offers insights into the multifaceted utility of MZI(Mach-Zehnder interferometer)and its relevance in the advancement of quantum technology.By integrating QEC with the capabilities of MZI,this study offers a holistic approach to advancing the precision and reliability of quantum technologies.展开更多
Quantum information is vulnerable to environmental noise and experimental imperfections,hindering the reli-ability of practical quantum information processors.Therefore,quantum error correction(QEC)that can pro-tect q...Quantum information is vulnerable to environmental noise and experimental imperfections,hindering the reli-ability of practical quantum information processors.Therefore,quantum error correction(QEC)that can pro-tect quantum information against noise is vital for universal and scalable quantum computation.Among many different experimental platforms,superconducting quantum circuits and bosonic encodings in superconducting microwave modes are appealing for their unprecedented potential in QEC.During the last few years,bosonic QEC is demonstrated to reach the break-even point,i.e.the lifetime of a logical qubit is enhanced to exceed that of any individual components composing the experimental system.Beyond that,universal gate sets and fault-tolerant operations on the bosonic codes are also realized,pushing quantum information processing towards the QEC era.In this article,we review the recent progress of the bosonic codes,including the Gottesman-Kitaev-Preskill codes,cat codes,and binomial codes,and discuss the opportunities of bosonic codes in various quantum applications,ranging from fault-tolerant quantum computation to quantum metrology.We also summarize the challenges associated with the bosonic codes and provide an outlook for the potential research directions in the long terms.展开更多
Measurement-based quantum computation with continuous variables,which realizes computation by performing measurement and feedforward of measurement results on a large scale Gaussian cluster state,provides a feasible w...Measurement-based quantum computation with continuous variables,which realizes computation by performing measurement and feedforward of measurement results on a large scale Gaussian cluster state,provides a feasible way to implement quantum computation.Quantum error correction is an essential procedure to protect quantum information in quantum computation and quantum communication.In this review,we briefly introduce the progress of measurement-based quantum computation and quantum error correction with continuous variables based on Gaussian cluster states.We also discuss the challenges in the fault-tolerant measurement-based quantum computation with continuous variables.展开更多
Quantum error correction technology is an important solution to solve the noise interference generated during the operation of quantum computers.In order to find the best syndrome of the stabilizer code in quantum err...Quantum error correction technology is an important solution to solve the noise interference generated during the operation of quantum computers.In order to find the best syndrome of the stabilizer code in quantum error correction,we need to find a fast and close to the optimal threshold decoder.In this work,we build a convolutional neural network(CNN)decoder to correct errors in the toric code based on the system research of machine learning.We analyze and optimize various conditions that affect CNN,and use the RestNet network architecture to reduce the running time.It is shortened by 30%-40%,and we finally design an optimized algorithm for CNN decoder.In this way,the threshold accuracy of the neural network decoder is made to reach 10.8%,which is closer to the optimal threshold of about 11%.The previous threshold of 8.9%-10.3%has been slightly improved,and there is no need to verify the basic noise.展开更多
For the quantum error correction and noisy intermediate-scale quantum algorithms to function with high efficiency,the raw fidelity of quantum logic gates on physical qubits needs to satisfy strict requirements.The neu...For the quantum error correction and noisy intermediate-scale quantum algorithms to function with high efficiency,the raw fidelity of quantum logic gates on physical qubits needs to satisfy strict requirements.The neutral atom quantum computing equipped with Rydberg blockade gates has made impressive progress recently,which makes it worthwhile to explore its potential in the two-qubit entangling gates,including the controlledphase gate,and in particular,the CZ gate.Provided the quantum coherence is well preserved,improving the fidelity of Rydberg blockade gates calls for special mechanisms to deal with adverse effects caused by realistic experimental conditions.Here,the heralded very-high-fidelity Rydberg blockade controlled-phase gate is designed to address these issues,which contains self-correction and projection as the key steps.This trailblazing method builds upon the previously established buffer-atom-mediated gate framework,with a special form of symmetry under parity–time transformation playing a crucial role in the process.We further analyze the performance with respect to a few typical sources of imperfections.This procedure can also be regarded as quantum hardware error correction or mitigation.While this paper by itself does not cover every single subtle issue and still contains many oversimplifications,we find it reasonable to anticipate a very-high-fidelity two-qubit quantum logic gate operated in the sense of heralded but probabilistic,whose gate error can be reduced to the level of 10^(-4)–10^(-6)or even lower with reasonably high possibilities.展开更多
