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In quantum metrology, achieving optimal simultaneous multiparameter estimation is of great significance but remains highly challenging. The research approach involving evolution on SU(N)$\mathrm{SU}(N)$ dynamics provides a framework to investigate simultaneous multiparameter estimation within graph states. For single‐parameter estimation, it is observed that the precision limit exceeds the Heisenberg limit in higher‐dimensional SU(2)$\mathrm{SU}(2)$ spin systems. For multiparameter estimation, two scenarios are considered: one with commutative Hamiltonian operators and another with non‐commutative Hamiltonian operators. The results demonstrate that the global estimation precision exceeds the local estimation precision. Under the conditions of parameter limit, the precision of parameter estimation for simultaneously estimating each parameter is equal to that of single‐parameter estimation. Furthermore, a precision‐enhancement scheme has been identified that depends on the dynamics of SU(N)$\mathrm{SU}(N)$. The smaller the value of N$N$ in the dynamic evolution, the higher the precision of the parameter estimation. Finally, it is demonstrated that graph states serve as optimal states in quantum metrology. A set of optimal measurement bases is also identified, and it is illustrated that the precision limit of multiparameter estimation can attain the quantum Cramér‐Rao bound. [ABSTRACT FROM AUTHOR]

In Quantum Information theory, graph states are quantum states defined by graphs. In this work we exhibit a correspondence between orbits of graph states and orbits in the variety of binary symmetric principal minors, under the action of SL (2 , F 2) × n ⋊ S n . First we study the orbits of maximal abelian subgroups of the n-fold Pauli group under the action of C n loc ⋊ S n , where C n loc is the n-fold local Clifford group, and we show that this action corresponds to the natural action of SL (2 , F 2) × n ⋊ S n on the variety Z n ⊂ P (F 2 2 n) of principal minors of binary symmetric n × n matrices: the crucial step is in translating the action of SL (2 , F 2) × n into an action of the local symplectic group Sp 2 n loc (F 2) . We conclude by showing how the former action restricts onto stabilizer groups, stabilizer states and graph states. [ABSTRACT FROM AUTHOR]

Protocols are proposed for controlled remote implementation of operations here. Sharing a (2N+1)$(2N+1)$‐partite graph state, 2N participants collaborate to prepare the stator and realize the operation ⊗j=1Nexp[iαjσnOj]$\otimes _{j=1}^N\exp {[i\alpha _j\sigma _{n_{O_j}}]}$ on N unknown states for distributed systems Oj$O_j$, only with the permission of a controller. The control power analysis shows that without the controller's permission, eight specified operations can be implemented with the success rate of 50%, other operations can be realized with the success rate of 25%, and thus the control power is reliable. All the implementation requirements of this protocol can be satisfied by means of local operations and classical communications, and the experimental feasibility is presented according to current techniques. The entanglement requirement of this protocol is characterized in terms of geometric measure of entanglement. It turns out to be economic to realize the control function from the perspective of entanglement cost. [ABSTRACT FROM AUTHOR]

Abstract Graph states are one of the most significant classes of entangled states, serving as the quantum resources for quantum technologies. Recently, integrated quantum photonics is becoming a promising platform for quantum information processing, enabling the generation, manipulation, and measurement of photonic quantum states. This article summarizes state-of-the-art experimental progress and advances in the chip-based photonic graph states.

Entanglement represents “the” key resource for several applications of quantum information processing, ranging from quantum communications to distributed quantum computing. Despite its fundamental importance, deterministic generation of maximally entangled qubits represents an on- going open problem. Here, we design a novel generation scheme exhibiting two attractive features, namely, i) deterministically generating different classes - particularly, GHZ-like, W-like and graph states - of genuinely multipartite entangled states, ii) not requiring any direct interaction between the qubits. Indeed, the only necessary condition is the possibility of coherently controlling - according to the indefinite causal order framework - the causal order among the unitaries acting on the qubits. Through the paper, we analyze and derive the conditions on the unitaries for deterministic generation, and we provide examples for unitaries practical implementation. We conclude the paper by discussing the scalability of the proposed scheme to higher dimensional genuine multipartite entanglement (GME) states and by introducing some possible applications of the proposal for quantum networks.

