First and second quantized digital quantum simulations of bosonic systems

TL;DR

This paper compares first and second quantized quantum simulation methods for bosonic systems, finding that binary first quantized mappings are often more resource-efficient in terms of gates and qubits for realistic parameters.

Contribution

The study introduces and evaluates unary and binary first quantized mappings, demonstrating their advantages over second quantized methods in resource efficiency for bosonic simulations.

Findings

Binary first quantized mappings are more gate-efficient than second quantized mappings.

Unary first quantized mappings are the most qubit-efficient but less gate-efficient.

Binary first quantized mappings require fewer gates than second quantized ones for realistic N and M.

Abstract

We compare the basic resource requirements for first and second quantized bosonic mappings in a system consisting of $N$ particles in $M$ modes. In addition to the standard binary first quantized mapping, we investigate the unary first quantized mapping, which we show to be the most gate-efficient mapping for bosons in the general case, although less qubit-efficient than binary mappings. Our comparison focuses on the $k$-body reduced density matrix ($k$-RDM) as well as two standard bosonic Hamiltonians. The first quantized mappings use less resources for off-diagonal terms of the $k$-RDM by a factor of $ \sim N^k$, compared to the second quantized mappings. The number of gates for the first quantized binary mapping increases faster with $M$ compared to the other mappings. Nevertheless, a detailed numeric analysis reveals that the binary first quantized mapping still requires fewer gates than the binary and unary second quantized ones for realistic combinations of $N$ and $M$, while requiring exponentially fewer qubits than the unary mappings. Additionally, the number of CNOT and $R_z(\phi)$ gates necessary to express the exponential of the Hamiltonian in the binary first quantized mapping is comparable to the (overall most efficient) unary first quantized one when $M = 2^n$ for both the Bose-Hubbard model and the harmonic trap with short-range interactions. This suggests that this mapping can be both qubit- and gate-efficient for practical problems.