Digital Quantum Simulation of the Holstein-Primakoff Transformation on Noisy Qubits

TL;DR

This study advances the digital quantum simulation of many-body systems involving bosonic degrees of freedom on currently available cloud quantum processors and provides a framework that can be extended to more complex spin-boson and multimode cavity models.

Abstract

Quantum simulation of many-body systems offers a powerful approach to exploring collective quantum dynamics beyond classical computational reach. Although spin and fermionic models have been extensively simulated on digital quantum computers, the simulation of bosonic systems on programmable quantum processors is often hindered by the intrinsically large Hilbert space of bosonic modes. In this work, we study the digital quantum simulation of bosonic modes using the Holstein-Primakoff (HP) transformation and implement this protocol on a cloud-based superconducting quantum processor. Two representative models are realized on quantum hardware: (i) the driven harmonic oscillator and (ii) the Jaynes-Cummings model. Using data obtained from the quantum simulations, we systematically examine the interplay between algorithmic and hardware-induced errors to identify optimal simulation parameters. The dominant algorithmic errors arise from the finite number of qubits used in the HP mapping and the finite number of Trotter steps in the time evolution, while hardware errors mainly originate from gate infidelity, decoherence, and readout errors. This study advances the digital quantum simulation of many-body systems involving bosonic degrees of freedom on currently available cloud quantum processors and provides a framework that can be extended to more complex spin-boson and multimode cavity models.

Figures (8)

• Figure 1: Schematic workflow of the procedure from mapping a bosonic mode onto a qubit ensemble, circuit construction, IBMQ execution, to the analysis of the measured data. Both classical (noiseless) and quantum simulations are performed
• Figure 2: Circuit diagram for the digital simulation of the driven harmonic oscillator using six qubits, up to a global phase of $\pi$. The qubit labels (e.g., 81 and 54) indicate their physical indices on the IBM Torino. The circuit is transpiled into the native IBMQ gate set and followed by qubit measurements. The phases of the $R_z$ gates correspond to the parameters $\Delta = 1$, $F = 0.75$, and evolution time $t = \pi$, and are shown inside the blue boxes of the gates. All parameters are expressed in dimensionless units.
• Figure 3: Probabilities of the ground state (blue, top), first excited state (red, middle), and second excited state (green, bottom) of the driven harmonic oscillator as functions of $t/T_{0}$. Solid lines: the analytical results for an ideal harmonic oscillator $P_{\mathrm{b}}^{(n)}$; circles: the ideal qubit ensemble simulated with Qiskit Aer $P_{\mathrm{c}}^{(n)}$; and crosses: results from noisy qubits on IBM Torino $P_{\mathrm{q}}^{(n)}$. Panels (a-c) correspond to $F = 0.3$ with $N = 3,\,5,\,7$ qubits, while panels (d-f) correspond to $F = 0.75$ with $N = 6,\,11,\,16$ qubits, respectively. The detuning is $\Delta = 1$, and $T_{0} = \pi/\Omega$ denotes the oscillation period.
• Figure 4: (a, c) Probability difference $\Delta P_{\mathrm{e}}^{(n)}$ as a function of qubit number $N$ for $F = 0.3$ and $F = 0.75$, respectively. (b, d) Probability difference $\Delta P_{\mathrm{tot}}^{(n)}$ as a function of $N$ for $F = 0.3$ and $F = 0.75$, respectively. Top lines and circles (blue) correspond to the ground state; middle lines and crosses (red) correspond to the first excited state; and bottom lines and triangles (green) correspond to the second excited state. The lines are drawn to guide the eye, while the symbols (circles, crosses, and triangles) represent simulation results.
• Figure 5: Circuit diagram of a single Suzuki-Trotter step (up to a global phase of $\pi$) for the digital simulation of the JC model using two qubits to represent the cavity mode ($q_0=\sigma_1$, $q_1=\sigma_2$) and one qubit coupled to the cavity mode ($q_2=\tau$). The circuit is transpiled into the native IBMQ gate set and followed by qubit measurements. The solid bars connecting pairs of qubits denote CZ gates. The phases of the $R_z$ gates correspond to the parameters $\omega_0 = \omega_z = 1.0$, $g = 0.1$, and evolution time $t = \pi$.