Kelvin Yip of the University of California, Merced, and colleagues have digitally simulated bosonic systems on a cloud-based superconducting quantum processor using the Holstein-Primakoff transformation, a technique overcoming the historically large Hilbert space required for these simulations. The researchers realized the driven harmonic oscillator and the Jaynes-Cummings model on quantum hardware, systematically examining how algorithmic limitations, specifically, qubit count and time evolution steps, interact with hardware-induced errors like gate infidelity and decoherence. This work advances digital quantum simulation of systems with bosonic degrees of freedom and provides a framework applicable to more complex models.

Holstein-Primakoff Transformation Maps Bosons to Qubit Ensembles

The Holstein-Primakoff transformation provides a pathway for representing a single bosonic mode using an ensemble of qubits, a technique that helps simulate complex quantum systems. This mapping allows quantum computers to emulate bosonic dynamics, a feat historically limited by the exponentially growing computational resources needed to represent the full bosonic Hilbert space. Collective spin operators, central to the HP transformation, are defined as sums of individual spin operators and adhere to angular-momentum commutation relations; these operators are then mapped to the operators of a single bosonic mode.

The resulting Hamiltonian, when applied to a qubit ensemble, simplifies to a form requiring only single-qubit quantum logic gates, a significant reduction in circuit complexity. Simulating these models on cloud-based superconducting quantum processors revealed that algorithmic errors arise from two primary constraints.

The finite number of qubits employed in the HP mapping introduces inaccuracies, as the qubit ensemble cannot perfectly represent the continuous bosonic mode. Equally impactful is the finite number of Trotter steps used to approximate time evolution, a standard technique in quantum simulation that introduces further discretization errors.

The researchers found that the Hamiltonian, after mapping onto qubit operators, becomes where Ω = √( ∆2 )2 + ( F 2√N )2, σiˆn denotes the Pauli operator along the rotated axis ˆn, and the constant term N∆ 2 has been neglected. This simplification is key to implementing the simulation with fewer quantum gates, but does not eliminate the errors stemming from qubit limitations and time evolution approximations.

The team’s approach allows for the emulation of a wide range of boson-related phenomena, but the fidelity of the simulation is directly tied to the scale of the qubit ensemble and the precision of the time evolution. The study highlights the delicate balance between these error sources in achieving meaningful quantum simulations of bosonic systems.

Simulating Driven Harmonic Oscillators with HP Transformation

Simulating the behavior of single bosonic modes on quantum hardware requires careful consideration of how to represent these systems within the constraints of available qubits; the work demonstrates a digital quantum simulation of the Holstein-Primakoff transformation, mapping a bosonic mode onto a spin ensemble for emulation. This approach directly addresses the challenge posed by the large Hilbert space intrinsically associated with bosonic systems, a factor historically limiting their simulation on programmable quantum processors.

These limitations, coupled with hardware noise sources like gate infidelity and readout errors, introduce inaccuracies into the simulation; the study details how these errors manifest in the emulation of bosonic dynamics and light-matter interaction. Specifically, the driven harmonic oscillator was analyzed using the HP mapping, with corresponding implemented on IBMQ hardware to assess the effects of qubit limitations and readout noise.

The Hamiltonian for a driven quantum harmonic oscillator, where ω0 is the oscillator frequency, ωd is the driving frequency, and F is the driving amplitude, was used as a benchmark. Applying the rotating-wave approximation, the Hamiltonian simplifies to a form where ∆ represents the detuning between the oscillator and the drive.

Impact of Qubit Count on Bosonic Mode Simulation

Mapping a bosonic mode onto an ensemble of superconducting qubits allows for the construction of that emulate the time evolution of that mode, as demonstrated by recent work using IBMQ, a cloud-based superconducting quantum processor. This technique moves beyond simple error metrics by providing a more detailed assessment of the simulation’s fidelity, acknowledging the complex interplay between algorithmic and hardware limitations.

