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arXiv:2008.11347 · Quantum Reports

Hybrid Quantum-Classical Eigensolver without Variation or Parametric Gates

Pejman Jouzdani, Stefan A. Bringuier

Published Jan 31, 2021• 2 citations
arXivHTMLPDF

Abstract

The use of near-term quantum devices that lack quantum error correction, for addressing quantum chemistry and physics problems, requires hybrid quantum-classical algorithms and techniques. Here, we present a process for obtaining the eigenenergy spectrum of electronic quantum systems. This is achieved by projecting the Hamiltonian of a quantum system onto a limited effective Hilbert space specified by a set of computational bases. From this projection, an effective Hamiltonian is obtained. Furthermore, a process for preparing short depth quantum circuits to measure the corresponding diagonal and off-diagonal terms of the effective Hamiltonian is given, whereby quantum entanglement and ancilla qubits are used. The effective Hamiltonian is then diagonalized on a classical computer using numerical algorithms to obtain the eigenvalues. The use case of this approach is demonstrated for ground state and excited states of BeH2 and LiH molecules, and the density of states, which agrees well with exact solutions. Additionally, hardware demonstration is presented using IBM quantum devices for H2 molecule.

DOI
10.3390/quantum3010008

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The use of near-term quantum devices that lack quantum error correction, for addressing quantum chemistry and physics problems, requires hybrid quantum-classical algorithms and techniques. Here, we present a process for obtaining the eigenenergy spectrum of electronic quantum systems. This is achieved by projecting the Hamiltonian of a quantum system onto a limited effective Hilbert space specified by a set of computational bases. From this projection, an effective Hamiltonian is obtained. Furthermore, a process for preparing short depth quantum circuits to measure the corresponding diagonal and off-diagonal terms of the effective Hamiltonian is given, whereby quantum entanglement and ancilla qubits are used. The effective Hamiltonian is then diagonalized on a classical computer using numerical algorithms to obtain the eigenvalues. The use case of this approach is demonstrated for ground state and excited states of BeH2 and LiH molecules, and the density of states, which agrees well with exact solutions. Additionally, hardware demonstration is presented using IBM quantum devices for H2 molecule.
Utilizing quantum computer to investigate quantum chemistry is an important research field nowadays. In addition to the ground-state problems that have been widely studied, the determination of excited-states plays a crucial role in the prediction and modeling of chemical reactions and other physical processes. Here, we propose a non-variational full circuit-based quantum algorithm for obtaining the excited-state spectrum of a quantum chemistry Hamiltonian. Compared with previous classical-quantum hybrid variational algorithms, our method eliminates the classical optimization process, reduces the resource cost caused by the interaction between different systems, and achieves faster convergence rate and stronger robustness against noise without barren plateau. The parameter updating for determining the next energy-level is naturally dependent on the energy measurement outputs of the previous energy-level and can be realized by only modifying the state preparation process of ancillary system, introducing little additional resource overhead. Numerical simulations of the algorithm with hydrogen, LiH, H2O and NH3 molecules are presented. Furthermore, we offer an experimental demonstration of the algorithm on a superconducting quantum computing platform, and the results show a good agreement with theoretical expectations. The algorithm can be widely applied to various Hamiltonian spectrum determination problems on the fault-tolerant quantum computers.
Determining the energy gap in a quantum many-body system is critical to understanding its behavior and is important in quantum chemistry and condensed matter physics. The challenge of determining the energy gap requires identifying both the excited and ground states of a system. In this work, we consider preparing the prior distribution and circuits for the eigenspectrum of structured, time-independent Hamiltonians (e.g., local, sparse, or symmetry-constrained), which can benefit both classical and quantum algorithms for solving eigenvalue problems. The proposed algorithm unfolds in three strategic steps: Hamiltonian transformation, parameter representation, and classical clustering. These steps are underpinned by two key insights: the use of quantum circuits to approximate the ground state of transformed Hamiltonians and the analysis of parameter representation to distinguish between eigenvectors. The algorithm is showcased through applications to the 1D Heisenberg system and the LiH molecular system, highlighting its potential for both near-term quantum devices and fault-tolerant quantum devices, restricted to structured Hamiltonians where approximate state preparation is feasible. The paper also explores the scalability of the method and its performance across various settings, setting the stage for more resource-efficient quantum computations that are both accurate and fast. The findings presented here mark a new insight into hybrid algorithms, offering a pathway to overcoming current computational challenges.