Quantum error-correcting codes are essential for fault-tolerant quantum computing,as they effectively detect and correct noise-induced errors by distributing information across multiple physical qubits.The subsystem s...Quantum error-correcting codes are essential for fault-tolerant quantum computing,as they effectively detect and correct noise-induced errors by distributing information across multiple physical qubits.The subsystem surface code with three-qubit check operators demonstrates significant application potential due to its simplified measurement operations and low logical error rates.However,the existing minimum-weight perfect matching(MWPM)algorithm exhibits high computational complexity and lacks flexibility in large-scale systems.Therefore,this paper proposes a decoder based on a graph attention network(GAT),representing error syndromes as undirected graphs with edge weights,and employing a multihead attention mechanism to efficiently aggregate node features and enable parallel computation.Compared to MWPM,the GAT decoder exhibits linear growth in computational complexity,adapts to different quantum code structures,and demonstrates stronger robustness under high physical error rates.The experimental results demonstrate that the proposed decoder achieves an overall accuracy of 89.95%under various small code lattice sizes(L=2,3,4,5),with the logical error rate threshold increasing to 0.0078,representing an improvement of approximately 13.04%compared to the MWPM decoder.This result significantly outperforms traditional methods,showcasing superior performance under small code lattice sizes and providing a more efficient decoding solution for large-scale quantum error correction.展开更多
Minimizing the effect of noise is essential for quantum computers.The conventional method to protect qubits against noise is through quantum error correction.However,for current quantum hardware in the so-called noisy...Minimizing the effect of noise is essential for quantum computers.The conventional method to protect qubits against noise is through quantum error correction.However,for current quantum hardware in the so-called noisy intermediate-scale quantum(NISQ)era,noise presents in these systems and is too high for error correction to be beneficial.Quantum error mitigation is a set of alternative methods for minimizing errors,including error extrapolation,probabilistic error cancella-tion,measurement error mitigation,subspace expansion,symmetry verification,virtual distillation,etc.The requirement for these methods is usually less demanding than error correction.Quantum error mitigation is a promising way of reduc-ing errors on NISQ quantum computers.This paper gives a comprehensive introduction to quantum error mitigation.The state-of-art error mitigation methods are covered and formulated in a general form,which provides a basis for comparing,combining and optimizing different methods in future work.展开更多
Error correction has long been suggested to extend the sensitivity of quantum sensors into the Heisenberg Limit. However, operations on logical qubits are only performed through universal gate sets consisting of finit...Error correction has long been suggested to extend the sensitivity of quantum sensors into the Heisenberg Limit. However, operations on logical qubits are only performed through universal gate sets consisting of finite-sized gates such as Clifford + T. Although these logical gate sets allow for universal quantum computation, the finite gate sizes present a problem for quantum sensing, since in sensing protocols, such as the Ramsey measurement protocol, the signal must act continuously. The difficulty in constructing a continuous logical op-erator comes from the Eastin-Knill theorem, which prevents a continuous sig-nal from being both fault-tolerant to local errors and transverse. Since error correction is needed to approach the Heisenberg Limit in a noisy environment, it is important to explore how to construct fault-tolerant continuous operators. In this paper, a protocol to design continuous logical z-rotations is proposed and applied to the Steane Code. The fault tolerance of the designed operator is investigated using the Knill-Laflamme conditions. The Knill-Laflamme condi-tions indicate that the diagonal unitary operator constructed cannot be fault tolerant solely due to the possibilities of X errors on the middle qubit. The ap-proach demonstrated throughout this paper may, however, find success in codes with more qubits such as the Shor code, distance 3 surface code, [15, 1, 3] code, or codes with a larger distance such as the [11, 1, 5] code.展开更多
In this article, we study the ability of error-correcting quantum codes to increase the fidelity of quantum states throughout a quantum computation. We analyze arbitrary quantum codes that encode all qubits involved i...In this article, we study the ability of error-correcting quantum codes to increase the fidelity of quantum states throughout a quantum computation. We analyze arbitrary quantum codes that encode all qubits involved in the computation, and we study the evolution of n-qubit fidelity from the end of one application of the correcting circuit to the end of the next application. We assume that the correcting circuit does not introduce new errors, that it does not increase the execution time (i.e. its application takes zero seconds) and that quantum errors are isotropic. We show that the quantum code increases the fidelity of the states perturbed by quantum errors but that this improvement is not enough to justify the use of quantum codes. Namely, we prove that, taking into account that the time interval between the application of the two corrections is multiplied (at least) by the number of qubits n (due to the coding), the best option is not to use quantum codes, since the fidelity of the uncoded state over a time interval n times smaller is greater than that of the state resulting from the quantum code correction.展开更多
