The set of problems that can be answered by a uniform family of polynomial-sized quantum circuits whose gates are drawn from a set K, and that return the correct answer with probability 1, and run in polynomial time with probability 1, and the allowed gates are drawn from a set K. K may be either finite or countable and enumerated. If S is a ring, the union of EQPK over all finite gate sets K whose amplitudes are in the ring R can be written EQPS.
Defined in [ADH97] in the special case of a finite set of 1-qubit gates controlled by a second qubit. It was shown there that transcendental gates may be replaced by algebraic gates without decreasing the size of EQPK.
[FR98] show that EQPQ is in LWPP. The proof can be generalized to any finite, algebraic gate set K.
The hidden shift problem for a vector space over Z/2 is in EQPQ by Simon's algorithm. The discrete logarithm problem over Z/p is in EQPQ-bar using infinitely many gates [MZ03].
The same as BQP, except that the quantum algorithm must return the correct answer with probability 1, and run in polynomial time with probability 1. Unlike bounded-error quantum computing, there is no theory of universal QTMs for exact quantum computing models. In the original definition in [BV97], each language in EQP is computed by a single QTM, equivalently to a uniform family of quantum circuits with a finite gate set K whose amplitudes can be computed in polynomial time. See EQPK. However, some results require an infinite gate set. The official definition here is that the gate set should be finite.
Without loss of generality, the amplitudes in the gate set K are algebraic numbers [ADH97].
There is an oracle that separates EQP from NP [BV97], indeed from Δ2P [GP01]. There is also an oracle relative to which EQP is not in ModpP where p is prime [GV02]. On the other hand, EQP is in LWPP [FR98].
P||NP[2k] is contained in EQP||NP[k], where "||NP[k]" denotes k nonadaptive oracle queries to NP (queries that cannot depend on the results of previous queries) [BD99].
See also ZBQP.