  We present a quantum algorithm that analyzes risk more efficiently than Monte
Carlo simulations traditionally used on classical computers. We employ quantum
amplitude estimation to evaluate risk measures such as Value at Risk and
Conditional Value at Risk on a gate-based quantum computer. Additionally, we
show how to implement this algorithm and how to trade off the convergence rate
of the algorithm and the circuit depth. The shortest possible circuit depth -
growing polynomially in the number of qubits representing the uncertainty -
leads to a convergence rate of $O(M^{-2/3})$. This is already faster than
classical Monte Carlo simulations which converge at a rate of $O(M^{-1/2})$. If
we allow the circuit depth to grow faster, but still polynomially, the
convergence rate quickly approaches the optimum of $O(M^{-1})$. Thus, for
slowly increasing circuit depths our algorithm provides a near quadratic
speed-up compared to Monte Carlo methods. We demonstrate our algorithm using
two toy models. In the first model we use real hardware, such as the IBM Q
Experience, to measure the financial risk in a Treasury-bill (T-bill) faced by
a possible interest rate increase. In the second model, we simulate our
algorithm to illustrate how a quantum computer can determine financial risk for
a two-asset portfolio made up of Government debt with different maturity dates.
Both models confirm the improved convergence rate over Monte Carlo methods.
Using simulations, we also evaluate the impact of cross-talk and energy
relaxation errors.
