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A solvable model for graph state decoherence dynamics

J. Houdayer, Haggai Landa, Grégoire Misguich

Published Feb 23, 2024• 3 citations
Publisher

Abstract

We present an exactly solvable toy model for the continuous dissipative dynamics of permutation-invariant graph states of N N qubits. Such states are locally equivalent to an N N -qubit Greenberger-Horne-Zeilinger (GHZ) state, a fundamental resource in many quantum information processing setups. We focus on the time evolution of the state governed by a Lindblad master equation with the three standard single-qubit jump operators, the Hamiltonian part being set to zero. Deriving analytic expressions for the expectation values of observables expanded in the Pauli basis at all times, we analyze the nontrivial intermediate-time dynamics. Using a numerical solver based on matrix product operators, we simulate the time evolution for systems with up to 64 qubits and verify a numerically exact agreement with the analytical results. We find that the evolution of the operator space entanglement entropy of a bipartition of the system manifests a plateau whose duration increases logarithmically with the number of qubits, whereas all Pauli-operator products have expectation values decaying at most in constant time.

DOI
10.21468/scipostphyscore.7.1.009

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We present an exactly solvable toy model for the continuous dissipative dynamics of permutation-invariant graph states of N N qubits. Such states are locally equivalent to an N N -qubit Greenberger-Horne-Zeilinger (GHZ) state, a fundamental resource in many quantum information processing setups. We focus on the time evolution of the state governed by a Lindblad master equation with the three standard single-qubit jump operators, the Hamiltonian part being set to zero. Deriving analytic expressions for the expectation values of observables expanded in the Pauli basis at all times, we analyze the nontrivial intermediate-time dynamics. Using a numerical solver based on matrix product operators, we simulate the time evolution for systems with up to 64 qubits and verify a numerically exact agreement with the analytical results. We find that the evolution of the operator space entanglement entropy of a bipartition of the system manifests a plateau whose duration increases logarithmically with the number of qubits, whereas all Pauli-operator products have expectation values decaying at most in constant time.
We present an exactly solvable toy model for the continuous dissipative dynamics of permutation-invariant graph states of NNN qubits. Such states are locally equivalent to an NNN-qubit Greenberger-Horne-Zeilinger (GHZ) state, a fundamental resource in many quantum information processing setups. We focus on the time evolution of the state governed by a Lindblad master equation with the three standard single-qubit jump operators, the Hamiltonian part being set to zero. Deriving analytic expressions for the expectation values of observables expanded in the Pauli basis at all times, we analyze the nontrivial intermediate-time dynamics. Using a numerical solver based on matrix product operators, we simulate the time evolution for systems with up to 64 qubits and verify a numerically exact agreement with the analytical results. We find that the evolution of the operator space entanglement entropy of a bipartition of the system manifests a plateau whose duration increases logarithmically with the number of qubits, whereas all Pauli-operator products have expectation values decaying at most in constant time.
We present a comprehensive analytical and numerical study of time-dependent decoherence in open quantum systems, focusing on a single-qubit model with time-varying Lindblad dissipation rates. Employing a master equation framework, we derive exact solutions for the evolution of off-diagonal density matrix elements and characterize the emergence of robust pointer states. Our findings reveal an initial rapid coherence decay followed by a gradual slowdown, mirroring experimental trends observed in superconducting qubits influenced by quasiparticle dynamics and in photonic cavities subject to fluctuating environments. By incorporating real experimental parameters and validating the model through quantum simulations using QuTiP, we provide a robust quantitative tool for analyzing decoherence in quantum technologies. This extended model addresses limitations in static-rate approximations, offering insights into non-Markovian effects without resolving the measurement problem, and suggests extensions to multi-qubit systems.
We investigate entanglement properties of multipartite states under the influence of decoherence. We show that the lifetime of (distillable) entanglement for Greenberger-Horne-Zeilinger (GHZ) -type superposition states decreases with the size of the system, while for a class of other states---namely, all graph states with constant degree---the lifetime is independent of the system size. We show that these results are largely independent of the specific decoherence model and are in particular valid for all models which deal with individual couplings of particles to independent environments, described by some quantum optical master equation of Lindblad form. For GHZ states, we derive analytic expressions for the lifetime of distillable entanglement and determine when the state becomes fully separable. For all graph states, we derive lower and upper bounds on the lifetime of entanglement. The lower bound is based on a specific distillation protocol, while upper bounds are obtained by showing that states resulting from decoherence in general become nondistillable or even separable after a finite time. This is done using