Wave equations provide a natural testbed for near-term quantum simulation of partial differential equations, but hardware demonstrations have remained limited in spatial dimension, equation class, system size, and physically meaningful output. We develop and benchmark structure-preserving, Fourier-based quantum circuits for the one- and two dimensional acoustic wave equations and Dirac dynamics with variable mass on the Quantinuum H2-2 trapped-ion processor. The experiments include one-dimensional grids with up to \(1024\) points and \(32\times32\) two-dimensional grids, corresponding to an encoded state-space dimension of up to \(4096\). Rather than reconstructing the full fields, we estimate subdomain kinetic energies directly from measurement samples. Across all tested acoustic and Dirac dynamics problems, the H2-2 results track the classical kinetic-energy dynamics with mean absolute errors between \(5.9\times10^{-3}\) and \(2.4\times10^{-2}\). At fixed retained bandwidth, the compiled gate counts grow approximately quadratically with the number of grid qubits; the acoustic circuit sizes are essentially independent of evolution time, whereas the cost also grows with the number of product-formula steps. These results provide hardware-level evidence that accurate observable dynamics can remain resolvable for structured wave problems with thousands of encoded degrees of freedom on a present-day trapped-ion processor.
Wave equations provide a natural testbed for near-term quantum simulation of partial differential equations, but hardware demonstrations have remained limited in spatial dimension, equation class, system size, and physically meaningful output. We develop and benchmark structure-preserving, Fourier-based quantum circuits for the one- and two dimensional acoustic wave equations and Dirac dynamics with variable mass on the Quantinuum H2-2 trapped-ion processor. The experiments include one-dimensional grids with up to (1024) points and (32\times32) two-dimensional grids, corresponding to an encoded state-space dimension of up to (4096). Rather than reconstructing the full fields, we estimate subdomain kinetic energies directly from measurement samples. Across all tested acoustic and Dirac dynamics problems, the H2-2 results track the classical kinetic-energy dynamics with mean absolute errors between (5.9\times10^{-3}) and (2.4\times10^{-2}). At fixed retained bandwidth, the compiled gate counts grow approximately quadratically with the number of grid qubits; the acoustic circuit sizes are essentially independent of evolution time, whereas the cost also grows with the number of product-formula steps. These results provide hardware-level evidence that accurate observable dynamics can remain resolvable for structured wave problems with thousands of encoded degrees of freedom on a present-day trapped-ion processor.
We investigate one-dimensional quantum wave-packet dynamics using a grid-based encoding of the wave function onto qubit registers. Time evolution is implemented using a split operator approach, with the kinetic energy operator realized via the Quantum Fourier Transform (QFT) and arbitrary discretized potentials represented by commuting Pauli-Z gates. The algorithm is benchmarked against classical grid-based simulations and validated using a noiseless Qiskit Statevector simulator before being implemented on current quantum hardware (IBM Quantum and IonQ) for qubit counts ranging from two to five. Good agreement with benchmark calculations is obtained for two- and three-qubit implementations, while deviations increase at larger qubit counts due to hardware noise and increased circuit depth. Among the devices examined, the algorithm maintains closer agreement with the benchmark on the trapped-ion IonQ Forte processor as the number of qubits increases.
We design and implement quantum circuits for the simulation of the one-dimensional wave equation on the Quantinuum H1-1 quantum computer. The circuit depth of our approach scales as O ( n 2 ) for n qubits representing the solution on 2 n grid points, and leads to infidelities of O ( 2 − 4 n t 2 ) for simulation time t assuming smooth initial conditions. By varying the qubit count we study the interplay between the algorithmic and physical gate errors to identify the optimal working point of minimum total error. Our approach to simulating the wave equation can be used with appropriate state preparation algorithms across different quantum processors and serve as an application-oriented benchmark. Published by the American Physical Society 2024
The quantum lattice Boltzmann scheme of Succi and Dellar advances a four-component Dirac spinor on a lattice by a fixed sequence of local, exactly norm-preserving operations: a basis rotation, a collision, a streaming shift, and the inverse rotation. While this correspondence has been noted qualitatively, it has remained unclear whether it can be realized exactly and unitarily as a gate-model quantum circuit. This unitarity is a structural property of the scheme, which should, in principle, enable an operator-by-operator port onto quantum hardware. We give a gate-level construction of every operation of the three-dimensional Dirac QLB scheme: the fixed rotation gates, the collision gate, the streaming shift as a controlled increment on a position register, the position-dependent potential as a phase oracle, and periodic and reflecting boundary conditions as unitary circuits. Together, these constructions provide the first complete demonstration that the scheme is exactly representable in the gate model without any approximation beyond the classical method itself. We then compose them into single-axis, two- and three-dimensional time steps. On a state-vector emulator the resulting circuits reproduce the classical QLB solver to machine precision, thus verifying our quantum implementation. Our aim is representability and correctness rather than computational speedup; questions of state preparation, asymptotic cost, and hardware noise are identified as open avenues for future work. Appendices restore physical units, derive the scheme's Zitterbewegung in closed form and compare it with a trapped-ion simulation, and execute a reduced two-site instance on a Rigetti superconducting processor, where choosing the qubits from the live calibration lowers the state-preparation error. All operators, circuits, and figures are reproducible from the open-source quantumKineticMethods library.
