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arXiv:2511.10124

First and second quantized digital quantum simulations of bosonic systems

Mathias Mikkelsen, Hubert Okadome Valencia

quant-phSubmitted Nov 13, 2025• Announced Nov 14, 2025• 0 citations• Revised Jul 21, 2026 (v2)
v1 · Nov 13, 2025v2 · Jul 21, 2026
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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. Our comparison focuses on the $k$-body reduced density matrix ($k$-RDM) and 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 $R_z(φ)$ gates necessary to express a single Trotter step of the Hamiltonian in the binary first quantized mapping is comparable to the (most efficient for a single Trotter step) unary first quantized one when $M = 2^n$ for both the Bose-Hubbard model and the harmonic trap with short-range interactions. Additionally the binary mapping leads to lower one-norms than the unary mapping making it the overall most efficient choice for qubitization-based quantum phase estimation.

Comments
14 pages, 7 figures (updated version which included a trotter error analysis and a 1-norm comparison for the BHM and HO hamiltonians in order to better quantify total resource usage, as well as a variety of minor fixes)

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We compare the basic resource requirements for first and second quantized bosonic mappings in a system consisting of NNN particles in MMM modes. In addition to the standard binary first quantized mapping, we investigate the unary first quantized mapping. Our comparison focuses on the kkk-body reduced density matrix (kkk-RDM) and two standard bosonic Hamiltonians. The first quantized mappings use less resources for off-diagonal terms of the kkk-RDM by a factor of ∼Nk \sim N^k∼Nk, compared to the second quantized mappings. The number of gates for the first quantized binary mapping increases faster with MMM 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 NNN and MMM, while requiring exponentially fewer qubits than the unary mappings. Additionally, the number of CNOT and Rz(φ)R_z(φ)Rz​(φ) gates necessary to express a single Trotter step of the Hamiltonian in the binary first quantized mapping is comparable to the (most efficient for a single Trotter step) unary first quantized one when M=2nM = 2^nM=2n for both the Bose-Hubbard model and the harmonic trap with short-range interactions. Additionally the binary mapping leads to lower one-norms than the unary mapping making it the overall most efficient choice for qubitization-based quantum phase estimation.
A primary objective of quantum computation is to efficiently simulate quantum physics. Scientifically and technologically important quantum Hamiltonians include those with spin-s, vibrational, photonic, and other bosonic degrees of freedom, i.e. problems composed of or approximated by d-level particles (qudits). Recently, several methods for encoding these systems into a set of qubits have been introduced, where each encoding's efficiency was studied in terms of qubit and gate counts. Here, we build on previous results by including effects of hardware connectivity. To study the number of SWAP gates required to Trotterize commonly used quantum operators, we use both analytical arguments and automatic tools that optimize the schedule in multiple stages. We study the unary (or one-hot), Gray, standard binary, and block unary encodings, with three connectivities: linear array, ladder array, and square grid. Among other trends, we find that while the ladder array leads to substantial efficiencies over the linear array, the advantage of the square over the ladder array is less pronounced. These results are applicable in hardware co-design and in choosing efficient qudit encodings for a given set of near-term quantum hardware. Additionally, this work may be relevant to the scheduling of other quantum algorithms for which matrix exponentiation is a subroutine.
Quantum simulations based on first-quantized approaches can offer lower space and gate complexities than second-quantized approaches. The gate requirements for these simulations can be further reduced by employing pseudopotentials, a technique widely used in classical quantum chemistry calculations to decrease the number of simulated electrons. However, implementing pseudopotentials in quantum circuits remains challenging due to their non-local nature. In this study, we proposed a method to efficiently implement non-local potentials using matrix product state (MPS) encoding. MPS encoding allows us to efficiently implement a Trotter step operator with a circuit depth of O(deg(Vloc)ndeg(Vloc)+8Lnnatomχ2+2n2)O(\mathrm{deg}(V_{\mathrm{loc}}) n^{\mathrm{deg}(V_{\mathrm{loc}})} + 8Lnn_{\mathrm{atom}}χ^2 + 2n^2)O(deg(Vloc​)ndeg(Vloc​)+8Lnnatom​χ2+2n2), where deg(Vloc)\mathrm{deg}(V_{\mathrm{loc}})deg(Vloc​) is the degree of the polynomial local potential, natomn_{\mathrm{atom}}natom​ is the number of atoms in the system, LLL is the maximum number of projector functions across all atoms, and nnn is the number of qubits used to represent the binary-encoded wave function in one-dimensional space. We demonstrated the effectiveness of this approach by applying it to a model of the ionization of a one-dimensional hydrogen atom driven by an ultrashort laser pulse. Our results showed that MPS encoding significantly reduces the required circuit depth compared to a direct unitary decomposition, providing a pathway for practical first-quantized simulations.
Abstract Simulation of quantum systems is expected to be one of the most important applications of quantum computing, with much of the theoretical work so far having focused on fermionic and spin- 12\frac{1}{2}21​ 1 2 systems. Here, we instead consider encodings of d -level (i.e., qudit) quantum operators into multi-qubit operators, studying resource requirements for approximating operator exponentials by Trotterization. We primarily focus on spin- s and truncated bosonic operators in second quantization, observing desirable properties for approaches based on the Gray code, which to our knowledge has not been used in this context previously. After outlining a methodology for implementing an arbitrary encoding, we investigate the interplay between Hamming distances, sparsity patterns, bosonic truncation, and other properties of local operators. Finally, we obtain resource counts for five common Hamiltonian classes used in physics and chemistry, while modeling the possibility of converting between encodings within a Trotter step. The most efficient encoding choice is heavily dependent on the application and highly sensitive to d , although clear trends are present. These operation count reductions are relevant for running algorithms on near-term quantum hardware because the savings effectively decrease the required circuit depth. Results and procedures outlined in this work may be useful for simulating a broad class of Hamiltonians on qubit-based digital quantum computers.
Simulating an open quantum system on a digital quantum computer often involves the use of Trotter-Suzuki (TS) Product Formulas (PF) to approximate the system's time evolution. Precise estimates for the required number of Trotter steps (and hence the overall gate count) can be crucial for minimising the computational cost of these methods. Building on established theoretical guarantees, we derive analytic bounds for the First- and Second-Order Deterministic and Randomised TS-PF, directly relating the number of Trotter steps to the model parameters, evolution time and precision. These bounds enable concrete resource estimation for each method. We then present a computationally efficient classical algorithm that uses diamond norm estimates of individual Liouvillian terms and a binary search to significantly reduce the Trotter steps required for a target precision. Our numerical results on two prototypical models - an XX-Spin Chain with boundary driving and local dephasing, and a Transverse-Field Ising Model - show that the theoretical (analytic) bounds are often overly conservative, whereas the empirical (optimised) bounds yield a significantly smaller number of Trotter steps for the same precision. Among the methods investigated, the Second-Order Randomised TS-PF typically achieves the lowest resource demands, especially for larger systems. These findings emphasise the significance of empirical bounding strategies in achieving more resource-efficient simulations of Markovian open quantum systems.
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