Proving the power of postselection

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 $\mathsf{NL}$ would equal $\mathsf{RL}$ if the randomized machines had the postselection capability.Comment: 26 pages. This is a heavily improved version of arXiv:1102.066

Exact affine counter automata

© F. Blanchet-Sadri & S. Osborne. We introduce an affine generalization of counter automata, and analyze their ability as well as affine finite automata. Our contributions are as follows. We show that there is a language that can be recognized by exact realtime affine counter automata but by neither 1-way deterministic pushdown automata nor realtime deterministic k-counter automata. We also show that a certain promise problem, which is conjectured not to be solved by two-way quantum finite automata in polynomial time, can be solved by Las Vegas affine finite automata. Lastly, we show that how a counter helps for affine finite automata by showing that the language MANYTWINS, which is conjectured not to be recognized by affine, quantum or classical finite state models in polynomial time, can be recognized by affine counter automata with one-sided bounded-error in realtime

One Time-traveling Bit is as Good as Logarithmically Many

We consider computation in the presence of closed timelike curves (CTCs), as proposed by Deutsch. We focus on the case in which the CTCs carry classical bits (as opposed to qubits). Previously, Aaronson and Watrous showed that computation with polynomially many CTC bits is equivalent in power to PSPACE. On the other hand, Say and Yakaryilmaz showed that computation with just 1 classical CTC bit gives the power of "postselection", thereby upgrading classical randomized computation (BPP) to the complexity class BPP_path and standard quantum computation (BQP) to the complexity class PP. It is natural to ask whether increasing the number of CTC bits from 1 to 2 (or 3, 4, etc.) leads to increased computational power. We show that the answer is no: randomized computation with logarithmically many CTC bits (i.e., polynomially many CTC states) is equivalent to BPP_path. (Similarly, quantum computation augmented with logarithmically many classical CTC bits is equivalent to PP.) Spoilsports with no interest in time travel may view our results as concerning the robustness of the class BPP_path and the computational complexity of sampling from an implicitly defined Markov chain

Classical and quantum Merlin-Arthur automata

We introduce Merlin-Arthur (MA) automata as Merlin provides a single certificate and it is scanned by Arthur before reading the input. We define Merlin-Arthur deterministic, probabilistic, and quantum finite state automata (resp., MA-DFAs, MA-PFAs, MA-QFAs) and postselecting MA-PFAs and MA-QFAs (resp., MA-PostPFA and MA-PostQFA). We obtain several results using different certificate lengths. We show that MA-DFAs use constant length certificates, and they are equivalent to multi-entry DFAs. Thus, they recognize all and only regular languages but can be exponential and polynomial state efficient over binary and unary languages, respectively. With sublinear length certificates, MA-PFAs can recognize several nonstochastic unary languages with cutpoint 1/2. With linear length certificates, MA-PostPFAs recognize the same nonstochastic unary languages with bounded error. With arbitrarily long certificates, bounded-error MA-PostPFAs verify every unary decidable language. With sublinear length certificates, bounded-error MA-PostQFAs verify several nonstochastic unary languages. With linear length certificates, they can verify every unary language and some NP-complete binary languages. With exponential length certificates, they can verify every binary language.Comment: 14 page

Quantum Logarithmic Space and Post-Selection

Post-selection, the power of discarding all runs of a computation in which an undesirable event occurs, is an influential concept introduced to the field of quantum complexity theory by Aaronson (Proceedings of the Royal Society A, 2005). In the present paper, we initiate the study of post-selection for space-bounded quantum complexity classes. Our main result shows the identity PostBQL = PL, i.e., the class of problems that can be solved by a bounded-error (polynomial-time) logarithmic-space quantum algorithm with post-selection (PostBQL) is equal to the class of problems that can be solved by unbounded-error logarithmic-space classical algorithms (PL). This result gives a space-bounded version of the well-known result PostBQP = PP proved by Aaronson for polynomial-time quantum computation. As a by-product, we also show that PL coincides with the class of problems that can be solved by bounded-error logarithmic-space quantum algorithms that have no time bound