Scalable quantum dynamics compilation via quantum machine learning

TL;DR

For the first time, VQC is extended to systems on two-dimensional strips with a quasi-1D treatment, demonstrating a significant resource advantage over standard Trotterization methods and highlighting the method's promise for advancing quantum simulation tasks on near-term quantum processors.

Abstract

Quantum dynamics compilation is an important task for improving quantum simulation efficiency: It aims to synthesize multi-qubit target dynamics into a circuit consisting of as few elementary gates as possible. Compared to deterministic methods such as Trotterization, variational quantum compilation (VQC) methods employ variational optimization to reduce gate costs while maintaining high accuracy. In this work, we explore the potential of a VQC scheme by making use of out-of-distribution generalization results in quantum machine learning (QML): By learning the action of a given many-body dynamics on a small data set of product states, we can obtain a unitary circuit that generalizes to highly entangled states such as the Haar random states. The efficiency in training allows us to use tensor network methods to compress such time-evolved product states by exploiting their low entanglement features. Our approach exceeds state-of-the-art compilation results in both system size and accuracy in one dimension ($1$D). For the first time, we extend VQC to systems on two-dimensional (2D) strips with a quasi-1D treatment, demonstrating a significant resource advantage over standard Trotterization methods, highlighting the method's promise for advancing quantum simulation tasks on near-term quantum processors.

Figures (9)

• Figure 1: Tensor-network diagrams for different VQC cost functions. (a) The most naive cost function for computing two unitaries involves computing $\Tr[U^\dag V]$. This cost function, however, cannot be directly computed on a quantum computer. Additionally, for classical simulations, $U^\dag$ can be costly to write down as a matrix product operator (MPO). (b) The HST circuit is introduced to compute $\Tr[U^\dag V]$ on a quantum computer, and is sometimes considered as a classical cost function mizuta2022local.Green squares represent Hadamard gates. This circuit, however, requires doubling the number of qubits of the systems and introduces highly non-local gates. (c) In this work, we consider a QML-generated cost function, Eq. \ref{['eq:empirical_risk']}. The cost function avoids the non-locality introduced by HST and has only a linear dependence on the number of samples $N_s$, which makes it suitable for both NISQ implementations and numerical simulations.In this work, we focus on classical simulation as a primary tool for developing and testing our methods. However, it would be of future interest to extend these techniques to perform compilation on a quantum computer.
• Figure 2: Geometry of the PQC.$1$D (left) and $2$D (right) ansatz used for the PQCs are shown here, both with depth $\tau=3$. For $2$D, we use a "snake" indexing to treat the state as a quasi-1D chain. Each block represents an $\mathrm{SU}(4)$ unitary and unitaries with the same color are considered within the same circuit layer. For the translation-invariant (TI) circuit ansätze, we set the parameters in each layer equal to each other.
• Figure 3: In-distribution generalization. (a) We first train a PQC with no translation invariance (TI) on $N_s$ samples drawn i.i.d. from the random product state ensemble and test it on 100 states drawn i.i.d. from the same distribution. The training and testing loss results are shown on the solid and dashed lines respectively. As we increase the circuit depth $\tau$, the training and testing loss both get lower. As we increase $N_s$, we see that the training loss curves converge, and the testing loss curve approaches the training one. This indicates that an increased number of data samples improves the in-distribution generalization accuracy. (b) We compare the testing loss results between PQCs with translation invariance (TI, left) and ones without (right), as defined in Fig. \ref{['fig:cir']}. The faster convergence of the training loss curves of the translation invariant PQCs can be reasoned from its smaller number of variational parameters. Additionally, despite the Hamiltonian's open boundary condition, both PQCs converge to similar test losses. (c) Fixing the PQC depth at $\tau=4$, we now train our PQC on various system sizes. We observe that, as we increase the system size, the "per-site" cost, $\Tilde{C}_{\mathcal{D}_\mathcal{Q}}$, as defined in Eq. \ref{['eq:cost_local']} remains relatively unchanged for $n>20$. This shows our VQC results are scalable.
• Figure 4: Out-of-distribution generalization. To test out out-of-distribution generalization, we repeatedly apply a VQC trained at time $t=0.1$ to perform a long-time evolution of a computational basis state for total evolution time $T = 20$. We see that we can obtain accurate dynamics at long times for an $n=40$ (a) Heisenberg chain and (b) Heisenberg chain with random fields on each spin. The fidelity of the variational simulation drops from about $\approx 0.99999$ to $\approx 0.99$ between $T=0.1$ and $T=20$, respectively. (c) Using the HST, we compare the actual unitary infidelity to the upper and lower bounds of Prop. \ref{['tho:equiv']} for the $n=40$ Heisenberg model with $t=0.5$. We see that the test cost is close to the true Haar random risk in practice.
• Figure 5: Dynamics of $2$D Heisenberg model on a cylinder with a variationally compiled circuit. Here we target the simulation of the expansion dynamics of ultracold hardcore bosons on a quasi-2D optical lattice after suddenly turning off a strong confining potential jreissaty2013hen2010strongly. We use the mapping between the hard core boson and spin operators, $S^+ = b^{\dagger}, S^- = b, S^z = b^{\dagger} b - 1/2$. We compile the time evolution of a $t=0.1$ Heisenberg model on a $3\times 21$ lattice with periodic boundaries in the $x$ direction. Then, we quench the state $\ket{0\ldots111\ldots0}$, where the $\ket{1}$ states are exactly in the middle of the lattice corresponding to the confined bosonic cloud. (Left) In real space, we see that the hardcore bosons rapidly expand away from the center of the lattice on both directions. (Right) We calculate the structure factor $S_k = \sum_{\boldsymbol{r}} e^{i \boldsymbol{r}\cdot \boldsymbol{k}} \expval{S^+_j S^-_j}$ for all time steps and show that the dynamics of the momentum distribution exhibits two peaks corresponding to the expanding wings moving up and down in the lattice at roughly constant velocity.

Theorems & Definitions (7)

• Proposition 2.1: Equivalence of in- and out-of-distribution risks
• Definition A.1: Locally scrambling unitary ensemble
• Definition A.2: Locally scrambling ensemble of states
• Theorem A.3: Equivalence of Locally Scrambled Risks
• Corollary A.4
• Theorem B.1: Shallow $\mathrm{U}(1)$ RQCs are not locally scrambling ensemble
• proof