  The standard setting of quantum computation for continuous problems uses
deterministic queries and the only source of randomness for quantum algorithms
is through measurement. This setting is related to the worst case setting on a
classical computer in the sense that the number of qubits needed to solve a
continuous problem must be at least equal to the logarithm of the worst case
information complexity of this problem. Since the number of qubits must be
finite, we cannot solve continuous problems on a quantum computer with infinite
worst case information complexity. This can even happen for continuous problems
with small randomized complexity on a classical computer. A simple example is
integration of bounded continuous functions. To overcome this bad property that
limits the power of quantum computation for continuous problems, we study the
quantum setting in which randomized queries are allowed. This type of query is
used in Shor's algorithm. The quantum setting with randomized queries is
related to the randomized classical setting in the sense that the number of
qubits needed to solve a continuous problem must be at least equal to the
logarithm of the randomized information complexity of this problem.
