Solving strongly correlated electron models on a quantum computer (1506.05135v2)

Abstract: One of the main applications of future quantum computers will be the simulation of quantum models. While the evolution of a quantum state under a Hamiltonian is straightforward (if sometimes expensive), using quantum computers to determine the ground state phase diagram of a quantum model and the properties of its phases is more involved. Using the Hubbard model as a prototypical example, we here show all the steps necessary to determine its phase diagram and ground state properties on a quantum computer. In particular, we discuss strategies for efficiently determining and preparing the ground state of the Hubbard model starting from various mean-field states with broken symmetry. We present an efficient procedure to prepare arbitrary Slater determinants as initial states and present the complete set of quantum circuits needed to evolve from these to the ground state of the Hubbard model. We show that, using efficient nesting of the various terms each time step in the evolution can be performed with just $\mathcal{O}(N)$ gates and $\mathcal{O}(\log N)$ circuit depth. We give explicit circuits to measure arbitrary local observables and static and dynamic correlation functions, both in the time and frequency domain. We further present efficient non-destructive approaches to measurement that avoid the need to re-prepare the ground state after each measurement and that quadratically reduce the measurement error.

• The paper introduces a framework combining adiabatic state preparation and refined quantum phase estimation to accurately approximate the ground state of the Hubbard model.
• It details efficient circuit designs and non-destructive measurement methods that reduce gate complexity and improve simulation fidelity.
• The findings pave the way for practical quantum simulations of complex electron systems, offering insights for high-temperature superconductivity studies.

Solving Strongly Correlated Electron Models on a Quantum Computer: A Summary

The paper "Solving strongly correlated electron models on a quantum computer" by Dave Wecker et al. presents a comprehensive framework for utilizing quantum computers to simulate quantum models, focusing on the Hubbard model as a quintessential example. This research is anchored in leveraging the potential of quantum computing to overcome challenges in simulating systems with strong electron correlations—a task that is notably arduous on classical computers due to the computational complexity and the sign problem associated with stochastic methods.

Overview of the Approach

The approach consists of several crucial steps aligning quantum computing capabilities with the requirements of simulating strongly correlated systems:

1. Adiabatic State Preparation: The paper outlines strategies to initialize a quantum state that is a good approximation of the ground state for the Hubbard model. This involves starting from a mean-field state or a resonating valence bond (RVB) state and employing adiabatic evolution to transition to the ground state of the interaction-abundant Hamiltonian.
2. Quantum Phase Estimation (QPE): After preparing a trial state, QPE is employed to refine this to the precise ground state by utilizing coherent evolution. The paper enhances traditional QPE methods, offering improvements in gate complexity, notably halving the required controlled rotations and thus depth.
3. Non-Destructive Measurements: Recognizing the cost of repeatedly preparing ground states due to destructive measurements, the authors develop methods to perform efficient non-destructive measurements. These include leveraging the HeLLMan-Feynman theorem and implementing a recovery map to maintain the ground state post-measurement.
4. Simulation of Time Evolution: The paper addresses the simulation of time evolution by presenting efficient circuits for individual Hamiltonian terms, ensuring that these can be realized on quantum computers using minimal depth relative to system size. This results in a depth of $\mathcal{O}(\log N)$ per Trotter step for systems involving $N$ sites.

Numerical Results and Implications

Strong results are obtained when applying adiabatic state preparation to small systems, demonstrating the efficacy and fidelity of the method. For quantum circuits, the paper verifies that a series of optimizations leads to efficient implementations for the Hubbard model simulations.

Implications of the Research:

• Theoretical Advancements: Providing detailed algorithms and circuits adapts the abstract methods of quantum simulation to practical applications. This fills a critical gap in realizing quantum simulations for models that are computationally intractable classically.
• Practical Impact: As the quantum hardware advances, these methods could allow for the extraction of physical insights regarding high-critical temperature superconductors and other materials of significant experimental and technological interest.
• Future Directions: While the paper focuses on closed systems and pure state QPE, exploring extensions to mixed states, thermal fields, and open quantum systems marks a natural progression. Moreover, applying these insights to more complex models such as the $t-J$ model or incorporating external fields can expand the scope of quantum simulations.

In conclusion, the paper furnishes a methodological bridge from conceptual quantum algorithms to the realization of quantum simulations for complex electron models, portraying a vital advancement towards solving long-standing physics problems with quantum computers. The meticulous articulation of circuit designs, along with robust error and efficiency analyses, underscores a pivotal step in rendering quantum computational simulations feasible within the constraints of contemporary quantum hardware advancements.