Simulation of n-qubit quantum systems. II. Separability and entanglement

Studies on the entanglement of n-qubit quantum systems have attracted a lot of interest during recent years. Despite the central role of entanglement in quantum information theory, however, there are still a number of open problems in the theoretical characterization of entangled systems that make symbolic and numerical simulation on n-qubit quantum registers indispensable for present-day research.To facilitate the investigation of the separability and entanglement properties of n-qubit quantum registers, here we present a revised version of the Feynman program in the framework of the computer algebra system Maple. In addition to all previous capabilities of this Maple code for defining and manipulating quantum registers, the program now provides various tools which are necessary for the qualitative and quantitative analysis of entanglement in n-qubit quantum registers. A simple access, in particular, is given to several algebraic separability criteria as well as a number of entanglement measures and related quantities. As in the previous version, symbolic and numeric computations are equally supported.

Program summary

Title of program:FeynmanCatalogue identifier:ADWE_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWE_v2_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:NoneComputers for which the program is designed: All computers with a license of the computer algebra system Maple [Maple is a registered trademark of Waterloo Maple Inc.]Operating systems under which the program has been tested: Linux, MS Windows XPProgramming language used:Maple 10Typical time and memory requirements:Most commands acting on quantum registers with five or less qubits take ?10 seconds of processor time (on a Pentium 4 with ?2 GHz or equivalent) and 5-20 MB of memory. However, storage and time requirements critically depend on the number of qubits, n, in the quantum registers due to the exponential increase of the associated Hilbert space.No. of lines in distributed program, including test data, etc.:3107No. of bytes in distributed program, including test data, etc.:13 859Distribution format:tar.gzReasons for new version:The first program version established the data structures and commands which are needed to build and manipulate quantum registers. Since the (evolution of) entanglement is a central aspect in quantum information processing the current version adds the capability to analyze separability and entanglement of quantum registers by implementing algebraic separability criteria and entanglement measures and related quantities.Does this version supersede the previous version: YesNature of the physical problem: Entanglement has been identified as an essential resource in virtually all aspects of quantum information theory. Therefore, the detection and quantification of entanglement is a necessary prerequisite for many applications, such as quantum computation, communications or quantum cryptography. Up to the present, however, the multipartite entanglement of n-qubit systems has remained largely unexplored owing to the exponential growth of complexity with the number of qubits involved.Method of solution: Using the computer algebra system Maple, a set of procedures has been developed which supports the definition and manipulation of n-qubit quantum registers and quantum logic gates [T. Radtke, S. Fritzsche, Comput. Phys. Comm. 173 (2005) 91]. The provided hierarchy of commands can be used interactively in order to simulate the behavior of n-qubit quantum systems (by applying a number of unitary or non-unitary operations) and to analyze their separability and entanglement properties.Restrictions onto the complexity of the problem: The present version of the program facilitates the setup and the manipulation of quantum registers by means of (predefined) quantum logic gates; it now also provides the tools for performing a symbolic and/or numeric analysis of the entanglement for the quantum states of such registers. Owing to the rapid increase in the computational complexity of multi-qubit systems, however, the time and memory requirements often grow rapidly, especially for symbolic computations. This increase of complexity limits the application of the program to about 6 or 7 qubits on a standard single processor (Pentium 4 with ?2 GHz or equivalent) machine with ?1 GB of memory.Unusual features of the program: The Feynman program has been designed within the framework of Maple for interactive (symbolic or numerical) simulations on n-qubit quantum registers with no other restriction than given by the memory and processor resources of the computer. Whenever possible, both representations of quantum registers in terms of their state vectors and/or density matrices are equally supported by the program. Apart from simulating quantum gates and quantum operations, the program now facilitates also investigations on the separability and the entanglement properties of quantum registers.     

