⚛️ quantum physics

A remark on the Brown-Susskind conjecture

Building on the Brown-Susskind conjecture, this paper demonstrates that the dimension of the set of $n$ -qubit unitaries generated by a fixed number of $2$-qubit gates strictly increases when an additional pair of qubits is included, provided the pairs are chosen appropriately at each step.

Original authors: Jean-Luc Brylinski, Ranee Brylinski

Published 2026-09-22

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CC BY 4.0

Original authors: Jean-Luc Brylinski, Ranee Brylinski

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern physics, there is a growing interest in understanding how complex a system can become when we build it up piece by piece. Imagine a machine made of tiny, interconnected switches, where each switch can be flipped in specific ways to change the state of the whole device. Scientists call these switches qubits, and the operations that flip them are known as gates. When we string these gates together in a sequence, we create a circuit that performs a calculation. A central question in this field is how the complexity of the final result grows as we add more steps to the sequence. For a long time, researchers believed that if you kept adding random steps, the complexity would rise steadily and predictably, eventually reaching a peak that is unimaginably large relative to the size of the system. This idea, known as the Brown-Susskind conjecture, suggests that complexity grows in a straight line until it hits a ceiling determined by the sheer number of possible configurations.

Two recent proofs confirmed that this linear growth does happen on average for random circuits. However, a new paper by Ranee Brylinski and Jean-Luc Brylinski asks a more precise question: does the complexity always increase, or are there moments where adding a step changes nothing? The authors investigate a specific scenario where we have a fixed set of allowed connections between pairs of these switches. They look at the collection of all possible outcomes that can be reached by multiplying a certain number of these allowed operations together. Their goal is to determine if there is always a way to choose the next operation so that the set of reachable outcomes gets strictly larger.

The researchers focused on a mathematical structure that describes these collections of outcomes. They treated the set of all possible operations as a geometric shape, where the size of the shape represents the complexity or "dimension" of the system. If the shape grows, it means we have gained new capabilities; if it stays the same size, we have merely retraced our steps. The paper proves a fundamental property about these shapes: as long as the network of allowed connections between the switches is linked together in a single piece, the total set of operations that can be generated eventually fills the entire space of possibilities. This means that if you keep adding steps, you will eventually be able to reach any possible state of the machine, provided the connections between the switches allow you to travel from any one switch to any other.

The core finding of the paper is that this growth is not just a possibility, but a certainty under the right conditions. The authors demonstrate that if the current set of outcomes has not yet reached its maximum possible size, there is always at least one choice for the next pair of switches to connect that will make the set of outcomes strictly larger. In other words, you can never get stuck in a loop where adding another step fails to expand your reach, as long as you are allowed to pick which pair of switches to use next. This result holds true even though the authors admit they cannot yet provide a simple rule for which specific pair to choose to guarantee this growth. They know such a choice exists, but finding the best one remains an open question.

This work refines our understanding of how quantum systems evolve. While previous studies showed that complexity grows linearly on average, this paper establishes that the growth is strictly monotonic for at least one path forward at every single step. The researchers used advanced tools from algebra and geometry to prove that the shape formed by these operations cannot stay the same size if it is not yet full. They showed that if the shape were to stop growing, it would imply that the entire system is trapped in a smaller, isolated part of the mathematical space, which contradicts the fact that the connections between the switches are linked. Therefore, the system must expand.

The paper also touches on a simpler version of this problem involving two specific, repeating patterns of operations. In this restricted case, the authors suggest that the complexity increases by exactly one unit with each new step, up to a certain limit. This aligns with the intuition that each new operation adds a distinct layer of capability. However, the authors note that this precise behavior relies on the operations being periodic, or repeating in a regular cycle. If the operations do not repeat, the mathematical tools used to prove the result become more difficult to apply, and the exact behavior remains less clear.

Ultimately, this research provides a rigorous guarantee that quantum circuits do not stagnate. It confirms that as long as the underlying network of connections is intact, there is always a way to push the system forward into new territory. The work does not solve the practical problem of how to find the best sequence of operations for a specific task, but it removes the fear that the system might hit a dead end where no further progress is possible. It assures us that the landscape of possibilities is always expanding, waiting for the right choice to reveal the next step.

Technical Summary: A Remark on the Brown-Susskind Conjecture

Problem Statement
This paper investigates the growth of the dimension of the set of $n$ -qubit unitaries ( $U(2^n)$ ) that can be constructed as a product of a fixed number of 2-qubit gates selected from a specific set of qubit pairs. The work is situated within the context of the Brown-Susskind conjecture, which posits that the quantum complexity of a local random circuit grows linearly with the number of gates until reaching a value exponential in the system size. While the conjecture was previously proven in [3] and [4] (with additional proofs by Zhi Li [4]), those works rely on fixed, repeating architectures (e.g., staircase or wheel-spoke topologies). This paper addresses the scenario where the sequence of 2-qubit pairs evolves freely within a fixed alphabet $S$ of qubit pairs, rather than following a predetermined recursive pattern.

