Researchers from Trinity College Dublin, Trinity Quantum Alliance, and Algorithmiq Limited have developed a quantum algorithm to estimate the density of states (DOS) on a digital quantum computer. Inspired by the classical kernel polynomial method (KPM), the algorithm computes moments of the expansion on quantum hardware. The team implemented their algorithm on the Quantinuum H11 trapped ion quantum simulator, approximating the DOS of a non-integrable spin chain for up to 18 qubits. This represents one of the first uses of near-term quantum computers for calculations in statistical mechanics.
Quantum Computing and Statistical Mechanics
A team of researchers from Trinity College Dublin, Trinity Quantum Alliance, and Algorithmiq Limited have developed a quantum algorithm to estimate the density of states (DOS) on a digital quantum computer. The algorithm is inspired by the kernel polynomial method (KPM), a classical method that allows for the sampling of spectral functions via a Chebyshev polynomial expansion. The team’s algorithm computes moments of the expansion on quantum hardware using a combination of random-state preparation for stochastic trace evaluation and a controlled unitary operator.
Quantum Simulation and Quantum Computers
The concept of using one quantum system to simulate another efficiently was proposed by physicist Richard Feynman over 40 years ago. This idea, known as quantum simulation, is expected to be one of the first real applications of the current generation of quantum computers. Recent progress has been made in simulating the dynamics of strongly correlated many-body systems on current devices. The hope is that the achievable system sizes will eventually become large enough to surpass what is classically possible.
Quantum Algorithms and Many-Body Systems
Regarding using quantum simulators to extract eigenenergies of many-body systems, early ideas include algorithms based on quantum Fourier transform, such as quantum phase estimation and adiabatic state preparation. The development of algorithms for extracting ground-state energies is central to the promise of performing quantum chemistry and materials simulations on quantum computers. Ground state energy calculation is a target of many variational quantum algorithms.
Quantum Computers and Statistical Mechanics
The idea of using quantum computers to do statistical mechanics is a topic that is gaining traction. In this work, the researchers focus on developing an algorithm that gives a coarse-grained estimate of the DOS based on the classical KPM. The KPM provides a reconstruction of a spectral function using a Chebyshev polynomial expansion weighted by suitable kernels to dampen the Gibbs oscillations that occur due to finite series truncation. Chebyshev moments are computed iteratively by applying Hamiltonian functions to some initial states. This step is a challenge to implement on quantum hardware.
Quantum Algorithm and Chebyshev Polynomials
In this work, the researchers devise a hybrid algorithm that uses a combination of pseudorandom-state preparation, Hadamard-test, and Suzuki-Trotter (ST) decomposition to evaluate Chebyshev moments. These moments are then used in the standard KPM expansion. The team uses an arccosine approximation of the Hamiltonian to implement Chebyshev polynomials from standard ST decomposition.
Quantum Simulator and Density of States
The team implemented their algorithm on the Quantinuum H11 trapped ion quantum simulator. They could approximate the DOS of a non-integrable spin chain for up to 18 qubits using a single ancillary qubit. Their simulations represent one of the first explorations of using near-term quantum computers for calculations in statistical mechanics.
Classical Kernel Polynomial Method for the Density of States
The classical KPM is able to approximate the DOS with memory scaling as O2^L when combined with a stochastic evaluation of the trace. The KPM provides an approximation of a function by a finite series of Chebyshev polynomials. The KPM expansion is reconstructed by computing the corresponding Chebyshev moments. The KPM can be adapted to estimate general spectral functions related to a given quantum mechanical Hamiltonian.
The article titled “Calculating the many-body density of states on a digital quantum computer” was published on January 26, 2024. The authors of this research paper are Alessandro Summer, Cecilia Chiaracane, Mark T. Mitchison, and John Goold. The paper discusses the complex calculations involved in determining the many-body density of states using a digital quantum computer.
Source: https://doi.org/10.1103/physrevresearch.6.013106
