Schr\"odinger-Heisenberg Variational Quantum Algorithms

Recent breakthroughs have opened the possibility to intermediate-scale quantum computing with tens to hundreds of qubits, and shown the potential for solving classical challenging problems, such as in chemistry and condensed matter physics. However, the extremely high accuracy needed to surpass classical computers poses a critical demand to the circuit depth, which is severely limited by the non-negligible gate infidelity, currently around 0.1-1%. Here, by incorporating a virtual Heisenberg circuit, which acts effectively on the measurement observables, to a real shallow Schr\"odinger circuit, which is implemented realistically on the quantum hardware, we propose a paradigm of Schr\"odinger-Heisenberg variational quantum algorithms to resolve this problem. We choose a Clifford virtual circuit, whose effect on the Hamiltonian can be efficiently and classically implemented according to the Gottesman-Knill theorem. Yet, it greatly enlarges the state expressivity, realizing much larger unitary t-designs. Our method enables accurate quantum simulation and computation that otherwise is only achievable with much deeper and more accurate circuits conventionally. This has been verified in our numerical experiments for a better approximation of random states and a higher-fidelity solution to the ground state energy of the XXZ model. Together with effective quantum error mitigation, our work paves the way for realizing accurate quantum computing algorithms with near-term quantum devices.