Digital-Analog Quantum Computing with Qudits

TL;DR

The paper generalizes digital-analog quantum computing to qudits by leveraging the Weyl–Heisenberg operator basis to conjugate an analog two-body Hamiltonian with local qudit unitaries, enabling universal simulation of arbitrary two-body qudit Hamiltonians using a fixed resource Hamiltonian. A constructive procedure translates target couplings into a linear system M vec t = T vec(h_p/h_s), with the number of analog blocks scaling as O(d^4 n^2); proofs are provided in the appendices. The authors illustrate the approach with a qutrit example (spin-1 BLBQ Ising-like model) and support the framework with numerical results showing favorable fidelity for banged-DAQC in certain parameter regimes compared to digital circuits. Overall, the work broadens the DAQC toolkit to higher-dimensional systems, offering potential advantages in resource efficiency and noise resilience for NISQ-era quantum simulations and applications such as quadrupolar spin dynamics and gauge-theory-inspired models.

Abstract

Digital-analog quantum computing with two-level systems is a computational paradigm that combines an analog Hamiltonian with single-qubit gates to achieve universality. We extend this framework to $d$-level systems by conjugating an analog Hamiltonian block with single-qudit gates drawn from the Weyl-Heisenberg basis, which provides a natural set of operations for qudit architectures. More specifically, we propose a protocol to simulate arbitrary two-body Hamiltonians with at most $O(d^4 n^2)$ analog blocks. The power of this approach is illustrated by the simulation of many-body qudit spin Hamiltonians including magnetic quadrupolar terms.

Figures (4)

• Figure 1: Example of the structure described in Eq. (\ref{['eq:SIS_Exp_Sum']}), where $G_1 = \sigma_x^1$ and $G_k = \bigotimes_{i=1}^n \sigma_x^i$ illustrate single-body local operators that conjugate the Hamiltonian $H_\text{S}$. The operators $G_m$ may include any combination of products of the Pauli matrices $\{\sigma_x, \sigma_y, \sigma_z\}$.
• Figure 2: Analytical example of the simulation of $H_\text{P}(\theta)$ for $n = 2$. Each row $i$ of matrix $M$ identifies the two-body operator in the Hamiltonian, while each column $j$ specifies the phase acquired by the operator upon conjugation. The total number of digital–analog blocks shown here is illustrative; the final implementation will not require nine blocks.
• Figure 3: Total simulation time and fidelities for bDAQC and DQC. Panel (a) compares the ideal total simulation time $t_A$ with the effective time $t^r_A$ after discarding blocks whose duration $t_i$ is too short to be implemented. Panel (b) shows the fidelity difference when simulating $H_\text{P}(\theta)$ using bDAQC and digital circuits. Four regimes appear in bDAQC: the initial point $\theta = 0$ and its adjacent points, the region $\theta < \pi/2$, the interval $\pi/2 < \theta < \pi$, and the endpoint $\theta = \pi$ and its adjacent points. The number of single-qutrit gates in these regimes is $0$, $18$, $27$, and $9$, respectively, which accounts for the observed variations in fidelity.
• Figure 4: Brick-wall circuit structure used to simulate $H_\text{P}(\theta)$ with the Hamiltonian $H$ for $n = 6$.