  Accurately predicting response properties of molecules such as the dynamic
polarizability and hyperpolarizability using quantum mechanics has been a
long-standing challenge with widespread applications in material and drug
design. Classical simulation techniques in quantum chemistry are hampered by
the exponential growth of the many-electron Hilbert space as the system size
increases. In this work, we propose an algorithm for computing linear and
nonlinear molecular response properties on quantum computers, by first
reformulating the target property into a symmetric expression more suitable for
quantum computation via introducing a set of auxiliary quantum states, and then
determining these auxiliary states via solving the corresponding linear systems
of equations on quantum computers. On one hand, we prove that using the quantum
linear system algorithm [Harrow et al., Phys. Rev. Lett. 103, 150502 (2009)] as
a subroutine the proposed algorithm scales only polynomially in the system size
instead of the dimension of the exponentially large Hilbert space, and hence
achieves an exponential speedup over existing classical algorithms. On the
other hand, we introduce a variational hybrid quantum-classical variant of the
proposed algorithm, which is more practical for near-term quantum devices.
