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Noise and Reachability Deficits:1
Challenging Quantum Supremacy Claims2
Daniyel Yaacov Bilar3
Norwich University, 158 Harmon Drive, Northfield Vermont 05663, USA4
ABSTRACT
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This short note examines recent research challenging Google’s 2019 claim of achieving quantum
supremacy. We analyze three studies that raise significant concerns about the reliability and scalability
of noisy intermediate-scale quantum (NISQ) computers, particularly in the context of random circuit
sampling and quantum optimization. These studies highlight the complex and often underestimated
impact of noise, its interplay with computational complexity, and the limitations of current noise models
and simulations. The findings emphasize the urgent need for more robust error correction methods and
improved hardware quality to overcome these challenges and pave the way for a true and demonstrable
quantum advantage.
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INTRODUCTION14
Google’s 2019 claim of achieving quantum supremacy using their 53-qubit Sycamore processor sparked
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intense debate and scrutiny within the scientific community Arute et al. (2019a,b). While Google’s
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experiment demonstrated a computational task (random circuit sampling) seemingly infeasible for classical
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computers at the time, subsequent research has challenged both the claimed classical intractability and the
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reliability of the quantum computations themselves.19
1 NOISE AS A FUNDAMENTAL LIMITATION20
Gil Kalai, a prominent critic of quantum supremacy claims since at least 2014, has argued that noise
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is not merely a technological obstacle but a fundamental physical constraint on NISQ computers Kalai
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(2018,2020,2023). Empirical findings from several independent recent studies provide support for
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his arguments.The statistical analysis of Google’s experimental data Kalai et al. (2023), analysis of
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reachability deficits in QAOA Akshay et al. (2020), and quantum tomography of the Sycamore gate
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AbuGhanem and Eleuch (2024) all point to the significant impact of noise on NISQ computer performance.
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A statistical investigation by Kalai et al. (2023) revealed a substantial deviation between the actual
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distribution of bitstrings produced in Google’s experiment and the expected distribution based on their
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noise model. This disparity suggests a more intricate and less predictable noise behavior than initially
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assumed. For example, for circuits with 12 qubits, the
χ2
value obtained was around 40,000, while the
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expected value for samples according to Google’s noise model is around 4,000.31
Further investigation using Fourier analysis in Kalai et al. (2024) reveals that gate errors, which are
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errors in executing quantum operations, have a substantial impact on high-degree Fourier coefficients
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Kalai et al. (2024). These coefficients represent complex correlations between multiple qubits, and their34
amplification by gate errors implies a less reliable output. This effect, however, is not observed in Google’s
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experimental data, raising concerns about the accuracy of current noise models and simulations. The
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study found that in simulations, gate errors significantly increased the estimated value of the effective
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readout error – a metric that combines the impact of readout and gate errors. For 12-qubit simulations, the
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effective readout error was 39% higher than the physical readout error, and for 14-qubit simulations, it
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was 24% higher. However, this effect was not observed in the actual Google experimental data.40
2 REACHABILITY DEFICITS AND SCALABILITY41
Akshay et al. (2020) focus on Google’s use of the Quantum Approximate Optimization Algorithm (QAOA)
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for demonstrating quantum advantage Akshay et al. (2020). They introduce the concept of “reachability43
deficits” arguing that QAOA’s performance is highly sensitive to the density of the problem graph (ratio
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of edges to nodes). As density increases, representing more complex problems, the effectiveness of
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fixed-depth QAOA diminishes rapidly, requiring deeper circuits with more quantum gates to maintain
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accuracy.47
Supporting this observation, AbuGhanem and Eleuch (2024) employed quantum tomography to
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analyze the fidelity of Google’s Sycamore gate on IBM’s quantum computers. Their study demonstrated a
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steep decline in fidelity when transitioning from idealized simulations to actual quantum hardware, even for
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a relatively straightforward five-qubit circuit. This suggests that the impact of noise escalates significantly
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in real-world quantum systems. They found a state fidelity of 97.72% in noise-free simulations, which
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dropped to 51.55% in noisy simulations and plummeted further to 15.95% when executed on a real IBM
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quantum computer. This highlights the rapid accumulation of errors in larger, more complex circuits
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and further emphasizes the need for deeper, and therefore more noise-prone, QAOA circuits to address