Quantum error correction is a technique that enhances a system’s ability to combat noise by encoding logical information into additional quantum bits,which plays a key role in building practical quantum computers.The...Quantum error correction is a technique that enhances a system’s ability to combat noise by encoding logical information into additional quantum bits,which plays a key role in building practical quantum computers.The XZZX surface code,with only one stabilizer generator on each face,demonstrates significant application potential under biased noise.However,the existing minimum weight perfect matching(MWPM)algorithm has high computational complexity and lacks flexibility in large-scale systems.Therefore,this paper proposes a decoding method that combines graph neural networks(GNN)with multi-classifiers,the syndrome is transformed into an undirected graph,and the features are aggregated by convolutional layers,providing a more efficient and accurate decoding strategy.In the experiments,we evaluated the performance of the XZZX code under different biased noise conditions(bias=1,20,200)and different code distances(d=3,5,7,9,11).The experimental results show that under low bias noise(bias=1),the GNN decoder achieves a threshold of 0.18386,an improvement of approximately 19.12%compared to the MWPM decoder.Under high bias noise(bias=200),the GNN decoder reaches a threshold of 0.40542,improving by approximately 20.76%,overcoming the limitations of the conventional decoder.They demonstrate that the GNN decoding method exhibits superior performance and has broad application potential in the error correction of XZZX code.展开更多
A （n, n）-threshold scheme of multiparty quantum secret sharing of classical or quantum message is proposed based on the discrete quantum Fourier transform. In our proposed scheme, the secret message, which is encode...A （n, n）-threshold scheme of multiparty quantum secret sharing of classical or quantum message is proposed based on the discrete quantum Fourier transform. In our proposed scheme, the secret message, which is encoded by using the forward quantum Fourier transform and decoded by using the reverse, is split and shared in such a way that it can be reconstructed among them only if all the participants work in concert. Fhrthermore, we also discuss how this protocol must be carefully designed for correcting errors and checking eavesdropping or a dishonest participant. Security analysis shows that our scheme is secure. Also, this scheme has an advantage that it is completely compatible with quantum computation and easier to realize in the distributed quantum secure computation.展开更多
For quantum sparse graph codes with stabilizer formalism, the unavoidable girth-four cycles in their Tanner graphs greatly degrade the iterative decoding performance with standard belief-propagation （BP） algorithm. ...For quantum sparse graph codes with stabilizer formalism, the unavoidable girth-four cycles in their Tanner graphs greatly degrade the iterative decoding performance with standard belief-propagation （BP） algorithm. In this paper, we present a jointly-check iterative algorithm suitable for decoding quantum sparse graph codes efficiently. Numerical simulations show that this modified method outperforms standard BP algorithm with an obvious performance improvement.展开更多
Entanglement-assisted quantum error correction codes(EAQECCs)play an important role in quantum communications with noise.Such a scheme can use arbitrary classical linear code to transmit qubits over noisy quantum chan...Entanglement-assisted quantum error correction codes(EAQECCs)play an important role in quantum communications with noise.Such a scheme can use arbitrary classical linear code to transmit qubits over noisy quantum channels by consuming some ebits between the sender(Alice)and the receiver(Bob).It is usually assumed that the preshared ebits of Bob are error free.However,noise on these ebits is unavoidable in many cases.In this work,we evaluate the performance of EAQECCs with noisy ebits over asymmetric quantum channels and quantum channels with memory by computing the exact entanglement fidelity of several EAQECCs.We consider asymmetric errors in both qubits and ebits and show that the performance of EAQECCs in entanglement fidelity gets improved for qubits and ebits over asymmetric channels.In quantum memory channels,we compute the entanglement fidelity of several EAQECCs over Markovian quantum memory channels and show that the performance of EAQECCs is lowered down by the channel memory.Furthermore,we show that the performance of EAQECCs is diverse when the error probabilities of qubits and ebits are different.In both asymmetric and memory quantum channels,we show that the performance of EAQECCs is improved largely when the error probability of ebits is reasonably smaller than that of qubits.展开更多
We address the problem of encoding entanglement-assisted （EA） quantum error-correcting codes （QECCs） and of the corresponding complexity. We present an iterative algorithm from which a quantum circuit composed of ...We address the problem of encoding entanglement-assisted （EA） quantum error-correcting codes （QECCs） and of the corresponding complexity. We present an iterative algorithm from which a quantum circuit composed of CNOT, H, and S gates can be derived directly with complexity O（n2） to encode the qubits being sent. Moreover, we derive the number of each gate consumed in our algorithm according to which we can design EA QECCs with low encoding complexity. Another advantage brought by our algorithm is the easiness and efficiency of programming on classical computers.展开更多
