Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Classification of Reversible Bit Operations   We present a complete classification of all possible sets of classical
reversible gates acting on bits, in terms of which reversible transformations
they generate, assuming swaps and ancilla bits are available for free. Our
classification can be seen as the reversible-computing analogue of Post's
lattice, a central result in mathematical logic from the 1940s. It is a step
toward the ambitious goal of classifying all possible quantum gate sets acting
on qubits. Our theorem implies a linear-time algorithm (which we have
implemented), that takes as input the truth tables of reversible gates G and H,
and that decides whether G generates H. Previously, this problem was not even
known to be decidable. The theorem also implies that any n-bit reversible
circuit can be "compressed" to an equivalent circuit, over the same gates, that
uses at most 2^n*poly(n) gates and O(1) ancilla bits; these are the first upper
bounds on these quantities known, and are close to optimal. Finally, the
theorem implies that every non-degenerate reversible gate can implement either
every reversible transformation, or every affine transformation, when
restricted to an "encoded subspace." Briefly, the theorem says that every set
of reversible gates generates either all reversible transformations on n-bit
strings (as the Toffoli gate does); no transformations; all transformations
that preserve Hamming weight (as the Fredkin gate does); all transformations
that preserve Hamming weight mod k for some k; all affine transformations (as
the Controlled-NOT gate does); all affine transformations that preserve Hamming
weight mod 2 or mod 4, inner products mod 2, or a combination thereof; or a
previous class augmented by a NOT or NOTNOT gate. Ruling out the possibility of
additional classes, not in the list, requires some arguments about polynomials,
lattices, and Diophantine equations.

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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph