DOCSLIB.ORG

Quantum Computing with Cluster States

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Computing with Cluster States Quantum Computing with Cluster States Gelo Noel M. Tabia ∗ Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5 Institute for Quantum Computing and University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1 15 April 2011 Summary In cluster state quantum computing, we start with a given fixed entangled state of many qubits called a cluster state and perform a quantum computation by applying a time-ordered sequence of single-qubit measurements to designated qubits in designated bases. The choice of basis for later measurements generally depend on the outcome of earlier measurements and the final result of the computation is determined from the classical information involving all measurement outcomes. This is in contrast to the more conventional network gate model of quantum computing, in which computational steps are unitary transformations. We describe how the underlying computational model of cluster state is different from the network model, in particular emphasizing how cluster state computers have no true notion of input and output registers. Nonetheless, we show that the two quantum computing models are equivalent by demonstrating how cluster states can be used to simulate quantum circuits efficiently. At the end, we include a short account on how to produce optical cluster states. ∗Email: [email protected] (UW ID: 20294659) Table of Contents 1 Introduction 1 2 What are cluster states? 1 3 Universality of cluster-state quantum computing 3 3.1 Simulating quantum circuits . 3 3.2 Random measurement outcomes . 8 3.3 Sequence of simulated quantum gates . 8 3.4 Efficiency of simulating quantum gates . 9 4 Computational model for the cluster-state quantum computer 11 4.1 Non-network character of cluster state computing . 11 4.2 A cluster-state computational model . 12 5 Optical implementation of cluster states 13 5.1 Cluster states and the Knill-Laflamme-Milburn (KLM) proposal . 13 5.2 Practical optical cluster state . 15 6 Concluding remarks 16 1 Introduction A quantum computer promises efficient information processing of some computational tasks that are believed to be intractable with classical computers. Standard quantum computing models are based on the paradigm of quantum logic networks, where a sequence of unitary quantum gates form the compu- tational circuit. The one-way quantum computer proposed by Raussendorf and Briegel [1] provides an alternative model for scalable quantum computing in which a specific form of a highly entangled multi- particle state, called a cluster state, plays the pivotal role of being a universal resource for quantum computation. To perform a quantum algorithm with a cluster state, individual qubits are measured in a temporal sequence of adaptive single-qubit measurements with classical feed-forward processing of outcomes. The classical readout for such a quantum computation essentially involves all measurement outcomes of the cluster qubits. In this project, we provide some basic details about cluster state quantum computing. Firstly, we define what cluster states are and how they can be realized from a Hamiltonian. Secondly, we show that cluster states are resources for universal quantum computing by showing how they can be imprinted with a simulated quantum circuit using single qubit rotations and CNOT gates. After that, we show how the random outcomes of individual qubit measurements do not affect the ability of a cluster state to perform an algorithm efficiently and how any necessary correction of Pauli errors due to these random outcomes can be delayed until the end of the computation. We follow that with a discussion on the non-network aspects of cluster state computing and provide a different computational model for cluster states that involves the temporal ordering of single-qubit measurements and an information flow vector that tracks the evolution of logical qubits. Lastly, we describe how cluster states can be created experimentally using polarization of photons and standard linear optical devices. 2 What are cluster states? A cluster state is a set of qubits arranged in a d-dimensional lattice at sites a 2 Zd that couple via some short-range interaction with Hamiltonian X 1 + Za 1 − Za0 H = g(t) f(a − a0) (1) int ~ 2 2 a;a0 where Zaj0ia = j0ia;Zaj1ia = −|1ia (2) defines the Pauli-Z operator. With respect to the entanglement properties of cluster states relevant to quantum computing, this Hamiltonian is equivalent to the quantum Ising model at low temperatures: X 0 HIsing = − ~g(t)f(a − a )ZaZa0 : (3) a;a0 They are equivalent in the sense that the states generated by Hint and HIsing from a given initial state are identical up to local unitary transformations applied to individual qubits. In principle, only next-neighbor interactions are needed to generate a cluster state from eq. (1). In such a case, Hint realizes simultaneous conditional phase gates between qubits at adjacent sites a and 0 a . For example, if we have a linear chain of N qubits then we can produce a cluster state using Hint 0 with f(a − a ) = δa;a0+1. 