Simulation of n-qubit quantum systems. V. Quantum measurements

The Feynman program has been developed during the last years to support case studies on the dynamics and entanglement of n-qubit quantum registers. Apart from basic transformations and (gate) operations, it currently supports a good number of separability criteria and entanglement measures, quantum channels as well as the parametrizations of various frequently applied objects in quantum information theory, such as (pure and mixed) quantum states, hermitian and unitary matrices or classical probability distributions. With the present update of the Feynman program, we provide a simple access to (the simulation of) quantum measurements. This includes not only the widely-applied projective measurements upon the eigenspaces of some given operator but also single-qubit measurements in various pre- and user-defined bases as well as the support for two-qubit Bell measurements. In addition, we help perform generalized and POVM measurements. Knowing the importance of measurements for many quantum information protocols, e.g., one-way computing, we hope that this update makes the Feynman code an attractive and versatile tool for both, research and education.

New version program summary

Program title: FEYNMANCatalogue identifier: ADWE_v5_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWE_v5_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 27 210No. of bytes in distributed program, including test data, etc.: 1 960 471Distribution format: tar.gzProgramming language: Maple 12Computer: Any computer with Maple software installedOperating system: Any system that supports Maple; the program has been tested under Microsoft Windows XP and LinuxClassification: 4.15Catalogue identifier of previous version: ADWE_v4_0Journal reference of previous version: Comput. Phys. Commun. 179 (2008) 647Does the new version supersede the previous version?: YesNature of problem: During the last decade, the field of quantum information science has largely contributed to our understanding of quantum mechanics, and has provided also new and efficient protocols that are used on quantum entanglement. To further analyze the amount and transfer of entanglement in n-qubit quantum protocols, symbolic and numerical simulations need to be handled efficiently.Solution method: Using the computer algebra system Maple, we developed a set of procedures in order to support the definition, manipulation and analysis of n-qubit quantum registers. These procedures also help to deal with (unitary) logic gates and (nonunitary) quantum operations and measurements that act upon the quantum registers. All commands are organized in a hierarchical order and can be used interactively in order to simulate and analyze the evolution of n-qubit quantum systems, both in ideal and noisy quantum circuits.Reasons for new version: Until the present, the FEYNMAN program supported the basic data structures and operations of n-qubit quantum registers [1], a good number of separability and entanglement measures [2], quantum operations (noisy channels) [3] as well as the parametrizations of various frequently applied objects, such as (pure and mixed) quantum states, hermitian and unitary matrices or classical probability distributions [4]. With the current extension, we here add all necessary features to simulate quantum measurements, including the projective measurements in various single-qubit and the two-qubit Bell basis, and POVM measurements. Together with the previously implemented functionality, this greatly enhances the possibilities of analyzing quantum information protocols in which measurements play a central role, e.g., one-way computation.Running time: Most commands require ?10 seconds of processor time on a Pentium 4 processor with ?2 GHz RAM or newer, if they work with quantum registers with five or less qubits. Moreover, about 5-20 MB of working memory is typically needed (in addition to the memory for the Maple environment itself). However, especially when working with symbolic expressions, the requirements on the CPU time and memory critically depend on the size of the quantum registers owing to the exponential growth of the dimension of the associated Hilbert space. For example, complex (symbolic) noise models, i.e. with several Kraus operators, may result in very large expressions that dramatically slow down the evaluation of e.g. distance measures or the final-state entropy, etc. In these cases, Maple's assume facility sometimes helps to reduce the complexity of the symbolic expressions, but more often than not only a numerical evaluation is feasible. Since the various commands can be applied to quite different scenarios, no general scaling rule can be given for the CPU time or the request of memory.References:

[1] T. Radtke, S. Fritzsche, Comput. Phys. Commun. 173 (2005) 91.

[2] T. Radtke, S. Fritzsche, Comput. Phys. Commun. 175 (2006) 145.

[3] T. Radtke, S. Fritzsche, Comput. Phys. Commun. 176 (2007) 617.

[4] T. Radtke, S. Fritzsche, Comput. Phys. Commun. 179 (2008) 647.

