Shor's algorithm is one of the most important quantum algorithm proposed by Peter Shor [Proceedings of the 35th Annual Symposium on Foundations of Computer Science, 1994, pp. 124--134]. Shor's algorithm can factor a large integer with certain probability and costs polynomial time in the length of the input integer. The key step of Shor's algorithm is the order-finding algorithm, the quantum part of which is to estimate $s/r$, where $r$ is the ``order" and $s$ is some natural number that less than $r$. {{Shor's algorithm requires lots of qubits and a deep circuit depth, which is unaffordable for current physical devices.}} In this paper, to reduce the number of qubits required and circuit depth, we propose a quantum-classical hybrid distributed order-finding algorithm for Shor's algorithm, which combines the advantages of both quantum processing and classical processing. {{ In our distributed order-finding algorithm, we use two quantum computers with the ability of quantum teleportation separately to estimate partial bits of $s/r$.}} The measuring results will be processed through a classical algorithm to ensure the accuracy of the results. Compared with the traditional Shor's algorithm that uses multiple control qubits, our algorithm reduces nearly $L/2$ qubits for factoring an $L$-bit integer and reduces the circuit depth of each computer.
Shor算法是Peter Shor提出的最重要的量子算法之一[Proceedings of the 35 Annual Symposium on Foundations of Computer Science, 1994, pp. 124—134]。Shor算法能够以一定的概率分解一个大整数，并且在输入整数的长度上花费多项式的时间。Shor算法的关键步骤是寻序算法，其中量子部分是估计$s/r$，其中$r$为“阶数”，$s$为小于$r$的自然数。{{肖尔的算法需要大量的量子比特和较深的电路深度，这对于当前的物理设备来说是无法承受的。在本文中，为了减少所需的量子比特数和电路深度，我们针对Shor算法提出了一种量子-经典混合分布式寻序算法，该算法结合了量子处理和经典处理的优点。{{在我们的分布式寻序算法中，我们分别使用两台具有量子隐形传态能力的量子计算机来估计$s/r$的部分比特。测量结果将通过经典算法进行处理，保证测量结果的准确性。与传统的使用多个控制量子位的Shor算法相比，我们的算法减少了近$L/2$量子位来分解一个$L$位的整数，并且减少了每台计算机的电路深度。
{"title":"Distributed Shor's algorithm","authors":"Li Xiao, Daowen Qiu, Leon Luo, P. Mateus","doi":"10.26421/qic23.1-2-3","DOIUrl":"https://doi.org/10.26421/qic23.1-2-3","url":null,"abstract":"Shor's algorithm is one of the most important quantum algorithm proposed by Peter Shor [Proceedings of the 35th Annual Symposium on Foundations of Computer Science, 1994, pp. 124--134]. Shor's algorithm can factor a large integer with certain probability and costs polynomial time in the length of the input integer. The key step of Shor's algorithm is the order-finding algorithm, the quantum part of which is to estimate $s/r$, where $r$ is the ``order\" and $s$ is some natural number that less than $r$. {{Shor's algorithm requires lots of qubits and a deep circuit depth, which is unaffordable for current physical devices.}} In this paper, to reduce the number of qubits required and circuit depth, we propose a quantum-classical hybrid distributed order-finding algorithm for Shor's algorithm, which combines the advantages of both quantum processing and classical processing. {{ In our distributed order-finding algorithm, we use two quantum computers with the ability of quantum teleportation separately to estimate partial bits of $s/r$.}} The measuring results will be processed through a classical algorithm to ensure the accuracy of the results. Compared with the traditional Shor's algorithm that uses multiple control qubits, our algorithm reduces nearly $L/2$ qubits for factoring an $L$-bit integer and reduces the circuit depth of each computer.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"16 1","pages":"27-44"},"PeriodicalIF":0.0,"publicationDate":"2022-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72780833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The research of ternary quantum system has gradually come into the attention of scholars in recent years. In $2018$, Guangcan Guo and his colleagues showed that in a qutrit-qutrit system they can observe quantum nonlocality and quantum contextuality at the same time. In $2019$, international cooperation team led by Anton Zelinger of Austrian Academy of Sciences and Jianwei Pan of University of science and technology of China, they have succeeded in teleporting complex high-dimensional quantum states. The work of the above scholars makes us clearly realize the importance of the study of ternary quantum system, but there is a few research results on this aspect. Furthermore, the quantum Fourier transform (QFT) offers an interesting way to perform arithmetic operations on a quantum computer. So, the paper extends the QFT to symmetric ternary quantum system and gives its applications. First, a set of quantum gates is defined for symmetric ternary quantum system. It is worth noting that in binary system, qubit flipping is realized by Not gate. Therefore, we need to extend Not gate to symmetric ternary system to realize qutrit flipping, which is called M-S gate. And then, by decomposing single-qutrit unitary gate in symmetric ternary quantum system, the universal gates are given. It means that any unitary operation on $n$ qutrits can be accurately implemented by single-qutrit symmetric ternary quantum gates and two-qutrit symmetric ternary M-S gates. By extending the QFT to the symmetric ternary quantum system, the paper successfully use some symmetric ternary quantum gates to construct the circuit which can realize symmetric ternary quantum Fourier transform (STQFT). Finally, the circuit