different methods, namely, (i) the map describing the decoherence process (e.g., the action of a thermal bath on the system) becomes entanglement breaking, (ii) the resulting state becomes separable, and (iii) the partial transposition with respect to certain partitions becomes positive. To this aim, we establish a method to calculate the spectrum of the partial transposition for all mixed states which are diagonal in a graph-state basis. We also consider entanglement between different groups of particles and determine the corresponding lifetimes as well as the change of the kind of entanglement with time. This enables us to investigate the behavior of entanglement under rescaling and in the limit of large number of particles $N\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}$. Finally we investigate the lifetime of encoded quantum superposition states and show that one can define an effective time in the encoded system which can be orders of magnitude smaller than the physical time. This provides an alternative view on quantum error correction and examples of states whose lifetime of entanglement (between groups of particles) in fact increases with the size of the system.
Abstract Accurately predicting quantum decoherence time is a core bottleneck in scaling up noisy intermediate-scale quantum (NISQ) devices. Traditional frameworks relying on the Lindblad master equation suffer from the curse of dimensionality (O(4^n )) and obscure the intuitive physical mechanisms of decoherence behind complex system-environment coupling assumptions. Based on graph algebraic connectivity, this paper proposes a deterministic topological model of quantum decoherence. We rigorously map multi-particle quantum entanglement to the algebraic connectivity (Fiedler value, λ_2) of an underlying structural graph and redefine decoherence as a Poissonian random edge-deletion process. By modeling this topological evolution as a continuous-time Markov chain (CTMC), we derive a unified and exact recursive analytical equation for the expected hitting time of complete decoherence for arbitrary asymmetric topological states. The analytical solutions strictly prove the relative robustness of prototypical entangled states (T_"Bell" >T_W≫T_"GHZ" ). Furthermore, the model demonstrates unprecedented computational advantages by instantly yielding the exact analytical decoherence time for a 10-body asymmetric complex network, which is perfectly validated by 50,000 Monte Carlo simulations with a relative error of <0.5%. We calibrate the core parameter γ (single-edge fracture rate) using the latest public experimental data of superconducting quantum chips from IBM and Google, and the theoretical predictions are in high agreement with the experimental results within the measurement error range (<3%). This topological downgrade mechanism reduces the computational complexity of decoherence calculations to O(n^3 ), providing a powerful theoretical and algorithmic tool for the optimization of quantum processing unit (QPU) underlying architectures and the design of fault-tolerant multi-particle quantum networks.
We study the entanglement dynamics of multi-qubit systems coupled to a common dissipative environment, focusing on systems with one or two initially excited qubits. Using the Lindblad master equation, we derive the time evolution of the density matrix and analyze the entanglement between qubit pairs via the concurrence measure. The solution method involves applying the Lindblad super-operator to the initial density matrix, generating a subspace of non-duplicate states. We extend our analysis to nnn-qubit systems, including three-qubit and four-qubit configurations, and explore the effects of thermal noise on entanglement dynamics. Our results demonstrate how initial conditions, system size, and environmental interactions shape entanglement, providing valuable insights for quantum information processing applications.
Accurately predicting quantum decoherence time is a core bottleneck in scaling up noisy intermediate-scale quantum (NISQ) devices. Traditional frameworks relying on the Lindblad master equation suffer from the curse of dimensionality and obscure the intuitive physical mechanisms of decoherence behind complex system-environment coupling assumptions. Based on graph algebraic connectivity, this paper proposes a deterministic topological model of quantum decoherence. We rigorously map multi-particle quantum entanglement to the algebraic connectivity (Fiedler value, ) of an underlying structural graph and redefine decoherence as a Poissonian random edge-deletion process. By modeling this topological evolution as a continuous-time Markov chain (CTMC), we derive a unified and exact recursive analytical equation for the expected hitting time of complete decoherence for arbitrary asymmetric topological states. The analytical solutions strictly prove the relative robustness of prototypical entangled states . Furthermore, the model demonstrates unprecedented computational advantages by instantly yielding the exact analytical decoherence time for a 10-body asymmetric complex network, which is perfectly validated by 50,000 Monte Carlo simulations with a relative error of . We calibrate the core parameter (single-edge fracture rate) using the latest public experimental data of superconducting quantum chips from IBM and Google, and the theoretical predictions are in high agreement with the experimental results within the measurement error range . This topological downgrade mechanism reduces the computational complexity of decoherence calculations to , providing a powerful theoretical and algorithmic tool for the optimization of quantum processing unit (QPU) underlying architectures and the design of fault-tolerant multi-particle quantum networks.