Qubit-based quantum simulators naturally target two-level systems, whereas many quantum many-body problems are intrinsically d level. Encodings from qudits to qubits then enlarge the Hilbert space and can introduce unphysical states that interfere with variational optimization. We formulate a variational framework for encoded d -level models that suppresses these illegitimate states with penalty terms and benchmark it for spin-1 and spin-3/2 bilinear-biquadratic Heisenberg chains. We compare binary encoding, which minimizes the qubit overhead, with symmetry encoding, which preserves the relevant spin symmetries and enables symmetry-conserving ansätze. Although binary encoding is more qubit efficient, its hardware-efficient ansatz is harder to train and less effective at exploiting conserved quantities. In contrast, symmetry encoding requires more qubits but reaches substantially higher fidelities, converges faster, and exhibits better trainability than the binary hardware-efficient ansatz. These results identify symmetry-preserving encodings as a practical route to simulating higher-spin models on existing qubit platforms.
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
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=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.
We propose a scheme for encoding many qubits in a single rotor, that is, a continuous and periodic degree of freedom. A key feature of this scheme is its ability to manipulate and entangle the encoded qubits with a single operation on the system. We also show, using quantum error-correcting codes, how to protect the qubits against small errors in angular position and momentum which may affect the rotor. We then discuss the feasibility of this scheme and suggest several candidates for its implementation. The proposed scheme is immediately generalizable to qudits of any finite dimension.
We describe methods for simulating general second-quantized Hamiltonians using the compact encoding, in which qubit states encode only the occupied modes in physical occupation number basis states. These methods apply to second-quantized Hamiltonians composed of a constant number of interactions, i.e., linear combinations of ladder operator monomials of fixed form. Compact encoding leads to qubit requirements that are optimal up to logarithmic factors. We show how to use sparse Hamiltonian simulation methods for second-quantized Hamiltonians in compact encoding, give explicit implementations for the required oracles, and analyze the methods. We also describe several example applications including the free boson and fermion theories, the ${\ensuremath{\phi}}^{4}$-theory, and the massive Yukawa model, all in both equal-time and light-front quantization. Our methods provide a general-purpose tool for simulating second-quantized Hamiltonians, with optimal or near-optimal scaling with error and model parameters.
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-
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.
Quantum computing provides a promising framework for the simulation of quantum field theories, where the computational cost depends both on the quantum algorithm employed and on the representation of the Hamiltonian. We investigate a formulation of the
φ4 model based on the harmonic-oscillator basis in coordinate space. We derive the lattice
φ4 Hamiltonian in this representation and analyze the structure of the resulting one-body matrix and interaction tensor. We show that both exhibit an effective band-diagonal structure, allowing controlled truncations of the Hamiltonian while preserving the low-energy spectrum. We validate this formulation by comparing low-energy observables obtained from numerical diagonalization with those computed in the standard harmonic-oscillator momentum-space representation. Finally, we estimate the resources required to encode the Hamiltonian on a quantum computer using both binary and unary boson-to-qubit mappings. By exploiting effective locality, the coordinate-space representation reduces the resources required for quantum simulation over a broad range of parameters.