1 Suppose we have a linear chain of qubits all initially in the state j0ia + j1ia j+ia = p (4) 2 then for ' = R g(t) dt, the unitary evolution ! X 1 + Za 1 − Za+1 U(') = exp −i' (5) 2 2 a yields a disentangled chain for ' = 2(k − 1)π and a `maximally entangled' one for ' = (2k − 1)π, where k 2 Z+. For ' = π, the cluster state for a linear chain CN of N qubits can be written in compact notation as N 1 O jφiC = p (j0iaZa+1 + j1ia) (6) N N 2 a=1 where ZN+1 = 1l by definition, since it is impossible to entangle with an empty site. Examples of the smallest linear cluster states are as follows: 1 jφiC = p (j0ij+i + j1i|−i) ; (7) 2 2 1 jφiC = p (j+ij0ij+i + |−i|1i|−i) ; (8) 3 2 1 jφi = (j+ij0ij+ij0i + j+ij0i|−i|1i + |−i|1i|−i|0i + |−i|1ij+ij1i) ; (9) C4 2 which up to local unitary transformations on single qubits are equivalent to l:u: 1 jφiC = p (j0ij0i + j1ij1i) ; (10) 2 2 l:u: 1 jφiC = p (j0ij0ij0i + j1ij1ij1i) ; (11) 3 2 1 jφi l:u:= (j0ij0ij0ij0i + j0ij0ij1ij1i + j1ij1ij0ij0i − j1ij1ij1ij1i) : (12) C4 2 Observe that jφiC2 is effectively a Bell state, jφiC3 is effectively a 3-qubit GHZ state but jφiC4 is not the same as the jGHZ i = p1 (j0000i + j1111i). 4 2 So far we have only considered a linear chain of qubits. For higher-dimensional lattices (d ≥ 2), every lattice site is now specified by a vector of d integers a. Each site a has 2d neighbors. If neighboring sites are occupied by qubits, those qubits interact with the qubit in a. Let the set A specify all the lattice sites occupied by qubits. Two sites a and a0 are connected (topologically) if there exists a sequence of neighboring occupied sites, that is, N 0 fajgj=1 ; such that a1 = a; aN = a : (13) In this case, a cluster C ⊂ A is a set of connected lattice sites such that the corresponding state of the cluster qubits is given by ! O O (c+x) jφiC = j0ic Z + j1ic (14) c2C x2Γ where Γ refers to a set of steps fxg needed to move from site a to any adjacent site a0 = a + x (that 2 Figure 1: Examples of cluster states: linear, square (or rectangular), and cubic. The yellow dots represent qubits while the lines represent entanglement between nearest neighbors. is, the set ηa = fa + xj x 2 Γg (15) is the set of neighboring sites of a) and Zc+x = 1l when c+x 62 C. Some examples of clusters of different lattice dimensions is shown in fig. (1). Two important entanglement properties [2] characterizing cluster states are: (i) Maximal connectedness: Any two qubits of CN can be projected into a pure Bell state by local measurements on all the other qubits. (ii) Persistency: Pent(j i) denotes the minimum number of local measurement of qubits necessary to disentangle the state j i. It turns out that this is equivalent to the Schmidt rank, the fewest number of terms in the generalized Schmidt decomposition [3]. For cluster states, N P (jφi ) = : (16) ent CN 2 We note that a convenient, compact way to define a cluster state is via a set of eigenvalue equations KajφiC = κajφiC (17) where O Ka = Xa Za+x; κa = ±1: (18) x2Γ It turns out that in the stabilizer formalism [4, 5], the operators Ka are stabilizer generators and cluster states are their corresponding eigenstates. 3 Universality of cluster-state quantum computing 3.1 Simulating quantum circuits One can think of using the cluster state as a substrate on which quantum circuits can be imprinted on [6]. To describe this computational circuit, it is important to distinguish between the physical qubits that make up the cluster state and the logical qubits on which quantum information processing is to take place. Due to the entanglement properties of a cluster state, no individual physical qubit carries information that corresponds to an input state|rather information is contained in correlations between 3 qubits. However, for simplicity we first consider the case where an encoded input state is read into the cluster. Suppose we start with an array of qubits arranged in a square lattice and whose state is given by jφiC . Then to perform a quantum computation, we do the following: (a) Remove the subset of qubits not needed for the calculation by measuring these qubits in the Z- 0 0 basis. If the calculation uses qubits arranged in some network C , then jφiC 7! jSiCnC0 ⊗ jφ iC0 , 0 jφ iC is local unitarily equivalent to jφiC0 , which is just a smaller cluster state.