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complex problems.56
3 THE CALIBRATION PROBLEM57
Kalai’s work also raises concerns about the extensive calibration process required in Google’s experiment,
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which involves adjusting gate parameters based on experimental feedback. This reliance on calibration, he
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argues, undermines the claim of having a ”programmable” quantum computer, suggesting a lack of control
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and predictability in the system Kalai et al. (2023). The tomography study supports this by showing
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that the fidelity of uncalibrated circuits drops significantly, highlighting the crucial role of calibration
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in achieving even modest performance. For instance, removing the calibration adjustments for a single
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two-qubit gate in the 12-qubit circuit often slashed the fidelity close to zero.64
4 IMPLICATIONS FOR QUANTUM SUPREMACY65
These combined findings cast significant doubt on the robustness and generalizability of Google’s quantum
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supremacy claims. The cumulative evidence points towards:67
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Oversimplified Noise Models: Existing noise models fail to capture the complex and nuanced be-
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havior of noise in NISQ devices.70
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Unrealistic Extrapolation: Performance results from small-scale experiments cannot be reliably extrapo-
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lated to larger, more complex quantum computations.73
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Limited Programmability: The heavy reliance on calibration indicates a lack of true ”programma-
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bility” in current NISQ systems.76
5 BACK TO THE FUTURE: LEVIN’S PRESCIENCE77
Leonid Levin, a renowned computer scientist known for his foundational work in computational complex-
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ity theory, expressed deep skepticism about the practicality of building large-scale, fault-tolerant quantum
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computers as early as 2003 in his paper ”The Tale of One-Way Functions” Levin (2003). He argued that
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the inherent sensitivity of quantum systems to noise, coupled with the unrealistic precision required for
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quantum amplitudes in algorithms like Shor’s factoring algorithm, posed insurmountable challenges.82
Levin particularly emphasized the issue of noise sensitivity, highlighting that “tiny backgrounds of
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neutrinos, gravitational waves, and other exotics, cannot be shielded” and that their effects on quantum
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amplitudes could readily disrupt the delicate superposition states required for quantum computation
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Levin (2003). He foresaw that even seemingly minor noise sources could lead to significant decoherence
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problems, especially as quantum computers scaled up in size and complexity.87
This prescient observation is strikingly reflected in the findings of Kalai et al. (2023,2024), where the
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discrepancy between simulated and experimental noise behavior and the observed noise magnification in
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larger circuits underscore the profound challenge of controlling noise in real-world quantum systems.90
Furthermore, Levin pointedly questioned the feasibility of maintaining the extreme precision required
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for quantum amplitudes, arguing that “we have never seen a physical law valid to over a dozen decimal”
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and that pushing quantum mechanics to hundreds or thousands of decimal places was highly speculative
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Levin (2003). He raised fundamental doubts about the physical meaning of such precise amplitudes and
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whether the very laws of physics would hold at such levels of accuracy.95
These concerns are echoed in the work of AbuGhanem and Eleuch (2024), who demonstrated
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a dramatic drop in fidelity when running quantum circuits on real hardware compared to noise-free
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simulations. The low fidelities observed, even for a modest five-qubit circuit, highlight the difficulties of
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maintaining high precision in realistic quantum computations, lending credence to Levin’s skepticism
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about achieving the accuracy required for fault-tolerant quantum computing.100
Summa summarum: The observed noise sensitivity, the discrepancy between simulated and experi-
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mental noise behavior, and the dramatic fidelity drops in real-world quantum computers all echo Levin’s
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early skepticism about the feasibility of fault-tolerant quantum computation. His prescience underscores
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the importance of critically assessing the fundamental limitations of physical systems.104
TERMS OF ART105
Quantum Supremacy is the point at which a quantum computer can perform a task that is demonstrably
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beyond the capabilities of any existing classical computer. This doesn’t necessarily mean solving
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practically useful problems, but rather demonstrating a clear computational advantage in a well-defined
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task. The task chosen for Google’s claim was random circuit sampling.109
NISQ (Noisy Intermediate-Scale Quantum) Computers are the current generation of quantum
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computers, characterized by a limited number of qubits (typically less than 100) and susceptibility to noise.