Quantum error correction is a crucial technology for realizing quantum computers.These computers achieve faulttolerant quantum computing by detecting and correcting errors using decoding algorithms.Quantum error corre...Quantum error correction is a crucial technology for realizing quantum computers.These computers achieve faulttolerant quantum computing by detecting and correcting errors using decoding algorithms.Quantum error correction using neural network-based machine learning methods is a promising approach that is adapted to physical systems without the need to build noise models.In this paper,we use a distributed decoding strategy,which effectively alleviates the problem of exponential growth of the training set required for neural networks as the code distance of quantum error-correcting codes increases.Our decoding algorithm is based on renormalization group decoding and recurrent neural network decoder.The recurrent neural network is trained through the ResNet architecture to improve its decoding accuracy.Then we test the decoding performance of our distributed strategy decoder,recurrent neural network decoder,and the classic minimum weight perfect matching(MWPM)decoder for rotated surface codes with different code distances under the circuit noise model,the thresholds of these three decoders are about 0.0052,0.0051,and 0.0049,respectively.Our results demonstrate that the distributed strategy decoder outperforms the other two decoders,achieving approximately a 5%improvement in decoding efficiency compared to the MWPM decoder and approximately a 2%improvement compared to the recurrent neural network decoder.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.12325501,12047503,and 12247104)the Chinese Academy of Sciences(Grant No.ZDRW-XX-2022-3-02)P.Z.is partially supported by the Innovation Program for Quantum Science and Technology(Grant No.2021ZD0301900).
文摘Quantum error correction is essential for realizing fault-tolerant quantum computing,where both the efficiency and accuracy of the decoding algorithms play critical roles.In this work,we introduce the implementation of the PLANAR algorithm,a software framework designed for fast and exact decoding of quantum codes with a planar structure.The algorithm first converts the optimal decoding of quantum codes into a partition function computation problem of an Ising spin glass model.Then it utilizes the exact Kac–Ward formula to solve it.In this way,PLANAR offers the exact maximum likelihood decoding in polynomial complexity for quantum codes with a planar structure,including the surface code with independent code-capacity noise and the quantum repetition code with circuit-level noise.Unlike traditional minimumweight decoders such as minimum-weight perfect matching(MWPM),PLANAR achieves theoretically optimal performance while maintaining polynomial-time efficiency.In addition,to demonstrate its capabilities,we exemplify the implementation using the rotated surface code,a commonly used quantum error correction code with a planar structure,and show that PLANAR achieves a threshold of approximately p_(uc)≈0.109 under the depolarizing error model,with a time complexity scaling of O(N^(0.69)),where N is the number of spins in the Ising model.
文摘For applying the perfect code to transmit quantum information over a noise channel,the standard protocol contains four steps:the encoding,the noise channel,the error-correction operation,and the decoding.In present work,we show that this protocol can be simplified.The error-correction operation is not necessary if the decoding is realized by the so-called complete unitary transformation.We also offer a quantum circuit,which can correct the arbitrary single-qubit errors.
基金Project supported by the National Natural Science Foundation of China(Grant No.61873251)。
文摘Quantum metrology provides a fundamental limit on the precision of multi-parameter estimation,called the Heisenberg limit,which has been achieved in noiseless quantum systems.However,for systems subject to noises,it is hard to achieve this limit since noises are inclined to destroy quantum coherence and entanglement.In this paper,a combined control scheme with feedback and quantum error correction(QEC)is proposed to achieve the Heisenberg limit in the presence of spontaneous emission,where the feedback control is used to protect a stabilizer code space containing an optimal probe state and an additional control is applied to eliminate the measurement incompatibility among three parameters.Although an ancilla system is necessary for the preparation of the optimal probe state,our scheme does not require the ancilla system to be noiseless.In addition,the control scheme in this paper has a low-dimensional code space.For the three components of a magnetic field,it can achieve the highest estimation precision with only a 2-dimensional code space,while at least a4-dimensional code space is required in the common optimal error correction protocols.
基金Project supported by the National Natural Science Foundation of China （Grant No 10504042）.Acknowledgments We would like to thank Liu Wei-Tao, Wu Wei and Gao Ming for useful discussions.
文摘The loss of a quantum channel leads to an irretrievable particle loss as well as information. In this paper, the loss of quantum channel is analysed and a method is put forward to recover the particle and information loss effectively using universal quantum error correction. Then a secure direct communication scheme is proposed, such that in a loss channel the information that an eavesdropper can obtain would be limited to arbitrarily small when the code is properly chosen and the correction operation is properly arranged.
