⚛️ quantum physics

Rational Quantum Mechanics: Testing Quantum Theory with Quantum Computers

Motivated by Wheeler's "it-from-bit" concept, this paper proposes Rational Quantum Mechanics (RaQM), a theory where gravity discretizes Hilbert space to impose a finite information capacity limit on qubits (estimated between 200 and 1,000), predicting that the exponential advantage of quantum algorithms like Shor's will saturate at this threshold, thereby rendering the factorization of large integers impossible even for quantum computers.

Original authors: Tim Palmer

Published 2026-02-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tim Palmer

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

The Big Idea: The Universe Might Be "Pixelated"

Imagine you are looking at a high-definition movie on a screen. From a distance, the image looks perfectly smooth and continuous. But if you zoom in with a magnifying glass, you see that the image is actually made of tiny, distinct squares called pixels.

Standard Quantum Mechanics (the physics we use today) assumes the universe is like that smooth, perfect movie. It says that the "state" of a particle (where it is and how it's moving) can be any number, no matter how precise. It's like saying a color can be any shade of blue, down to an infinite number of decimal places.

Tim Palmer's new theory, "Rational Quantum Mechanics" (RaQM), suggests the universe is actually more like a digital video game. It proposes that the "smoothness" is an illusion. In reality, the universe is made of discrete "pixels" of information. You can't have any number; you can only have specific, "rational" numbers (fractions like 1/2, 3/4, 5/8).

The Problem: The "Pixel Limit"

Here is where things get tricky for quantum computers.

In a standard quantum computer, we use qubits. As you add more qubits, the amount of information the system can hold grows exponentially.

1 qubit = 2 states.
10 qubits = 1,024 states.
100 qubits = A number so huge it's bigger than the number of atoms in the universe.

Standard theory says this growth can go on forever. But Palmer argues that because the universe is "pixelated" (discrete), there is a hard limit to how much information you can squeeze into a system.

Think of it like a backpack:

The backpack is your quantum computer (the qubits).
The items you want to put in are the complex mathematical states required by quantum physics.
The size of the backpack is limited by the "pixels" of the universe.

Palmer calculates that for current technology, the backpack can only hold about 200 to 400 items before it bursts. If you try to force 1,000 items in, the "pixels" of the universe simply don't have enough space to describe the state. The math breaks down.

The Culprit: Gravity

Why is the universe pixelated? Palmer suggests the culprit is Gravity.

In our daily lives, gravity is weak. But at the tiniest scales, gravity might act like a "graininess" filter. Just as you can't divide a grain of sand infinitely, you can't divide the information in a quantum state infinitely because gravity puts a "floor" under it.

Analogy: Imagine trying to draw a perfect circle on a piece of paper. If the paper is smooth, you can draw a perfect curve. But if the paper is made of coarse sandpaper, your line will eventually look jagged. Gravity is the sandpaper. It prevents the "smooth" curve of quantum mechanics from existing at the smallest scales.

The Big Prediction: Quantum Computers Can't Crack RSA

This is the most exciting (and scary) part for the tech world.

For decades, we've been told that quantum computers will eventually be able to break the encryption (RSA) that protects our bank accounts and secrets. To do this, they need to run a specific algorithm (Shor's algorithm) that requires a massive number of entangled qubits—roughly 2,000 to 3,000 perfect qubits.

Palmer's theory says: This will never happen.

Here is the logic:

To break RSA, you need to use the full power of the quantum computer's "exponential growth."
But the universe has a "pixel limit" (around 1,000 qubits).
Once you hit that limit, the quantum computer stops behaving like a quantum machine and starts acting like a very expensive, very slow classical computer.
Result: A quantum computer will never be able to factor a 2,048-bit number any faster than a normal laptop. The encryption is safe, not because our computers aren't good enough yet, but because the laws of physics forbid it.

How Can We Test This?

You might think, "Well, we can't build 1,000 perfect qubits yet, so how do we know?"

Palmer suggests we don't need to wait that long. We can test this in the next 5 years.

As we build better quantum computers, we will try to run these complex algorithms with increasing numbers of qubits.

Standard Theory (QM) predicts: The computer gets faster and faster as we add qubits, eventually crushing the encryption.
Palmer's Theory (RaQM) predicts: The computer will get faster up to a point (around 200–400 qubits), but then it will hit a "wall." Adding more qubits won't help; the performance will plateau or crash.

If we see that "wall" appear before we reach 1,000 qubits, it proves that the universe is indeed "pixelated" and that gravity limits our computing power.

Summary in a Nutshell

The Theory: The universe isn't smooth; it's made of tiny information "pixels" (Rational Quantum Mechanics).
The Cause: Gravity creates a limit on how much information can exist in a single system.
The Consequence: Quantum computers have a hard ceiling. They cannot grow infinitely powerful.
The Impact: We will likely never be able to use quantum computers to break modern encryption.
The Test: We might find out if this is true within 5 years by watching how quantum computers perform as they get bigger.

If Palmer is right, it means the universe has a "maximum resolution," and while that limits our ability to break codes, it might help us finally understand how to unite Quantum Mechanics and Gravity into one beautiful, unified theory.

1. Problem Statement

Standard Quantum Mechanics (QM) relies on a continuous complex Hilbert Space, implying that quantum states can possess irrational amplitudes and phases, requiring an infinite amount of information to describe precisely. While environmental decoherence poses practical limits to quantum computing, standard QM imposes no fundamental limit on the number of qubits ( $N$ ) that can be coherently entangled. Consequently, theoretical models suggest that sufficiently large quantum computers could break RSA encryption (e.g., factoring 2048-bit integers) via algorithms like Shor's.