     

An improved quantum watermarking scheme using small-scale quantum circuits and color scrambling

In order to solve the problem of embedding the watermark into the quantum color image, in this paper, an improved scheme of using small-scale quantum circuits and color scrambling is proposed. Both color carrier image and color watermark image are represented using novel enhanced quantum representation. The image sizes for carrier and watermark are assumed to be \(2^{n+1}\times 2^{n+2}\) and \(2^{n}\times 2^{n}\), respectively. At first, the color of pixels in watermark image is scrambled using the controlled rotation gates, and then, the scrambled watermark with \(2^n\times 2^n\) image size and 24-qubit gray scale is expanded to an image with \(2^{n+1}\times 2^{n+2}\) image size and 3-qubit gray scale. Finally, the expanded watermark image is embedded into the carrier image by the controlled-NOT gates. The extraction of watermark is the reverse process of embedding it into carrier image, which is achieved by applying operations in the reverse order. Simulation-based experimental results show that the proposed scheme is superior to other similar algorithms in terms of three items, visual quality, scrambling effect of watermark image, and noise resistibility.     

Symmetric quantum fully homomorphic encryption with perfect security

Suppose some data have been encrypted, can you compute with the data without decrypting them? This problem has been studied as homomorphic encryption and blind computing. We consider this problem in the context of quantum information processing, and present the definitions of quantum homomorphic encryption (QHE) and quantum fully homomorphic encryption (QFHE). Then, based on quantum one-time pad (QOTP), we construct a symmetric QFHE scheme, where the evaluate algorithm depends on the secret key. This scheme permits any unitary transformation on any $n$ -qubit state that has been encrypted. Compared with classical homomorphic encryption, the QFHE scheme has perfect security. Finally, we also construct a QOTP-based symmetric QHE scheme, where the evaluate algorithm is independent of the secret key.     

多用户网络量子密码术的改进

针对经典的利用EPR粒子纠缠态互换的量子密钥传输协议存在的问题,它提出了一个在多用户传输网络中,基于3个粒子的最大纠缠态GHZ安全的量子密钥传输协议.改进的量子密钥传输协议在通信节点与控制中心之间通过多个GHZ对完成该密码的安全分配系统.与经典的利用EPR粒子纠缠态互换的量子密钥传输协议相比,在传输网络中,窃听者Eve如果参与了3方的通信,要获得有用信息,必然要不断的引入错误,于是该网络的节点和控制中心将会发现Eve,保证了改进的多用户网络安全性.     

Efficient arbitrated quantum signature and its proof of security

In this paper, an efficient arbitrated quantum signature scheme is proposed by combining quantum cryptographic techniques and some ideas in classical cryptography. In the presented scheme, the signatory and the receiver can share a long-term secret key with the arbitrator by utilizing the key together with a random number. While in previous quantum signature schemes, the key shared between the signatory and the arbitrator or between the receiver and the arbitrator could be used only once, and thus each time when a signatory needs to sign, the signatory and the receiver have to obtain a new key shared with the arbitrator through a quantum key distribution protocol. Detailed theoretical analysis shows that the proposed scheme is efficient and provably secure.     

An efficient scheme for five-party quantum state sharing of an arbitrary m-qubit state using multiqubit cluster states

We present an efficient scheme for five-party quantum state sharing (QSTS) of an arbitrary m-qubit state with multiqubit cluster states. Unlike the three-partite QSTS schemes using the same quantum channel [Phys. Rev. A 78, 062333 (2008)], our scheme for sharing of quantum information among five parties utilizing a cluster state as an entangled resource. It is found that the six-partite cluster state can be used for QSTS of an entangled state, the five-partite cluster state can be used for QSTS of an arbitrary two-qubit state and also can be used for QSTS of an arbitrary m-qubit state. It involves two-qubit Bell-basis or three-qubit GHZ-basis measurements, not multipartite joint measurements, which makes it more convenient than some previous schemes. In addition, the total efficiency really approaches the maximal value.     

