Abstract The objective of Quantum State Verification (QSV) is to evaluate whether an unknown quantum state is sufficiently similar to a predetermined target state at a certain confidence level. The required number of identical and independent copies, N, of an unknown state for such a task is inversely proportional to the spectral gap of a verification operator. Stabilizer states form a foundational class of quantum states, extensively utilized in quantum computing. Defined through a group of stabilizer operators constructed from Pauli matrices in qubit systems, their structural properties not only simplify the mathematical representation and simulation of quantum states but also make them particularly suitable for verification protocols that rely on local measurements. Restricting the verification operator to be those that can be achieved by local measurements, previous researches had determined an upper bound for the spectral gap to be 2/3 in the cases of qubit stabilizer states. In this research, we show a similar upper bound of d/(d+1) in the cases of graph states of d-level systems by a technique from [1] but adapted to d-level graph states. In particular, we show that d-level graph states are maximally entangled across any bipartition of the graph, which allows us to reduce the problem to the case of a bipartite maximally entangled d-level systems for which the maximal spectral gap is known.
Abstract The objective of Quantum State Verification (QSV) is to evaluate whether an unknown quantum state is sufficiently similar to a predetermined target state at a certain confidence level. The required number of identical and independent copies, N, of an unknown state for such a task is inversely proportional to the spectral gap of a verification operator. Stabilizer states form a foundational class of quantum states, extensively utilized in quantum computing. Defined through a group of stabilizer operators constructed from Pauli matrices in qubit systems, their structural properties not only simplify the mathematical representation and simulation of quantum states but also make them particularly suitable for verification protocols that rely on local measurements. Restricting the verification operator to be those that can be achieved by local measurements, previous researches had determined an upper bound for the spectral gap to be 2/3 in the cases of qubit stabilizer states. In this research, we show a similar upper bound of d/(d+1) in the cases of graph states of d-level systems by a technique from [1] but adapted to d-level graph states. In particular, we show that d-level graph states are maximally entangled across any bipartition of the graph, which allows us to reduce the problem to the case of a bipartite maximally entangled d-level systems for which the maximal spectral gap is known.
Statistical verification of a quantum state aims to certify whether a given unknown state is close to the target state with confidence. So far, sample-optimal verification protocols based on local measurements have been found only for disparate groups of states: bipartite pure states, GHZ states, and antisymmetric basis states. In this work, we investigate systematically optimal verification of entangled stabilizer states using Pauli measurements. First, we provide a lower bound on the sample complexity of any verification protocol based on separable measurements, which is independent of the number of qubits and the specific stabilizer state. Then we propose a simple algorithm for constructing optimal protocols based on Pauli measurements. Our calculations suggest that optimal protocols based on Pauli measurements can saturate the above bound for all entangled stabilizer states, and this claim is verified explicitly for states up to seven qubits. Similar results are derived when each party can choose only two measurement settings, say X and Z. Furthermore, by virtue of the chromatic number, we provide an upper bound for the minimum number of settings required to verify any graph state, which is expected to be tight. For experimentalists, optimal protocols and protocols with the minimum number of settings are explicitly provided for all equivalent classes of stabilizer states up to seven qubits. For theorists, general results on stabilizer states (including graph states in particular) and related structures derived here may be of independent interest beyond quantum state verification.
Abstract 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.
Fathoming out quantum state space is a challenging endeavor due to its exponentially growing dimensionality. At the expense of being bound in its expressiveness, the discrete and finite subspace of graph states is easier to investigate via a pictorial framework accompanied with a theoretical toolkit from the stabilizer formalism. Analyzing hand-drawn graphs is a tedious and time-consuming task and imposes limitations to the problem sizes that can be addressed. Similarly, algorithmic studies using adjacency matrices alone lack the benefit of a visual representation of the states. We argue that applying visual analytics to investigate graph states can be advantageous. To this end, we introduce GRAPHSTATEVIS, a web-based application for the visual analysis of qubit graph states and their stabilizer groups. Our tool facilitates the interactive construction of a graph through multiple components supported by linking and brushing. The user can explore graph-state-specific properties, including the Pauli-weight distribution of its stabilizer operators and noise thresholds for entanglement criteria. We propose a use case in the context of near-term quantum algorithms to illustrate the capabilities of our prototype. We provide access to GRAPHSTATEVIS as an open-source project and invite the broader quantum computing and engineering communities to take advantage of this tool and further boost its development.
Characterizing large noisy multiparty quantum states using genuine multipartite entanglement is a challenging task. In this paper, we calculate lower bounds of genuine multipartite entanglement localized over a chosen multiparty subsystem of multiqubit stabilizer states in the noiseless and noisy scenarios. In the absence of noise, adopting a graph-based technique, we perform the calculation for arbitrary graph states as representatives of the stabilizer states and show that the graph operations required for the calculation have a polynomial scaling with the system size. As demonstrations, we compute the localized genuine multipartite entanglement over subsystems of large graphs having linear, ladder, and square structures. We also extend the calculation for graph states subjected to single-qubit Markovian or non-Markovian Pauli noise on all qubits and demonstrate, for a specific lower bound of the localizable genuine multipartite entanglement corresponding to a specific Pauli measurement setup, the existence of a critical noise strength beyond which all of the postmeasured states are biseparable. The calculation is also useful for arbitrary large stabilizer states under noise due to the local unitary connection between stabilizer states and graph states. We demonstrate this by considering a toric code defined on a square lattice and computing a lower bound of localizable genuine multipartite entanglement over a nontrivial loop of the code. Similar to the graph states, we show the existence of the critical noise strength in this case also and discuss its interesting features.
We have generalised the concept of graph states to what we have called mixed graph states, which we define in terms of mixed graphs, that is graphs with both directed and undirected edges, as the density matrix stabilized by the associated stabilizer matrix defined by the mixed graph. We can interpret this matrix as a quantum object by making it part of a larger fully commuting matrix, i.e. where we choose the environment appropriately, and this will imply that our quantum object is a mixed state. We prove that, in the same way as (pure) graph states, the density matrix of a parent of mixed graph state can be written as sum of a few Pauli matrices, well defined from the mixed graph. We have proven that the set of matrices that appear in this sum is fully pair-wise commuting, and form a multiplicative group up to global constants, which is always of maximum size. Furthermore, the cardinality of the set depends solely of the miminum possible number of extension columns/rows, and the number of nodes of the mixed graph. We prove a formula for this cardinality. Finally, in the case of purely undirected graphs, this corresponds to the usual pure graph state. Also, we have developed a way of finding maximal commutative group of such Pauli matrices as a linear subspace problem, for any given mixed graph. We also have proven how the structure of maximal commutative groups is independent of the direction of the arrows of the mixed graph, and also of the undirected edges; this allows the simplification of the problem of finding these groups in general to finding them for a much smaller set of graphs.
The class of entangled N-qubit states known as graph states, and the corresponding stabilizer groups of N-qubit Pauli observables, have found a wide range of applications in quantum information processing and the foundations of quantum mechanics. A review of the properties of graph states is given and core spaces of graph states are introduced and discussed. A bonding model of entanglement for generalized graph states is then presented, in which the presence or absence of a bond between two qubits unequivocally specifies whether or not they are entangled. A physical interpretation of these bonds is given, along with a characterization of how they can be created or destroyed by entangling unitary operations and how they can be destroyed by local Pauli measurements. It is shown that local unitary operations do not affect the bond structure of a graph state, and therefore that if two graph states have nonisomorphic bond structures, then local unitary operations and/or reordering of qubits cannot change one into the other. Color multigraphs are introduced to depict the bond structures of graph states and to make some of their properties more apparent.