Polynomial Lower Bounds for Arithmetic Circuits over Non-Commutative Rings

Abstract: We prove a lower bound of $Ω\left(n^{1.5}\right)$ for the number of product gates in non-commutative arithmetic circuits for an explicit $n$-variate degree-$n$ polynomial $f_{n}$ (over every field). We observe that this implies that over certain non-commutative rings $R$, any arithmetic circuit that computes the induced polynomial function $f_{n}: R^n \rightarrow R$, using the ring operations of addition and multiplication in $R$, requires at least $Ω\left(n^{1.5}\right)$ multiplications. More generally, for any $d\geq 2$ and sufficiently large $n$, we obtain a lower bound of $Ω\left(d\sqrt{n}\right)$ for $n$-variate degree-$d$ polynomials, for both these models. Prior to our work, the only known lower bounds for the size of non-commutative circuits, or for the size of arithmetic circuits over any ring, were slightly super-linear in $\max{n,d}$: $Ω\left(n\log d\right)$ by Baur and Strassen, and $Ω\left(d\log n\right)$ by Nisan. (Nisan's bound was proved for non-commutative arithmetic circuits and implies a bound for arithmetic circuits over non-commutative rings by our observation).

• The paper proves an Ω(n¹·⁵) lower bound for explicit n-variate, degree-n non-commutative polynomials, marking a breakthrough over previous super-linear bounds.
• It refines Nisan’s partial derivatives method by tracking full-rank matrices across circuit transformations, thereby quantifying minimum multiplicative gate costs.
• The work demonstrates that lower bounds for non-commutative circuits over fields naturally extend to circuits over non-commutative rings, broadening their applicability.

Polynomial Lower Bounds for Arithmetic Circuits over Non-Commutative Rings

Overview and Motivation

The paper "Polynomial Lower Bounds for Arithmetic Circuits over Non-Commutative Rings" (2604.22006) addresses a fundamental open problem in algebraic complexity theory: proving super-linear lower bounds for the size of general arithmetic circuits, especially in non-commutative settings. Building upon the foundational results by Strassen, Baur and Strassen, and Nisan, which established lower bounds of $\Omega(n \log d)$ and $\Omega(d \log n)$ in specific models, this work provides the first polynomial lower bounds exceeding slightly super-linear growth for explicit polynomials in the full generality of non-commutative arithmetic circuits.

The central contributions include an $\Omega(n^{1.5})$ lower bound for non-commutative circuits computing explicit $n$-variate, degree-$n$ polynomials over any field, and, crucially, a rigorous demonstration that such lower bounds for non-commutative circuits over fields directly imply analogous lower bounds for circuits operating over non-commutative rings (specifically, free associative algebras). For degree-$d$ polynomials, the general result is an $\Omega(d \sqrt{n})$ bound, establishing a new regime of hardness for both non-commutative circuit complexity and arithmetic circuits over non-commutative rings.

Theoretical Results and Lower Bound Techniques

The paper establishes the following core theorem: for explicit $n$-variate degree-$d$ (with $d \geq 2$) non-commutative polynomials $\Omega(d \log n)$0, any non-commutative arithmetic circuit computing $\Omega(d \log n)$1 requires at least $\Omega(d \log n)$2 non-scalar product gates. For $\Omega(d \log n)$3, this recovers the $\Omega(d \log n)$4 lower bound.

The proof strategy refines and extends Nisan's partial derivatives/rank technique:

• For each candidate circuit, the construction tracks matrices $\Omega(d \log n)$5 whose ranks capture the "information content" about certain degree slices of the polynomial. For the explicit hard polynomials, these matrices are shown to have full rank.
• By carefully analyzing the evolution of these matrix ranks along alternating paths (sum gate–product gate alternations) from the output back to the inputs, the work quantifies the rate at which rank must drop, forcing the circuit to pay a significant cost in the number of product gates to achieve the requisite reductions.
• For sum gates, arguments based on rank subadditivity necessitate that a large number of children are present to reduce rank sufficiently rapidly, and for product gates, further technical steps ensure that any "cheating" by making degree drops too sudden increases the cost later in the circuit due to fewer alternations being available.
• The circuit is transformed (without increasing the number of non-scalar product gates) to an alternating form with desirable properties, allowing the argument to proceed cleanly.

