In computational complexity theory, PostBQP is a complexity class consisting of all of the computational problems solvable in polynomial time on a quantum Turing machine with postselection and bounded error (in the sense that the algorithm is correct at least 2/3 of the time on all inputs). Postselection is not considered to be a feature that a realistic computer (even a quantum one) would possess, but nevertheless postselecting machines are interesting from a theoretical perspective. Removing either one of the two main features (quantumness, postselection) from PostBQP gives the following two complexity classes, both of which are subsets of PostBQP:
BQP is the same as PostBQP except without postselection BPPpath is the same as PostBQP except that instead of quantum, the algorithm is a classical randomized algorithm (with postselection) The addition of postselection seems to make quantum Turing machines much more powerful: Scott Aaronson proved PostBQP is equal to PP, a class which is believed to be relatively powerful, whereas BQP is not known even to contain the seemingly smaller class NP. Using similar techniques, Aaronson also proved that small changes to the laws of quantum computing would have significant effects. As specific examples, under either of the two following changes, the "new" version of BQP would equal PP:
if we broadened the definition of 'quantum gate' to include not just unitary operations but linear operations, or if the probability of measuring a basis state | x ⟩ {\displaystyle |x\rangle } was proportional to | α x | p {\displaystyle |\alpha _{x}|^{p}} instead of | α x | 2 {\displaystyle |\alpha _{x}|^{2}} for any even integer p > 2.
Basic properties In order to describe some of the properties of PostBQP we fix a formal way of describing quantum postselection. Define a quantum algorithm to be a family of quantum circuits (specifically, a uniform circuit family). We designate one qubit as the postselection qubit P and another as the output qubit Q. Then PostBQP is defined by postselecting upon the event that the postselection qubit is | 1 ⟩ {\displaystyle |1\rangle } . Explicitly, a language L is in PostBQP if there is a quantum algorithm A so that after running A on input x and measuring the two qubits P and Q,
P = 1 with nonzero probability if the input x is in L then Pr[Q = 1|P = 1] ≥ 2/3 if the input x is not in L then Pr[Q = 0|P = 1] ≥ 2/3. One can show that allowing a single postselection step at the end of the algorithm (as described above) or allowing intermediate postselection steps during the algorithm are equivalent. Here are three basic properties of PostBQP (which also hold for BQP via similar proofs):
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