Load more

Recommended publications
  • Quantum Entanglement Concentration Based on Nonlinear Optics for Quantum Communications
    Entropy 2013, 15, 1776-1820; doi:10.3390/e15051776 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Review Quantum Entanglement Concentration Based on Nonlinear Optics for Quantum Communications Yu-Bo Sheng 1;2;* and Lan Zhou 2;3 1 Institute of Signal Processing Transmission, Nanjing University of Posts and Telecommunications, Nanjing 210003, China 2 Key Lab of Broadband Wireless Communication and Sensor Network Technology, Nanjing University of Posts and Telecommunications, Ministry of Education, Nanjing 210003, China 3 College of Mathematics & Physics, Nanjing University of Posts and Telecommunications, Nanjing 210003, China; E-Mail: [email protected] (L.Z.) * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel./Fax: +86-025-83492417. Received: 14 March 2013; in revised form: 3 May 2013 / Accepted: 8 May 2013 / Published: 16 May 2013 Abstract: Entanglement concentration is of most importance in long distance quantum communication and quantum computation. It is to distill maximally entangled states from pure partially entangled states based on the local operation and classical communication. In this review, we will mainly describe two kinds of entanglement concentration protocols. One is to concentrate the partially entangled Bell-state, and the other is to concentrate the partially entangled W state. Some protocols are feasible in current experimental conditions and suitable for the optical, electric and quantum-dot and optical microcavity systems. Keywords: quantum entanglement; entanglement concentration; quantum communication 1. Introduction Quantum communication and quantum computation have attracted much attention over the last 20 years, due to the absolute safety in the information transmission for quantum communication and the super fast factoring for quantum computation [1,2].
    [Show full text]
  • Quantum Macroscopicity Versus Distillation of Macroscopic
    Quantum macroscopicity versus distillation of macroscopic superpositions Benjamin Yadin1 and Vlatko Vedral1, 2 1Atomic and Laser Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK 2Centre for Quantum Technologies, National University of Singapore, Singapore 117543 (Dated: October 7, 2018) We suggest a way to quantify a type of macroscopic entanglement via distillation of Greenberger- Horne-Zeilinger states by local operations and classical communication. We analyze how this relates to an existing measure of quantum macroscopicity based on the quantum Fisher information in sev- eral examples. Both cluster states and Kitaev surface code states are found to not be macroscopically quantum but can be distilled into macroscopic superpositions. We look at these distillation pro- tocols in more detail and ask whether they are robust to perturbations. One key result is that one-dimensional cluster states are not distilled robustly but higher-dimensional cluster states are. PACS numbers: 03.65.Ta, 03.67.Mn, 03.65.Ud I. INTRODUCTION noted here by N ∗. We will not impose any cut-off above which a value of N ∗ counts as macroscopic, but instead Despite the overwhelming successes of quantum me- consider families of states parameterized by some obvious chanics, one of its greatest remaining problems is to ex- size quantity N (e.g. the number of qubits). Then the plain why it appears to break down at the macroscopic relevant property of the family is the scaling of N ∗ with scale. In particular, macroscopic objects are never seen N. The case N ∗ = O(N) is ‘maximally macroscopic’ [9]. in quantum superpositions.