基金Project supported by the Natural Science Foundation of Shandong Province,China(Grant No.ZR2021MF049)Joint Fund of Natural Science Foundation of Shandong Province(Grant Nos.ZR2022LLZ012 and ZR2021LLZ001)the Key R&D Program of Shandong Province,China(Grant No.2023CXGC010901)。
文摘Quantum computing has the potential to solve complex problems that are inefficiently handled by classical computation.However,the high sensitivity of qubits to environmental interference and the high error rates in current quantum devices exceed the error correction thresholds required for effective algorithm execution.Therefore,quantum error correction technology is crucial to achieving reliable quantum computing.In this work,we study a topological surface code with a two-dimensional lattice structure that protects quantum information by introducing redundancy across multiple qubits and using syndrome qubits to detect and correct errors.However,errors can occur not only in data qubits but also in syndrome qubits,and different types of errors may generate the same syndromes,complicating the decoding task and creating a need for more efficient decoding methods.To address this challenge,we used a transformer decoder based on an attention mechanism.By mapping the surface code lattice,the decoder performs a self-attention process on all input syndromes,thereby obtaining a global receptive field.The performance of the decoder was evaluated under a phenomenological error model.Numerical results demonstrate that the decoder achieved a decoding accuracy of 93.8%.Additionally,we obtained decoding thresholds of 5%and 6.05%at maximum code distances of 7 and 9,respectively.These results indicate that the decoder used demonstrates a certain capability in correcting noise errors in surface codes.
文摘The present study delves into the application of investigating quantum state behaviour,particularly focusing on coherent and superposition states.These states,characterized by their remarkable stability and precision,have found extensive utility in various domains of quantum mechanics and quantum information processing.Coherent states are valuable for manipulating quantum systems with accuracy.Superposition states allow quantum systems to exist in numerous configurations at the same time,which paves the way for quantum computing’s capacity for parallel processing.The research accentuates the crucial role of quantum error correction(QEC)in ensuring the stability and reliability of quantum information processing systems.Quantum systems are prone to errors from decoherence and environmental noise,making QEC essential for ensuring accurate results by employing the Shor code,an error-correcting code devised by Peter Shor,it becomes feasible to detect and rectify errors that may arise during quantum computations.The Shor code detects and corrects both bit-flip and phase-flip errors,greatly enhancing the robustness of quantum information systems.This research offers insights into the multifaceted utility of MZI(Mach-Zehnder interferometer)and its relevance in the advancement of quantum technology.By integrating QEC with the capabilities of MZI,this study offers a holistic approach to advancing the precision and reliability of quantum technologies.
基金This work was supported by National Key Research and Development Program of China(Grant No.2017YFA0304303)the National Natu-ral Science Foundation of China(Grant No.11925404 and 11874235,11874342 and 11922411)+1 种基金Anhui Initiative in Quantum Information Technologies(AHY130200)a grant from the Institute for Guo Qiang(No.2019GQG1024),Tsinghua University.
文摘Quantum information is vulnerable to environmental noise and experimental imperfections,hindering the reli-ability of practical quantum information processors.Therefore,quantum error correction(QEC)that can pro-tect quantum information against noise is vital for universal and scalable quantum computation.Among many different experimental platforms,superconducting quantum circuits and bosonic encodings in superconducting microwave modes are appealing for their unprecedented potential in QEC.During the last few years,bosonic QEC is demonstrated to reach the break-even point,i.e.the lifetime of a logical qubit is enhanced to exceed that of any individual components composing the experimental system.Beyond that,universal gate sets and fault-tolerant operations on the bosonic codes are also realized,pushing quantum information processing towards the QEC era.In this article,we review the recent progress of the bosonic codes,including the Gottesman-Kitaev-Preskill codes,cat codes,and binomial codes,and discuss the opportunities of bosonic codes in various quantum applications,ranging from fault-tolerant quantum computation to quantum metrology.We also summarize the challenges associated with the bosonic codes and provide an outlook for the potential research directions in the long terms.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11834010,11804001,and 11904160)the Natural Science Foundation of Anhui Province,China(Grant No.1808085QA11)+1 种基金the Program of Youth Sanjin Scholar,National Key R&D Program of China(Grant No.2016YFA0301402)the Fund for Shanxi"1331 Project"Key Subjects Construction.
文摘Measurement-based quantum computation with continuous variables,which realizes computation by performing measurement and feedforward of measurement results on a large scale Gaussian cluster state,provides a feasible way to implement quantum computation.Quantum error correction is an essential procedure to protect quantum information in quantum computation and quantum communication.In this review,we briefly introduce the progress of measurement-based quantum computation and quantum error correction with continuous variables based on Gaussian cluster states.We also discuss the challenges in the fault-tolerant measurement-based quantum computation with continuous variables.
