Subspace Variational Quantum Simulator

TL;DR

SVQS introduces a quantum–classical hybrid method to simulate time-independent Hamiltonian dynamics on NISQ devices by partial diagonalization of the target Hamiltonian using SSVQE and performing subspace time evolution via phase rotations in a mapped computational subspace. The approach reduces circuit depth relative to full diagonalization or VQS-based methods while preserving accuracy within the chosen low-lying eigensubspace, as demonstrated on a two-qubit hydrogen molecule system with subspace process fidelities of $0.88$–$0.98$. Experimental results include SSVQE validation of low-lying hydrogen eigenstates and SVQS-based subspace evolution characterized by subspace process fidelity and Pauli-transfer matrices, yielding a speed error of $1.1\%$ and axis error of $19.3^{\circ}$ in the effective subspace Hamiltonian. Extensions via ancilla-assisted subspace expansion and controlled-SVQS are discussed, offering a path toward scaling SVQS to larger systems and broader quantum-classical hybrid applications on NISQ devices.

Abstract

Quantum simulation is one of the key applications of quantum computing, which accelerates research and development in the fields such as chemistry and material science. The recent development of noisy intermediate-scale quantum (NISQ) devices urges the exploration of applications without the necessity of quantum error correction. In this paper, we propose an efficient method to simulate quantum dynamics driven by a static Hamiltonian on NISQ devices, named subspace variational quantum simulator (SVQS). SVQS employs the subspace-search variational quantum eigensolver (SSVQE) to find a low-lying eigensubspace and extends it to simulate dynamics within the subspace with lower overhead compared to the existing schemes. We experimentally simulate the time-evolution operator in a low-lying eigensubspace of a hydrogen molecule. We also define the subspace process fidelity as a measure between two quantum processes in a subspace. The subspace time evolution mimicked by SVQS shows the subspace process fidelity of $0.88$-$0.98$.

Figures (10)

• Figure 1: Quantum circuit for SVQS. $P(\xi)$ denotes a phase gate with the rotation angle $\xi$. The quantum circuit shown on the left approximates the time-evolution operator within the eigensubspace $\mathcal{S}_{\mathrm{\parallel}}$.
• Figure 2: Experimental results of (a) the single-qubit simultaneous randomized benchmarking and (b) the two-qubit interleaved randomized benchmarking. To align the execution times of all the Clifford gates, the single-qubit and two-qubit Clifford gates are constructed based on Euler decomposition mckay2017efficient and KAK decomposition e15061963, respectively. Therefore, each single-qubit Clifford gate is implemented with two $R_{X}(\pi/2)$ and three $R_{Z}(\theta)$ gates, and each two-qubit Clifford gate is implemented with 28 $R_{X}(\pi/2)$, 27 $R_{Z}(\theta)$, and 6 $R_{ZX}(\pi/4)$ gates except the ones for the interleaved $R_{ZX}(\pi/2)$ gates. We take $10$ random circuits for each Clifford sequence length and have $10^4$ sampling of measurements for each random circuit to obtain a single data point. The ground state populations of each qubit $p_0$ and two qubits $p_{00}$ are fitted to the exponential decay.
• Figure 3: (a) Hardware efficient ansatz used in the SSVQE experiments and (b) its Hermitian conjugate used in the SVQS experiments, where $U_3(\bm{\theta})$ ($\bm{\theta} = \{ \bm{\theta_i} \}$; $i =1, \dots, 6$) represents a parameterized single-qubit rotation implemented with two $R_{X}(\pi/2)$ gates and three parameterized $RZ$ gates mckay2017efficient. Each $\bm{\theta_i}$ consists of three phase parameters.
• Figure 4: SSVQE for different atomic distances. The blue and orange dots represent the experimentally obtained ground and first-excited eigenenergies, respectively. The error bars indicate the standard deviation of the eigenenergies, where the shot number is $10^4$. The black dashed lines depict the theoretical eigenenergies of the hydrogen molecule.
• Figure 5: SVQS for the evolution time from $0$ to $5.4~h/{\rm Ha}$. The blue and orange dashed lines represent the PIF and SPIF between the ideal and numerically calculated time-evolution operators mimicked by SVQS, respectively. The data points represent the corresponding experimental results.