Graph states are one of the most significant classes of entangled states, serving as the quantum resources for quantum technologies. Recently, integrated quantum photonics is becoming a promising platform for quantum information processing, enabling the generation, manipulation, and measurement of photonic quantum states. This article summarizes state-of-the-art experimental progress and advances in the chip-based photonic graph states. [ABSTRACT FROM AUTHOR]

Constant‐depth quantum circuits that prepare and measure graph states on 2D grids are proved to possess a computational quantum advantage over their classical counterparts due to quantum nonlocality and are also well suited for demonstrations on current superconducting quantum processor architectures. To simulate the partial or full sampling of 2D graph states, a practical two‐stage classical strategy that can exactly generate any number of samples (bit strings) from such circuits is proposed. The strategy is inspired by exploiting specific properties of a hidden linear function problem solved by the target quantum circuit, which in particular combines traditional classical parallel algorithms and an explicit gate‐based constant‐depth classical circuit together. A theoretical analysis reveals that on average each sample can be obtained in nearly constant time for sampling specific circuit instances of large size. Moreover, the feasibility of the theoretical model is demonstrated by implementing typical instances up to 25 qubits on a moderate field programmable gate array platform. Therefore, the strategy can be used as a practical tool for verifying experimental results obtained from shallow quantum circuits of this type. [ABSTRACT FROM AUTHOR]

Bell nonlocality—the existence of quantum correlations that cannot be explained by classical means—is certainly one of the most striking features of quantum mechanics. Its range of applications in device-independent protocols is constantly growing. Many relevant quantum features can be inferred from violations of Bell inequalities, including entanglement detection and quantification, and state certification applicable to systems of arbitrary number of particles. A complete characterisation of nonlocal correlations for many-body systems is, however, a computationally intractable problem. Even if one restricts the analysis to specific classes of states, no general method to tailor Bell inequalities to be violated by a given state is known. In this work we provide a general construction of Bell expressions tailored to the graph states of any prime local dimension. These form a broad class of multipartite quantum states that have many applications in quantum information, including quantum error correction. We analytically determine their maximal quantum values, a number of high relevance for device-independent applications of Bell inequalities. Importantly, the number of expectation values to determine in order to test the violation of our inequalities scales only linearly with the system size, which we expect to be the optimal scaling one can hope for in this case. Finally, we show that these inequalities can be used for self-testing of multi-qutrit graph states such as the well-known four-qutrit absolutely maximally entangled state AME(4,3).

We quantify the entanglement and its variations via generalized concurrence, Meyer–Wallach measure and its generalizations. Then we investigate the interactions between qubits on complete graphs, tree graphs, chain graphs and loop graphs with 3 to 8 qubits. It is observed that the amount of all entanglement measures in tree graph in the first group of considered graphs, are equal. The complete equivalent graph is equal and the entanglement measures for these graphs change altering the number of qubits. Furthermore, it can be seen that the entanglement measures are increased augmenting the number of qubits in the chain graph and loop graph. The entanglement of loop graphs is equal to or greater than the entanglement of chain graphs assuming the same number of qubits. Also there is a significant relation between entanglement, number of qubits, and interactions between qubits for the second group of considered graphs. As the number of interactions (presented by edges) increases between the qubits, the entanglements increase in these states. It is shown that the employed entanglement measures reveal the different values changing the number of qubits (vertices). [ABSTRACT FROM AUTHOR]

Based on the topological features of quantum graph states,a quantum secret sharing scheme based on Chinese remainder theorem with a vivid graphic description was proposed.The dealer extracts sub-secrets according to Chinese remainder theorem over finite field,which were imbedded with quantum graph states and transmitted to the legal participants with unitary operations.Group-recovery protocols were used in the secret recovering processing through rebuilding sub-secrets among legal cooperative participants.Analysis shows that it could provide better security and capability of the information.

Analyzing pseudo-telepathy graph games, we propose a way to build contextuality scenarios exhibiting the quantum supremacy using graph states. We consider the combinatorial structures generating equivalent scenarios. We introduce a new tool called multipartiteness width to investigate which scenarios are hard to decompose and show that there exist graphs generating scenarios with a linear multipartiteness width. [ABSTRACT FROM AUTHOR]

We propose a new classification for the entanglement in graph states based on generalized concurrence. The numerical results indicate that the eight different three-qubit graph states in three categories, 64 four-qubit graph states in five categories and 1024 five-qubit graph states in 10 classes. We also compare this classification with equivalence classes of these graph states under local complementation (LC) operator, and the obtained result suggests that classification by generalized concurrence is not in contradiction with the LC-rule. Furthermore, our results suggest that the generalized concurrence is a suitable measure of entanglement for multi-qubit graph states. [ABSTRACT FROM AUTHOR]

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A multiparty quantum blind signature scheme is proposed based on the principle of graph state, in which the unitary operations of graph state particles can be applied to generate the quantum blind signature and achieve verification. Different from the classical blind signature based on the mathematical difficulty, the scheme could guarantee not only the anonymity but also the unconditionally security. The analysis shows that the length of the signature generated in our scheme does not become longer as the number of signers increases, and it is easy to increase or decrease the number of signers. [ABSTRACT FROM AUTHOR]

