Dynamical simulation via quantum machine learning with provable generalization

TL;DR

A framework for using QML methods to simulate quantum dynamics on near-term quantum hardware and uses generalization bounds, which bound the error a machine learning model makes on unseen data, to rigorously analyze the training data requirements of an algorithm within this framework.

Abstract

Much attention has been paid to dynamical simulation and quantum machine learning (QML) independently as applications for quantum advantage, while the possibility of using QML to enhance dynamical simulations has not been thoroughly investigated. Here we develop a framework for using QML methods to simulate quantum dynamics on near-term quantum hardware. We use generalization bounds, which bound the error a machine learning model makes on unseen data, to rigorously analyze the training data requirements of an algorithm within this framework. This provides a guarantee that our algorithm is resource-efficient, both in terms of qubit and data requirements. Our numerics exhibit efficient scaling with problem size, and we simulate 20 times longer than Trotterization on IBMQ-Bogota.

Figures (3)

• Figure 1: QML framework for dynamical simulation. Our general framework (left panel) consists of using quantum training data, e.g., composed of input-output state pairs and/or input-output observable pairs, to train a time-dependent QNN, $V_{t}(\boldsymbol{\alpha})$. Typically the training occurs at a short time $\Delta t$, resulting in the trained QNN, $V_{\Delta t}(\boldsymbol{\alpha}_{\rm opt})$. The evolution of the system at some longer time $T$ is extrapolated via $V_{T}(\boldsymbol{\alpha}_{\rm opt})$. The Resource Efficient Fast Forwarding (REFF) algorithm (right panel) is an illustrative example of this framework. The training data consists of Haar-random product states as inputs and then the time-evolved states as outputs, i.e., evolved under $U_{\Delta t} \approx \exp(- i H \Delta t)$. The time-dependent QNN is a parameterized quantum circuit in a diagonal form: $W(\boldsymbol{\theta}) D_t(\boldsymbol{\gamma}) W^\dagger(\boldsymbol{\theta})$. Hence training the QNN amounts to approximately diagonalizing the short-time evolution $U_{\Delta t}$, resulting in the trained QNN: $W(\boldsymbol{\theta}_{\rm opt}) D_{\Delta t}(\boldsymbol{\gamma}_{\rm opt}) W^\dagger(\boldsymbol{\theta}_{\rm opt})$. This model can simulate the evolution of arbitrary input states up to time $T$ via $W(\boldsymbol{\theta}_{\rm opt}) D_T(\boldsymbol{\gamma}_{\rm opt}) W^\dagger(\boldsymbol{\theta}_{\rm opt})$.
• Figure 2: Numerical simulations. a) REFF is used to diagonalize the 4-qubit Heisenberg Hamiltonian with periodic boundary conditions. Only 5 Haar-random product training states are required to generalize over the whole Hilbert space, measured by the decrease in simulation infidelity $1-\overline{\mathcal{F}}_1$. The inset shows that the average fidelity $\overline{\mathcal{F}}_M \equiv \overline{\mathcal{F}} (\boldsymbol{\alpha}_{\rm opt}, M \Delta t)$ in this case remains over 0.95 for 2000 time steps. b) REFF is applied to increasing sizes of the XY model. For all sizes tested, a single Haar-random product training state was sufficient to achieve a machine precision simulation infidelity. For each system size, the ansatz is saved upon reaching $C^{\rm L}_{\rm REFF} = 10^{-14}$ and used to fast-forward the evolution, as shown in the inset.
• Figure 3: Quantum hardware implementation. a) The 2-qubit XY Hamiltonian is diagonalized via REFF on ibmq_bogota using $2^{16}$ measurement shots per circuit. The noisy $C^{\rm G}_{\rm REFF}$ cost is measured on ibmq_bogota, whereas the noise-free $C^{\rm G}_{\rm REFF}$ cost and $1-\overline{\mathcal{F}}_1$ are computed classically. b) After training, the fast-forwarded performance is compared to the iterated Trotter method for 5 Haar-random product states and we plot the mean and standard deviations.

Theorems & Definitions (1)

• Theorem 1: Simulation error for product-state training -- Informal