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arXiv:1810.09434 · Physical Review Research

Subspace-search variational quantum eigensolver for excited states

Ken M. Nakanishi, Kosuke Mitarai, Keisuke Fujii

Published Oct 30, 2019• 335 citations
arXivHTMLPDF

Abstract

The authors propose an algorithm, the subspace-search variational quantum eigensolver (SSVQE) that searches a low energy subspace by supplying orthogonal input states to the variational ansatz and relies on the unitarity of transformations to ensure the orthogonality of output states. This work extends the applicable domain of the Variational Quantum Eigensolver to excited states and their related properties.

DOI
10.1103/physrevresearch.1.033062

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The authors propose an algorithm, the subspace-search variational quantum eigensolver (SSVQE) that searches a low energy subspace by supplying orthogonal input states to the variational ansatz and relies on the unitarity of transformations to ensure the orthogonality of output states. This work extends the applicable domain of the Variational Quantum Eigensolver to excited states and their related properties.
The weighted subspace-search variational quantum eigensolver (SSVQE) is a prominent algorithm for calculating excited-state properties of molecular quantum systems. In this work, we elaborate on some of its fundamental features with the aim of improving its practical realization. First, we demonstrate that the initial ansätze for various excited states could be prepared into a single pure state through a minimal number of ancilla qubits, followed by the optimization of a subsequent global unitary rotation in the targeted subspace. Since the ancillas' sole purpose is to purify an underlying ensemble ρwρ_{\boldsymbol{w}}ρw​ state with spectral weights w\boldsymbol{w}w, their measurement would just collapse ρwρ_{\boldsymbol{w}}ρw​ with probabilities wjw_jwj​ to one of its eigenstates ∣Ψj⟩|Ψ_j \rangle∣Ψj​⟩. We thus observe that our realization of SSVQE is equivalent to the original SSVQE improved by importance sampling. Then, we elaborate by numerical means on the potential influence of the auxiliary weights w\boldsymbol{w}w on the accuracy of the sought-after eigenstates and eigenenergies. Clear trends are discovered which are contrasted with some recent mathematical results concerning the ensemble variational principle that underlies SSVQE.
Quantum systems in excited states are attracting significant interest with the advent of noisy intermediate-scale quantum (NISQ) devices. While ground states of small molecular systems are typically explored using hybrid variational algorithms like the variational quantum eigensolver (VQE), the study of excited states has received much less attention, partly due to the absence of efficient algorithms. In this work, we introduce the subspace search quantum imaginary time evolution (SSQITE) method, which calculates excited states using quantum devices by integrating key elements of the subspace search variational quantum eigensolver (SSVQE) and the variational quantum imaginary time evolution (VarQITE) method. The effectiveness of SSQITE is demonstrated through calculations of low-lying excited states of benchmark model systems including H 2 and LiH molecules. A toy Hamiltonian is also employed to demonstrate that the robustness of VarQITE in avoiding local minima extends to its use in excited state algorithms. With this robustness in avoiding local minima, SSQITE shows promise for advancing quantum computations of excited states across a wide range of applications.
Subspace-Search Variational Quantum Eigen solver (SSVQE) is a searching method of multiple states and relies on the unitarity of transformations to ensure the orthogonality of output states for multiple states. Therefore, this method is thought to be a promising method for quantum chemistry because ordinary Variational Quantum Eigen solver (VQE) can only calculate the excited states step by step from the ground state based on Variational Quantum deflation (VQD). We compare the advantage of VQE, SSVQE with/without the constraint term and/or Tabu search term, which is added by the Lagrange multiplier method so as to calculate the desired energy levels. We evaluated the advantage by calculating each level of H2 and HeH, respectively. As there simulation results, the accuracy calculated by constrained VQE with Tabu search indicates higher accuracy than that of our other algorithm, for analysis on H2. The accuracy calculated by constrained SSVQE indicates higher that of the constrained VQE with Tabu search. We found it is beneficial for enhancing the accuracy to use constraint terms decreasing convergence times to use Tabu search terms according to the nature of molecules. We demonstrate that constraint and Tabu search terms contribute to the accuracy and convergence time on quantum chemical calculating.
Accurate excited state energies are essential for interpreting fluorescence and photochemical processes, yet variational quantum eigensolvers (VQEs) on noisy intermediate-scale quantum (NISQ) hardware have largely been confined to ground states. Subspace-search VQE (SSVQE) offers a NISQ-friendly route to excited states without ancilla-based overlap measurements, however it has lacked an experimental demonstration since it requires the coherent preparation and manipulation of multiple orthogonal basis states under a shared variational unitary. Here we realize SSVQE on a photonic platform by encoding orthogonal qudit states in the orbital angular momentum of single photons and optimizing a shared variational circuit. We estimate excited state energies for a four-dimensional HeH+\rm{HeH^+}HeH+ molecule and an eight-dimensional Heisenberg chain, thereby demonstrating scalable excited state eigensolving in a high-dimensional Hilbert space. Our results establish photonic qudits as a resource-efficient pathway to excited state quantum simulation.
Variational quantum eigensolver (VQE) is a promising method for ground-state calculations on current noisy intermediate-scale quantum computers. However, the research progress of excited-state calculations on quantum computers is relatively slow. In order to extend the framework of VQE for excited-state calculations, we propose a new algorithm, orthogonal state reduction variational eigensolver (OSRVE), to determine the energies of excited states. Theoretical derivations prove that the optimized state in the OSRVE method can ensure the energy minimum and orthogonality constraint simultaneously, and OSRVE is also applicable for the degenerate state. The performance of OSRVE is demonstrated by numerical calculations of the H 4 and H 2 O molecules. Compared with other excited-state calculation algorithms, OSRVE has obvious advantages in calculating lower-order excited states. This work extends the VQE algorithm to excited-state calculations, and OSRVE can be implemented on near-term quantum computers.
We propose a new approach to improve the accuracy of ground state estimates in Variational Quantum Eigensolver (VQE) algorithms by employing subspace representations with soft-coded orthogonality constraints. As in other subspace-based VQE methods, such as the Subspace-Search VQE (SSVQE) and Multistate Contracted VQE (MCVQE), once the parameters are optimized to maximize the subspace overlap with the low-energy sector of the Hamiltonian, one diagonalizes the Hamiltonian restricted to the subspace. Unlike these methods, where \emph{hard-coded} orthogonality constraints are enforced at the circuit level among the states spanning the subspace, we consider a subspace representation where orthogonality is \emph{soft-coded} via penalty terms in the cost function. We show that this representation allows for shallower quantum circuits while maintaining high fidelity when compared to single-state (standard VQE) and multi-state (SSVQE or MCVQE) representations, on two benchmark cases: a 3×33\times 33×3 transverse-field Ising model and random realizations of the Edwards--Anderson spin-glass model on a 4×44\times 44×4 lattice.