Stabilizer states form an important class of states in quantum information, and are of central importance in quantum error correction. Here, we provide an algorithm for deciding whether one stabilizer (target) state can be obtained from another stabilizer (source) state by single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM) and classical communication (CC) between sites holding the individual qubits. What is more, we provide a recipe to obtain the sequence of LC + LPM + CC operations which prepare the desired target state from the source state, and show how these operations can be applied in parallel to reach the target state in constant time. Our algorithm has applications in quantum networks, quantum computing, and can also serve as a design tool-for example, to find transformations between quantum error correcting codes. We provide a software implementation of our algorithm that makes this tool easier to apply. A key insight leading to our algorithm is to show that the problem is equivalent to one in graph theory, which is to decide whether some graph G' is a vertex-minor of another graph G. The vertex-minor problem is, in general, NP-Complete, but can be solved efficiently on graphs which are not too complex. A measure of the complexity of a graph is the rank-width which equals the Schmidt-rank width of a subclass of stabilizer states called graph states, and thus intuitively is a measure of entanglement. Here, we show that the vertex-minor problem can be solved in time O(|G|³), where |G| is the size of the graph G, whenever the rank-width of G and the size of G' are bounded. Our algorithm is based on techniques by Courcelle for solving fixed parameter tractable problems, where here the relevant fixed parameter is the rank width. The second half of this paper serves as an accessible but far from exhausting introduction to these concepts, that could be useful for many other problems in quantum information. This article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'. [ABSTRACT FROM AUTHOR]

We present a simple protocol for certifying graph states in quantum networks using stabiliser measurements. The certification statements can easily be applied to different protocols using graph states. We see, for example, how it can be used for measurement based verified quantum computation, certified sampling of random unitaries, quantum metrology and sharing quantum secrets over untrusted channels.

Topological quantum error correcting codes have emerged as leading candidates towards the goal of achieving large-scale fault-tolerant quantum computers. However, quantifying entanglement in these systems of large size in the presence of noise is a challenging task. In this paper, we provide two different prescriptions to characterize noisy stabilizer states, including the surface and the color codes, in terms of localizable entanglement over a subset of qubits. In one approach, we exploit appropriately constructed entanglement witness operators to estimate a witness-based lower bound of localizable entanglement, which is directly accessible in experiments. In the other recipe, we use graph states that are local unitary equivalent to the stabilizer state to determine a computable measurement-based lower bound of localizable entanglement. If used experimentally, this translates to a lower bound of localizable entanglement obtained from single-qubit measurements in specific bases to be performed on the qubits outside the subsystem of interest. Towards computing these lower bounds, we discuss in detail the methodology of obtaining a local unitary equivalent graph state from a stabilizer state, which includes a new and scalable geometric recipe as well as an algebraic method that applies to general stabilizer states of arbitrary size. Moreover, as a crucial step of the latter recipe, we develop a scalable graph-transformation algorithm that creates a link between two specific nodes in a graph using a sequence of local complementation operations. We develop open-source Python packages for these transformations, and illustrate the methodology by applying it to a noisy topological color code, and study how the witness and measurement-based lower bounds of localizable entanglement varies with the distance between the chosen qubits.

An important class of contextuality arguments in quantum foundations are the all-versus-nothing (AvN) proofs, generalizing a construction originally due to Mermin. We present a general formulation of AvN arguments and a complete characterization of all such arguments that arise from stabilizer states. We show that every AvN argument for an n-qubit stabilizer state can be reduced to an AvN proof for a three-qubit state that is local Clifford-equivalent to the tripartite Greenberger--Horne--Zeilinger state. This is achieved through a combinatorial characterization of AvN arguments, the AvN triple theorem, whose proof makes use of the theory of graph states. This result enables the development of a computational method to generate all the AvN arguments in Z2 on n-qubit stabilizer states. We also present new insights into the stabilizer formalism and its connections with logic. This article is part of the themed issue 'Second quantum revolution: foundational questions'. [ABSTRACT FROM AUTHOR]

