On the de-randomization of space-bounded approximate counting problems

Eliminating Intermediate Measurements in Space-Bounded Quantum Computation

“…, n}, where m = poly(n) and σ 1 (A j 1 ,j 2 ) ≤ poly(n) for 1 ≤ j 1 ≤ j 2 ≤ m. We then define the function poly-itmatprod : Proof. By [18,Theorem 14] (and the discussion following it), the functions R(poly-matinv(•)) and |poly-matinv(•)| each have a FLQ U AS (to be precise, the aforementioned theorem considered only the case in which A ∈ Pos(n), the general case then follows from the fact that the reduction from MATINV to MATINV + given by Lemma 23 preserves the value of the corresponding entry of the inverse matrix, not merely its magnitude); this improved upon the earlier result of Ta-Shma [45], which showed that these functions each have a FLQAS [16]. Notice that the reduction…”

“…By the preceding theorem, poly-conditioned-ITMATPROD ∈ BQ U L, which implies BQL = BQ U L; Theorem 1, which states the more general equivalence for any larger (space-constructible) space bound, then follows from a standard padding argument. In Section 3.3, we continue the study of fully logarithmic approximation schemes, initiated by Doron and Ta-Shma [16], and show that the BQL vs. BPL question is equivalent to several distinct questions involving the relative power of quantum and probabilistic fully logarithmic approximation schemes.…”

“…In this section, we study the class of functions that are well-approximable in quantum logspace, following (essentially) the notation and definitions of [16]. In particular, we work with the general (resp.…”

“…We then consider the poly-conditioned matrix inversion function poly-matinv : D(poly-matinv) → C, given by poly-matinv( A, s, t ) = A −1 [s, t]. For consistency with [16], as well as to make the relationship between our results and certain logspace counting classes more clear, we then consider R(poly-matinv(•)) : D(poly-matinv) → R (resp. |poly-matinv(•)| : D(poly-matinv) → R ≥0 ), which are given by the real part (resp.…”

“…Note that, following [16], we have defined fully logarithmic (quantum, randomized, etc.) approximation schemes with respect to additive error ǫ; that is to say, we approximate f (x) by a value y such that Pr[|f (x) − y| ≥ ǫ] ≤ δ.…”