Using quantum devices supported by classical computational resources is a promising approach to quantum-enabled computation. One powerful example of such a hybrid quantum-classical approach optimized for classically intractable eigenvalue problems is the variational quantum eigensolver, built to utilize quantum resources for the solution of eigenvalue problems and optimizations with minimal coherence time requirements by leveraging classical computational resources. These algorithms have been placed as leaders among the candidates for the first to achieve supremacy over classical computation. Here, we provide evidence for the conjecture that variational approaches can automatically suppress even nonsystematic decoherence errors by introducing an exactly solvable channel model of variational state preparation. Moreover, we develop a more general hierarchy of measurement and classical computation that allows one to obtain increasingly accurate solutions by leveraging additional measurements and classical resources. We demonstrate numerically on a sample electronic system that this method both allows for the accurate determination of excited electronic states as well as reduces the impact of decoherence, without using any additional quantum coherence time or formal error-correction codes.
We propose quantum-selected configuration interaction, a class of hybrid quantum-classical algorithms for calculating the ground- and excited-state energies of many-electron Hamiltonians on noisy quantum devices. Suppose that an approximate ground state can be prepared on a quantum computer either by variational quantum eigensolver or by some other method. Then, by sampling the state in the computational basis, which is hard for classical computation in general, one can identify the electron configurations that are important for reproducing the ground state. The Hamiltonian in the subspace spanned by those important configurations is diagonalized on classical computers to output the ground-state energy and the corresponding eigenvector. The excited-state energies can be obtained similarly. The result is robust against statistical and physical errors because the noisy quantum devices are used only to define the subspace, and the resulting ground-state energy strictly satisfies the variational principle even in the presence of such errors. The expectation values of various other operators can also be estimated for obtained eigenstates with no additional quantum cost, since the explicit eigenvectors in the subspaces are known. We verified our proposal by numerical simulations and demonstrated it on a quantum device for an eight-qubit molecular Hamiltonian. The proposed algorithms are potentially feasible to tackle some challenging molecules by exploiting quantum devices with several tens of qubits, assisted by high-performance classical computing resources for diagonalization.
The variational quantum eigensolver (VQE) is currently the flagship algorithm for solving electronic structure problems on near-term quantum computers. This hybrid quantum/classical algorithm involves implementing a sequence of parameterized gates on quantum hardware to generate a target quantum state, and then measuring the expectation value of the molecular Hamiltonian. Due to finite coherence times and frequent gate errors, the number of gates that can be implemented remains limited on current quantum devices, preventing accurate applications to systems with significant entanglement, such as strongly correlated molecules. In this work, we propose an alternative algorithm (which we refer to as ctrl-VQE) where the quantum circuit used for state preparation is removed entirely and replaced by a quantum control routine which variationally shapes a pulse to drive the initial Hartree-Fock state to the full CI target state. As with VQE, the objective function optimized is the expectation value of the qubit-mapped molecular Hamiltonian. However, by removing the quantum circuit, the coherence times required for state preparation can be drastically reduced by directly optimizing the pulses. We demonstrate the potential of this method numerically by directly optimizing pulse shapes which accurately model the dissociation curves of the hydrogen molecule (covalent bond) and helium hydride ion (ionic bond), and we compute the single point energy for LiH with four transmons.
Determining the spectral structure of molecular Hamiltonians is one of the most important and challenging research fields in quantum chemistry. In the near future, such a problem may be solvable by means of quantum simulation. However, constrained by a handful of qubits and noisy quantum gates available in the near-term quantum devices, hybrid quantum-classical algorithms based on variational methods such as variational quantum eigensolver have become a preferable choice. Here we propose an interesting variational quantum algorithm for obtaining the excited states of many-body Hamiltonians. To obtain the kth excited states, the previous variational quantum deflation algorithm requires the use of k different variational circuits, which is now reduced to only two circuits with our algorithm. The feasibility of the algorithm was numerically tested on the hydrogen molecule with and without random noise, where the chemical accuracy is reached for both cases. This algorithm is expected to significantly reduce the resource requirements of quantum simulation.