The theory of variational hybrid quantum-classical algorithms (1509.04279v1)

Abstract: Many quantum algorithms have daunting resource requirements when compared to what is available today. To address this discrepancy, a quantum-classical hybrid optimization scheme known as "the quantum variational eigensolver" was developed with the philosophy that even minimal quantum resources could be made useful when used in conjunction with classical routines. In this work we extend the general theory of this algorithm and suggest algorithmic improvements for practical implementations. Specifically, we develop a variational adiabatic ansatz and explore unitary coupled cluster where we establish a connection from second order unitary coupled cluster to universal gate sets through relaxation of exponential splitting. We introduce the concept of quantum variational error suppression that allows some errors to be suppressed naturally in this algorithm on a pre-threshold quantum device. Additionally, we analyze truncation and correlated sampling in Hamiltonian averaging as ways to reduce the cost of this procedure. Finally, we show how the use of modern derivative free optimization techniques can offer dramatic computational savings of up to three orders of magnitude over previously used optimization techniques.

• The paper introduces a variational adiabatic ansatz that extends VQE through dynamically adjustable quantum state preparation.
• It establishes a formal link between unitary coupled cluster methods and universal gate sets to enhance quantum computing efficiency.
• The study demonstrates quantum variational error suppression and derivative-free optimizations, significantly reducing computational costs.

Overview of Variational Hybrid Quantum-Classical Algorithms

The paper discusses the theoretical underpinnings and advancements of variational hybrid quantum-classical algorithms, with a focus on the Quantum Variational Eigensolver (VQE). This algorithm was developed as a response to the impractical resource demands of many quantum algorithms, aiming to leverage minimal quantum resources combined with classical routines. The work explores improvements to the VQE, such as the development of a variational adiabatic ansatz, connections to unitary coupled cluster methodologies, and quantum variational error suppression techniques, among other enhancements.

Core Contributions

1. Variational Adiabatic Ansatz: The paper introduces a variational adiabatic ansatz, extending the scope of VQE by exploring adiabatic state preparation with parameterized paths. This modification allows for adaptable quantum state preparation influenced by dynamically adjustable parameters, broadening the practical applicability of variational algorithms.
2. Unitary Coupled Cluster (UCC) for Quantum Computation: A significant portion of the paper is dedicated to connecting UCC theory with quantum computation. By establishing a formal connection from second-order unitary coupled clusters to universal gate sets via relaxation of exponential splitting, the authors provide a method for employing UCC in quantum computing contexts efficiently.
3. Quantum Variational Error Suppression: This new approach to error suppression allows certain errors to be mitigated naturally within the framework of VQE, making it particularly suitable for pre-threshold quantum devices where full error correction isn't feasible.
4. Hamiltonian Averaging Techniques: The work further addresses computational costs by suggesting methods such as truncation and correlated sampling in Hamiltonian averaging. These techniques aim to reduce measurement overhead and enhance the practical feasibility of deploying variational algorithms.
5. Optimization Strategies: The authors discuss advanced, derivative-free optimization strategies that can lead to computational savings of up to three orders of magnitude over traditional techniques. This development is crucial for making VQE implementations more efficient.

Theoretical and Practical Implications

On a theoretical level, the advancements presented in the paper push the boundaries of what is possible with current quantum computing technology, particularly on near-term devices. The unification of UCC methods with universal gate theory via parameter relaxation offers a promising direction for quantum state preparation and manipulation.

Practically, these developments provide pathways to solving highly complex eigenvalue and optimization problems that are otherwise inaccessible to classical computation or alternative quantum algorithms due to resource constraints. The ability to variationally suppress quantum errors and the reduction of computational costs associated with Hamiltonian averaging are significant steps toward making VQE a viable option on early quantum computers.

Future Outlook

The future of AI and quantum computing can benefit substantially from the methodological innovations discussed in this paper. There is potential for further exploration into the intersection of quantum computing and machine learning, where such hybrid algorithms could open new paradigms for data processing and analysis.

As quantum hardware continues to evolve, the outlined strategies may serve as foundational components in developing scalable, efficient quantum algorithms with practical applicability to real-world problems across industries. The possibility of quantum advantages achieved through algorithms like VQE signifies a substantial milestone on the path to fully functional quantum computing solutions.