Quantum-Selected Configuration Interaction: Classical Diagonalization of Hamiltonians in Subspaces Selected by Quantum Computers

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Introduction

Quantum computing has opened up a realm of possibilities, especially in the area of simulating and understanding complex quantum systems in areas like quantum chemistry and condensed matter physics. Recent paper by QunaSys describes a promising approach- Quantum-Selected Configuration Interaction (QSCI) algorithm. It offers a novel and more efficient way to find ground-state and excited-state energies in quantum systems. In particular, the new approach circumvents effects of the errors that can spoil the variational nature of VQE: that is, the energy estimated by quantum devices is not guaranteed to give an upper bound on the exact ground-state energy. This is problematic because lowering the resulting energy of VQE does not necessarily mean approaching to the exact ground state. As we will learn QSCI solves this problem and yields results obeying the variational principle. In this blog post, we’ll delve into the key components and advantages of the QSCI algorithm, and how it is paving the way towards useful quantum computing. Below we describe the algorithm in general terms.

Algorithm Overview

Schematic description of the QSCI algorithm for finding the ground state.

1. Start with an Approximate Ground State: The QSCI journey begins with the preparation of an approximate ground state. Quantum computers have the unique capability to generate these states, often using techniques such as Variational Quantum Eigensolvers (VQE) or other
   quantum algorithms. This is the foundation on which QSCI builds. Note that our recent algorithm, ADAPT-QSCI, enables us to avoid the dependence on VQE. This will be discussed in the next article.
2. Identifying Crucial Configurations: Once we have our approximate ground state, the next step is to identify the vital computational basis states or electron configurations that are essential for expressing the ground state accurately. This is where the quantum magic happens. The quantum state is repeatedly measured to pinpoint these key configurations, a task that would be incredibly challenging for classical computers.
3. Truncated Hamiltonian Matrix Diagonalization: With the important configurations in hand, it’s time to perform the diagonalization, but here’s the twist — it happens on classical computers. The Hamiltonian matrix, which describes the energy of the quantum system, is truncated to include only the identified configurations. This considerably reduces the computational workload, making it classically tractable. As the final result one obtains the smallest eigenvalue and its corresponding eigenvector, which approximates the ground-state energy.
4. Extending the Algorithm for Excited States: QSCI isn’t limited to ground state calculations. It can also be used to find excited states by expanding the subspace or by repeating the procedure for each energy eigenstate. This versatility is a significant advantage of the algorithm.
5. A Strict Upper Bound on Ground-State Energy: The Hamiltonian matrix elements in the computational basis can be accurately calculated using classical computers. Therefore, the diagonalization step yields an energy value that serves as a strict upper bound on the exact ground-state energy. Regardless of the quality of the subspace spanned by the identified configurations, this bound remains valid. The quality mainly affects how close the bound is to the exact value.
6. Mitigating Errors with Symmetries: In quantum systems with symmetries and conserved quantities like particle number, the post-selection of computational basis states in the sampling outcome can help mitigate bit-flip errors. This adds robustness to the algorithm in the
   presence of physical and statistical errors.
7. Beyond Ground State - Quantum Eigenstate Tomography: QSCI isn’t just about ground state calculations. It can be advantageous for eigenstate tomography, allowing classical estimation of the expectation values of various observables at no additional quantum cost. This efficiency stems from having a classical representation of the state, which enables the computation of expectation values efficiently.

Schematic descriptions of the QSCI algorithms for finding the ground state and the first excited state: (a) single diagonalization scheme, and (b) sequential diagonalization scheme.

Conclusion

In the world of quantum computing, the Quantum-Selected Configuration Interaction (QSCI) algorithm stands out as a stepping stone approach for finding ground-state and excited-state energies efficiently. Its ability to harness quantum computing for identifying crucial configurations and then performing classical diagonalization reduces computational complexity, making it an attractive tool for tackling complex quantum systems. Moreover, its utility extends beyond ground state calculations, offering a versatile solution for eigenstate tomography. As quantum computing continues to evolve, QSCI is poised to play a pivotal role in solving challenging quantum problems.