The paper addresses the question: Is there a fundamental, physical constraint preventing quantum computers from scaling indefinitely, specifically regarding the factorization of large integers? The author challenges the assumption that the continuum nature of Hilbert Space is fundamental, proposing instead that it is an idealization of a deeper, discrete reality.

2. Methodology and Theoretical Framework

The paper proposes a new theory called Rational Quantum Mechanics (RaQM). The methodology involves:

Discretization of Hilbert Space: RaQM restricts the definition of quantum states to bases where the squared amplitudes ( $\cos^2(\theta/2)$ ) and phases ( $\phi/2\pi$ ) are rational numbers of the form $m/L$ and $n/L$ , where $L$ is a large integer and $m, n$ are whole numbers.
Information-Theoretic Representation: Using a constructive discretization of the Riemann Sphere (based on $p$ -adic metrics and Cantor sets), the author represents an $N$ -qubit state not as a continuous vector, but as $N$ correlated bit strings of finite length $L$ .
The Qubit Information Capacity (QIC): The theory defines a finite capacity $N_{max}$ $N_{ma x}$ .
- In QM, the degrees of freedom for $N$ qubits grow exponentially: $D_{QM} = 2^{N+1} - 2$ .
- In RaQM, the available information bits grow linearly: $D_{RaQM} = N \times L$ .
- A breakdown occurs when $2^{N+1} - 2 > N \times L$ . At this point, there is insufficient information in the $N$ qubits to allocate even one bit to each degree of freedom required by the continuum theory.
Role of Gravity: The discretization scale $L$ $L$ is hypothesized to be determined by gravity. The author equates the time required for a quantum state to reduce to a classical state (via a chaotic shift map on the Cantor Set) with the Diósi-Penrose collapse time ( $\tau_{DP} = \hbar / E_G$ $τ_{D P} = ℏ/ E_{G}$ ), where $E_G$ $E_{G}$ is the gravitational self-energy.
- This yields the relationship: $L \approx \lceil E_P / E_G \rceil$ , where $E_P$ is the Planck energy.
- Standard QM is viewed as the singular limit where $G \to 0$ (and thus $L \to \infty$ ).

3. Key Contributions

Formulation of RaQM: A mathematically consistent framework where quantum states are defined only on rational bases, rendering irrational states (and counterfactual measurements involving them) undefined rather than merely improbable.
Derivation of $N_{max}$ : The derivation of a hard upper bound on the number of entangled qubits a physical system can support before the exponential growth of Hilbert Space outstrips the linear growth of available information bits.
Reinterpretation of Measurement: State reduction is described not as a modification of the Schrödinger equation, but as a chaotic instability in a discretized state space (a magnification of a Cantor Set), consistent with local realism and violating Bell's Measurement Independence assumption only for exact, uncontrollable settings.
Falsifiable Prediction: The theory predicts that the exponential advantage of quantum algorithms (like Shor's) will saturate at a specific qubit count, regardless of error correction or noise isolation.

4. Results and Estimates

The author estimates $L$ and the resulting $N_{max}$ for various qubit technologies using the Diósi-Penrose gravitational self-energy:

Quantum Dot Qubits (Electrons):
- Based on electron mass and superposition distances, $L \approx 10^{64} - 10^{70}$ .
- Result: $N_{max} \approx 200$ .
Photonic Qubits:
- Based on relativistic mass-energy of infrared photons.
- Result: $N_{max} \approx 300$ .
Ion Trap Qubits:
- Based on hyperfine splitting energy.
- Result: $N_{max} \approx 400$ .
Theoretical Upper Bound:
- Considering the lowest possible frequency photon (wavelength $\approx$ age of universe) superposed over a Planck length.
- Result: $N_{max}$ can never exceed 1,000.

Implication for Cryptography:

Factoring a 2048-bit RSA integer requires a quantum computer with $N > 2048$ qubits (or a massive number of noisy qubits).
Since $N_{max} \le 1,000$ , RaQM predicts that it is fundamentally impossible for a quantum computer to factor 2048-bit RSA integers, even with perfect error correction. The exponential advantage of Shor's algorithm will saturate well before reaching the necessary scale.

5. Significance and Implications

Fundamental Physics: If verified, RaQM suggests that the continuum of Hilbert Space is an approximation, and that gravity plays a constitutive role in the structure of quantum theory, not just as a field to be quantized. It offers a path to a locally realistic theory of quantum mechanics that satisfies Bell's theorem constraints via "holism" rather than non-locality.
Quantum Computing Roadmap: The paper challenges the prevailing optimism that quantum computers will inevitably break modern encryption. It suggests that the "quantum advantage" has a hard ceiling determined by the interplay of information theory and gravity.
Experimental Test: The theory is falsifiable within approximately 5 years. As quantum hardware scales toward 1,000 qubits, RaQM predicts that the performance of algorithms requiring maximal entanglement (like Shor's) will degrade or fail to scale exponentially, whereas standard QM predicts continued exponential growth.
Conceptual Shift: It reframes quantum weirdness (entanglement, non-commutativity) as artifacts of the singular limit of a discrete, rational theory, potentially resolving paradoxes like the measurement problem without invoking wavefunction collapse as a physical discontinuity.

In summary, the paper posits that gravity imposes a finite information capacity on quantum systems, creating a fundamental barrier to the scalability of quantum computers that standard QM fails to predict. This barrier would render the breaking of current RSA encryption via quantum computing impossible.