  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.
