The class of languages accepted by a BQP machine subject to the constraint that at every time step t, the machine's state is exponentially close to a tree state -- that is, a state expressible by a polynomial-size tree of additions and tensor products (together with complex constants and |0> and |1> leaf nodes).
More formally, a uniform classical polynomial-time algorithm generates a sequence of gates g(1), ... ,g(p(n)). Each g(t) can be either be selected from some finite universal basis of unitary gates (the choice turns out not to matter), or can be a 1-qubit measurement. When we perform a measurement, the state evolves to one of two possible pure states, with the usual probabilities, rather than to a mixed state. We require that the final gate g(p(n)) is a measurement of the first qubit. If at least one intermediate state was more than distance 2-Ω(n) away from the nearest state of tree size at most p(n), then the outcome of the final measurement is chosen adversarially; otherwise it is given by the usual Born probabilities. The measurement must return 1 with probability at least 2/3 if the input is in the language, and with probability at most 1/3 otherwise.
Contains BPP, and is contained in BQP.
Defined in [Aar03b], where it was also shown that TreeBQP iscontained in the third level of PH, which might provide weak evidence thatTreeBQP does not equal BQP.
Similar to TreeBQP except that the quantum computer's state at each time step is restricted to being exponentially close to a state in AmpP (that is, a state for which the amplitudes are computable by a classical polynomial-size circuit).
Defined in [Aar03b], where it was also observed that AmpP-BQP is contained in the third level of PH, just as TreeBQP is.