Continuous-variable Graph States Research Articles

Graph states of atomic ensembles engineered by photon-mediated entanglement

Graph states are a broad family of entangled quantum states, each defined by a graph composed of edges representing the correlations between subsystems. Such states constitute versatile resources for quantum computation and quantum-enhanced measurement. Their generation and engineering require a high level of control over entanglement. Here we report on the generation of continuous-variable graph states of atomic spin ensembles, which form the nodes of the graph. We program the entanglement structure encoded in the graph edges by combining global photon-mediated interactions in an optical cavity with local spin rotations. By tuning the entanglement between two subsystems, we either localize correlations within each subsystem or enable Einstein–Podolsky–Rosen steering—a strong form of entanglement that enables the extraction of precise information from one subsystem through measurements on the other. We further engineer a four-mode square graph state, highlighting the flexibility of our approach. Our method is scalable to larger and more complex graphs, laying groundwork for measurement-based quantum computation and advanced protocols in quantum metrology.

Flow conditions for continuous variable measurement-based quantum computing

In measurement-based quantum computing (MBQC), computation is carried out by a sequence of measurements and corrections on an entangled state. Flow, and related concepts, are powerful techniques for characterising the dependence of the corrections on previous measurement results. We introduce flow-based methods for quantum computation with continuous-variable graph states, which we call CV-flow. These are inspired by, but not equivalent to, the notions of causal flow and g-flow for qubit MBQC. We also show that an MBQC with CV-flow approximates a unitary arbitrarily well in the infinite-squeezing limit, addressing issues of convergence which are unavoidable in the infinite-dimensional setting. In developing our proofs, we provide a method for converting a CV-MBQC computation into a circuit form, analogous to the circuit extraction method of Miyazaki et al, and an efficient algorithm for finding CV-flow when it exists based on the qubit version by Mhalla and Perdrix. Our results and techniques naturally extend to the cases of MBQC for quantum computation with qudits of prime local dimension.

Sharing Classical Secrets with Continuous-Variable Entanglement: Composable Security and Network Coding Advantage

Secret sharing is a multi-party cryptographic primitive that can be applied to a network of partially distrustful parties for encrypting data that is both sensitive (it must remain secure) and important (it must not be lost or destroyed). When sharing classical secrets (as opposed to quantum states), one can distinguish between protocols that leverage bi-partite quantum key distribution (QKD) and those that exploit multi-partite entanglement. The latter class are known to be vulnerable to so-called participant attacks and, while progress has been made recently, there is currently no analysis that quantifies their performance in the composable, finite-size regime which has become the gold standard for QKD security. Given this - and the fact that distributing multi-partite entanglement is typically challenging - one might well ask: Is there any virtue in pursuing multi-partite entanglement based schemes? Here, we answer this question in the affirmative for a class of secret sharing protocols based on continuous variable graph states. We establish security in a composable framework and identify a network topology, specifically a bottleneck network of lossy channels, and parameter regimes within the reach of present day experiments for which a multi-partite scheme outperforms the corresponding QKD based method in the asymptotic and finite-size setting. Finally, we establish experimental parameters where the multi-partite schemes outperform any possible QKD based protocol. This one of the first concrete compelling examples of multi-partite entangled resources achieving a genuine advantage over point-to-point protocols for quantum communication and represents a rigorous, operational benchmark to assess the usefulness of such resources.

Continuous-variable graph states for quantum metrology

Graph states are a unique resource for quantum information processing, such as measurement-based quantum computation. Here, we theoretically investigate the use of continuous-variable graph states for single-parameter quantum metrology, including both phase and displacement sensing. We identified the optimal graph states for the two sensing modalities and showed that Heisenberg scaling of the accuracy for both phase and displacement sensing can be achieved with local homodyne measurements.

