45 research outputs found
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
From Complexity to Algebra and Back: Digraph Classes, Collapsibility, and the PGP
Inspired by computational complexity results for the quantified constraint satisfaction problem, we study the clones of idem potent polymorphisms of certain digraph classes. Our first results are two algebraic dichotomy, even "gap", theorems. Building on and extending [Martin CP'11], we prove that partially reflexive paths bequeath a set of idem potent polymorphisms whose associated clone algebra has: either the polynomially generated powers property (PGP), or the exponentially generated powers property (EGP). Similarly, we build on [DaMM ICALP'14] to prove that semi complete digraphs have the same property. These gap theorems are further motivated by new evidence that PGP could be the algebraic explanation that a QCSP is in NP even for unbounded alternation. Along the way we also effect a study of a concrete form of PGP known as collapsibility, tying together the algebraic and structural threads from [Chen Sicomp'08], and show that collapsibility is equivalent to its Pi2-restriction. We also give a decision procedure for k-collapsibility from a singleton source of a finite structure (a form of collapsibility which covers all known examples of PGP for finite structures). Finally, we present a new QCSP trichotomy result, for partially reflexive paths with constants. Without constants it is known these QCSPs are either in NL or Pspace-complete [Martin CP'11], but we prove that with constants they attain the three complexities NL, NP-complete and Pspace-complete
stateQIP = statePSPACE
Complexity theory traditionally studies the hardness of solving classical
computational problems. In the quantum setting, it is also natural to consider
a different notion of complexity, namely the complexity of physically preparing
a certain quantum state. We study the relation between two such state
complexity classes: statePSPACE, which contains states that can be generated by
space-uniform polynomial-space quantum circuits, and stateQIP, which contains
states that a polynomial-time quantum verifier can generate by interacting with
an all-powerful untrusted quantum prover. The latter class was recently
introduced by Rosenthal and Yuen (ITCS 2022), who proved that statePSPACE
stateQIP.
Our main result is the reverse inclusion, stateQIP statePSPACE,
thereby establishing equality of the two classes and providing a natural
state-complexity analogue to the celebrated QIP = PSPACE theorem of Jain, et
al. (J. ACM 2011). To prove this, we develop a polynomial-space quantum
algorithm for solving a large class of exponentially large "PSPACE-computable"
semidefinite programs (SDPs), which also prepares an optimiser encoded in a
quantum state. Our SDP solver relies on recent block-encoding techniques from
quantum algorithms, demonstrating that these techniques are also useful for
complexity theory.
Using similar techniques, we also show that optimal prover strategies for
general quantum interactive protocols can be implemented in quantum polynomial
space. We prove this by studying an algorithmic version of Uhlmann's theorem
and establishing an upper bound on the complexity of implementing Uhlmann
transformations.Comment: 61 page