Transmon qutrit-based simulation of spin-1 AKLT systems

TL;DR

This work demonstrates transmon qutrits as a hardware-efficient platform for simulating spin-1 AKLT physics by calibrating high-fidelity $0$-$1$ and $1$-$2$ manifold gates and compiling qutrit circuits that prepare open-boundary AKLT ground states and measure a topological Berry phase under bond perturbations. It combines hardware experiments with scalable noisy tensor-network simulations to compare qutrit and qubit approaches, showing a robustness advantage for qutrit-based encodings in realistic noise. The results establish a practical pathway for exploring spin-1 physics, topological order, and related phenomena in chemistry and magnetism using transmon qutrits, including scalable methods for state preparation and Berry-phase computation. The work also introduces algorithmic and circuit-assembly strategies (MPS-based, Hadamard-test) to enable larger spin-1 systems on near-term quantum hardware.

Abstract

Qutrit-based quantum circuits could help reduce the overall circuit depths, and hence the effect of noise, when the system of interest has a local dimension of three. Accessing second excited states in superconducting transmons provides a straightforward hardware realization of qutrits useful for such ternary encoding. In this work, we successfully calibrate microwave pulse gates to a low error rate to operate transmon qutrits. We use these qutrits to simulate one-dimensional spin-1 AKLT states (Affleck, Kennedy, Lieb, and Tasaki), which exhibit a multitude of interesting phenomena, such as topologically protected ground states, string order, and the existence of a robust Berry phase. We demonstrate the efficacy of qutrit-based simulation by preparing high-fidelity ground states of the AKLT Hamiltonian with open boundaries for various chain lengths. We then use ground state preparations of the perturbed AKLT Hamiltonian with periodic boundaries to calculate the Berry phase and illustrate non-trivial ground state topology. To establish the advantage of qutrits over qubits in the presence of noise, we present scalable methods for preparing the AKLT state and computing its Berry phase using tensor network simulations. Our work provides a pathway toward more general spin-1 physics simulations using transmon qutrits, with applications in chemistry, magnetism, and topological phases of matter.

Figures (22)

• Figure 1: The Figure describes a broad overview of this work. (A) We realize qutrits by accessing the second excited states of the transmons by applying calibrated microwave pulse gates driven at $\omega_{12}$ frequency (Section \ref{['qutrit gate section']}). These pulse gates are used to implement qutrit-based circuits compiled classically (B) using their corresponding pulse schedule (C). (D) We elucidate the pertinency of these transmon qutrit-based circuits by simulating spin-1 systems (AKLT (Section \ref{['Theory']}) in this work) in Sections \ref{['OBC']}, \ref{['PBC']}, and \ref{['BP section']} which would be costlier in terms of depth and number of entangling gates in the qubit setting. The advantage of using qutrits over qubits is further elucidated by our noisy simulations in a simplified noisy setting.
• Figure 2: In spin-1 systems, each spin-1 site can be represented by a tensor product of a pair of spin-1/2 sites (a $\&$ b) followed by their projection to the spin-1 sector. In AKLT ground states, these adjacent spin-1 sites decompose as pairs of two spin-1/2 sites (a $\&$ c) that form singlets (valence bond states) between them.
• Figure 3: Circuit ansatzes for OBC AKLT ground states and their hardware Hellinger fidelities: (A-C) capture all four ground states of $n=2,3$ and $4$ OBC AKLT. The exact parameters and sequence of pulse gates used in these single-site gates can be found in the Appendix \ref{['github']} (D) Table contains the Hellinger fidelities (See S2) of all four ground states corresponding to $n=2,3$, and $4$ except for the one ground state of $n=2$, which is fully confined to the $0-1$ manifold and has $99\% - 100 \%$ Hellinger fidelity.
• Figure 4: Ground states of Perturbed PBC AKLT ground states The ground state vector $\alpha(\theta)$$\ket{02}$ + $\beta(\theta)$$\ket{11}$ + $\gamma(\theta)$$\ket{20}$'s dependence on the perturbing angle $\theta$ is described in Figures A and B. We note that $|\alpha| = |\gamma|$ for all perturbation angles. At $\theta = \pi$, we observe that $\beta = 0$, coinciding with the point where the phase difference experiences a sudden discontinuity. (C) $\mathbf{n=2}$PBC ansatz captures the perturbed PBC ground states of $n=2$ AKLT for all perturbations. However, for $\theta = \pi$, ansatz with just a single entangling gate exists (See Fig. \ref{['fig:obc_hardware']} A). (D) Table contains the Hellinger fidelities of the $n=2$ perturbed PBC AKLT states for various perturbation angles. These angles are used for the computation of the Berry Phase in Section \ref{['berry_hardware']}.
• Figure 5: (A) Scalable AKLT state preparation. The ground state of the PBC AKLT model with perturbation angle $\theta$ is generated from an equal superposition of the $\ket{00}$ and $\ket{11}$ post-selected states. Note that for qubit-based implementation, each bulk qutrit must be encoded into two qubits. The $H$ and $RZ$ gates are standard qubit gates on a qubit device and the equivalent 0-1 manifold qutrit gates on a qutrit device. The entangling operation $V$ can be derived from the tensors of the MPS representation of the ground state. (B) Simulation of state preparation in the presence of noise Qutrit-based implementation consistently has higher fidelity than qubit-based implementation. The projected noise model uses $T_1$ and $T_2$ times that are five times longer than the ones measured on the physical device.