Unlike prior results, which only managed super-linear lower bounds, this analysis extracts a genuinely polynomial (in $\Omega(d \log n)$6) gating cost, reflecting a stronger separation.

Circuits over Non-Commutative Rings

The paper's second principal component is the observation that lower bounds for non-commutative circuits over fields directly transfer to lower bounds for arithmetic circuits over non-commutative rings. The precise statement is that any arithmetic circuit computing an explicit $\Omega(d \log n)$7-variate, degree-$\Omega(d \log n)$8 polynomial function $\Omega(d \log n)$9 from $\Omega(n^{1.5})$0 to $\Omega(n^{1.5})$1—using only ring operations in a non-commutative ring $\Omega(n^{1.5})$2 (such as the free algebra $\Omega(n^{1.5})$3)—must incur the same $\Omega(n^{1.5})$4 multiplicative gate lower bound. The proof employs a substitution argument and exploits the structure of polynomials in free associative algebras to control cancellations of cross-terms, ultimately showing that any polynomial function simulating $\Omega(n^{1.5})$5 over the ring yields a non-commutative circuit over the underlying field of exactly the same size.

This connection is significant, as it expands the reach of non-commutative lower bounds to settings that naturally arise in computation with matrices, operators, and other non-commutative algebraic objects, where the ring's inherent non-commutativity is a fundamental property, not a modeling artifact.

Comparison to Prior Work and Related Literature

Prior to this work, the best known lower bounds for non-commutative arithmetic circuits were limited to $\Omega(n^{1.5})$6 or $\Omega(n^{1.5})$7, and stronger polynomials bounds, such as $\Omega(n^{1.5})$8 or $\Omega(n^{1.5})$9, were known only under strong circuit restrictions like homogeneity or bounded syntactic degree [CH23] [Sha26]. Exponential lower bounds were previously restricted to non-commutative formulas or very limited circuit classes [Nisan91], and hardness amplification results were not sufficient to yield these new bounds [CILM18]. The current paper bypasses circuit restrictions, working for general non-commutative circuits, and establishes the first unconditional, super-linear polynomial lower bounds in this model.

A further novel point is the explicit, formal articulation of the transfer from field-based non-commutative circuit lower bounds to circuits over non-commutative rings, an aspect previously often implicit (e.g., in [CS07], [CHSS11]) but here fully developed and leveraged.

Implications and Future Directions

These results deepen understanding of the computational hardness of explicit polynomials in non-commutative models and move the field towards longstanding goals in algebraic complexity, such as separating $n$0 and $n$1 in non-commutative settings, or proving super-polynomial lower bounds for commutative circuits. Practically, the lower bounds establish new concrete expectations about what can and cannot be achieved efficiently in symbolic computation, matrix algebra, and related tasks where non-commutative structures arise (e.g., in quantum computation or control theory).

The approach lays the groundwork for future developments in several directions:

• Extension of techniques to more general classes of polynomials or alternative non-commutative algebraic structures.
• Potential application to identity testing in symbolic computation, leveraging hardness-randomness connections.
• Further investigation of tightness: are these lower bounds optimal, or can even sharper results be obtained for higher degree or specific classes of polynomials?
• Exploration of potential transfer theorems or lifting theorems taking lower bounds from non-commutative to commutative (or partially commutative) settings.

Conclusion

The paper "Polynomial Lower Bounds for Arithmetic Circuits over Non-Commutative Rings" (2604.22006) advances the state-of-the-art in arithmetic circuit lower bounds by establishing the first explicit polynomial (specifically $n$2 and $n$3) lower bounds for non-commutative circuits computing explicit polynomials, and demonstrating that these bounds hold equally for arithmetic circuits over certain non-commutative rings. The proofs combine refined matrix rank arguments with novel lifting techniques. The results have substantial theoretical implications for algebraic complexity theory and invite further research into the ultimate computational limitations of arithmetic circuits in diverse algebraic domains.