of adder in symmetric ternary quantum system are designed based on the STQFT and the universal quantum gates. ternary quantum Fourier transform and its application (pp733-754) Hao Dong, Dayong Lu, and Xiaoyun Sun doi: https://doi.org/10.26421/QIC22.9-10-2 Abstracts: The research of ternary quantum system has gradually come into the attention of scholars in recent years. In $2018$, Guangcan Guo and his colleagues showed that in a qutrit-qutrit system they can observe quantum nonlocality and quantum contextuality at the same time. In $2019$, international cooperation team led by Anton Zelinger of Austrian Academy of Sciences and Jianwei Pan of University of science and technology of China, they have succeeded in teleporting complex high-dimensional quantum states. The work of the above scholars makes us clearly realize the importance of the study of ternary quantum system, but there is a few research results on this aspect. Furthermore, the quantum Fourier transform (QFT) offers an interesting way to perform arithmetic operations on a quantum computer. So, the paper extends the QFT to symmetric ternary quantum system and gives its applications. First, a set of quantum gates is defined for symmetric ternary quantum system. It is worth noting that in binary system, qubit
{"title":"Symmetric ternary quantum Fourier transform and its application","authors":"Hao Dong, Dayong Lu, Xiaoyun Sun","doi":"10.26421/qic22.9-10-2","DOIUrl":"https://doi.org/10.26421/qic22.9-10-2","url":null,"abstract":"The research of ternary quantum system has gradually come into the attention of scholars in recent years. In $2018$, Guangcan Guo and his colleagues showed that in a qutrit-qutrit system they can observe quantum nonlocality and quantum contextuality at the same time. In $2019$, international cooperation team led by Anton Zelinger of Austrian Academy of Sciences and Jianwei Pan of University of science and technology of China, they have succeeded in teleporting complex high-dimensional quantum states. The work of the above scholars makes us clearly realize the importance of the study of ternary quantum system, but there is a few research results on this aspect. Furthermore, the quantum Fourier transform (QFT) offers an interesting way to perform arithmetic operations on a quantum computer. So, the paper extends the QFT to symmetric ternary quantum system and gives its applications. First, a set of quantum gates is defined for symmetric ternary quantum system. It is worth noting that in binary system, qubit flipping is realized by Not gate. Therefore, we need to extend Not gate to symmetric ternary system to realize qutrit flipping, which is called M-S gate. And then, by decomposing single-qutrit unitary gate in symmetric ternary quantum system, the universal gates are given. It means that any unitary operation on $n$ qutrits can be accurately implemented by single-qutrit symmetric ternary quantum gates and two-qutrit symmetric ternary M-S gates. By extending the QFT to the symmetric ternary quantum system, the paper successfully use some symmetric ternary quantum gates to construct the circuit which can realize symmetric ternary quantum Fourier transform (STQFT). Finally, the circuit of adder in symmetric ternary quantum system are designed based on the STQFT and the universal quantum gates. ternary quantum Fourier transform and its application (pp733-754) Hao Dong, Dayong Lu, and Xiaoyun Sun doi: https://doi.org/10.26421/QIC22.9-10-2 Abstracts: The research of ternary quantum system has gradually come into the attention of scholars in recent years. In $2018$, Guangcan Guo and his colleagues showed that in a qutrit-qutrit system they can observe quantum nonlocality and quantum contextuality at the same time. In $2019$, international cooperation team led by Anton Zelinger of Austrian Academy of Sciences and Jianwei Pan of University of science and technology of China, they have succeeded in teleporting complex high-dimensional quantum states. The work of the above scholars makes us clearly realize the importance of the study of ternary quantum system, but there is a few research results on this aspect. Furthermore, the quantum Fourier transform (QFT) offers an interesting way to perform arithmetic operations on a quantum computer. So, the paper extends the QFT to symmetric ternary quantum system and gives its applications. First, a set of quantum gates is defined for symmetric ternary quantum system. It is worth noting that in binary system, qubit ","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"76 1","pages":"733-754"},"PeriodicalIF":0.0,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76098056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
On the space of mixed quantum states, several alternative fidelities have been proposed besides the standard Uhlmann-Jozsa fidelity. It has been known that several properties of the Uhlmann-Jozsa fidelity still hold true for these alternative fidelities. The aim of this paper is to give positive answers to some questions about the metric properties of functionals of alternative quantum fidelities raised by Y. C. Liang {it et al.} in cite{LYM}. Our method is to use the non-negativity of the Gram determinant of three vectors constructed from the quantum states to prove the triangle inequality for the modified Bures angle.