While quantum theory cannot be described by a local hidden variable model, it is nevertheless possible to construct such models that exhibit features commonly associated with quantum mechanics. These models are also used to explore the question of $$\psi $$ -ontic versus $$\psi $$ -epistemic theories for quantum mechanics. Spekkens' toy theory is one such model. It arises from classical probabilistic mechanics via a limit on the knowledge an observer may have about the state of a system. The toy theory for the simplest possible underlying system closely resembles stabilizer quantum mechanics, a fragment of quantum theory which is efficiently classically simulable but also non-local. Further analysis of the similarities and differences between those two theories can thus yield new insights into what distinguishes quantum theory from classical theories, and $$\psi $$ -ontic from $$\psi $$ -epistemic theories. In this paper, we develop a graphical language for Spekkens' toy theory. Graphical languages offer intuitive and rigorous formalisms for the analysis of quantum mechanics and similar theories. To compare quantum mechanics and a toy model, it is useful to have similar formalisms for both. We show that our language fully describes Spekkens' toy theory and in particular, that it is complete: meaning any equality that can be derived using other formalisms can also be derived entirely graphically. Our language is inspired by a similar graphical language for quantum mechanics called the ZX-calculus. Thus Spekkens' toy bit theory and stabilizer quantum mechanics can be analysed and compared using analogous graphical formalisms. [ABSTRACT FROM AUTHOR]

Society relies and depends increasingly on information exchange and communication. In the quantum world, security and privacy is a built-in feature for information processing. The essential ingredient for exploiting these quantum advantages is the resource of entanglement, which can be shared between two or more parties. The distribution of entanglement over large distances constitutes a key challenge for current research and development. Due to losses of the transmitted quantum particles, which typically scale exponentially with the distance, intermediate quantum repeater stations are needed. Here we show how to generalise the quantum repeater concept to the multipartite case, by describing large-scale quantum networks, i.e. network nodes and their long-distance links, consistently in the language of graphs and graph states. This unifying approach comprises both the distribution of multipartite entanglement across the network, and the protection against errors via encoding. The correspondence to graph states also provides a tool for optimising the architecture of quantum networks.

We present an experimentally feasible method, based on currently available cavity QED technology, to generate n-partite linear cluster and graph states in external degree of freedom of atoms. The scheme is based on first tagging n two-level atoms with the respective cavity fields in momentum space. Later on an effective Ising interaction between such tagged atoms, realized through consecutive resonant and dispersive interactions of auxiliary atoms with the remanent cavity fields, can generate the desired atomic momenta states. The procedure is completed when the auxiliary atoms after passing through Ramsey zones are detected in either of their internal states. We also briefly explain the generation of weighted graph states in the atomic external degree of freedom. [ABSTRACT FROM AUTHOR]

A set of necessary and sufficient conditions are derived for the equivalence of an arbitrary pure state and a graph state on n qubits under stochastic local operations and classical communication (SLOCC), using the stabilizer formalism. Because all stabilizer states are equivalent to graph states by local unitary transformations, these conditions constitute a classical algorithm for the determination of SLOCC-equivalence of pure states and stabilizer states. This algorithm provides a distinct advantage over the direct solution of the SLOCC-equivalence condition |ψ〉 = S|g〉 for an unknown invertible local operator S, as it usually allows for easy detection of states that are not SLOCC-equivalent to graph states. [ABSTRACT FROM AUTHOR]

Generalizations of the bent property of a Boolean function are presented, by proposing spectral analysis with respect to a well-chosen set of local unitary transforms. Quadratic Boolean functions are related to simple graphs and it is shown that the orbit generated by successive local complementations on a graph can be found within the transform spectra under investigation. The flat spectra of a quadratic Boolean function are related to modified versions of its associated adjacency matrix. [ABSTRACT FROM AUTHOR]

The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics (QM), meaning any pure state, unitary operation and post-selected pure projective measurement can be expressed in the ZX-calculus. The calculus is also sound , i.e. any equality that can be derived graphically can also be derived using matrix mechanics. Here, we show that the ZX-calculus is complete for pure qubit stabilizer QM, meaning any equality that can be derived using matrices can also be derived pictorially. The proof relies on bringing diagrams into a normal form based on graph states and local Clifford operations.

• Graph states generated by operator of evolution with Ising Hamiltonian are examined. • Entanglement of a spin with other spins in graph state is found on quantum computer. • The results of quantum calculations are in good agreement with the theoretical ones. • Entanglement of graph states is related with properties of the corresponding graphs. We consider graph states generated by operator of evolution with Ising Hamiltonian. The geometric measure of entanglement of a spin with other spins in the graph state is obtained analytically and quantified on IBM's quantum computer, IBM Q Valencia. The results of quantum calculations are in good agreement with the theoretical ones. We conclude that the geometric measure of entanglement of a spin with other spins in the graph state is related with degree of vertex representing the spin in the corresponding graph. [ABSTRACT FROM AUTHOR]