Resource-efficient verification of quantum computing using Serfling\u2019s bound

Verifying quantum states is central to certifying the correct operation of various quantum information processing tasks. In particular, in measurement-based quantum computing, checking whether correct graph states are generated is essential for reliable quantum computing. Several verification protocols for graph states have been proposed, but none of these are particularly resource efficient: multiple copies are required to extract a single state that is guaranteed to be close to the ideal one. The best protocol currently known requires O(n15) copies of the state, where n is the size of the graph state. In this paper, we construct a significantly more resource-efficient verification protocol for graph states that only requires O(n5 log n) copies. The key idea is to employ Serfling’s bound, which is a probability inequality in classical statistics. Utilizing Serfling’s bound also enables us to generalize our protocol for qudit and continuous-variable graph states. Constructing a resource-efficient verification protocol for them is non-trivial. For example, the previous verification protocols for qubit graph states that use the quantum de Finetti theorem cannot be generalized to qudit and continuous-variable graph states without tremendously increasing the resource overhead. This is because the overhead caused by the quantum de Finetti theorem depends on the local dimension. On the other hand, in our protocol, the resource overhead is independent of the local dimension, and therefore generalizing to qudit or continuous-variable graph states does not increase the overhead. The flexibility of Serfling’s bound also makes our protocol robust: our protocol accepts slightly noisy but still useful graph states.

Tailoring Non-Gaussian Continuous-Variable Graph States.

Graph states are the backbone of measurement-based continuous-variable quantum computation. However, experimental realizations of these states induce Gaussian measurement statistics for the field quadratures, which poses a barrier to obtain a genuine quantum advantage. In this Letter, we propose mode-selective photon addition and subtraction as viable and experimentally feasible pathways to introduce non-Gaussian features in such continuous-variable graph states. In particular, we investigate how the non-Gaussian properties spread among the vertices of the graph, which allows us to show the degree of control that is achievable in this approach.

Quantum anonymous voting with unweighted continuous-variable graph states

Motivated by the revealing topological structures of continuous-variable graph state (CVGS), we investigate the design of quantum voting scheme, which has serious advantages over the conventional ones in terms of efficiency and graphicness. Three phases are included, i.e., the preparing phase, the voting phase and the counting phase, together with three parties, i.e., the voters, the tallyman and the ballot agency. Two major voting operations are performed on the yielded CVGS in the voting process, namely the local rotation transformation and the displacement operation. The voting information is carried by the CVGS established before hand, whose persistent entanglement is deployed to keep the privacy of votes and the anonymity of legal voters. For practical applications, two CVGS-based quantum ballots, i.e., comparative ballot and anonymous survey, are specially designed, followed by the extended ballot schemes for the binary-valued and multi-valued ballots under some constraints for the voting design. Security is ensured by entanglement of the CVGS, the voting operations and the laws of quantum mechanics. The proposed schemes can be implemented using the standard off-the-shelf components when compared to discrete-variable quantum voting schemes attributing to the characteristics of the CV-based quantum cryptography.

Generation of cluster states in optomechanical quantum systems

We consider an optomechanical quantum system composed of a single cavity mode interacting with N mechanical resonators. We propose a scheme for generating continuous-variable graph states of arbitrary size and shape, including the so-called cluster states for universal quantum computation. The main feature of this scheme is that, differently from previous approaches, the graph states are hosted in the mechanical degrees of freedom rather than in the radiative ones. Specifically, via a 2N-tone laser drive, we engineer a linear Hamiltonian which is instrumental to dissipatively drive the system to the desired target state. The robustness of this scheme is assessed against finite interaction times and mechanical noise, confirming it as a valuable approach towards quantum state engineering for continuous-variable computation in a solid-state platform.

Quantum network dense coding via continuous-variable graph states

We present a dense coding network based on continuous-variable graph state along with its corresponding protocol. A scheme to distill bipartite entanglement between two arbitrary modes in a graph state is provided in order to realize the dense coding network. We also analyze the capacity of network dense coding and provide a method to calculate its maximum mutual information. As an application, we analyze the performance of dense coding in a square lattice graph state network. The result showed that the mutual information of the dense coding is not largely affected by the complexity of the network. We conclude that the performance of dense coding network is very optimistic.

Quantum secret sharing with continuous variable graph state

In this paper, we study several physically feasible quantum secret sharing (QSS) schemes using continuous variable graph state (CVGS). Their implementation protocols are given, and the estimation error formulae are derived. Then, we present a variety of results on the theory of QSS with CVGS. Any \((k,n)\) threshold protocol of the three specific schemes satisfying \(\frac{n}{2}<k\le n\), where \(n\) denotes the total number of players and \(k\) denotes the minimum number of players who can collaboratively access the secret, can be implemented by certain weighted CVGS. The quantum secret is absolutely confidential to any player group with number less than threshold. Besides, the effect of finite squeezing to these results is properly considered. In the end, the duality between two specific schemes is investigated.

