New Quantum Algorithm Proves Robustness: Dissipative Transverse Field Ising Model Simulated with High-Order Precision
The Transverse Field Ising Model
The detailed simulation of the dissipative Transverse Field Ising Model (TFIM) successfully demonstrates the capabilities of a recent advancement in variational quantum simulation (VQS) methodology, which offers a much more reliable and accurate approach to modelling the dynamics of open quantum systems. Combining high-order stochastic Magnus expansion (SME) with unraveled Lindblad dynamics, this new framework represents a significant advancement in using near-term quantum hardware to tackle challenging physics and chemistry challenges.
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Understanding the Dissipative TFIM
Many-body interactions and quantum phase transitions are studied using the Transverse-Field Ising Model (TFIM), a fundamental physics model. It is a key test bed for dissipative quantum dynamics, where the Lindblad master equation (LME) and damping regulate the system.
The dissipative dynamics of a two-site TFIM were investigated in this work. The following is the general Hamiltonian (Hs) that describes the dynamics of the internal system the transverse magnetic field supplied to the system, and J is the coupling strength between adjacent spins. Lindblad jump operators (L_k) are introduced to simulate the dissipation or interaction with the environment.
Overcoming Simulation Limitations
Since the many-body density operator scales exponentially (4 L) with the number of qubits (L), it is very challenging to simulate open quantum systems traditionally. By focusing solely on the wavefunction and lowering the problem dimensionality to 2 L, variational quantum simulation (VQS) aims to get around this. It does this by “unravelling” the LME into a stochastic differential equation (SDE) for the wavefunction, specifically using the Quantum State Diffusion (QSD) technique.
However, in order to handle the accumulating variation inherent in the stochastic sampling, traditional VQS algorithms based on QSD frequently lack resilience and require either extremely tiny time steps or a large number of simulation trajectories samples. This is addressed by the suggested algorithm, which produces high-order exponential integrators by solving the QSD equation using the stochastic Magnus expansion. Even with greater time increments, this methodical high-order technique offers better precision and stability.
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Numerical Validation: Higher Order Means Higher Accuracy
To illustrate the two advantages provided by the new framework, the dissipative TFIM simulation was essential. Using an ensemble of Ntraj= 103 wavefunction trajectories and a comparatively large time step of \Δ= 0.25 tJ, the researchers propagated the system dynamics up to a stopping time of T=25 tJ. Bare drawn from the numerical specifically ∣00⟩, ∣01⟩, and ∣11⟩
- Systematic Improvement with Order: When compared to the first-order scheme (Scheme I), the higher-order Magnus approach (Scheme II) consistently shown superior agreement with the actual solution. This demonstrated that the methodical use of high-order expansions improved accuracy, which is particularly important when using a bigger time step.
- Superiority of Nonlinear Unraveling: The simulation compared outcomes utilising nonlinear QSD and linear QSD. When compared to the linear method, the nonlinear unravelling showed better accuracy. This benefit results from the nonlinear scheme’s norm-preserving characteristic, which improves overall resilience by preventing the ensemble’s variance from increasing unnecessarily across discrete time steps.
The TFIM simulation uses the Hamiltonian Variational Ansatz (HVA), designed for a two-site system with three layers (m=3). It has quantum gates for σz1σz2, σz1, σz2, σy1, σy2, σx1, This demonstrated that the well-established VQS approach could create high-order stochastic Magnus integrators.
In conclusion
The dissipative TFIM simulation successfully demonstrated that this new algorithm offers a methodical way to improve the accuracy and robustness of VQS for Lindblad dynamics, providing useful avenues for simulating intricate open quantum systems on Noisy Intermediate-Scale Quantum (NISQ) devices. It is anticipated that the techniques developed will be applicable to a larger class of open quantum systems than the particular examples examined.
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