Methodology and Mathematical Framework
The authors employ tools from real-algebraic geometry and Lie theory to analyze the structure of the reachable unitary sets.

Graph-Theoretic Generation: The authors define a graph $\Gamma$ where nodes represent qubits and edges represent the available pairs of qubits ( $S$ ). They establish that the subgroups of 2-qubit gates ( $G_\alpha$ ) generate the full unitary group $U(2^n)$ if and only if $\Gamma$ is connected. The proof utilizes the Lie algebra $\mathfrak{u}(2^n)$ , demonstrating that if $\Gamma$ is connected, the Lie algebra generated by the 2-qubit gates contains all Pauli strings, thereby spanning the entire algebra.
Algebraic Varieties and Semialgebraic Sets: The set of unitaries reachable by a sequence of $k$ gates, denoted $U_k$ , is defined as the image of a construction map $F_k: G_{\alpha_1} \times \dots \times G_{\alpha_k} \to U(2^n)$ . The authors note that while $U(2^n)$ and the gate subgroups are real-algebraic varieties, the image $U_k$ is generally a semialgebraic set (a finite union of sets defined by polynomial equations and inequalities), not necessarily an algebraic subvariety.
Dimension Analysis via Tarski–Seidenberg: The analysis relies on the Tarski–Seidenberg theorem, which ensures that the image of a polynomial map between real-algebraic varieties is a semialgebraic set. The authors utilize the concept of the Zariski closure and the dimension of algebraic sets to analyze the growth of $U_k$ .

Key Contributions and Results
The central result is Theorem 1.1, which addresses the dimensionality of the reachable set at each step of the circuit construction:

Existence of Dimension-Increasing Choices: The theorem states that given a sequence of qubit pairs $\tau = \alpha_1, \dots, \alpha_k$ such that the resulting set $U_k$ does not yet have the maximal dimension (i.e., $\dim(U_k) < \dim(U(2^n))$ ), there exists a choice of a subsequent pair $\alpha_{k+1}$ from the alphabet $S$ such that $\dim(U_{k+1}) > \dim(U_k)$ .
Proof Mechanism: The proof proceeds by contradiction. It assumes that for a given $U_k$ , no choice of $\alpha_{k+1}$ increases the dimension. Under this assumption, the inclusion $U_k \subseteq U_{k+1}$ would imply equality ( $U_k = U_{k+1}$ ) because both would be irreducible algebraic varieties of the same dimension. This equality implies that $U_k$ is invariant under the right action of any $G_{k+1}$ in the alphabet. Since the subgroups in $S$ generate $U(2^n)$ , this would force $U_k$ to be equal to the entire group $U(2^n)$ , contradicting the initial assumption that $U_k$ was not maximal.
Irreducibility: A critical lemma establishes that the real-algebraic variety $U_k$ is irreducible, meaning it cannot be decomposed into the union of two proper Zariski closed subsets. This property is essential for concluding that equal dimensions imply set equality.
Non-Constructive Nature: The authors explicitly note that while they prove the existence of a choice that increases the dimension, they do not know how to determine or construct the specific next qubit pair $\alpha_{k+1}$ to achieve this. They speculate that a good choice might involve a pair "far away" from recently used edges, but this remains unproven. This contrasts with works like [3] and [4], which use fixed architectures where the next gate is predetermined.

Specific Case: One-Parameter Subgroups
The paper also examines a specific case involving two one-parameter subgroups $T_1$ and $T_2$ generated by Hamiltonians $H_1$ and $H_2$ .

Conjecture 1 (attributed to Seth Lloyd): If $iH_1$ and $iH_2$ generate the Lie algebra of skew-Hermitian operators and the subgroups are periodic, then for $k \le 2^{2n}$ , the dimension of the set of $k$ -fold products is exactly $k$ .
Support: The authors argue that under the periodicity assumption, the subgroups are closed real-algebraic subvarieties. Consequently, the dimension must increase by at least 1 at each step (as one must switch between subgroups), and since it cannot increase by more than 1, the growth is exactly linear.

Significance and Limitations
The paper provides a theoretical guarantee that for any sequence of gates that has not yet reached the full unitary group, there exists at least one choice of the next gate from the available alphabet that will strictly increase the dimension of the reachable set. This result relies on the irreducibility of the reachable sets and the connectivity of the underlying qubit interaction graph.

However, the authors are explicit about the limitations of their constructive capability:

Existence vs. Construction: The paper proves the existence of a qubit pair $\alpha_{k+1}$ that increases the dimension but explicitly states that it does not provide an algorithm or method to choose this pair. The authors note that a heuristic choice might involve selecting a pair "far away" from recently used edges, but this remains speculation.
Contrast with Prior Work: The authors highlight that this uncertainty regarding the specific choice of the next gate does not arise in works like [3] and [4], which utilize recursive, fixed architectures where the next gate is predetermined.

In summary, the paper provides a theoretical guarantee that dimension growth is possible at every step in a free-evolving architecture, relying on the irreducibility of the reachable sets and the connectivity of the underlying qubit interaction graph, without offering a constructive strategy for achieving this growth.

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