Quantum Imaginary Time Evolution, Implemented Via Unitary Evolution, Efficiently Finds Hamiltonian Ground States for Computationally Hard Problems

Many challenging computational problems translate into finding the lowest energy state of a complex system, describable by a Hamiltonian. Andreu Anglés-Castillo, Ion Luca, and Pandit Tanmoy, alongside Gomez-Lurbe Rafael, Martínez Rodrigo, and Garcia-March Miguel Angel, investigate a powerful technique called Imaginary Time Evolution, or QITE, which offers a promising route to finding these crucial ground states. This method cleverly approximates the process of reaching the ground state by using a series of steps that can be directly implemented on quantum computing hardware, sidestepping some of the limitations of traditional approaches. The team presents a thorough review of this technique, alongside a complete and readily available computer program, offering a significant resource for researchers exploring quantum solutions to complex problems and providing a practical tool for advancing the field.

Variational and Imaginary Time Evolution Algorithms Compared

Scientists have made significant advances in finding the lowest energy state of complex systems, employing two distinct quantum algorithms, QITE and varQITE. The research team developed QITE, which approximates a process called imaginary time evolution using a series of quantum operations implementable on a quantum computer. Experiments demonstrate that, under specific conditions, the number of quantum operations required for one step of QITE is comparable to a key operation within the varQITE algorithm. Achieving convergence in energy, however, requires multiple steps, with the initial fidelity of the solution being a key consideration.

The team also investigated varQITE, a variational algorithm that constructs a quantum state and updates its parameters to minimize energy. Unlike QITE, varQITE primarily requires calculating the energy of the system, reducing computational demands. The researchers implemented both algorithms using a 16-qubit system on a desktop computer, utilizing specialized software for solving the linear equations within QITE. The performance of varQITE is heavily influenced by the chosen structure of the quantum state, or ansatz, and its initial parameter configuration, as no single optimal design currently exists.

The study did not observe evidence of the barren plateau phenomenon, a common obstacle in quantum algorithms where gradients vanish, but the team acknowledges that this does not guarantee convergence to the true lowest energy state. The research highlights a fundamental trade-off between the ability to effectively train the algorithm and the capacity of the quantum state to accurately represent the target solution. To potentially combine the strengths of both approaches, scientists propose a hybrid method, using QITE to approach the ground state and then using the resulting state as the starting point for varQITE, potentially avoiding barren plateaus.

Ground State Finding With Quantum Algorithms

This research presents two algorithms, Quantum Imaginary Time Evolution (QITE) and variational QITE (varQITE), designed to find the ground state of a problem Hamiltonian, a crucial step in solving many computationally challenging problems. QITE achieves this by approximating imaginary time evolution using quantum operations suitable for implementation on a universal quantum computer, while varQITE constructs a parameterized quantum state and optimizes its parameters using a natural gradient descent approach. The team demonstrated both algorithms on the transverse-field Ising model, highlighting the strengths and weaknesses of each method. The study acknowledges a fundamental trade-off between the trainability of the algorithm and its expressivity, its capacity to accurately represent the target ground state.

While simulations did not reveal evidence of the barren plateau problem, where gradients vanish hindering optimization, the researchers caution that this does not guarantee convergence to the true ground state, particularly as system size increases. They suggest a promising avenue for future work involves a hybrid approach, leveraging QITE to obtain an initial state close to the ground state, then using this as a starting point for varQITE to mitigate barren plateau issues. This research contributes to the ongoing development of quantum algorithms for tackling complex computational problems and identifies key considerations for maximizing performance and scalability.

Advancing Quantum Simulations of Complex Systems

Scientists have achieved significant advances in finding the lowest energy state of complex systems, employing two distinct quantum algorithms, QITE and varQITE. The research team developed QITE, which approximates a process called imaginary time evolution using a series of quantum operations implementable on a quantum computer. Experiments demonstrate that, under specific conditions, the number of quantum operations required for one step of QITE is comparable to a key operation within the varQITE algorithm. However, achieving convergence in energy requires multiple steps, with the initial fidelity of the solution being a key consideration.

The team also investigated varQITE, a variational algorithm that constructs a quantum state and updates its parameters to minimize energy. Unlike QITE, varQITE primarily requires calculating the energy of the system, reducing computational demands. The researchers implemented both algorithms using a 16-qubit system on a desktop computer, utilizing specialized software for solving the linear equations within QITE. The performance of varQITE is heavily influenced by the chosen structure of the quantum state, or ansatz, and its initial parameter configuration, as no single optimal design currently exists.

The study did not observe evidence of the barren plateau phenomenon, a common obstacle in quantum algorithms where gradients vanish, but the team acknowledges that this does not guarantee convergence to the true lowest energy state. The research highlights a fundamental trade-off between the ability to effectively train the algorithm and the capacity of the quantum state to accurately represent the target solution. To potentially combine the strengths of both approaches, scientists propose a hybrid method, using QITE to approach the ground state and then using the resulting state as the starting point for varQITE, potentially avoiding barren plateaus.