Post BQP - Basic Properties

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>. 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 ≥ 2/3
• if the input x is not in L then Pr ≥ 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):

1. PostBQP is closed under complement. Given a language L in PostBQP and a corresponding deciding circuit family, create a new circuit family by flipping the output qubit after measurement, then the new circuit family proves the complement of L is in PostBQP.

2. You can do probability amplification in PostBQP. The definition of PostBQP is not changed if we replace the 2/3 value in its definition by any other constant strictly between 1/2 and 1. As an example, given a PostBQP algorithm A with success probability 2/3, we can construct another algorithm which runs three independent copies of A, outputs a postselection bit equal to the conjunction of the three "inner" ones, and outputs an output bit equal to the majority of the three "inner" ones; the new algorithm will be correct with conditional probability, greater than the original 2/3.

3. PostBQP is closed under intersection. Suppose we have PostBQP circuit families for two languages L1 and L2, with respective postselection qubits and output qubits P1, P2, Q1, Q2. We may assume by probability amplification that both circuit families have success probability at least 5/6. Then we create a composite algorithm where the circuits for L1 and L2 are run independently, and we set P to the conjunction of P1 and P2, and Q to the conjunction of Q1 and Q2. It is not hard to see by a union bound that this composite algorithm correctly decides membership in with (conditional) probability at least 2/3.

More generally, combinations of these ideas show that PostBQP is closed under union and BQP truth-table reductions.