{"title":"Metric properties of alternative fidelities","authors":"V. T. Khoi, Ho Minh Toan","doi":"10.26421/qic22.9-10-5","DOIUrl":"https://doi.org/10.26421/qic22.9-10-5","url":null,"abstract":"On the space of mixed quantum states, several alternative fidelities have been proposed besides the standard Uhlmann-Jozsa fidelity. It has been known that several properties of the Uhlmann-Jozsa fidelity still hold true for these alternative fidelities. The aim of this paper is to give positive answers to some questions about the metric properties of functionals of alternative quantum fidelities raised by Y. C. Liang {it et al.} in cite{LYM}. Our method is to use the non-negativity of the Gram determinant of three vectors constructed from the quantum states to prove the triangle inequality for the modified Bures angle.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"1 1","pages":"790-799"},"PeriodicalIF":0.0,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82005700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The point of building a quantum computer is that it allows to model living things with predictive power and gives the opportunity to control life. Its scaling means not just the improvement of the instrument part, but also, mainly, mathematical and software tools, and our understanding of the QC problem. The first principle of quantum modeling is the reduction of reality to finite-dimensional models similar to QED in optical cavities. The second principle is a strict limitation of the so-called Feynman principle, the number of qubits in the standard formulation of the QC. This means treating decoherence exclusively as a limitation of the memory of a classical modeling computer, and introducing corresponding progressive restrictions on the working area of the Hilbert space of quantum states as the model expands. The third principle is similarity in processes of different nature. The quantum nature of reality is manifested in this principle; its nature is quantum nonlocality, which is the main property that ensures the prospects of quantum physical devices and their radical advantage over classical ones.
{"title":"Three principles of quantum computing","authors":"Y. Ozhigov","doi":"10.26421/QIC22.15-16-2","DOIUrl":"https://doi.org/10.26421/QIC22.15-16-2","url":null,"abstract":"The point of building a quantum computer is that it allows to model living things with predictive power and gives the opportunity to control life. Its scaling means not just the improvement of the instrument part, but also, mainly, mathematical and software tools, and our understanding of the QC problem. The first principle of quantum modeling is the reduction of reality to finite-dimensional models similar to QED in optical cavities. The second principle is a strict limitation of the so-called Feynman principle, the number of qubits in the standard formulation of the QC. This means treating decoherence exclusively as a limitation of the memory of a classical modeling computer, and introducing corresponding progressive restrictions on the working area of the Hilbert space of quantum states as the model expands. The third principle is similarity in processes of different nature. The quantum nature of reality is manifested in this principle; its nature is quantum nonlocality, which is the main property that ensures the prospects of quantum physical devices and their radical advantage over classical ones.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"99 1","pages":"1280-1288"},"PeriodicalIF":0.0,"publicationDate":"2022-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82751383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We will fix an error in the proof of Theorem 2 of the work On the extremal points of the $Lambda $-polytopes and classical simulation of quantum computation with magic states by the current authors, published in Quantum Information and Computation Vol.21 No.13&14, 1533-7146 (2021). The theorem as it is stated is still correct, however there is a gap in the proof that needs to be filled.