    [Show full text]
  • Bidirectional Quantum Controlled Teleportation by Using Five-Qubit Entangled State As a Quantum Channel
    Bidirectional Quantum Controlled Teleportation by Using Five-qubit Entangled State as a Quantum Channel Moein Sarvaghad-Moghaddam1,*, Ahmed Farouk2, Hussein Abulkasim3 to each other simultaneously. Unfortunately, the proposed Abstract— In this paper, a novel protocol is proposed for protocol required additional quantum and classical resources so implementing BQCT by using five-qubit entangled states as a that the users required two-qubit measurements and applying quantum channel which in the same time, the communicated users global unitary operations. There are many approaches and can teleport each one-qubit state to each other under permission prototypes for the exploitation of quantum principles to secure of controller. The proposed protocol depends on the Controlled- the communication between two parties and multi-parties [18, NOT operation, proper single-qubit unitary operations and single- 19, 20, 21, 22]. While these approaches used different qubit measurement in the Z-basis and X-basis. The results showed that the protocol is more efficient from the perspective such as techniques for achieving a private communication among lower shared qubits and, single qubit measurements compared to authorized users, but most of them depend on the generation of the previous work. Furthermore, the probability of obtaining secret random keys [23, 2]. In this paper, a novel protocol is Charlie’s qubit by eavesdropper is reduced, and supervisor can proposed for implementing BQCT by using five-qubit control one of the users or every two users. Also, we present a new entanglement state as a quantum channel. In this protocol, users method for transmitting n and m-qubits entangled states between may transmit an unknown one-qubit quantum state to each other Alice and Bob using proposed protocol.
    [Show full text]
  • Arxiv:Quant-Ph/0512071 V1 9 Dec 2005 Contents I.Ipoeet Ntekmprotocol KLM the on Improvements III
    Linear optical quantum computing Pieter Kok,1,2, ∗ W.J. Munro,2 Kae Nemoto,3 T.C. Ralph,4 Jonathan P. Dowling,5,6 and G.J. Milburn4 1Department of Materials, Oxford University, Oxford OX1 3PH, UK 2Hewlett-Packard Laboratories, Filton Road Stoke Gifford, Bristol BS34 8QZ, UK 3National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan 4Centre for Quantum Computer Technology, University of Queensland, St. Lucia, Queensland 4072, Australia 5Hearne Institute for Theoretical Physics, Dept. of Physics and Astronomy, LSU, Baton Rouge, Louisiana 6Institute for Quantum Studies, Department of Physics, Texas A&M University (Dated: December 9, 2005) Linear optics with photo-detection is a prominent candidate for practical quantum computing. The protocol by Knill, Laflamme and Milburn [Nature 409, 46 (2001)] explicitly demonstrates that efficient scalable quantum computing with single photons, linear optical elements, and projec- tive measurements is possible. Subsequently, several improvements on this protocol have started to bridge the gap between theoretical scalability and practical implementation. We review the original proposal and its improvements, and we give a few examples of experimental two-qubit gates. We discuss the use of realistic components, the errors they induce in the computation, and how they can be corrected. PACS numbers: 03.67.Hk, 03.65.Ta, 03.65.Ud Contents References 36 I. Quantum computing with light 1 A. Linear quantum optics 2 B. N-port interferometers and optical circuits 3 C. Qubits in linear optics 4 I. QUANTUM COMPUTING WITH LIGHT D. Early optical quantum computers and nonlinearities 5 Quantum computing has attracted much attention over II.