基金the National Natural Science Foundation of China(Grant Nos.11975132 and 61772295)the Natural Science Foundation of Shandong Province,China(Grant No.ZR2019YQ01)the Project of Shandong Province Higher Educational Science and Technology Program,China(Grant No.J18KZ012).
文摘Quantum error correction technology is an important solution to solve the noise interference generated during the operation of quantum computers.In order to find the best syndrome of the stabilizer code in quantum error correction,we need to find a fast and close to the optimal threshold decoder.In this work,we build a convolutional neural network(CNN)decoder to correct errors in the toric code based on the system research of machine learning.We analyze and optimize various conditions that affect CNN,and use the RestNet network architecture to reduce the running time.It is shortened by 30%-40%,and we finally design an optimized algorithm for CNN decoder.In this way,the threshold accuracy of the neural network decoder is made to reach 10.8%,which is closer to the optimal threshold of about 11%.The previous threshold of 8.9%-10.3%has been slightly improved,and there is no need to verify the basic noise.
基金supported by the Science and Technology Commission of Shanghai Municipality(Grant No.24DP2600202)the National Key R&D Program of China(Grant No.2024YFB4504002)the National Natural Science Foundation of China(Grant No.92165107)。
文摘For the quantum error correction and noisy intermediate-scale quantum algorithms to function with high efficiency,the raw fidelity of quantum logic gates on physical qubits needs to satisfy strict requirements.The neutral atom quantum computing equipped with Rydberg blockade gates has made impressive progress recently,which makes it worthwhile to explore its potential in the two-qubit entangling gates,including the controlledphase gate,and in particular,the CZ gate.Provided the quantum coherence is well preserved,improving the fidelity of Rydberg blockade gates calls for special mechanisms to deal with adverse effects caused by realistic experimental conditions.Here,the heralded very-high-fidelity Rydberg blockade controlled-phase gate is designed to address these issues,which contains self-correction and projection as the key steps.This trailblazing method builds upon the previously established buffer-atom-mediated gate framework,with a special form of symmetry under parity–time transformation playing a crucial role in the process.We further analyze the performance with respect to a few typical sources of imperfections.This procedure can also be regarded as quantum hardware error correction or mitigation.While this paper by itself does not cover every single subtle issue and still contains many oversimplifications,we find it reasonable to anticipate a very-high-fidelity two-qubit quantum logic gate operated in the sense of heralded but probabilistic,whose gate error can be reduced to the level of 10^(-4)–10^(-6)or even lower with reasonably high possibilities.
基金Project supported by the Natural Science Foundation of Shandong Province,China(Grant No.ZR2021MF049)the Joint Fund of the Natural Science Foundation of Shandong Province,China(Grant Nos.ZR2022LLZ012 and ZR2021LLZ001)the Key Research and Development Program of Shandong Province,China(Grant No.2023CXGC010901)。
文摘Quantum error-correcting codes are essential for fault-tolerant quantum computing,as they effectively detect and correct noise-induced errors by distributing information across multiple physical qubits.The subsystem surface code with three-qubit check operators demonstrates significant application potential due to its simplified measurement operations and low logical error rates.However,the existing minimum-weight perfect matching(MWPM)algorithm exhibits high computational complexity and lacks flexibility in large-scale systems.Therefore,this paper proposes a decoder based on a graph attention network(GAT),representing error syndromes as undirected graphs with edge weights,and employing a multihead attention mechanism to efficiently aggregate node features and enable parallel computation.Compared to MWPM,the GAT decoder exhibits linear growth in computational complexity,adapts to different quantum code structures,and demonstrates stronger robustness under high physical error rates.The experimental results demonstrate that the proposed decoder achieves an overall accuracy of 89.95%under various small code lattice sizes(L=2,3,4,5),with the logical error rate threshold increasing to 0.0078,representing an improvement of approximately 13.04%compared to the MWPM decoder.This result significantly outperforms traditional methods,showcasing superior performance under small code lattice sizes and providing a more efficient decoding solution for large-scale quantum error correction.
基金This work is supported by the National Natural Science Foundation of China(Grant Nos.11875050 and 12088101)NSAF(Grant No.U1930403).
