Counting, fanout and the complexity of quantum ACC

F. Green; S. Homer; C. Moore; C. Pollett

doi:10.26421/qic2.1-3

Counting, fanout and the complexity of quantum ACC

Quantum Information and Computation ◽

10.26421/qic2.1-3 ◽

2002 ◽

Vol 2 (1) ◽

pp. 35-65

Author(s):

F. Green ◽

S. Homer ◽

C. Moore ◽

C. Pollett

Keyword(s):

Complexity Classes ◽

Constant Depth ◽

Classical Counterpart ◽

Classical Class ◽

Polynomial Size ◽

Constant Depth Circuits ◽

Language Classes ◽

Threshold Circuits ◽

Quantum Analog ◽

Quantum Complexity

We propose definitions of QAC^0, the quantum analog of the classical class AC^0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and QACC[q], the analog of the class ACC[q] where Mod_q gates are also allowed. We prove that parity or fanout allows us to construct quantum MOD_q gates in constant depth for any q, so QACC[2] = QACC. More generally, we show that for any q,p > 1, MOD_q is equivalent to MOD_p (up to constant depth and polynomial size). This implies that QAC^0 with unbounded fanout gates, denoted QACwf^0, is the same as QACC[q] and QACC for all q. Since \ACC[p] \ne ACC[q] whenever p and q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC^0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We define classes of languages closely related to QACC[2] and show that restricted versions of them can be simulated by polynomial-size circuits. With further restrictions, language classes related to QACC[2] operators can be simulated by classical threshold circuits of polynomial size and constant depth.

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Uniform derandomization from pathetic lower bounds

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2011.0318 ◽

2012 ◽

Vol 370 (1971) ◽

pp. 3512-3535 ◽

Cited By ~ 1

Author(s):

Eric Allender ◽

V. Arvind ◽

Rahul Santhanam ◽

Fengming Wang

Keyword(s):

Word Problem ◽

Lower Bounds ◽

Black Box ◽

Arithmetic Circuits ◽

List Type ◽

Probabilistic Algorithms ◽

Constant Depth ◽

Subexponential Time ◽

Polynomial Size ◽

Threshold Circuits

The notion of probabilistic computation dates back at least to Turing, who also wrestled with the practical problems of how to implement probabilistic algorithms on machines with, at best, very limited access to randomness. A more recent line of research, known as derandomization, studies the extent to which randomness is superfluous. A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e. superpolynomial, or even nearly exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic. Here, we present two instances where ‘pathetic’ lower bounds of the form n 1+ ϵ would suffice to derandomize interesting classes of probabilistic algorithms. We show the following: — If the word problem over S 5 requires constant-depth threshold circuits of size n 1+ ϵ for some ϵ >0, then any language accepted by uniform polynomial size probabilistic threshold circuits can be solved in subexponential time (and, more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size). — If there are no constant-depth arithmetic circuits of size n 1+ ϵ for the problem of multiplying a sequence of n 3×3 matrices, then, for every constant d , black-box identity testing for depth- d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC 0 circuits of subexponential size).

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Some Results on Uniform Arithmetic Circuit Complexity

DAIMI Report Series ◽

10.7146/dpb.v20i343.7969 ◽

1991 ◽

Vol 20 (343) ◽

Author(s):

Gudmund Skovbjerg Frandsen ◽

Mark Valence ◽

David Mix Barrington

Keyword(s):

Arithmetic Function ◽

Circuit Complexity ◽

Arithmetic Circuits ◽

Arithmetic Functions ◽

Constant Depth ◽

Arithmetic Circuit ◽

Arithmetic Complexity ◽

Polynomial Size ◽

Threshold Circuits ◽

Class Of Functions

We introduce a natural set of arithmetic expressions and define the complexity class AE to consist of all those arithmetic functions (over the fields F_(2)n) that are described by these expressions. We show that AE coincides with the class of functions that are computable with constant depth and polynomial size unbounded fan-in arithmetic circuits satisfying a natural uniformity constraint (DLOGTIME-uniformity). A 1-input and 1-output arithmetic function over the fields F_(2)n may be identified with an <em>n</em>-input and an n-output Boolean function when field elements are represented as bit strings. We prove that if some such representation is X-uniform (where X is P or DLOGTIME) then the arithmetic complexity of a function (measured with X-uniform unbounded fan-in arithmetic circuits) is identical to the Boolean complexity of this function (measured with X-uniform threshold circuits). We show the existence of a P-uniform representation and we give partial results concerning the existence of representations with more restrictive uniformity properties.

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