Demonstration of Adiabatic Variational Quantum Computing with a Superconducting Quantum Coprocessor

Ming-Cheng Chen, Ming Gong, Xiaosi Xu, Xiao Yuan, Jian-Wen Wang, Can Wang, Chong Ying, Jin Lin, Yu Xu, Yulin Wu, Shiyu Wang, Hui Deng, Futian Liang, Cheng-Zhi Peng, Simon C. Benjamin, Xiaobo Zhu, Chao-Yang Lu, and Jian-Wei Pan

Phys. Rev. Lett. 125, 180501 – Published 26 October 2020

Abstract

Adiabatic quantum computing enables the preparation of many-body ground states. Realization poses major experimental challenges: Direct analog implementation requires complex Hamiltonian engineering, while the digitized version needs deep quantum gate circuits. To bypass these obstacles, we suggest an adiabatic variational hybrid algorithm, which employs short quantum circuits and provides a systematic quantum adiabatic optimization of the circuit parameters. The quantum adiabatic theorem promises not only the ground state but also that the excited eigenstates can be found. We report the first experimental demonstration that many-body eigenstates can be efficiently prepared by an adiabatic variational algorithm assisted with a multiqubit superconducting coprocessor. We track the real-time evolution of the ground and excited states of transverse-field Ising spins with a fidelity that can reach about 99%.

Received 31 July 2019
Accepted 22 September 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.180501

Physics Subject Headings (PhySH)

Adiabatic quantum optimization Quantum algorithms Quantum simulation

Superconducting qubits

Quantum Information, Science & Technology

Authors & Affiliations

Ming-Cheng Chen^1,2, Ming Gong ^1,2, Xiaosi Xu ³, Xiao Yuan³, Jian-Wen Wang^1,2, Can Wang^1,2, Chong Ying ^1,2, Jin Lin^1,2, Yu Xu^1,2, Yulin Wu^1,2, Shiyu Wang^1,2, Hui Deng^1,2, Futian Liang ^1,2, Cheng-Zhi Peng^1,2, Simon C. Benjamin³, Xiaobo Zhu^1,2, Chao-Yang Lu ^1,2, and Jian-Wei Pan^1,2

¹Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China
²CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
³Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom

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Vol. 125, Iss. 18 — 30 October 2020

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Images

Figure 1
Variational quantum simulation of dynamics. (a) Sketch of the variational algorithm. A shallow circuit ansatz $U (\vec{θ}) | 0 ⟩$ is used to approximate the true wave function $| ψ (t) ⟩$ with a polynomial number of parameters $\vec{θ} (t)$ . The Schrödinger equation is mapped to an ordinary differential equation (ODE) according to $M \cdot (\partial / \partial t) \vec{θ} (t) = V (t)$ . The ODE’s coefficients $M$ and $V (t)$ are estimated on a quantum coprocessor and the ODE is numerically integrated on a classical computer. (b) The Hadamard test circuits are used to estimate $M_{i j}$ and $V_{i} (t)$ on the quantum coprocessor, respectively. In the circuit, $H$ is the Hadamard gate and $S$ is the phase gate. The unitary operator $A = (\partial_{θ_{i}} U^{†}) (\partial_{θ_{j}} U)$ and $B_{l} = (\partial_{θ_{i}} U^{†}) P_{l} U$ , where $U$ is the circuit of ansatz and $P_{l}$ is the Pauli term in the time-dependent Hamiltonian $H (t) = \sum_{l = 1}^{K} h_{l} (t) P_{l}$ .
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Figure 2
The quantum Ising model simulated in the experiment. (a) The adiabatic change of system Hamiltonians. (b) The circuit ansatze for the eigenstates of the two-spin system. (i) is used for preparing the ground and the third-excited states, and (ii) is used to prepare for the first- and second-excited states. (c) The circuit ansatz for the ground-state problem of the fully coupled three-spin system.
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Figure 3
Experimental results of adiabatic variational dynamics of two spins. (a) The measured $M_{12}$ . Note that $M_{12}$ is only dependent on $θ_{1}$ after the circuit compiling; $M_{11}$ and $M_{22}$ are always zero and $M_{21} = - M_{12}$ due to the symmetry from Eq. (3). (b),(c) The measured $V_{1} (t)$ and $V_{2} (t)$ at time $t = T / 2$ . (d) The simulated parameter trajectories of all four eigenstates. (e),(f) The calculated energies and quantum state fidelities during the dynamics. The error bars are produced from 100 Monte Carlo simulations of the statistical fluctuation of 5000 samples of each circuit.
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Figure 4
Experimental results of the adiabatic variational dynamics of three spins. (a) The measured $M_{12}$ . $M_{12}$ is only dependent on $θ_{1}$ after the circuit compiling; $M_{11}$ and $M_{22}$ are always zero and $M_{21} = - M_{12}$ due to the symmetry from the Eq. (3). (b),(c) The measured $V_{1} (t)$ and $V_{2} (t)$ at time $t = T / 2$ , respectively. (d) The simulated parameter trajectory. (e),(f) The calculated evolution of ground-state energy and fidelity. The error bars are produced from 100 Monte Carlo simulations of the statistical fluctuation of 5000 samples of each circuit. We observe an interesting phenomenon that the quantum state fidelity recovers at the end of the dynamics.
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