Universal digital quantum computers are able to approximate any Hamiltonian evolution as a digital quantum circuit consisting of discrete time steps. Strongly interacting physics models, for example in high-energy physics or materials science, make this a challenging task for near-term devices as they place a high demand on the quantum resources required [1]. One alternative is analog quantum simulation where the native evolution of a quantum system can be tailored to mimic that of the model of interest. While more efficient at specific problems, there are limits on the types of evolution available and on the observables that can be measured. In this conference paper we present first results towards unifying these two concepts in a trapped-ion experiment. Trapped atomic ions are one of the most advanced experimental platforms for both analog quantum simulation and digital quantum computing. Typically, the internal spin degrees of freedom are used qubits or for the simulation of spin models. The harmonic oscillator modes given by the motion of the ions in the trap are only employed transiently to create quantum gates or spin-spin interactions. However, these motional modes constitute a large Hilbert space themselves and present a bosonic degree of freedom, which is costly to model with qubits. Many systems of interest, for example Lattice gauge theories, require both spin and boson degrees of freedom. It was recently shown that they can be encoded more efficiently using the harmonic oscillator modes directly [2].
Explicit quantum circuits for the wave equation, built by Hamiltonian simulation, are restricted to closed domains, in which outgoing waves reflect off the edge of the computational region and return. We lift that restriction with absorbing boundaries of the convolution type - a complex-frequency-shifted perfectly matched layer realised through exponential-kernel memory variables - and make the resulting non-unitary dynamics quantum-implementable by Schrodingerisation. We obtain three results. First, explicit gate-level circuits for the absorbing evolution, obtained by extending a Bell- basis term-evolution circuit to projector-valued operator strings and Trotterising to second order; these run end to end on seventeen to twenty-one qubits and agree with exact references to within a tenth of a percent to a percent. Second, a structural obstruction: the memory form of the absorbing generator carries an irreducibly indefinite Hermitian part, whose largest eigenvalue grows as the square root of the absorption strength divided by the grid spacing and survives any diagonal rescaling of the memory fields. The known recovery threshold for Schrodingerisation then makes the post-selection cost grow exponentially in the simulated time. Third, a remedy: a Lyapunov symmetrizer, precomputed classically, renders the transformed generator dissipative and replaces that time-dependent penalty with a time-independent conditioning factor of several hundred, measured on grids of up to four thousand unknowns and saturating under mesh refinement. The crossover is early: past it the symmetrized recovery is cheaper by four to twenty-six orders of magnitude in post-selection cost, and at the longest horizons tested it is the only recovery that works.
Abstract The ability to efficiently load functions on quantum computers with high fidelity is essential for many quantum algorithms, including those for solving partial differential equations and Monte Carlo estimation. In this work, we introduce the Fourier series loader (FSL) method for preparing quantum states that exactly encode multi-dimensional Fourier series using linear-depth quantum circuits. Specifically, the FSL method prepares a (Dn)-qubit state encoding the 2 Dn -point uniform discretization of a D-dimensional function specified by a D-dimensional Fourier series. A free parameter, m, which must be less than n, determines the number of Fourier coefficients, 2 D ( m + 1 ) , used to represent the function. The FSL method uses a quantum circuit of depth at most 2 ( n − 2 ) + ⌈ log 2 ( n − m ) ⌉ + 2 D ( m + 1 ) + 2 − 2 D ( m + 1 ) , which is linear in the number of Fourier coefficients, and linear in the number of qubits (Dn) despite the fact that the loaded function’s discretization is over exponentially many (2 Dn ) points. The FSL circuit consists of at most D n + 2 D ( m + 1 ) + 1 − 1 single-qubit and D n ( n + 1 ) / 2 + 2 D ( m + 1 ) + 1 − 3 D ( m + 1 ) − 2 two-qubit gates; we present a classical compilation algorithm with runtime O ( 2 3 D ( m + 1 ) ) to determine the FSL circuit for a given Fourier series. The FSL method allows for the highly accurate loading of complex-valued functions that are well-approximated by a Fourier series with finitely many terms. We report results from noiseless quantum circuit simulations, illustrating the capability of the FSL method to load various continuous 1D functions, and a discontinuous 1D function, on 20 qubits with infidelities of less than 10−6 and 10−3, respectively. We also demonstrate the practicality of the FSL method for near-term quantum computers by presenting experiments performed on the Quantinuum H1-1 and H1-2 trapped-ion quantum computers: we loaded a complex-valued function on 3 qubits with a fidelity of over 95 % , as well as various 1D real-valued functions on up