The class VP is the algebraic analog of P; it is the class of polynomials of polynomial degree that have polynomial size circuits over a fixed field The class VNP is the analog of NP . VNP can be thought of as the class of polynomials

**VP vs. VNP** conjecture is whether** VP** =** VP,** where** VP** is the class of families of polynomials that are of polynomial degree** and** can be computed by arithmetic circuits of polynomial size,** and VP** is the class of families of polynomials that are of polynomial degree and can be approximated.

Abstract: One fundamental question in the context of the geometric complexity theory approach to the** VP vs. VNP** conjecture is whether** VP** = , where** VP** is the class of families of polynomials that are of polynomial degree and can be computed by arithmetic circuits of polynomial size, and is the class of families of polynomials that are of polynomial degree and can be approximated infinitesimally.

One fundamental question in the context of the geometric complexity theory approach to the** VP vs. VNP** conjecture is whether** VP** = $overline{textrm{VP}}$, where** VP** is the class of families of polynomials that are of polynomial degree and can be computed by arithmetic circuits of polynomial size, and $overline{textrm{VP}}$ is the class of families of polynomials that are of polynomial degree and can.

## Is there any quantum analog of the VP vs. VNP problem?

From Wikipedia:** V P:** The class** VP** is the algebraic analog of P; it is the class of polynomials f of polynomial degree that have polynomial size circuits over a fixed field K. V N P: The class** VNP** is the analog of** NP.**.

2-depth arithmetic circuits and VP vs VNP. Ask Question Asked 6 years, 7 months ago. Active 6 years, 7 months ago. Viewed 98 times 0 $begingroup$ the field of arithmetic circuit complexity is undergoing major discoveries in recent years as mentioned by Fortnow. am looking for a more layman-readable summary: is this new paper Sums of.

what is an outline/ overview of an intuition that VP=?VNP is connected or not (closely?) connected to the P=?NP problem? am looking for “not complex” or “straightforward” sketches or constructions, not necessarily “rigorous”. The field of arithmetic circuit complexity has lately seen a flurry of results. Several of these results are centered homogeneous depth four circuits, and come tantalizingly close to.

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Abstract. One fundamental question in the context of the geometric complexity theory approach to the VP vs. VNP conjecture is whether VP = $overline {textrm {VP}}$, where VP is the class of families of polynomials that are of polynomial degree and can be computed by arithmetic circuits of polynomial size, and $overline {textrm {VP}}$ is the class of families of polynomials that are of.

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## Arithmetic circuit complexity

The class VP is the algebraic analog of P; it is the class of polynomials of polynomial degree that have polynomial size circuits over a fixed field . The class VNP is the analog of NP. VNP can be thought of as the class of polynomials f {displaystyle f} of polynomial degree such that given a monomial we can determine its coefficient in f {displaystyle f} efficiently, with a polynomial size circuit.

In this setting the VP versus VNP problem is posed as a problem of separating the orbit closures of two forms in projective space equipped with a natural action of the general linear group. Viewed through the lens of GCT, the VP versus VNP problem makes connections to classical invariant theory, representation theory and algebraic geometry.

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## Demystifying the border of depth

VP [Mul12a,GMQ16]. Conﬁrmation or refutation of this question has multiple consequences, both in the algebraic complexity and at the frontier of algebraic geometry. If VP = VP, then any proof of VP 6= VNP will in fact also show that VNP 6 VP, as conjectured in [Mul12b]; however a refutation would imply that any realistic approach to the VP vs.

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