“Under these conditions, it is appropriate to quantify the simulation accuracy using the averaged probability differences,” the researchers state, emphasizing the importance of a robust metric for evaluating the simulation’s performance. This focus on detailed calibration and validation procedures highlights the practical challenges of translating theoretical quantum simulations into tangible results on noisy intermediate-scale quantum computers.

Trotter Steps and Algorithmic Errors in Time Evolution

The study systematically examined how these algorithmic limitations interact with hardware-induced errors on cloud-based quantum processors, identifying optimal simulation parameters for currently available devices. Researchers quantified the deviation between ideal simulations and those run on actual hardware, defining a circuit error, ∆Pe, to reflect the impact of gate infidelity, decoherence and readout errors.

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.

Jaynes-Cummings Model Simulation via Suzuki-Trotter

Researchers assessed the impact of two-qubit gate errors on simulation performance by employing both the Suzuki-Trotter decomposition and a synthesized unitary approach to model the quantum dynamics of the Jaynes-Cummings model. This approach revealed the interplay between algorithmic errors and those stemming from the hardware itself, a key consideration for achieving accurate results.

The study builds on previous work in the field, citing A. Burger, L. C. Kwek and D. Poletti’s 2022 Entropy paper on digital quantum simulation of the spin-boson model, as well as F. Mei, V. M. Stojanovi´c, I. Siddiqi and L. Tian’s 2013 Physical Review B work on analog superconducting quantum simulation of Holstein polarons. These earlier studies provided foundational insights into simulating bosonic systems. The detailed analysis of error sources and the development of a framework for assessing simulation fidelity represent a step toward realizing the potential of quantum computers for simulating complex physical systems.

Hardware Errors: Gate Fidelity and Readout Effects

Cloud-based quantum platforms, such as IBM Quantum, are now enabling systematic benchmarking of simulation algorithms against the realities of hardware limitations like finite qubit coherence and gate errors. Researchers are focusing on how these physical imperfections interact with algorithmic approximations during digital quantum simulations, specifically when modeling systems with bosonic degrees of freedom. The circuit error, defined as the deviation between ideal qubit ensembles and results from actual hardware, generally increases with circuit depth, reflecting the accumulation of gate and measurement errors.

Conversely, algorithmic error, resulting from a finite number of qubits used in the Holstein-Primakoff transformation and limited Trotter steps, decreases as the system size grows and the Trotter approximation improves. However, as the number of qubits increases, the circuit error grows monotonically, indicating the accumulation of hardware imperfections, particularly readout inaccuracies.

This accumulation creates a competition with the diminishing algorithmic error, resulting in a minimum total deviation between the simulation and hardware results around an ensemble size of 12 for a driving strength of 0. 75. The researchers found that the algorithmic error vanishes when the number of qubits significantly exceeds the number of bosons being simulated.

This suggests that optimizing both algorithmic techniques and hardware fidelity is key for achieving accurate quantum simulations of complex bosonic systems. The study’s findings demonstrate the need to carefully balance the trade-off between algorithmic and hardware-induced errors to maximize simulation accuracy.

Balancing Algorithmic and Hardware Errors for Fidelity

Researchers systematically examined how finite qubit counts in the Holstein-Primakoff transformation and the number of Trotter steps combine with hardware noise, including gate and readout errors, to affect simulation fidelity. However, this increase occurs alongside a decrease in algorithmic error, stemming from the finite qubit representation of bosons and the approximation inherent in time evolution methods. This suggests an optimal system size exists for minimizing overall error. “∆Pe generally increases with KT as the circuit depth (and thus accumulated gate error) grows, whereas ∆Pa decreases with KT as the Trotter approximation improves,” the researchers report.

👉 More information
🗞 Digital quantum simulation of the Holstein-Primakoff transformation on noisy qubits
✍️ Kelvin Yip, Alessandro Monteros, Sahel Ashhab and Lin Tian
🧠 DOI: http://link.aps.org/doi/10.1103/67yq-8h8v

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