Universal Euler-Cartan Circuits for Quantum Field Theories

TL;DR

The presented algorithm relies on a universal parametrized quantum circuit ansatz based on Euler and Cartan's decompositions of single and two-qubit operators for the computation of non-perturbative characteristics of quantum field theories.

Abstract

Quantum computers can efficiently solve problems which are widely believed to lie beyond the reach of classical computers. In the near-term, hybrid quantum-classical algorithms, which efficiently embed quantum hardware in classical frameworks, are crucial in bridging the vast divide in the performance of the purely-quantum algorithms and their classical counterparts. Here, a hybrid quantum-classical algorithm is presented for the computation of non-perturbative characteristics of quantum field theories. The presented algorithm relies on a universal parametrized quantum circuit ansatz based on Euler and Cartan's decompositions of single and two-qubit operators. It is benchmarked by computing the energy spectra of lattice realizations of quantum field theories with both short and long range interactions. Low depth circuits are provided for false vacua as well as highly excited states corresponding to mesonic and baryonic excitations occurring in the analyzed models. The described algorithm opens a hitherto-unexplored avenue for the investigation of mass-ratios, scattering amplitudes and false-vacuum decays in quantum field theories.

Figures (3)

• Figure 1: (a) Schematic of the proposed algorithm initialized with a state $|{\rm \psi_i}\rangle$. The quantum processor (maroon dotted line) is involved in the application of $N$ layers of parametrized unitary operators yielding the state $|{\rm \psi_f}\rangle$, followed by measurements of a cost-function and relevant gradients. A classical processor (blue dotted line) verifies the iteration convergence criterion. This checks if the results for the cost-function are converged when compared to that obtained in the previous iteration step for a given $N$. If the iteration convergence criterion is not met, the parameters of the $N$ unitary operators are updated using an optimization routine. Some typical examples are BFGS and the quantum natural gradient (QNG). The updated set of parameters are used in the next iteration and this process is repeated until the desired iteration convergence criterion is reached. Subsequently, a second layer convergence criterion is checked by comparing the result to that obtained with $N-1$ layers. If the convergence criterion is satisfied, the entire process terminates, else the number of layers is grown by 1 and the previous steps are repeated. (b) Decomposition of each layer unitary operator in terms of general two-qubit unitary operators (orange rectangles) and those after Euler and Cartan's KAK decompositions and compression (orange ellipses). (c) Cartan's KAK decomposition of a general SU(4) two-qubit unitary operator in terms of four single-qubit SU(2) operators and the SU(4) entangler. (d) Euler's decomposition of a general SU(2) operator in terms of rotations by three angles about two non-parallel axes (chosen here to be $\hat{y}$ and $\hat{z}$). The SU(4) entangler is decomposed into three controlled-NOT gates and five single-qubit rotations. (e) The two-qubit operator after Euler's and Cartan's decompositions used for the optimization process. (f) Legends for the different symbols used. The symbol $R_\alpha$ stands for the SU(2) operator $e^{-i\theta\alpha}$, where $\alpha$ is one of the three Pauli matrices and $\theta$ is an unspecified angle determined by the optimization procedure. Two boxes of the same color perform different amounts of rotations about the same axis.
• Figure 2: (a) - (c) Results obtained for the ground state energies ($E_0$) of the Ising, 3-state Potts and the massive Schwinger models using the optimized Euler-Cartan circuits. A periodic chain of $L$ qubits is subject to $N$ layers for all three models. A translation-invariant Euler-Cartan circuit ansatz is optimized for panels (a, b), while the most general ansatz is considered in panel (c). The different values of $N~(L)$ are distinguished by the different colors (markers). For the lowest value of $N$, a uniform choice of $\theta_0 = 0.1$ was used as a starting guess for all of the angles being optimized. Once the iteration convergence criterion is met for a chosen value of $N$, the number of layers is increased by one: $N\rightarrow N+1$. While optimizing for $N+1$ layers, the previously obtained results for the $N$ layers was used as a starting guess, augmented by an initial guess of $\theta_0/10$ for the last layer. This process was continued until the layer-convergence criterion was reached. Both convergence criteria were chosen to be $5\times10^{-4}$. The learning rates were chosen to be $0.04$ for panels (a), (b) and 0.1 for panel (c). (d) Energies and fidelities to the corresponding target states for the smallest and largest values of $N$ for $L = 16$ are compared to the DMRG results. The smallest choice of $N=2$ layers yielded results within $1\%$ error of the target energies, while the additional layers caused changes in the third and fourth significant digits. While the precision for the massive Schwinger model is lower due to the all-to-all couplings in the spin Hamiltonian [Eq. \ref{['eq:H_MS']}] when compared to the other two models, increasing the number of layers leads to further improvement in the obtained results (see also Ref. Rogerson2024). Note that the exact values of the changes in the energies with layers depend on the specific choices of the coupling constants and the distances between the initial and target ground states.
• Figure 3: (a) - (b) Results for the first seven excited states obtained using optimized Euler-Cartan circuits for the Ising [(a)] and Potts [(b)] models encoded in a periodic chain of $L = 8$ qubits. The error percentages when compared to exact diagonalization results (black dotted lines) are shown in the top left insets. The variation of the obtained energies with number of layers is shown in the bottom right insets. The energies were obtained with a convergence criterion of $10^{-3}(10^{-4})$ for the Ising (Potts) models with the same learning rate of 0.02. In both cases, a translation-invariant circuit ansatz was used. The Euler-Cartan circuit for the $i^{\rm th}$ state was obtained by minimizing the Hamiltonian cost-function while orthogonalizing against the previously obtained states. Furthermore, in the Potts model, another term was added to the cost-function to obtain states with $+1$ eigenvalue for the operator $C$ (see maintext). The seventh translation-invariant excited state in the Ising (Potts) models corresponds to the $48^{\rm th}(42^{\rm nd})$ excited state in the spectrum. (c) - (d) Schematic of the confinement phenomena in the Ising and Potts models in the presence of a magnetic field. The latter breaks the two (three)-fold degeneracy of the ground state leading to one true ground state and one (two) false vacua in the Ising (Potts) case. As a consequence, the domain-walls experience a confining force (yellow lines with stripes), leading to the formation of mesonic and baryonic excitations. The non-monotonicity in the convergence for highly excited states can be improved by lowering the learning rates used in the circuit optimization process.