{"title":"Erratum: On the extremal points of the Lambda-polytopes and classical simulation of quantum computation with magic states","authors":"C. Okay, Michael Zurel, R. Raussendorf","doi":"10.26421/QIC22.7-8-4","DOIUrl":"https://doi.org/10.26421/QIC22.7-8-4","url":null,"abstract":"We will fix an error in the proof of Theorem 2 of the work On the extremal points of the $Lambda $-polytopes and classical simulation of quantum computation with magic states by the current authors, published in Quantum Information and Computation Vol.21 No.13&14, 1533-7146 (2021). The theorem as it is stated is still correct, however there is a gap in the proof that needs to be filled.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"68 1","pages":"627-628"},"PeriodicalIF":0.0,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82553065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Spatial search occurs in a connected graph if a continuous-time quantum walk on the adjacency matrix of the graph, suitably scaled, plus a rank-one perturbation induced by any vertex will unitarily map the principal eigenvector of the graph to the characteristic vector of the vertex. This phenomenon is a natural continuous-time analogue of Grover search. The spatial search is said to be optimal if it occurs with constant fidelity and in time inversely proportional to the shadow of the target vertex on the principal eigenvector. Extending a result of Chakraborty etal ({em Physical Review A}, {bf 102}:032214, 2020), we prove a simpler characterization of optimal spatial search. Based on this characterization, we observe that some families of distance-regular graphs, such as Hamming and Grassmann graphs, have optimal spatial search. We also show a matching lower bound on time for spatial search with constant fidelity, which extends a bound due to Farhi and Gutmann for perfect fidelity. Our elementary proofs employ standard tools, such as Weyl inequalities and Cauchy determinant formula.
{"title":"Of shadows and gaps in spatial search","authors":"Ada Chan, C. Godsil, C. Tamon, Weichen Xie","doi":"10.26421/qic22.13-14-2","DOIUrl":"https://doi.org/10.26421/qic22.13-14-2","url":null,"abstract":"Spatial search occurs in a connected graph if a continuous-time quantum walk on the adjacency matrix of the graph, suitably scaled, plus a rank-one perturbation induced by any vertex will unitarily map the principal eigenvector of the graph to the characteristic vector of the vertex. This phenomenon is a natural continuous-time analogue of Grover search. The spatial search is said to be optimal if it occurs with constant fidelity and in time inversely proportional to the shadow of the target vertex on the principal eigenvector. Extending a result of Chakraborty etal ({em Physical Review A}, {bf 102}:032214, 2020), we prove a simpler characterization of optimal spatial search. Based on this characterization, we observe that some families of distance-regular graphs, such as Hamming and Grassmann graphs, have optimal spatial search. We also show a matching lower bound on time for spatial search with constant fidelity, which extends a bound due to Farhi and Gutmann for perfect fidelity. Our elementary proofs employ standard tools, such as Weyl inequalities and Cauchy determinant formula.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"5 1","pages":"1110-1131"},"PeriodicalIF":0.0,"publicationDate":"2022-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81298706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Due to their rich algebraic structure, cyclic codes have a great deal of significance amongst linear codes. Duadic codes are the generalization of the quadratic residue codes, a special case of cyclic codes. The $m$-adic residue codes are the generalization of the duadic codes. The aim of this paper is to study the structure of the $m$-adic residue codes over the quotient ring $frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$. We determine the idempotent generators of the $m$-adic residue codes over $frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$. We obtain some parameters of optimal $m$-adic residue codes over $frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$ with respect to Griesmer bound for rings. Furthermore, we derive a condition for $m$-adic residue codes over $frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$ to contain their dual. By making use of a preserving-orthogonality Gray map, we construct a family of quantum error correcting codes from the Gray images of dual-containing $m$-adic residue codes over $frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$ and give some examples to illustrate our findings.