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These computers are not yet fault-tolerant, meaning errors accumulate during computations, limiting their
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capabilities.113
Qubit is the basic unit of information in a quantum computer, analogous to a bit in a classical computer.
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Qubits leverage quantum phenomena like superposition and entanglement, allowing them to exist in a
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combination of 0 and 1 states simultaneously, potentially enabling parallel processing.116
Quantum Gate is a logical operation that acts on qubits, similar to logic gates in classical computing.
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Examples include single-qubit gates like the Pauli-X gate (analogous to a NOT gate) and two-qubit gates
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like the controlled-NOT (CNOT) gate, which entangles two qubits.119
Fidelity is a measure of the accuracy of a quantum gate or computation. Higher fidelity means less
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error. Fidelity can be affected by various factors, including noise and imperfections in the quantum
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hardware.122
Random Circuit Sampling is a computational task where a quantum computer executes a randomly
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generated sequence of quantum gates and measures the resulting output bitstring. The distribution of
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these bitstrings is believed to be difficult to simulate classically, making it a candidate for demonstrating
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quantum supremacy.126
Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm designed to find
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approximate solutions to optimization problems, which are problems that involve finding the best solution
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from a set of possible options.129
Reachability Deficits refers to the phenomenon where the performance of QAOA degrades as the
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complexity (density) of the problem graph increases. This suggests that QAOA might not offer a significant
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advantage for highly complex problems, especially on noisy devices.132
Noise refers to unwanted disturbances in a quantum system that can lead to errors in computations.
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Noise can arise from various sources, including environmental interactions (e.g., stray electromagnetic
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fields, temperature fluctuations) and imperfections in the quantum hardware itself (e.g., variations in qubit
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properties, gate inaccuracies).136
Quantum Tomography is a set of techniques used to characterize the state of a quantum system
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or the operation of a quantum gate. It involves performing a series of measurements to reconstruct the
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quantum state or the process matrix that describes the operation.139
REFERENCES140
AbuGhanem, M. and Eleuch, H. (2024). Full quantum tomography study of google’s sycamore gate on
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ibm’s quantum computers. The European Physical Journal Quantum Technology, 10(1):25.142
Akshay, V., Philathong, H., Zacharov, I., and Biamonte, J. (2020). Reachability deficits implicit in
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google’s quantum approximate optimization of graph problems. arXiv preprint arXiv:2004.04197.144
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Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J. C., Barends, R., Biswas, R., Boixo, S., Brandao,
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F. G., Buell, D. A., et al. (2019a). Quantum supremacy using a programmable superconducting
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processor. Nature, 574(7779):505–510.147
Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J. C., Barends, R., Biswas, R., Boixo, S., Brandao,
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F. G., Buell, D. A., et al. (2019b). Supplementary information for “quantum supremacy using a
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programmable superconducting processor”. arXiv preprint arXiv:1910.11333.150
Kalai, G. (2018). Three puzzles on mathematics, computation and games. In Proceedings of the
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International Congress of Mathematicians 2018, Rio de Janeiro, Vol. I 2018, pages 551–606.152
Kalai, G. (2020). The argument against quantum computers. In Quantum, Probability, Logic: Itamar
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Pitowsky’s Work and Influence, pages 399–422. Springer.154
Kalai, G. (2023). The argument against quantum computers, the quantum laws of nature, and google’s
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supremacy claims. The Intercontinental Academia Laws: Rigidity and Dynamics.156
Kalai, G., Rinott, Y., and Shoham, T. (2023). Questions and concerns about google’s quantum supremacy
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claim. arXiv preprint arXiv:2305.01064.158
Kalai, G., Rinott, Y., and Shoham, T. (2024). Quantum advantage demonstrations via random circuit
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sampling: Fourier expansion and statistics. arXiv preprint arXiv:2404.00935.160
Levin, L. A. (2003). The tale of one-way functions. Problems of Information Transmission, 39(1):92–103.
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