    [Show full text]
  • G53NSC and G54NSC Non Standard Computation Research Presentations
    G53NSC and G54NSC Non Standard Computation Research Presentations March the 23rd and 30th, 2010 Tuesday the 23rd of March, 2010 11:00 - James Barratt • Quantum error correction 11:30 - Adam Christopher Dunkley and Domanic Nathan Curtis Smith- • Jones One-Way quantum computation and the Measurement calculus 12:00 - Jack Ewing and Dean Bowler • Physical realisations of quantum computers Tuesday the 30th of March, 2010 11:00 - Jiri Kremser and Ondrej Bozek Quantum cellular automaton • 11:30 - Andrew Paul Sharkey and Richard Stokes Entropy and Infor- • mation 12:00 - Daniel Nicholas Kiss Quantum cryptography • 1 QUANTUM ERROR CORRECTION JAMES BARRATT Abstract. Quantum error correction is currently considered to be an extremely impor- tant area of quantum computing as any physically realisable quantum computer will need to contend with the issues of decoherence and other quantum noise. A number of tech- niques have been developed that provide some protection against these problems, which will be discussed. 1. Introduction It has been realised that the quantum mechanical behaviour of matter at the atomic and subatomic scale may be used to speed up certain computations. This is mainly due to the fact that according to the laws of quantum mechanics particles can exist in a superposition of classical states. A single bit of information can be modelled in a number of ways by particles at this scale. This leads to the notion of a qubit (quantum bit), which is the quantum analogue of a classical bit, that can exist in the states 0, 1 or a superposition of the two. A number of quantum algorithms have been invented that provide considerable improvement on their best known classical counterparts, providing the impetus to build a quantum computer.
    [Show full text]
  • Quadratic Entanglement Criteria for Qutrits K
    Vol. 137 (2020) ACTA PHYSICA POLONICA A No. 3 Quadratic Entanglement Criteria for Qutrits K. Rosołeka, M. Wieśniaka;b;∗and L. Knipsc;d aInstitute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics, and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland bInternational Centre for Theories of Quantum Technologies, University of Gdańsk, Wita Swosza 57, 80-308 Gdańsk, Poland cMax-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany dDepartment für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany (Received September 10, 2019; revised version December 2, 2019; in final form December 5, 2019) The problem of detecting non-classical correlations of states of many qudits is incomparably more involved than in the case of qubits. One reason is that for qubits we have a convenient description of the system by the means of the well-studied correlation tensor allowing to encode the complete information about the state in mean values of dichotomic measurements. The other reason is the more complicated structure of the state space, where, for example, different Schmidt ranks or bound entanglement comes into play. We demonstrate that for three-dimensional quantum subsystems we are able to formulate nonlinear entanglement criteria of the state with existing formalisms. We also point out where the idea for constructing these criteria fails for higher-dimensional systems, which poses well-defined open questions. DOI: 10.12693/APhysPolA.137.374 PACS/topics: quantum information, quantum correlations, entanglements, qutrits 1. Introduction practical reasons. The task is very simple for pure states, which practically never occur in a real life. However, for For complex quantum systems, we distinguish between mixed states, it is still an open question.
    [Show full text]
  • Success Rates for Linear Optical Generation of Cluster States in Coincidence Basis
    Success rates for linear optical generation of cluster states in coincidence basis D B Uskov1,2*, P M Alsing3, M L Fanto3, L Kaplan2 and A M Smith3 1Brescia University, Owensboro, KY 42301 2Tulane University, New Orleans, LA 70118 3Air Force Research Lab, Information Directorate, 525 Brooks Road, Rome, NY 13440 *E-mail: [email protected] Abstract. We report on theoretical research in photonic cluster-state computing. Finding optimal schemes of generating non-classical photonic states is of critical importance for this field as physically implementable photon- photon entangling operations are currently limited to measurement-assisted stochastic transformations. A critical parameter for assessing the efficiency of such transformations is the success probability of a desired measurement outcome. At present there are several experimental groups which are capable of generating multi-photon cluster states carrying more than eight qubits. Separate photonic qubits or small clusters can be fused into a single cluster state by a probabilistic optical CZ gate conditioned on simultaneous detection of all photons with 1/9 success probability for each gate. This design mechanically follows the original theoretical scheme of cluster state generation proposed more than a decade ago by Raussendorf, Browne and Briegel. The optimality of the destructive CZ gate in application to linear optical cluster state generation has not been analyzed previously. Our results reveal that this method is far from the optimal one. Employing numerical optimization we have identified that maximal success probability of fusing n unentangled dual-rail optical qubits into a linear cluster state is equal to 12n1 ;m- tuple of photonic Bell pair states, commonly generated via spontaneous parametric down-conversion, can be fused into a single cluster with the maximal success probability of 14m1 .