基于量子密码的物联网RFID系统安全的研究

RFID系统的普及应用和计算机处理能力不断提高使得传统公钥密码体制的不足日益凸显。为了替代传统公钥密码体制,解决标签的安全问题,本文基于遍历矩阵构造多元二次多项式（Bisectional Multivariate Quadratic Equation,BMQE）的方法,建立一种新的基于量子计算机构造的公钥密码方案,并且给出物联网移动RFID安全协议模型。接着从密钥尺寸、加/解密速度等对该方案进行性能评估,表明该方案在RFID系统中应用的可行性。最后从各项攻击方法等进行分析,表明该方案的安全性。该研究成果对量子密码时代推进RFID的安全研究具有重要参考价值。     

A highly efficient scheme for joint remote preparation of multi-qubit W state with minimum quantum resource

We present a highly efficient scheme for perfect joint remote preparation of an arbitrary \( 2^{n} \)-qubit W state with minimum quantum resource. Both the senders Alice and Bob intend to jointly prepare one \( 2^{n} \)-qubit W state for the remote receiver Charlie. In the beginning, they help the remote receiver Charlie to construct one n-qubit intermediate state which is closely related to the target \( 2^{n} \)-qubit W state. Afterward, Charlie introduces auxiliary qubits and applies appropriate operations to obtain the target \( 2^{n} \)-qubit W state. Compared with previous schemes, our scheme requires minimum quantum resource and least amount of classical communication. Moreover, our scheme has a significant potential for being adapted to remote state preparation of other special states.     

基于Polar码改进的McEliece密码体制

随着量子计算机对计算能力的提高,RSA和椭圆曲线密码等经典密码方案在量子计算机时代已经不再安全,基于编码的密码方案具有抵抗量子计算的优势,在未来具有良好的应用前景。文章研究极化码的极化性质,改进密钥存储方法,提出了基于Polar码改进的McEliece密码体制。改进后的编码加密方案不再存储整个矩阵,而是存储冻结比特对应的矩阵,其密钥大小比原始密码方案减少约63.36%。采用连续消除(SC)译码算法,译码复杂度较低,并通过实验证明了提出的密码方案达到140bit的安全级别,可以抵抗目前已知存在的各种攻击。最后,文章进一步阐述了基于Polar码的密码方案未来的发展方向,拓宽了极化码在编码密码方案中的应用。     

应用量子盲签名和群签名的电子支付系统

提出了一个基于量子群签名和盲签名电子支付系统的实现方案。基于经典的签名的现有的电子支付系统不能保证无条件安全性。与经典的电子支付系统不同,提出的方案既能满足电子支付系统的需求,又能实现无条件安全。应用量子密钥分发技术,量子一次一密算法和审计机制实现了基于量子签名的电子支付系统。     

Quantum-chaotic key distribution in optical networks: from secrecy to implementation with logistic map

In a recent paper, the quantum-chaotic key distribution (QCKD) in optical networks was introduced. In the present work, we extend the QCKD theory in two ways: Firstly, we propose to use the dependent Bernoulli trials to model the key generation in QCKD. Using this model, we show that the key generated by QCKD is far from presenting the observed correlations in chaos-based cryptography, and it is very close to the maximum secrecy offered by ideal quantum cryptography. Secondly, we show a new optical scheme for QCKD in which the optical chaotic scheme using optoelectronic oscillators is substituted by nonlinear discrete equations running in computers and the information carrier used is the phase instead of the light polarization. These changes make much easier its implementation with today technology while keeping the same security level guaranteed by chaotic and quantum rules.     

Construction of general quantum channel for quantum teleportation

We investigate teleportation and controlled teleportation of an arbitrary $N$ -qubit state by using a multipartite entanglement channel. By establishing one-to-one correspondence between an $N$ -qubit quantum state and a high-dimension quantum state, we construct a general quantum channel for quantum teleportation and controlled teleportation of an arbitrary $N$ -qubit state. Furthermore, we generalize the definition of bipartite maximally entangled state for a multi-qubit system, and show that our teleportation protocols can be utilized not only to construct a variety of genuine multipartite entangled states, but also to identify and explore the capability of multipartite entanglement for quantum teleportation and controlled teleportation.     

