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arXiv:2505.15945 · Physical Review Research

Bloch oscillation with a diatomic tight-binding model on quantum computers

Peng Guo, Jaime Park, Frank X. Lee

Published Sep 8, 2025• 2 citations
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Abstract

We aim to explore a more efficient way to simulate few-body dynamics on quantum computers. Instead of mapping the second quantization of the system Hamiltonian to qubit Pauli gate representation via the Jordan-Wigner transform, we propose to use the few-body Hamiltonian matrix under the state-vector basis representation, which is more economical on the required number of quantum registers. For a single-particle excitation state on a one-dimensional chain, Γ qubits can simulate N = 2 Γ number of sites, in comparison to N qubits for N sites via the Jordan-Wigner approach. A two-band diatomic tight-binding model is used to demonstrate the effectiveness of the state-vector basis representation. Both one-particle and two-particle quantum circuits are constructed, and some numerical tests on IBM hardware are presented.

DOI
10.1103/271g-ntj6

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We aim to explore a more efficient way to simulate few-body dynamics on quantum computers. Instead of mapping the second quantization of the system Hamiltonian to qubit Pauli gate representation via the Jordan-Wigner transform, we propose to use the few-body Hamiltonian matrix under the state-vector basis representation, which is more economical on the required number of quantum registers. For a single-particle excitation state on a one-dimensional chain, Γ qubits can simulate N = 2 Γ number of sites, in comparison to N qubits for N sites via the Jordan-Wigner approach. A two-band diatomic tight-binding model is used to demonstrate the effectiveness of the state-vector basis representation. Both one-particle and two-particle quantum circuits are constructed, and some numerical tests on IBM hardware are presented.
Quantum simulation is an important application of future quantum computers with applications in quantum chemistry, condensed matter, and beyond. Quantum simulation of fermionic systems presents a specific challenge. The Jordan-Wigner transformation allows for representation of a fermionic operator by O(n) qubit operations. Here, we develop an alternative method of simulating fermions with qubits, first proposed by Bravyi and Kitaev [Ann. Phys. 298, 210 (2002); e-print arXiv:quant-ph/0003137v2], that reduces the simulation cost to O(log n) qubit operations for one fermionic operation. We apply this new Bravyi-Kitaev transformation to the task of simulating quantum chemical Hamiltonians, and give a detailed example for the simplest possible case of molecular hydrogen in a minimal basis. We show that the quantum circuit for simulating a single Trotter time step of the Bravyi-Kitaev derived Hamiltonian for H(2) requires fewer gate applications than the equivalent circuit derived from the Jordan-Wigner transformation. Since the scaling of the Bravyi-Kitaev method is asymptotically better than the Jordan-Wigner method, this result for molecular hydrogen in a minimal basis demonstrates the superior efficiency of the Bravyi-Kitaev method for all quantum computations of electronic structure.
The ability to perform classically intractable electronic structure calculations is often cited as one of the principal applications of quantum computing. A great deal of theoretical algorithmic development has been performed in support of this goal. Most techniques require a scheme for mapping electronic states and operations to states of and operations upon qubits. The two most commonly used techniques for this are the Jordan-Wigner transformation and the Bravyi-Kitaev transformation. However, comparisons of these schemes have previously been limited to individual small molecules. In this paper, we discuss resource implications for the use of the Bravyi-Kitaev mapping scheme, specifically with regard to the number of quantum gates required for simulation. We consider both small systems, which may be simulatable on near-future quantum devices, and systems sufficiently large for classical simulation to be intractable. We use 86 molecular systems to demonstrate that the use of the Bravyi-Kitaev transformation is typically at least approximately as efficient as the canonical Jordan-Wigner transformation and results in substantially reduced gate count estimates when performing limited circuit optimizations.
Simulating large-scale coupled-oscillator systems presents substantial computational challenges for classical algorithms, particularly when pursuing first-principles analyses in the thermodynamic limit. Motivated by the quantum algorithm framework proposed by Babbush et al. (2023), we present and implement a detailed quantum circuit construction for simulating one-dimensional spring–mass systems. Our approach incorporates key quantum subroutines, including block encoding, quantum singular value transformation (QSVT), and amplitude amplification, to realize the unitary time-evolution operator associated with simulating classical oscillators dynamics. In the uniform mass–spring setting, our circuit construction requires a gate complexity ofO(log⁡22N log⁡2(1/ε))\mathcal {O}\bigl (\log _{2}^{2} N\,\log _{2}(1/\varepsilon )\bigr )O(log22​Nlog2​(1/ε)), where N is the number of oscillators and$\varepsilon isthetargetaccuracyoftheapproximation.Formoregeneral,heterogeneousmass–springsystems,thetotalgatecomplexityisis the target accuracy of the approximation. For more general, heterogeneous mass–spring systems, the total gate complexity isisthetargetaccuracyoftheapproximation.Formoregeneral,heterogeneousmass–springsystems,thetotalgatecomplexityis\mathcal {O}\bigl (N\log _{2} N\,\log _{2}(1/\varepsilon )\bigr ).Bothsettingsrequire. Both settings require.Bothsettingsrequire\mathcal {O}(\log _{2} N)$qubits. Numerical simulations agree with classical solvers across all tested configurations, indicating that this circuit-based Hamiltonian simulation approach can substantially reduce computational costs and potentially enable larger-scale many-body studies on future quantum hardware.
Quantum simulation of fermionic systems is a promising application of quantum computers, but to program them, we need to map fermionic states and operators to qubit states and quantum gates. While quantum processors may be built as two-dimensional qubit networks with couplings between nearest neighbors, standard fermion-to-qubit mappings do not account for that kind of connectivity. In this work we concatenate the (one-dimensional) Jordan-Wigner transform with specific quantum codes defined under the addition of a certain number of auxiliary qubits. This yields a class of mappings with which any fermionic system can be embedded in a two-dimensional qubit setup, fostering scalable quantum simulation. Our technique is demonstrated on the two-dimensional Fermi-Hubbard model, which we transform into a local Hamiltonian. What is more, we adapt the Verstraete-Cirac transform and Bravyi-Kitaev superfast simulation to the square lattice connectivity and compare them to our mappings. An advantage of our approach in this comparison is that it allows us to encode and decode a logical state with a simple unitary quantum circuit.
We show in detail how the Jordan-Wigner transformation can be used to simulate any fermionic many-body Hamiltonian on a quantum computer. We develop an algorithm based on appropriate qubit gates that takes a general fermionic Hamiltonian, written as products of a given number of creation and annihilation operators, as input. To demonstrate the applicability of the algorithm, we calculate eigenvalues and eigenvectors of two model Hamiltonians, the well-known Hubbard model and a generalized pairing Hamiltonian. Extensions to other systems are discussed.
Background: The structure and dynamics of many-body systems are the result of a delicate interplay between underlying interactions. Fermionic pairing, for example, plays a central role in various physical systems, ranging from condensed matter to nuclear systems, where it can lead to collective phenomena such as superconductivity and superfluidity. In atomic nuclei, the interplay between pairing and particle-hole interactions leads to a high degree of complexity and intricate entanglement structures. Despite this apparent complexity, symmetries emerge and manifest themselves in observable regular patterns. These symmetries and their breakings have long been used to determine relevant degrees of freedom and simplify classical descriptions of many-body systems.Purpose: This work explores the potential utility of quantum computers with arrays of qudits in simulating interacting fermionic systems, when the qudits can naturally map the relevant degrees of freedom determined by an underlying symmetry group.Method: The Agassi model of fermions interacting via particle-hole and pairing interactions is based on an underlying so(5)\mathrm{so}(5)so(5) algebra. Such systems can intuitively be partitioned into pairs of modes with five basis states, which thus naturally map to arrays of d=5d=5d=5 qudits (qu5its). Classical noiseless simulations of the time evolution of systems with up to twelve qu5its are performed, by implementing quantum circuits that are developed herein, using python codes invoking Google's cirq software. The resource requirements of the qu5it circuits are analyzed and compared with two different mappings to qubit systems: a physics-aware Jordan-Wigner mapping requiring four qubits per mode pair and a state-to-state mapping requiring three qubits per mode pair.Results: While the dimensionality of Hilbert spaces in mappings to qu5it systems are less than those for the corresponding qubit systems, the number of entangling operations, depending on the available hardware, can either be greater or smaller than for the physics-aware Jordan-Wigner mapping. The state-to-state mapping, while having a smaller Hilbert space than Jordan-Wigner mappings, appears to be the least efficient in gate counts. Further, a previously unknown sign problem has been identified from Trotterization errors in time evolving high-energy excitations.Conclusions: There appear to be advantages in employing quantum computers with arrays of qudits to perform simulations of many-body dynamics that exploit the role of underlying symmetries, specifically in lowering the required quantum resources and in reducing anticipated errors that take the simulation out of the physical space. If the necessary entangling gates are not directly supported by the hardware, physics-aware mappings to qubits may, however, be advantageous for other aspects.