Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution (1901.07653v3)

Abstract: The accurate computation of Hamiltonian ground, excited, and thermal states on quantum computers stands to impact many problems in the physical and computer sciences, from quantum simulation to machine learning. Given the challenges posed in constructing large-scale quantum computers, these tasks should be carried out in a resource-efficient way. In this regard, existing techniques based on phase estimation or variational algorithms display potential disadvantages; phase estimation requires deep circuits with ancillae, that are hard to execute reliably without error correction, while variational algorithms, while flexible with respect to circuit depth, entail additional high-dimensional classical optimization. Here, we introduce the quantum imaginary time evolution and quantum Lanczos algorithms, which are analogues of classical algorithms for finding ground and excited states. Compared to their classical counterparts, they require exponentially less space and time per iteration, and can be implemented without deep circuits and ancillae, or high-dimensional optimization. We furthermore discuss quantum imaginary time evolution as a subroutine to generate Gibbs averages through an analog of minimally entangled typical thermal states. Finally, we demonstrate the potential of these algorithms via an implementation using exact classical emulation as well as through prototype circuits on the Rigetti quantum virtual machine and Aspen-1 quantum processing unit.

• The paper introduces QITE and QLanczos, novel quantum algorithms that calculate Hamiltonian ground, excited, and thermal states with exponential resource reduction.
• The paper demonstrates efficient simulation validations on both quantum hardware and classical emulators using models like Ising and Heisenberg.
• The paper highlights a clear path for leveraging near-term quantum devices for advanced quantum simulations and practical scientific computing.

Determining Eigenstates and Thermal States on a Quantum Computer Using Quantum Imaginary Time Evolution

The paper "Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution" presents significant advancements in quantum computing algorithms tailored to efficiently compute Hamiltonian ground, excited, and thermal states. The motivation for this work is rooted in addressing practical challenges faced by existing quantum algorithms like phase estimation and variational algorithms, specifically their demands for deep circuits and substantial classical optimization, respectively.

Key Algorithms Proposed

The authors introduce quantum analogues of classical algorithms, namely the Quantum Imaginary Time Evolution (QITE) and Quantum Lanczos (QLanczos) algorithms. These algorithms are designed to analogously function to their classical counterparts but with substantial reductions in space and time resources, achieved without reliance on deep quantum circuits or ancilla qubits:

1. Quantum Imaginary Time Evolution (QITE):
   • This algorithm approximates an imaginary-time evolution operation on quantum states, which results in convergence towards the ground-state of a Hamiltonian.
   • QITE's conceptual foundation lies in utilizing a unitary approximation method that applies a sequence of local unitary transformations in place of non-unitary imaginary-time evolution. By assuming a bounded correlation length, the QITE algorithm achieves an exponential reduction in computational resources over classical approaches.
2. Quantum Lanczos Algorithm:
   • QLanczos utilizes sequences of imaginary-time evolution vectors generated by QITE to form a basis that approximates the ground state within a reduced Krylov subspace.
   • This algorithm is especially efficient for calculating not only ground states but also excited states, serving as an economical option compared to full imaginary time evolution.
3. Quantum Minimally Entangled Typical Thermal States (QMETTS):
   • An extension using QITE as a subroutine to sample thermal averages without deep circuits by creating a Markov chain of states that approximate the thermal distribution.

Numerical and Experimental Validation

The researchers validate their proposed algorithms through both classical emulation and implementation on quantum hardware, notably the Rigetti quantum virtual machine and Aspen-1 quantum processing unit. They demonstrate how these methods can effectively tackle various models such as the Ising and Heisenberg models, providing promising results for convergence to ground and thermal states. Particularly, they find QITE and QLanczos outperform variational quantum eigensolver (VQE) approaches in terms of measurement resources in certain instances, highlighting their potential for near-term quantum devices.

Implications and Future Scope

Implications of this research are profound, suggesting that these quantum algorithms can efficiently approximate the eigenstates required for crucial problems in quantum simulation and quantum chemistry applications. The exponential efficiency gains indicate a tangible path forward for leveraging near-term quantum hardware in scientific computing, potentially reaching problems that were previously computationally prohibitive.

Looking ahead, it is crucial to explore this framework further in practical large-scale implementations and investigate the impacts of noise and quantum error mitigation techniques on these algorithms. Furthermore, these methods could be extended or adapted for other complex systems and applications in optimization and quantum machine learning.

Overall, the work showcases an elegant integration of classical computational strategies into quantum frameworks, providing a robust toolset for advancing the field of quantum computing.