We implement the Hadamard test with Trotterized time-evolution operators on IBM quantum computers to simultaneously extract ground- and excited-state energies of the transverse field Ising model (TFIM) and transverse longitudinal field Ising model (TLFIM). The Trotterization circuits for the TFIM admit constant-depth circuits (CDCs) for arbitrary time, allowing us to locate a large number of eigen-energies above the background noise for up to six spins. Via circuit synthesis we show that the three-spin TLFIM has constant-depth structure although it does not meet the known CDC criteria. The CDCs enable the extraction of the ground and first-excited state energies of the three-site TLFIM via its dynamics. We also address complications from the noisy background and discrete Fourier transform to enhance the reliability of the extraction process and compare the results from different generations of IBM hardware to highlight the improvement.
We implement the Hadamard test with Trotterized time-evolution operators on IBM quantum computers to simultaneously extract ground- and excited-state energies of the transverse field Ising model (TFIM) and transverse longitudinal field Ising model (TLFIM). The Trotterization circuits for the TFIM admit constant-depth circuits (CDCs) for arbitrary time, allowing us to locate a large number of eigen-energies above the background noise for up to six spins. Via circuit synthesis we show that the three-spin TLFIM has constant-depth structure although it does not meet the known CDC criteria. The CDCs enable the extraction of the ground and first-excited state energies of the three-site TLFIM via its dynamics. We also address complications from the noisy background and discrete Fourier transform to enhance the reliability of the extraction process and compare the results from different generations of IBM hardware to highlight the improvement.
Under suitable assumptions, the algorithms in [Lin, Tong, Quantum 2020] can estimate the ground state energy and prepare the ground state of a quantum Hamiltonian with near-optimal query complexities. However, this is based on a block encoding input model of the Hamiltonian, whose implementation is known to require a large resource overhead. We develop a tool called quantum eigenvalue transformation of unitary matrices with real polynomials (QET-U), which uses a controlled Hamiltonian evolution as the input model, a single ancilla qubit and no multi-qubit control operations, and is thus suitable for early fault-tolerant quantum devices. This leads to a simple quantum algorithm that outperforms all previous algorithms with a comparable circuit structure for estimating the ground state energy. For a class of quantum spin Hamiltonians, we propose a new method that exploits certain anti-commutation relations and further removes the need of implementing the controlled Hamiltonian evolution. Coupled with Trotter based approximation of the Hamiltonian evolution, the resulting algorithm can be very suitable for early fault-tolerant quantum devices. We demonstrate the performance of the algorithm using IBM Qiskit for the transverse field Ising model. If we are further allowed to use multi-qubit Toffoli gates, we can then implement amplitude amplification and a new binary amplitude estimation algorithm, which increases the circuit depth but decreases the total query complexity. The resulting algorithm saturates the near-optimal complexity for ground state preparation and energy estimating using a constant number of ancilla qubits (no more than 3).
Abstract Dynamic simulation of materials is a promising application for near-term quantum computers. Current algorithms for Hamiltonian simulation, however, produce circuits that grow in depth with increasing simulation time, limiting feasible simulations to short-time dynamics. Here, we present a method for generating circuits that are constant in depth with increasing simulation time for a specific subset of one-dimensional (1D) materials Hamiltonians, thereby enabling simulations out to arbitrarily long times. Furthermore, by removing the effective limit on the number of feasibly simulatable time-steps, the constant-depth circuits enable Trotter error to be made negligibly small by allowing simulations to be broken into arbitrarily many time-steps. For an N -spin system, the constant-depth circuit contains only O(N2) O ( N 2 ) CNOT gates. Such compact circuits enable us to successfully execute long-time dynamic simulation of ubiquitous models, such as the transverse field Ising and XY models, on current quantum hardware for systems of up to 5 qubits without the need for complex error mitigation techniques. Aside from enabling long-time dynamic simulations with minimal Trotter error for a specific subset of 1D Hamiltonians, our constant-depth circuits can advance materials simulations on quantum computers more broadly in a number of indirect ways.
The quantum phase transitions provide a paradigm for studying collective quantum phenomena that are a result of competing non-commuting interactions. This paper will study the ground state properties and quantum critical dynamics of the one-dimensional transverse field Ising model through a combined perspective that includes exact diagonalisation, variational quantum eigensolver (VQE) simulations, and simulations on realistic physical quantum devices. We focus on a lattice of four spins, where we calculate the ground-state energies, magnetic order parameters and correlation functions at uniformly applied conditions, which is repeated by all systems. Precise diagonalisation provides both a benchmark, which is symmetry-conserving, and a depth-two, physics inspired variational approximation, which provides simulations accessible to hardware. The circuits that have been optimised identically are then placed on the IQM Garnet quantum processor, using a resource-efficient batched protocol. We find that the ground-state energies of shallow variational circuits are reliably captured by the circuit over the entire parameter space; the magnetic arrangement parameters and observables sensitive to correlation signal significantly more noise. The error analysis of quantitative analysis reveals a strong broadening of critical crossover on hardware, which is consistent with the noise attenuation of long-range correlations. These findings highlight the current capabilities as well as the fundamental limitations of noisy intermediate-scale quantum systems in modelling quantum critical phenomena as a benchmark to future enhancements in obtaining quantum hardware and quantum algorithms development.
A higher-order Suzuki-Trotter decomposition or Trotterization can be exploited to mitigate the Trotter error in digital quantum simulation. This work revisits the second-order symmetric Trotterization in terms of the Trotter error, where quantum many-body spin dynamics of the transverse-field Ising model is simulated. While the work presents a pedagogical way to exploit a real quantum computer, the effectiveness of the symmetric Trotterization is evaluated in a prototype superconducting quantum device on IBM Quantum Experience. It turns out that the symmetric Trotterization does not provide higher accuracy than the first-order Trotterization in the testbed using the transverse-field Ising model. The result indicates that apart from the quantum errors, such as logical gate error and readout error, the use of a higher-order Trotterization should be circumspect, and the Trotter error would play an insignificant role in particular applications in an early stage of realized noisy intermediate-scale quantum (NISQ) devices.
The transverse-field Ising model serves as a paradigm for studying confinement and excitation spectra, particularly the emergence of E8 symmetry near criticality. However, experimentally resolving the Ising meson spectroscopy required to verify these symmetries is challenging on near-term quantum hardware due to the depth of circuits required for real-time evolution. Here, we demonstrate improved spectroscopy of confined excitations using two distinct error-resilient circuit construction techniques on the IBM Torino device: first-order Trotter decomposition utilizing native fractional gates, and a tensor-network-based circuit compression via Riemannian optimization. By analyzing the Fourier spectrum of error-mitigated time-series data, we successfully identify key signatures of E8 symmetry despite hardware noise. These results validate the viability of both circuit compression and hardware-efficient compilation for probing complex topological phenomena on NISQ devices.
Unitary evolution under a time-dependent Hamiltonian is a key component of simulation on quantum hardware. Synthesizing the corresponding quantum circuit is typically done by breaking the evolution into small time steps, also known as Trotterization, which leads to circuits the depth of which scales with the number of steps. When the circuit elements are limited to a subset of SU(4)---or equivalently, when the Hamiltonian may be mapped onto free fermionic models---several identities exist that combine and simplify the circuit. Based on this, we present an algorithm that compresses the Trotter steps into a single block of quantum gates using algebraic relations between adjacent circuit elements. This results in a fixed depth time evolution for certain classes of Hamiltonians. We explicitly show how this algorithm works for several spin models, and demonstrate its use for adiabatic state preparation of the transverse field Ising model.