Qutrit Circuits and Algebraic Relations: A Pathway to Efficient Spin-1 Hamiltonian Simulation

TL;DR

The turnover relation for the three-qutrit time propagator and its potential use in reducing circuit depth is introduced and whether this relation can be generalized for higher-dimensional quantum circuits, including a focused study on the one-dimensional spin-1 Heisenberg model is investigated.

Abstract

Quantum information processing has witnessed significant advancements through the application of qubit-based techniques within universal gate sets. Recently, exploration beyond the qubit paradigm to $d$-dimensional quantum units or qudits has opened new avenues for improving computational efficiency. This paper delves into the qudit-based approach, particularly addressing the challenges presented in the high-fidelity implementation of qudit-based circuits due to increased complexity. As an innovative approach towards enhancing qudit circuit fidelity, we explore algebraic relations, such as the Yang-Baxter-like turnover equation, that may enable circuit compression and optimization. The paper introduces the turnover relation for the three-qutrit time propagator and its potential use in reducing circuit depth. We further investigate whether this relation can be generalized for higher-dimensional quantum circuits, including a focused study on the one-dimensional spin-1 Heisenberg model. Our work outlines both rigorous and numerically efficient approaches to potentially achieve this generalization, providing a foundation for further explorations in the field of qudit-based quantum computing.

Figures (4)

• Figure 1: A list of qutrit unitary pairs $W_L(\theta, \cdots, \theta)$ and $W_R(\bm{\theta}_R)$, for which we test the approximate circuit relations through numerical minimization of the infidelity function \ref{['eq:CostFunction']}. The top 6 rows correspond to the circuit reflection for different Trotter schemes \ref{['eq:trot1']}-\ref{['eq:trot6']}. While the unitaries in the bottom row are shown to be identical through \ref{['turnover1']}, under the parameter constraint of \ref{['eq:param']}, their equivalence is also tested numerically as a proxy to measure numerical deviations and limitations of optimizers.
• Figure 2: The minimized infidelity, $\log_{10}\min_{\bm{\theta}_R}\mathcal{C}(\theta, \cdots, \theta, \bm{\theta}_R)$, is obtained through parameter optimization of $\bm{\theta}_R$ across various spin couplings $J \in \{0.1, 0.55, 1.0\}$ and Trotter schemes $T \in \{T_1, T_2, \cdots, T_6\}$. The displayed values are on the logarithmic base 10 scale. Circuit diagrams of parameterized unitaries $W_L(\theta, \cdots, \theta)$ and $W_R(\bm{\theta}_R)$ for each Trotter scheme are shown in Fig. \ref{['fig:trotforms']}. The parameter optimization was performed using the BFGS algorithm. We consider the minimized infidelity to be reasonably low if it closely matches the 'lower bound', which solely accounts for the numerical inaccuracy of the exact identity \ref{['eq:lbound']}.
• Figure 3: The return probability \ref{['eq:return_prob']} of the spin-1 XY model on a three-site lattice, starting and ending at the state $|202\rangle$, is displayed as a function of time within $0 \leq t < 5$. The spin coupling is set at $J=1.0$. The best Trotter scheme $(T_2, n_b = 5)$ as from Fig. \ref{['fig:Trotforms']} is compared to another scheme $(T_3, n_b=2)$, with its data points represented as red and green dots.
• Figure 4: The return probability \ref{['eq:return_prob']} of the spin-1 XY model on a three-site lattice, starting and ending at the state $|202\rangle$, is shown as a function of time within $0 \leq t < 5$. Each plot correspond to respective spin-couplings $J=0.1$, $0.55$, $1.0$. The dynamic simulation is represented as the blue line. We employ the Trotterization of the time-evolution operator \ref{['eq:H_XY']} over 200 steps, with a corresponding step size of $\theta = 0.025$. Then we apply the circuit compression strategy detailed in Section \ref{['ssec:approx_identities']} to reduce number of gates. For Trotter schemes $(T_3, n_b=2)$, $(T_2, n_b=4)$ and $(T_2, n_b=5)$, the resulting data points are indicated by small red dots. More generally, for a $(T_3, n_b=2)$ Trotter circuit with more than $n>3$ steps, the approximate count of reduced gates is $2n/3$. For a $(T_2, n_b=4)$ Trotter circuit with more than $n > 4$ steps, the approximate count of reduced gates is $\lfloor 2n/5\rfloor - 1$. For a $(T_2, n_b=5)$ Trotter circuit with more than $n > 5$ steps, the approximate count of reduced gates is $n/3 - 1$ when $n$ is a multiple of 6, and $2\lfloor n/6\rfloor$ otherwise.