How Real is Incomputability in Physics?

Abstract: A physical system is determined by a finite set of initial conditions and "laws" represented by equations. The system is computable if we can solve the equations in all instances using a "finite body of mathematical knowledge". In this case, if the laws of the system can be coded into a computer program, then given the initial conditions of the system, one can compute the system's evolution. Are there incomputable physical systems? This question has been theoretically studied in the last 30-40 years. In this paper, we experimentally show for the first time the strong incomputability of a quantum experiment, namely the outputs of a quantum random number generator. Moreover, the experimental results are robust and statistically significant.

• The paper demonstrates experimental evidence for physical incomputability by implementing a 3D-QRNG and validating its algorithmic randomness.
• It employs a superconducting transmon qutrit and CNN-based classification to achieve high-speed data generation with rigorous calibration protocols.
• Comparative tests using Z-liar metrics reveal statistically significant differences (p=0.0047) between the QRNG outputs and traditional PRNG sequences.

This paper, "How Real is Incomputability in Physics?" (2311.00908), explores whether the effects of incomputability in physical systems can be experimentally observed and practically justified, moving beyond purely theoretical proofs. The authors achieve this by implementing a specific type of Quantum Random Number Generator (QRNG) and testing the algorithmic randomness of its outputs using a method based on algorithmic information theory.

The core idea is rooted in the concept of value indefiniteness in quantum mechanics, linked to the Kochen-Specker theorem. In a quantum system, an observable is value definite if its outcome is predetermined (either 0 or 1) before measurement, and value indefinite if it is not. The Localised Kochen-Specker Theorem, applicable in Hilbert spaces of dimension 3 or higher ($\mathbb{C}^3$), provides a constructive way to identify such value indefinite observables under specific physical assumptions (Admissibility, Non-contextuality of definite values, and Eigenstate principle). Crucially, the authors rely on a further principle, the 'epr principle' (related to Einstein, Podolsky, Rosen), which states that if repeated measurements of an observable generate a computable sequence, then the observable must be value definite. Combining the Eigenstate and epr principles, it's theoretically proven that infinite sequences generated by measuring a quantum value indefinite observable are strongly incomputable, specifically "bi-immune" (no algorithm can generate infinitely many correct elements) and individually unpredictable.

The paper presents a physical realization of such a 3D-QRNG using a superconducting transmon system, which effectively acts as a qutrit (a 3-level quantum system). The implementation details involve mapping logical spin states to transmon energy eigenstates ($|0\rangle, |1\rangle, |2\rangle$). The system is prepared in a specific state, and then microwave pulses are applied to perform rotations corresponding to a measurement in the eigenbasis of an $S_x$ operator, which acts as the value indefinite observable in this setup. Measurements are performed using a standard dispersive readout scheme amplified by a Josephson parametric amplifier, and the resulting signals are classified using a convolutional neural network (CNN) to distinguish the three qutrit states with high fidelity.

The data generation process (Algorithm 1) involves repeatedly preparing the qutrit, applying control pulses, and measuring its state. The process is designed for high-speed operation (rate of 312.5 kHz) using an active reset protocol to quickly return the transmon to its ground state. To ensure robustness over large datasets (100 Gbit generated), the system incorporates intermittent checks of the CNN classification fidelity and triggers recalibration (Algorithm 2) if fidelity drops below a threshold. Measurement errors are identified as the dominant error source, typically resulting in fidelities of 95%, 88%, and 78% for the ground, first, and second excited states, respectively.

procedure RunGeneration(num_runs, measurements_per_run): calibrate_system() train_cnn() measurement_fidelity = assess_fidelity() low_fidelity_counter = 0 calibration_counter = 0 run_index = get_last_run_index() while run_index < num_runs: while measurement_fidelity < FIDELITY_THRESHOLD: low_fidelity_counter += 1 if low_fidelity_counter > LOW_FIDELITY_MAX_RETRIES: calibration_counter += 1 if calibration_counter > CALIBRATION_MAX_RETRIES: raise Error("Calibration failed too many times.") calibrate_system() low_fidelity_counter = 0 # Reset counter after calibration train_cnn(extra_cycles=20) # Train CNN more measurement_fidelity = assess_fidelity() # Fidelity is acceptable, generate data setup_protocol_pulses() results = [] for i in range(measurements_per_run): prepare_qutrit() apply_pulses() measure_state() results.append(classify_measurement()) active_reset() # Reset qutrit quickly store_measurements(results, run_index) run_index += 1 # Optional: Longer wait time between runs if needed for stability # wait(LONG_WAIT_TIME) end while end procedure procedure Calibrate(): # Detailed calibration steps for frequencies, pulses, and readout # (Based on Algorithm 2 in the paper) tune_frequencies() calibrate_pulses() calibrate_reset_pulse() optimize_readout_frequency() create_and_train_cnn(initial_cycles=50) end procedure

Testing the incomputability of the generated sequences is challenging because incomputability is an asymptotic property that finite samples cannot definitively prove or disprove. Standard statistical tests for randomness, like those checking for uniform distribution or lack of correlations, are also insufficient as they can be passed by computable sequences (e.g., the Champernowne sequence). To overcome this, the authors use an algorithmic test based on the Chaitin-Schwartz theorem. This theorem connects algorithmic randomness (Kolmogorov complexity) to the ability of a string to de-randomize probabilistic algorithms. Specifically, a sufficiently $c$-random string can de-randomize the Solovay-Strassen probabilistic primality test. The test works by using digits from the random string to generate bases for the Solovay-Strassen predicate $W(i,n)$. For a composite number $n$, the predicate $Z(s,n)$ (constructed from multiple applications of $W$) should ideally be false for a truly random string $s$. If $Z(s,n)$ is true for a composite $n$, $s$ is called a "Z-liar", indicating a lack of algorithmic randomness as it fails to reveal the compositeness of $n$.

The authors applied this test to long strings ($2^{32}$ bits each) generated by their 3D-QRNG, the Python3 Mersenne Twister PRNG, and the SHA3 cryptographic hash function. They tested against odd composite numbers less than 50. The metric used was the average number of Z-liars found for each composite number across ten generated strings. The results (Table 1 and Figure 1) showed that the 3D-QRNG generated significantly more Z-liars for composite numbers 9 and 15 compared to the PRNGs. While Z-liars were only consistently detected for smaller composites (up to 25), the difference observed for 9 and 15 was substantial. A Kolmogorov-Smirnov test (Table 2) confirmed that the distribution of Z-liar counts from the 3D-QRNG was statistically significantly different ($p=0.0047$) from both PRNGs.

This difference is interpreted as experimental evidence that the 3D-QRNG's output possesses a form of algorithmic randomness that distinguishes it from computable pseudo-random sequences, reflecting its underlying theoretical incomputability. The higher Z-liar count for the QRNG implies its outputs are less effective at "de-randomizing" the primality test for composites, which, counter-intuitively perhaps, aligns with the Chaitin-Schwartz theorem's statement about c-random strings revealing compositeness (i.e., not being Z-liars). The authors suggest this indicates the test is sensitive to repetitive structures in PRNG outputs, which are less present in the more algorithmically random QRNG outputs.

In conclusion, the paper provides the first experimental demonstration of the real effects of incomputability in a physical system, specifically a 3D-QRNG. By using a non-statistical test based on the Chaitin-Schwartz theorem, they show a statistically significant advantage of the implemented QRNG over PRNGs in terms of algorithmic randomness, thus providing an "Einsteinian justification" based on observable usefulness for the theoretical notion of incomputability in this context. The authors acknowledge that the computational cost of this specific test is high and highlights the need for more efficient methods to probe algorithmic randomness.