文摘Minimizing the effect of noise is essential for quantum computers.The conventional method to protect qubits against noise is through quantum error correction.However,for current quantum hardware in the so-called noisy intermediate-scale quantum(NISQ)era,noise presents in these systems and is too high for error correction to be beneficial.Quantum error mitigation is a set of alternative methods for minimizing errors,including error extrapolation,probabilistic error cancella-tion,measurement error mitigation,subspace expansion,symmetry verification,virtual distillation,etc.The requirement for these methods is usually less demanding than error correction.Quantum error mitigation is a promising way of reduc-ing errors on NISQ quantum computers.This paper gives a comprehensive introduction to quantum error mitigation.The state-of-art error mitigation methods are covered and formulated in a general form,which provides a basis for comparing,combining and optimizing different methods in future work.
文摘Error correction has long been suggested to extend the sensitivity of quantum sensors into the Heisenberg Limit. However, operations on logical qubits are only performed through universal gate sets consisting of finite-sized gates such as Clifford + T. Although these logical gate sets allow for universal quantum computation, the finite gate sizes present a problem for quantum sensing, since in sensing protocols, such as the Ramsey measurement protocol, the signal must act continuously. The difficulty in constructing a continuous logical op-erator comes from the Eastin-Knill theorem, which prevents a continuous sig-nal from being both fault-tolerant to local errors and transverse. Since error correction is needed to approach the Heisenberg Limit in a noisy environment, it is important to explore how to construct fault-tolerant continuous operators. In this paper, a protocol to design continuous logical z-rotations is proposed and applied to the Steane Code. The fault tolerance of the designed operator is investigated using the Knill-Laflamme conditions. The Knill-Laflamme condi-tions indicate that the diagonal unitary operator constructed cannot be fault tolerant solely due to the possibilities of X errors on the middle qubit. The ap-proach demonstrated throughout this paper may, however, find success in codes with more qubits such as the Shor code, distance 3 surface code, [15, 1, 3] code, or codes with a larger distance such as the [11, 1, 5] code.
文摘In this article, we study the ability of error-correcting quantum codes to increase the fidelity of quantum states throughout a quantum computation. We analyze arbitrary quantum codes that encode all qubits involved in the computation, and we study the evolution of n-qubit fidelity from the end of one application of the correcting circuit to the end of the next application. We assume that the correcting circuit does not introduce new errors, that it does not increase the execution time (i.e. its application takes zero seconds) and that quantum errors are isotropic. We show that the quantum code increases the fidelity of the states perturbed by quantum errors but that this improvement is not enough to justify the use of quantum codes. Namely, we prove that, taking into account that the time interval between the application of the two corrections is multiplied (at least) by the number of qubits n (due to the coding), the best option is not to use quantum codes, since the fidelity of the uncoded state over a time interval n times smaller is greater than that of the state resulting from the quantum code correction.
基金supported by the Natural Science Foundation of Shandong Province,China(Grant No.ZR2021MF049)the Joint Fund of Natural Science Foundation of Shandong Province,China(Grant Nos.ZR2022LL.Z012 and ZR2021LLZ001)the Key Research and Development Program of Shandong Province,China(Grant No.2023CXGC010901).
文摘Quantum error correction is a technique that enhances a system’s ability to combat noise by encoding logical information into additional quantum bits,which plays a key role in building practical quantum computers.The XZZX surface code,with only one stabilizer generator on each face,demonstrates significant application potential under biased noise.However,the existing minimum weight perfect matching(MWPM)algorithm has high computational complexity and lacks flexibility in large-scale systems.Therefore,this paper proposes a decoding method that combines graph neural networks(GNN)with multi-classifiers,the syndrome is transformed into an undirected graph,and the features are aggregated by convolutional layers,providing a more efficient and accurate decoding strategy.In the experiments,we evaluated the performance of the XZZX code under different biased noise conditions(bias=1,20,200)and different code distances(d=3,5,7,9,11).The experimental results show that under low bias noise(bias=1),the GNN decoder achieves a threshold of 0.18386,an improvement of approximately 19.12%compared to the MWPM decoder.Under high bias noise(bias=200),the GNN decoder reaches a threshold of 0.40542,improving by approximately 20.76%,overcoming the limitations of the conventional decoder.They demonstrate that the GNN decoding method exhibits superior performance and has broad application potential in the error correction of XZZX code.