    [Show full text]
  • Quantum Computing Cluster State Model
    Quantum Computing Cluster State Model Phillip Blakey, Alex Karapetyan, Thomas Ma December 2019 1 Introduction The traditional circuit-based model of quantum computation is strictly based on unitary evolution. Especially in view of the Principle of Deferred Measurement in the context of implementation of algorithms with the circuit model, the role of measurement is secondary to unitary quantum gates - a quantum circuit, sans the final measurements, is the application of a carefully crafted unitary transformation to a carefully chosen initial quantum state. Measurement is a non-unitary operation and thereby ruins quantum states. The cluster state model of quantum computing, also known as a \one-way quantum computer", is based on the idea of performing a sequence of single-qubit measurements on some large initial state, resulting in an output in the form of a several-qubit state. In particular, measurement is essential to the entirety of any computation, in contrast to the circuit model. This model was first introduced in [1] (see also [2]), and it has been studied from both the theoretical and experimental implementation points of view. The initial state of a computation in this model, called a cluster state, is a specially prepared entangled state associated to a particularly chosen graph. To perform a quantum computation on a cluster state, the rules of the game are: (a) the allowed operations are single-qubit measurements in any basis on any of the nodes of the graph, (b) the basis in which to measure at every stage of the sequence is conditioned on the outcome of the previous measurement, and (c) efficient classical processing is allowed on the measurement outcomes.
    [Show full text]
  • Logical Measurement-Based Quantum Computation in Circuit-QED
    www.nature.com/scientificreports OPEN Logical measurement-based quantum computation in circuit- QED Jaewoo Joo1,2*, Chang-Woo Lee 1,3, Shingo Kono4 & Jaewan Kim1 We propose a new scheme of measurement-based quantum computation (MBQC) using an error- correcting code against photon-loss in circuit quantum electrodynamics. We describe a specifc protocol of logical single-qubit gates given by sequential cavity measurements for logical MBQC and a generalised Schrödinger cat state is used for a continuous-variable (CV) logical qubit captured in a microwave cavity. To apply an error-correcting scheme on the logical qubit, we utilise a d-dimensional quantum system called a qudit. It is assumed that a three CV-qudit entangled state is initially prepared in three jointed cavities and the microwave qudit states are individually controlled, operated, and measured through a readout resonator coupled with an ancillary superconducting qubit. We then examine a practical approach of how to create the CV-qudit cluster state via a cross-Kerr interaction induced by intermediary superconducting qubits between neighbouring cavities under the Jaynes- Cummings Hamiltonian. This approach could be scalable for building 2D logical cluster states and therefore will pave a new pathway of logical MBQC in superconducting circuits toward fault-tolerant quantum computing. Measurement-based quantum computation (MBQC) ofers a new platform of quantum information (QI) pro- cessing. Quantum algorithms are performed by sequential single-qubit measurements in multipartite entangled states initially (e.g., cluster states1) instead of massive controls of individual qubits during the whole information processing2,3. Tis advantage is, however, only benefcial for QI processing if the specifc multipartite entangled state can be initially well-prepared and the capability of fast and precise single-qubit measurements are viable.
    [Show full text]
  • Large-Scale Cluster-State Entanglement in the Quantum Optical Frequency Comb
    Large-Scale Cluster-State Entanglement in the Quantum Optical Frequency Comb Moran Chen Nanchong, Sichuan, China B.S. Physics, Beijing Normal University, 2009 A Dissertation presented to the Graduate Faculty of the University of Virginia in Candidacy for the degree of Doctor of Philosophy Department of Physics University of Virginia May 2015 c Copyright All Rights Reserved Moran Chen Ph.D., May 2015 i Acknowledgments First of all, I would like to thank my adviser, Prof. Olivier Pfister. Without his help and guidance, all of the research and work described in this thesis would have been impossible. I am wholeheartedly grateful to have him as my adviser. He gave me enough freedom and trust to allow me to conduct research and independent thinking on my own, and his incredible patience and knowledge in both the theory and in the experiments also gave me confidence and reassurance when I came across obstacles and difficulties. He is also always encouraging and kind, and is the best adviser I could have hoped for. He has led me into the amazing field of quantum optics, and I am truly blessed to have him as my adviser. I want to give special thanks to our theoretical collaborator Nick Menicucci. Nick is an outstanding theorist and a very smart, quick and straightforward person, and it has always been a pleasure talking with him about research topics or other general issues. We had many good and interesting discussions and his genius ideas and simplified theoretical proposals helped make our experiment easier and more feasible. I am truly thankful to have him as our theoretical collaborator and friend.