Simulation of quantum key distribution in a secure star topology optimization in quantum channel

The field of quantum cryptography is mostly theoretical therefore in this paper we represent its implementation by means of virtual scenarios. The central issue in cryptography is the secure transmission of the key between nodes. Thus, in this paper we establish a secure channel using Quantum Key Distribution (QKD) for the transfer of the key material between the nodes and help to identify an eavesdropper in the channel. A graphical representation of the quantum channel traffic at the ideal state and also during network disruption has been established. Due to the complex nature of quantum networks and high cost of establishment, a physical implementation of the same is not feasible. Hence a simulation has been implemented via the use of NS-3 (Network Simulator Version 3) which has QKDNetSim module built into it. Finally, our simulation indicates the presence of an intruder by virtue of various network implementations within the quantum channel.     

One-dimensional quantum walks with absorbing boundaries

In this paper we analyze the behavior of quantum random walks. In particular, we present several new results for the absorption probabilities in systems with both one and two absorbing walls for the one-dimensional case. We compute these probabilities both by employing generating functions and by use of an eigenfunction approach. The generating function method is used to determine some simple properties of the walks we consider, but appears to have limitations. The eigenfunction approach works by relating the problem of absorption to a unitary problem that has identical dynamics inside a certain domain, and can be used to compute several additional interesting properties, such as the time dependence of absorption. The eigenfunction method has the distinct advantage that it can be extended to arbitrary dimensionality. We outline the solution of the absorption probability problem of a (D−1)-dimensional wall in a D-dimensional space.     

Massively parallel quantum computer simulator

We describe portable software to simulate universal quantum computers on massive parallel computers. We illustrate the use of the simulation software by running various quantum algorithms on different computer architectures, such as a IBM BlueGene/L, a IBM Regatta p690+, a Hitachi SR11000/J1, a Cray X1E, a SGI Altix 3700 and clusters of PCs running Windows XP. We study the performance of the software by simulating quantum computers containing up to 36 qubits, using up to 4096 processors and up to 1 TB of memory. Our results demonstrate that the simulator exhibits nearly ideal scaling as a function of the number of processors and suggest that the simulation software described in this paper may also serve as benchmark for testing high-end parallel computers.     

A quantum full adder for a scalable nuclear spin quantum computer

We demonstrate a strategy for implementation a quantum full adder in a spin chain quantum computer. As an example, we simulate a quantum full adder in a chain containing 201 spins. Our simulations also demonstrate how one can minimize errors generated by non-resonant effects.     

Secret sharing based on quantum Fourier transform

Secret sharing plays a fundamental role in both secure multi-party computation and modern cryptography. We present a new quantum secret sharing scheme based on quantum Fourier transform. This scheme enjoys the property that each share of a secret is disguised with true randomness, rather than classical pseudorandomness. Moreover, under the only assumption that a top priority for all participants (secret sharers and recovers) is to obtain the right result, our scheme is able to achieve provable security against a computationally unbounded attacker.     

Cyclic quantum teleportation

We propose a scheme of cyclic quantum teleportation for three unknown qubits using six-qubit maximally entangled state as the quantum channel. Suppose there are three observers Alice, Bob and Charlie, each of them has been given a quantum system such as a photon or spin-\(\frac{1}{2}\) particle, prepared in state unknown to them. We show how to implement the cyclic quantum teleportation where Alice can transfer her single-qubit state of qubit a to Bob, Bob can transfer his single-qubit state of qubit b to Charlie and Charlie can also transfer his single-qubit state of qubit c to Alice. We can also implement the cyclic quantum teleportation with \(N\geqslant 3\) observers by constructing a 2N-qubit maximally entangled state as the quantum channel. By changing the quantum channel, we can change the direction of teleportation. Therefore, our scheme can realize teleportation in quantum information networks with N observers in different directions, and the security of our scheme is also investigated at the end of the paper.