循环码由于其丰富的代数结构，在线性码中具有重要的意义。二元码是二次剩余码的推广，是循环码的一种特例。$m$进数剩余码是对二进码的推广。本文的目的是研究商环$frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$上的$m$ -进剩余码的结构。我们确定了$frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$上$m$ -进剩码的幂等生成器。对于环的Griesmer界，我们得到了$frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$上最优$m$ -进剩余码的一些参数。此外，我们推导了$frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$上的$m$ -adic剩余码包含其对偶的条件。我们利用保持正交的灰度映射，从$frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$上的双包含$m$ -adic残差码的灰度图像中构造了一组量子纠错码，并给出了一些例子来说明我们的发现。
{"title":"Applications to Quantum Codes","authors":"Ferhat Kuruz, Mustafa Sarı, M. Köroğlu","doi":"10.26421/qic22.5-6-4","DOIUrl":"https://doi.org/10.26421/qic22.5-6-4","url":null,"abstract":"Due to their rich algebraic structure, cyclic codes have a great deal of significance amongst linear codes. Duadic codes are the generalization of the quadratic residue codes, a special case of cyclic codes. The $m$-adic residue codes are the generalization of the duadic codes. The aim of this paper is to study the structure of the $m$-adic residue codes over the quotient ring $frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$. We determine the idempotent generators of the $m$-adic residue codes over $frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$. We obtain some parameters of optimal $m$-adic residue codes over $frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$ with respect to Griesmer bound for rings. Furthermore, we derive a condition for $m$-adic residue codes over $frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$ to contain their dual. By making use of a preserving-orthogonality Gray map, we construct a family of quantum error correcting codes from the Gray images of dual-containing $m$-adic residue codes over $frac{{{mathbb{F}_q}left[ v right]}}{{leftlangle {{v^s} - v} rightrangle }}$ and give some examples to illustrate our findings.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"45 1","pages":"427-439"},"PeriodicalIF":0.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75524109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the shallow sub-threshold regime, fault-tolerant quantum computation requires a tremendous amount of qubits. In this paper, we study the error correction in the deep sub-threshold regime. We estimate the physical error rate for achieving the logical error rates of $10^{-6} - 10^{-15}$ using few-qubit codes, i.e.~short repetition codes, small surface codes and the Steane code. Error correction circuits that are efficient for biased error operators are identified. Using the Steane code, when error operators are biased with a ratio of $10^{-3}$, the logical error rate of $10^{-15}$ can be achieved with the physical error rate of $10^{-5}$, which is much higher than the physical error rate of $10^{-9}$ for depolarising errors.
{"title":"Qubit Codes: Parity-check Circuits for Biased Error Operators","authors":"Dawei Jiao, Y. Li","doi":"10.26421/qic22.5-6-3","DOIUrl":"https://doi.org/10.26421/qic22.5-6-3","url":null,"abstract":"In the shallow sub-threshold regime, fault-tolerant quantum computation requires a tremendous amount of qubits. In this paper, we study the error correction in the deep sub-threshold regime. We estimate the physical error rate for achieving the logical error rates of $10^{-6} - 10^{-15}$ using few-qubit codes, i.e.~short repetition codes, small surface codes and the Steane code. Error correction circuits that are efficient for biased error operators are identified. Using the Steane code, when error operators are biased with a ratio of $10^{-3}$, the logical error rate of $10^{-15}$ can be achieved with the physical error rate of $10^{-5}$, which is much higher than the physical error rate of $10^{-9}$ for depolarising errors.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"18 1","pages":"408-426"},"PeriodicalIF":0.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81751589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A quantum secret sharing scheme is a method to share a quantum secret among participants in such a way that only certain specified subsets of the set of participants can combine to recover the secret. A quantum evolving secret sharing scheme is a recently introduced variant of secret sharing schemes where the sets of participants is not fixed and an unbounded and unspecified number of participants arrive one by one with time. The number of participants can be potentially infinite and and a quantum secret is shared and protected among the participants. The existing construction has some major drawbacks such as 1) the exponential quantum memory requirement and 2) the very high dimensions of the shares which makes the scheme difficult to implement. In this paper we overcome these drawbacks by constructing a scheme which uses quantum memory linear in the number of participants and significantly improves on the dimensions of the shares of the participants. This construction uses quantum secret redistribution and trap codes. The construction is flexible and can be modified to different types of access structures(subsets of participants which can recover the secret). Certain ramp properties can also be incorporated in the scheme.