基金supported in part by National Natural Science Foundation of China under Grant Nos.60573127,60773012,and 60873082Natural Science Foundation of Hunan Province under Grant Nos.07JJ3128 and 2008RS4016+1 种基金Scientific Research Fund of Hunan Provincial Education Department under Grant No.08B011Postdoctoral Science Foundation of China under Grant Nos.20070420184 and 200801341
文摘A （n, n）-threshold scheme of multiparty quantum secret sharing of classical or quantum message is proposed based on the discrete quantum Fourier transform. In our proposed scheme, the secret message, which is encoded by using the forward quantum Fourier transform and decoded by using the reverse, is split and shared in such a way that it can be reconstructed among them only if all the participants work in concert. Fhrthermore, we also discuss how this protocol must be carefully designed for correcting errors and checking eavesdropping or a dishonest participant. Security analysis shows that our scheme is secure. Also, this scheme has an advantage that it is completely compatible with quantum computation and easier to realize in the distributed quantum secure computation.
基金Project supported by the National Natural Science Foundation of China(Grant No.60972046)Grant from the National Defense Pre-Research Foundation of China
文摘For quantum sparse graph codes with stabilizer formalism, the unavoidable girth-four cycles in their Tanner graphs greatly degrade the iterative decoding performance with standard belief-propagation （BP） algorithm. In this paper, we present a jointly-check iterative algorithm suitable for decoding quantum sparse graph codes efficiently. Numerical simulations show that this modified method outperforms standard BP algorithm with an obvious performance improvement.
基金Project supported by the National Key R&D Program of China (Grant No.2022YFB3103802)the National Natural Science Foundation of China (Grant Nos.62371240 and 61802175)the Fundamental Research Funds for the Central Universities (Grant No.30923011014)。
文摘Entanglement-assisted quantum error correction codes(EAQECCs)play an important role in quantum communications with noise.Such a scheme can use arbitrary classical linear code to transmit qubits over noisy quantum channels by consuming some ebits between the sender(Alice)and the receiver(Bob).It is usually assumed that the preshared ebits of Bob are error free.However,noise on these ebits is unavoidable in many cases.In this work,we evaluate the performance of EAQECCs with noisy ebits over asymmetric quantum channels and quantum channels with memory by computing the exact entanglement fidelity of several EAQECCs.We consider asymmetric errors in both qubits and ebits and show that the performance of EAQECCs in entanglement fidelity gets improved for qubits and ebits over asymmetric channels.In quantum memory channels,we compute the entanglement fidelity of several EAQECCs over Markovian quantum memory channels and show that the performance of EAQECCs is lowered down by the channel memory.Furthermore,we show that the performance of EAQECCs is diverse when the error probabilities of qubits and ebits are different.In both asymmetric and memory quantum channels,we show that the performance of EAQECCs is improved largely when the error probability of ebits is reasonably smaller than that of qubits.
基金supported by the National Basic Research Program of China (Grant No.2010CB328300)the National Natural Science Foundation of China (Grant Nos.60972046 and 60902030)+4 种基金the Program for Changjiang Scholars and Innovative Research Team in University (Grant No.IRT0852)the Natural Science Foundation of Shaanxi Province (Grant No.2010JQ8025)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No.20100203120004)the 111 Program (Grant No.B08038)the China Scholarship Council (Grant No.[2008]3019)
文摘We address the problem of encoding entanglement-assisted （EA） quantum error-correcting codes （QECCs） and of the corresponding complexity. We present an iterative algorithm from which a quantum circuit composed of CNOT, H, and S gates can be derived directly with complexity O（n2） to encode the qubits being sent. Moreover, we derive the number of each gate consumed in our algorithm according to which we can design EA QECCs with low encoding complexity. Another advantage brought by our algorithm is the easiness and efficiency of programming on classical computers.
基金Project supported by Natural Science Foundation of Shandong Province,China (Grant Nos.ZR2021MF049,ZR2022LLZ012,and ZR2021LLZ001)。
文摘Quantum error correction is a crucial technology for realizing quantum computers.These computers achieve faulttolerant quantum computing by detecting and correcting errors using decoding algorithms.Quantum error correction using neural network-based machine learning methods is a promising approach that is adapted to physical systems without the need to build noise models.In this paper,we use a distributed decoding strategy,which effectively alleviates the problem of exponential growth of the training set required for neural networks as the code distance of quantum error-correcting codes increases.Our decoding algorithm is based on renormalization group decoding and recurrent neural network decoder.The recurrent neural network is trained through the ResNet architecture to improve its decoding accuracy.Then we test the decoding performance of our distributed strategy decoder,recurrent neural network decoder,and the classic minimum weight perfect matching(MWPM)decoder for rotated surface codes with different code distances under the circuit noise model,the thresholds of these three decoders are about 0.0052,0.0051,and 0.0049,respectively.Our results demonstrate that the distributed strategy decoder outperforms the other two decoders,achieving approximately a 5%improvement in decoding efficiency compared to the MWPM decoder and approximately a 2%improvement compared to the recurrent neural network decoder.