Quantum-cryptography network via continuous-variable graph states

We propose a quantum-cryptography network based on a continuous-variable graph state along with its corresponding quantum key distribution (QKD) protocol. It allows two arbitrary parties in the graph state to share a secret Gaussian key (any-to-any QKD). A mathematical model is established to determine an arbitrary graph state'!`\ifmmode\bar\else\textasciimacron\fi{}s properties, including the possibility of QKD and the relevant criteria. The general entangling cloner attack strategy is analyzed in detail employing Shannon information theory. Results show that the proposed network is secure against such attack if the graph state meets certain criteria.

Graphical calculus for Gaussian pure states

We provide a unified graphical calculus for all Gaussian pure states, including graph transformation rules for all local and semi-local Gaussian unitary operations, as well as local quadrature measurements. We then use this graphical calculus to analyze continuous-variable (CV) cluster states, the essential resource for one-way quantum computing with CV systems. Current graphical approaches to CV cluster states are only valid in the unphysical limit of infinite squeezing, and the associated graph transformation rules only apply when the initial and final states are of this form. Our formalism applies to all Gaussian pure states and subsumes these rules in a natural way. In addition, the term "CV graph state" currently has several inequivalent definitions in use. Using this formalism we provide a single unifying definition that encompasses all of them. We provide many examples of how the formalism may be used in the context of CV cluster states: defining the "closest" CV cluster state to a given Gaussian pure state and quantifying the error in the approximation due to finite squeezing; analyzing the optimality of certain methods of generating CV cluster states; drawing connections between this new graphical formalism and bosonic Hamiltonians with Gaussian ground states, including those useful for CV one-way quantum computing; and deriving a graphical measure of bipartite entanglement for certain classes of CV cluster states. We mention other possible applications of this formalism and conclude with a brief note on fault tolerance in CV one-way quantum computing.

Graphical rule of transforming continuous-variable graph states by local homodyne detection

Graphical rule, describing that any single-mode homodyne detection turns a given continuous-variable (CV) graph state into a new one, is presented. Employing two simple graphical rules---local complement operation and vertex deletion (single quadrature-amplitude $\mathrm{x\ifmmode \hat{}\else \^{}\fi{}}$ measurement)---the graphical rule for any single-mode quadrature component measurement can be obtained. The shape of CV weighted graph state may be designed and constructed easily from a given larger graph state by applying this graphical rule.

Quantum computing with continuous-variable clusters

Continuous-variable cluster states offer a potentially promising method of implementing a quantum computer. This paper extends and further refines theoretical foundations and protocols for experimental implementation. We give a cluster-state implementation of the cubic phase gate through photon detection, which, together with homodyne detection, facilitates universal quantum computation. In addition, we characterize the offline squeezed resources required to generate an arbitrary graph state through passive linear optics. Most significantly, we prove that there are universal states for which the offline squeezing per mode does not increase with the size of the cluster. Simple representations of continuous-variable graph states are introduced to analyze graph state transformations under measurement and the existence of universal continuous-variable resource states.

Universal teleportation via continuous-variable graph states

We present a universal protocol for continuous-variable graph-state teleportation. A mathematical model is established to determine an arbitrary graph state's properties, including the possibility of teleportation and the relevant analytical criteria. By investigating the properties of several fundamental graph-state types, we identify a general strategy for optimizing excess noise due to finite squeezing. As an application, a general scheme for teleportation between two arbitrary parties on a two-dimensional square lattice cluster state is given, along with a cheaper yet equivalent optimization for fixed source-target teleportation. Finally, a universal way to distill bipartite entanglement from a graph-state entanglement network is presented based on the equivalence between teleportation and entanglement distillation.

Local complementation rule for continuous-variable four-mode unweighted graph states

The local complementation rule is applied for continuous-variable (CV) graph states in the paper, which is an elementary graph transformation rule and successive application of which generates the orbit of any graph states. The corresponding local Gaussian transformations of local complementation for four-mode unweighted graph states were found, which do not mirror the form of the local Clifford unitary of qubit exactly. This work is an important step to characterize the local Gaussian equivalence classes of CV graph states.