{"title":"On quantum evolving schemes","authors":"S. S. Chaudhury","doi":"10.26421/qic22.5-6-2","DOIUrl":"https://doi.org/10.26421/qic22.5-6-2","url":null,"abstract":"A quantum secret sharing scheme is a method to share a quantum secret among participants in such a way that only certain specified subsets of the set of participants can combine to recover the secret. A quantum evolving secret sharing scheme is a recently introduced variant of secret sharing schemes where the sets of participants is not fixed and an unbounded and unspecified number of participants arrive one by one with time. The number of participants can be potentially infinite and and a quantum secret is shared and protected among the participants. The existing construction has some major drawbacks such as 1) the exponential quantum memory requirement and 2) the very high dimensions of the shares which makes the scheme difficult to implement. In this paper we overcome these drawbacks by constructing a scheme which uses quantum memory linear in the number of participants and significantly improves on the dimensions of the shares of the participants. This construction uses quantum secret redistribution and trap codes. The construction is flexible and can be modified to different types of access structures(subsets of participants which can recover the secret). Certain ramp properties can also be incorporated in the scheme.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"3 1","pages":"385-407"},"PeriodicalIF":0.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75156725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Many entanglement measures are first defined for pure states of a bipartite Hilbert space, and then extended to mixed states via the convex roof extension. In this article we alter the convex roof extension of an entanglement measure, to produce a sequence of extensions that we call $f$-$d$ extensions, for $d in mathbb{N}$, where $f:[0,1]to [0, infty)$ is a fixed continuous function which vanishes only at zero. We prove that for any such function $f$, and any continuous, faithful, non-negative function, (such as an entanglement measure), $mu$ on the set of pure states of a finite dimensional bipartite Hilbert space, the collection of $f$-$d$ extensions of $mu$ detects entanglement, i.e. a mixed state $rho$ on a finite dimensional bipartite Hilbert space is separable, if and only if there exists $d in mathbb{N}$ such that the $f$-$d$ extension of $mu$ applied to $rho$ is equal to zero. We introduce a quantum variational algorithm which aims to approximate the $f$-$d$ extensions of entanglement measures defined on pure states. However, the algorithm does have its drawbacks. We show that this algorithm exhibits barren plateaus when used to approximate the family of $f$-$d$ extensions of the Tsallis entanglement entropy for a certain function $f$ and unitary ansatz $U(theta)$ of sufficient depth. In practice, if additional information about the state is known, then one needs to avoid using the suggested ansatz for long depth of circuits.
首先对二部Hilbert空间的纯态定义了许多纠缠测度，然后通过凸顶扩展扩展到混合态。在本文中，我们改变了一个纠缠测度的凸顶扩展，以产生一系列扩展，我们称之为$f$ - $d$扩展，对于$d in mathbb{N}$，其中$f:[0,1]to [0, infty)$是一个固定的连续函数，它只在零处消失。我们证明了对于任何这样的函数$f$和任何连续的、忠实的、非负的函数(如纠缠测度)，$mu$在有限维二部希尔伯特空间的纯态集合上，$mu$的$f$ - $d$扩展集合检测到纠缠，即在有限维二部希尔伯特空间上的混合态$rho$是可分的。当且仅当存在$d in mathbb{N}$使得应用于$rho$的$mu$的$f$ - $d$扩展等于零。我们引入了一种量子变分算法，旨在近似定义在纯态上的纠缠测度的$f$ - $d$扩展。然而，该算法也有它的缺点。我们表明，当用于近似特定函数$f$的Tsallis纠缠熵的$f$ - $d$扩展和足够深度的一元分析z $U(theta)$时，该算法显示出荒芜的平台。在实践中，如果关于状态的附加信息是已知的，那么需要避免对长深度电路使用建议的ansatz。
{"title":"Variational Quantum Algorithm for Approximating Convex Roofs","authors":"G. Androulakis, Ryan McGaha","doi":"10.26421/QIC22.13-14-1","DOIUrl":"https://doi.org/10.26421/QIC22.13-14-1","url":null,"abstract":"Many entanglement measures are first defined for pure states of a bipartite Hilbert space, and then extended to mixed states via the convex roof extension. In this article we alter the convex roof extension of an entanglement measure, to produce a sequence of extensions that we call $f$-$d$ extensions, for $d in mathbb{N}$, where $f:[0,1]to [0, infty)$ is a fixed continuous function which vanishes only at zero. We prove that for any such function $f$, and any continuous, faithful, non-negative function, (such as an entanglement measure), $mu$ on the set of pure states of a finite dimensional bipartite Hilbert space, the collection of $f$-$d$ extensions of $mu$ detects entanglement, i.e. a mixed state $rho$ on a finite dimensional bipartite Hilbert space is separable, if and only if there exists $d in mathbb{N}$ such that the $f$-$d$ extension of $mu$ applied to $rho$ is equal to zero. We introduce a quantum variational algorithm which aims to approximate the $f$-$d$ extensions of entanglement measures defined on pure states. However, the algorithm does have its drawbacks. We show that this algorithm exhibits barren plateaus when used to approximate the family of $f$-$d$ extensions of the Tsallis entanglement entropy for a certain function $f$ and unitary ansatz $U(theta)$ of sufficient depth. In practice, if additional information about the state is known, then one needs to avoid using the suggested ansatz for long depth of circuits.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"3 1","pages":"1081-1109"},"PeriodicalIF":0.0,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88348179","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：