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The standard setting of quantum computation for continuous problems uses deterministic queries and the only source of randomness for quantum algorithms is through measurement. Without loss of generality we may consider quantum algorithms which use only one 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. Hence, there is also a limit to the power of the quantum setting with randomized queries since we cannot solve continuous problems with infinite randomized information complexity. An example is approximation of bounded continuous functions. We study the quantum setting with randomized queries for a number of problems in terms of the query and qubit complexities defined as the minimal number of queries/qubits needed to solve the problem to within 驴 by a quantum algorithm. We prove that for path integration we have an exponential improvement for the qubit complexity over the quantum setting with deterministic queries.

The Quantum Setting with Randomized Queries for Continuous Problems