Quantum simulation of real-time bosonic dynamics with binary and unary position encodings on superconducting and trapped-ion processors

Encoding Matters: Unary Beats Binary for Bosonic Simulations

Recent work by Hanada, Seo, Son, and Yoon (arXiv:2611.13079) tackles a practical obstacle in quantum simulation: how to map a continuous bosonic degree of freedom onto a finite set of qubits. The authors compare two straightforward discretizations—binary and unary position encodings—on two very different hardware platforms: IBM’s superconducting qubits and IonQ’s trapped‑ion system.

Binary vs. Unary Lattices

• Binary encoding stores the lattice index in the binary representation of a register of qubits. The kinetic term of the Hamiltonian is implemented via a quantum Fourier transform (QFT), which requires many two‑qubit gates.
• Unary encoding assigns one qubit to each lattice site. The potential term becomes a simple single‑qubit rotation, while the kinetic term is realized with excitation‑conserving exchange gates that act only between neighboring sites.

For a single bosonic mode, the unary formulation collapses to a free‑fermion problem, allowing the product‑formula evolution to be compressed exactly into a fixed‑size nearest‑neighbor circuit.

Experimental Results

The team ran both encodings on a symmetric double‑well potential. With roughly 200 two‑qubit gates, the unary circuits reproduced the expected tunneling dynamics on both platforms:

• IBM: right‑well probability dropped from 0.97 to 0.06.
• IonQ: right‑well probability dropped from 0.96 to 0.12.

In contrast, binary circuits required several thousand two‑qubit gates and failed to capture the transfer, highlighting the compilation overhead of the QFT‑based kinetic term.

Why the Gap?

• Gate‑count compression – Unary encoding leverages a fixed‑size circuit that scales linearly with lattice sites, whereas binary encoding’s QFT scales as O(n log n) in gate depth.
• Error mitigation – Post‑selection on the single‑excitation subspace removes leakage, and the symmetric nature of the grid ensures that measured population reversal directly bounds coherence between energy eigenstates.

Benchmarks Beyond the Simple Case

The authors also explored coupled‑oscillator systems where inter‑register interactions break the exact compression. These tests reveal that the optimal encoding depends on both lattice size and evolution time, suggesting a hybrid strategy may be needed for larger, interacting bosonic fields.

Takeaways for Practitioners

1. Encoding choice is a first‑order performance lever; unary can dramatically reduce circuit depth for low‑dimensional bosonic problems.
2. Hardware‑agnostic advantage – The same unary benefit appears on both superconducting and trapped‑ion devices, indicating that the effect is algorithmic rather than platform‑specific.
3. Post‑selection and symmetry provide inexpensive error‑mitigation pathways when the problem structure permits.

As quantum hardware continues to scale, such algorithm‑level optimizations will be essential for extracting meaningful physics before full error correction becomes available.

--- The study is available as a pre‑print on arXiv and includes detailed circuit constructions, gate‑count analyses, and experimental data for both platforms.