A hybrid method for quantum dynamics simulation

The gist The hybrid method combines Trotter-based quantum algorithm with classical dynamic mode decomposition to predict observables of a quantum state in long time by using data from short time measurements from a quantum computer

Method Overview

The proposed approach simulates quantum many body dynamics by combining Trotter based quantum algorithm with classical dynamic mode decomposition. The method aims to predict observables of a quantum state in the long time by using data from a set of short time measurements from a quantum computer. This method is particularly useful because the interest often lies in estimating observables rather than explicitly obtaining the wave function’s form. The upper bound for the global error of our method scales as O(t 3/2) with a fixed set of the measurement.

Hybrid Simulation Protocol

The simulation begins by time evolving a quantum state in a quantum computer using the Trotter decomposition of the evolution operator e-iHt. This is followed by simultaneous measurement of an observable O(x, t). The measured data is then utilized within the DMD framework to predict long time dynamics of the observable.

Dynamic Mode Decomposition (DMD)

The DMD procedure involves several key steps to approximate the infinite-dimensional operator K∆t with a finite-dimensional linear operator K. The process is summarized in Algorithm 1, which includes:

1. Recording time snapshots X1 = [O 1 O 2.. O m].

2. Recording time snapshots X2 = [O 2 O 3.. ⟨O m⟩].

3. Finding the matrix Km which approximates K∆t by solving a linear least squares problem min Km KmX1 − X2 squared F.

4. Obtaining the approximation of the Koopman operator Ke m by solving Ke m = X2X+1.

5. Constructing the solution of (3) at a later time step n > m as ⟨O(tn)⟩ = ΦmΛ(n-1)/m Φ†m ⟨O(t1)⟩.

Error Analysis

The error analysis for the method has two directions: how the error scales w.r.t time by keeping m, the number of snapshots we use in DMD, fixed. The global truncation error at the n-th time step (n > m) is given by Theorem 2 as ϵ n squared ≤ Φm squared Φ-1m squared [ϵ m squared + (n - m)ϵ m]. For long time n >> m, the global error scales as at most O(t 3/2). The local truncation error is bounded by Theorem 3 as ϵ m ≤ cm squared OF(1 + 2n∆t)H squared F-1/2.

Application to Quantum Systems

The method was applied to quench dynamics in Hubbard model and nearest neighbor spin systems. Specifically, the simulation involves starting from the ground state ψ0⟩ of a non-interacting Hamiltonian H0 on L sites and time evolving it with an interacting Hamiltonian H1. Observables O are chosen as density matrix operators ρpq = c†pσc qσ: (p, q) ∈ 1 to L as the set of observables. The results demonstrate that DMD gives good extrapolation for both Us in the Hubbard model simulations. The error estimate of the Trotter method can be found in [19] with time step size ∆t = 0.01, and the extrapolation results with m ≥ 200 are generally acceptable. The oscillation behavior of the error with respect to m is due to the nature of data-driven methods as we have observed before[38,39]. The error drops to a lower threshold as we go from L = 6 to L = 12 for the XXZ model (Fig. 3(d)) because having more observables in O with L = 12 improves the spatial resolution of the observables.

Conclusion

We have successfully combined DMD, a well-known data-driven predictive method with quantum data to predict long term observables in a quantum system. By utilizing predictive methods like DMD, we can run quantum simulations for a small number of time steps and predict future results. Our method can be applied to any other dynamical systems as well and will find application in other forms of property prediction for quantum systems. The method could be used in multiple dynamical problems to obtain meaningful results in the near term quantum devices.

Acknowledgements

The authors would like to thank Yizhi Shen and Nathan Wiebe for useful discussions. This work was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research through the Accelerated Research in Quantum Computing Program (NG, WAdJ). This work was also supported by the Center for Computational Study of Excited-State Phenomena in Energy Materials (C2SEPEM) at the Lawrence Berkeley National Laboratory, which is funded by the U. S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract No. DE-AC02-05CH11231.

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1. No explicit wave function reconstruction is required for long-time observable prediction, allowing AI systems to bypass computationally prohibitive Trotter-based quantum circuit depths by instead relying on data from a set of short time measurements from a quantum computer to predict observables.

2. The system can perform Hamiltonian simulations on NISQ devices for a limited number of steps and then use the DMD framework to predict long time values of the set using the reconstruction formula: ⟨O(t)⟩ = Φm exp(omegamt)Φ†m ⟨O(t1)⟩.

3. The AI system can achieve an error bound scaling of at most O(t3/2), which is derived from Theorem 3, providing a quantifiable guarantee on the accuracy of long-time predictions based on the number of snapshots 'm' used in DMD.

4. The system can be applied to predict observables for long time simulation in complex systems like Hubbard models and XXZ spin models by collecting expectation values of observables like density matrix elements or spin correlations, as demonstrated in Figure 3.