Programming Empty Space to Solve Hard Problems

In the paper's words

The exponential growth of Hilbert space dimensions presents a fundamental barrier to the classical simulation of quantum systems, necessitating a paradigm shift from algorithmic simulation to physical instantiation. This study validates the “Inverted Church-Turing-Deutsch” (CTD) framework, which posits that physical Hamiltonian systems can serve as efficient computational substrates for problems intractable to Turing machines. We operationalize this framework through two modalities: “Class A” discrete optimization using Ising models and “Class B” spectral engineering targeting the Riemann zeros.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Furthermore, we successfully engineer a 1D physical potential $V(x)$ that reproduces the first five Riemann zeros with a Mean Absolute Percentage Error of 0.033%, effectively functioning as a “Physical Oracle” for number theory.

   from A

2. We defined TTS as the expected wall-clock time required to find the ground state with a 99% probability of success.

   from 3.6 Performance Metrics

3. We initialized the potential $V(x)$ as a standard Harmonic Oscillator, $V_{init}(x) = 0.5 x^2 + 10.0$, shifted in energy to align roughly with the magnitude of the target zeros.

   from 4.6 Initial Conditions

4. The results, summarized in Table 1, demonstrate a Mean Absolute Percentage Error (MAPE) of just 0.033% across the first five energy levels, significantly outperforming the pre-defined success criterion of 5%.

   from 5.2 Class B: Spectral Accuracy

5. Engineered Eigenvalues Level Target (Riemann $\gamma_n$) Engineered ($E_n$) Error (%) $E_0$ 14.1347 14.1411 0.05% $E_1$ 21.0220 21.0201 0.01% $E_2$ 25.0109 25.0236 0.05% $E_3$ 30.4249 30.4335 0.03% $E_4$ 32.9351 32.9255 0.03%

   from 5.2 Class B: Spectral Accuracy

6. The synthetic success rate data indicated that the Ising configuration achieved a mean success probability of 85%, significantly outperforming the Heisenberg model (62%) and the Hubbard model (45%).

   from 5.6 Comparative

How it is done

1. This approach, termed “Hamiltonian Engineering,” treats the physical laws governing a system—specifically the potential landscape $V(x)$ and interaction terms $J_{ij}$—as the “source code” of the computation.

   from 1.2 The Inverted

2. We build upon these precedents by synthesizing two distinct historical threads: the “Ising machine” approach for combinatorial optimization and the “spectral synthesis” approach for mathematical constants.

   from 1.3 Historical Context and

3. We approach these questions through rigorous computational validation rather than physical experimentation, allowing us to isolate algorithmic scaling from hardware noise.

   from 1.4 Research Objectives

4. This approach allows us to explore scaling behaviors up to the classical simulation limit ($N \approx 12$ for full dynamics, $N \approx 60$ for 1D spectral problems) without the confounding variables of decoherence and fabrication error present in current hardware.

   from 1.5 Scope and Limitations

What would falsify it

1. In Class A, the observable is the spin configuration $\sigma_z$; in Class B, the observable is the energy spectrum $E_n$.

   from 2.4 The Unified

2. The Hamiltonian runtime data was best described by a polynomial function $T(N) \propto N^b$ with a fitted exponent of $b \approx 2.02$, closely matching the theoretical prediction of $O(N^2)$ scaling associated with the quadratic cost of programming the interaction matrix (Berns et al., 2025).

   from 5.1 Class A: Scaling Analysis

3. Both are inverse problems of the Schrödinger equation: given a target observable (ground state or spectrum), find the generator $\hat{H}$ (Sierra, 2019).

   from 6.4

Where the claim stops

1. We acknowledge that “in silico” validation cannot capture the full spectrum of experimental anomalies, such as control line crosstalk or thermal fluctuations, though we incorporate synthetic noise models to test robustness.

   from 1.5 Scope and Limitations

2. Second, our noise models, though robust, assumed simple log-normal distributions; real quantum hardware suffers from non-Markovian noise and correlated errors that are far more destructive.

   from 6.6 Limitations of the Study

3. This limitation forces a paradigm shift from the algorithmic representation of physical reality to the direct instantiation of physical systems as computational resources.

   from 1.1 Context: The

Where to read

1. A232 words · 1 min
2. 1.0 Introduction1,183 words · 5 min
3. 2.0 Theoretical Framework1,212 words · 5 min
4. 3.0 Methodology: Class A1,082 words · 5 min
5. 4.0 Methodology: Class B981 words · 4 min
6. 5.0 Results and Analysis1,129 words · 5 min
7. 6.0 Discussion and1,022 words · 4 min
8. 7.0 Conclusion and Future Work631 words · 3 min
9. Appendices468 words · 2 min