Proving the Power of Postselection

Abstract

It is a widely believed, though unproven, conjecture that the capability of postselection increases the language recognition power of both probabilistic and quantum polynomial-time computers. It is also unknown whether polynomial-time quantum machines with postselection are more powerful than their probabilistic counterparts with the same resource restrictions. We approach these problems by imposing additional constraints on the resources to be used by the computer, and are able to prove for the first time that postselection does augment the computational power of both classical and quantum computers, and that quantum does outperform probabilistic in this context, under simultaneous time and space bounds in a certain range. We also look at postselected versions of space-bounded classes, as well as those corresponding to error-free and one-sided error recognition, and provide classical characterizations. It is shown that NL would equal RL if the randomized machines had the postselection capability.keywords: postselection, quantum Turing machines, probabilistic Turing machines, space-bounded computation, one-sided error, zero error IntroductionThe notion of postselection as a mode of computation was introduced by Aaronson [Aar05]. Postselection is the (unrealistic) capability of discarding all branches of a computation in which a specific event does not occur, and focusing on the surviving branches for the final decision about the membership of the input string in the recognized language. Aaronson examined PostBQP, the class of languages recognized with bounded error by polynomial-time quantum computers with postselection, and showed it to be identical to the well-known classical complexity class PP. The corresponding class for probabilistic polynomial-time computers with postselection is known to equal BPP path . It is, however, still an open question whether postselection adds anything to the power of quantum or classical polynomial-time computation, since we do not know whether the standard classes (without postselection) for these models, that is, BQP and BPP, respectively, equal their postselected versions or not. It is also not known whether BPP path = PP, or polynomial-time quantum machines with postselection outperform their classical counterparts, as is conjectured to be the case between the standard versions. All these classes sit between P and PSPACE, and any proof of unequality between them would therefore be as hard as proving P = PSPACE. ⋆ Earlier versions of this paper appeared as [YS10b,YS11a] ⋆⋆

Quantum Logarithmic Space and Post-selection

“…A postselecting PTM (PostPTM) has the ability to discard some predetermined outcomes and then makes its decision with the rest of the outcomes, which is guaranteed to happen with non-zero probability (see [1,21]). Formally, a PostPTM is a modified PTM with three halting states.…”

“…Both machines recognize the same language with the same error bound since the restarting and postselecting mechanism can be used interchangeably [19,21], i.e., the accepting and rejecting probabilities by M and M ′ are the same on every input. Thus, we can conclude that PostBPL ⊆ BPL(exp).…”

“…In view of the impact of the concept of post-selection to quantum complexity theory and in view of the surge of recent activities on space-bounded quantum complexity classes, a natural question is investigating the power of post-selection for space-bounded quantum complexity classes. To our knowledge, this question has not been investigated so far in the literature (while the notion of post-selection was previously studied in quantum automata theory [21]). In this paper, we tackle this question and obtain the following result (here PostBQL denotes the class of problems that can be solved by a bounded-error polynomial-time logarithmic-space quantum Turing machine that uses post-selection -see Section 2 for a formal definition):…”

Uncountable classical and quantum complexity classes

“…So, the input is read from left to the right, then right to left, and then left to right, and so on. A very restricted version of sweeping models are realtime restarting models (see [24,26] for the details of restarting concept): the models have an additional state s i such that immediately after entering s i the overall computation is terminated and all computations start from the initial configuration. In this paper, we focus on restarting rtQCFAs.…”

Postselecting probabilistic finite state recognizers and verifiers

“…In [1], Aaronson introduced bounded-error postselecting quantum polynomial time and proved that it is identical to the unbounded-error probabilistic polynomial time. Later, postselecting quantum and probabilistic finite automata models have been investigated in [14,15,17,18]. It was proven that postselecting realtime finite automata are equivalent to a restricted variant of two-way finite automata, called restarting realtime automata [16].…”