    [Show full text]
  • Arxiv:Quant-Ph/0512071V2 14 Mar 2006 I Ulo:Byn Ieroptics Linear Beyond Outlook: VI
    Linear optical quantum computing 1,2, 2 3 4 5, 6 4 Pieter Kok, ∗ W.J. Munro, Kae Nemoto, T.C. Ralph, Jonathan P. Dowling, and G.J. Milburn 1Department of Materials, Oxford University, Oxford OX1 3PH, UK 2Hewlett-Packard Laboratories, Filton Road Stoke Gifford, Bristol BS34 8QZ, UK 3National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan 4Centre for Quantum Computer Technology, University of Queensland, St. Lucia, Queensland 4072, Australia 5Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, LSU, Baton Rouge LA, 70803, USA 6Institute for Quantum Studies, Department of Physics, Texas A&M University, 77843-4242, USA (Dated: February 1, 2008) Linear optics with photon counting is a prominent candidate for practical quantum computing. The protocol by Knill, Laflamme, and Milburn [Nature 409, 46 (2001)] explicitly demonstrates that efficient scalable quantum computing with single photons, linear optical elements, and projective measure- ments is possible. Subsequently, several improvements on this protocol have started to bridge the gap between theoretical scalability and practical implementation. We review the original theory and its improvements, and we give a few examples of experimental two-qubit gates. We discuss the use of realistic components, the errors they induce in the computation, and how these errors can be corrected. PACS numbers: 03.67.Hk, 03.65.Ta, 03.65.Ud Contents I. QUANTUM COMPUTING WITH LIGHT I. Quantum computing with light 1 Quantum computing has attracted much attention over A. Linear quantum optics 2 B. N port interferometers and optical circuits 4 the last ten to fifteen years, partly because of its promise C.
    [Show full text]
  • Arxiv:2010.14422V1 [Quant-Ph] 27 Oct 2020
    Deterministic multi-mode gates on a scalable photonic quantum computing platform Mikkel V. Larsen,∗ Xueshi Guo, Casper R. Breum, Jonas S. Neergaard-Nielsen, and Ulrik L. Andersen† Center for Macroscopic Quantum States (bigQ), Department of Physics, Technical University of Denmark, Fysikvej, 2800 Kgs. Lyngby, Denmark (Dated: October 27, 2020) Quantum computing can be realized with nu- dred single-mode gates6. In CV quantum comput- merous different hardware platforms and com- ing2,12,20,21, information is encoded and processed in putational protocols. A highly promising ap- bosonic harmonic oscillators—e.g. the optical field—that proach to foster scalability is to apply a photonic are described by states in infinite-dimensional Hilbert platform combined with a measurement-induced spaces22,23. Although the idea of using CVs for quantum quantum information processing protocol where computing dates back more than 20 years20, it is only gate operations are realized through optical mea- within the last few years that feasible models for fault- surements on a multipartite entangled quantum tolerant large-scale CV MBQC were conceived13,14,24–26. state—a so-called cluster state1,2. Heretofore, Our demonstration represents a critical step towards a few quantum gates on non-universal or non- these CV computing models. It constitutes the first real- scalable cluster states have been realized3–10, but ization of a fully deterministic and programmable multi- a full set of gates for universal scalable quan- mode computation platform for MBQC. tum computing has not been realized. We pro- pose and demonstrate the deterministic imple- Architecture and hardware mentation of a multi-mode set of measurement- In CV MBQC, quantum information processing is real- induced quantum gates in a large two-dimensional ized by teleporting the constituent gates through a com- (2D) optical cluster state using phase-controlled putationally universal cluster state, comprising quadra- 2,11 continuous variable quadrature measurements .
    [Show full text]