Quantum Computational Advantage (QCA)
Secured Quantum Computational Advantage: Boson Sampling Intractability Is Proved by the Logarithmic Noise Threshold
By Hyunseok Jeong, Changhun Oh, and Byeongseon Go. The pursuit of proving Quantum Computational Advantage (QCA), the point at which a quantum device completes a particular work that is beyond the capabilities of conventional supercomputers, has reached a crucial point. Boson Sampling (BS), one of the most promising methods for achieving QCA, has strong complexity-theoretical support for its capacity to withstand a large amount of experimental noise while maintaining its basic classical intractability.
Favoured for its compelling proof of computational complexity and experimental viability, boson sampling has recently achieved system sizes large enough to assert QCA in a number of experiments. A significant challenge, meanwhile, is the inevitable existence of physical flaws in near-term quantum devices, such as photon loss and incomplete distinguishability of photons. Quantum Classical algorithms can effectively imitate the process if the noise rate is too high, “ruling out” the quantum advantage.
Therefore, using near-term noisy devices to illustrate QCA requires a precise characterization of the limit of noise rates where the traditional intractability is preserved.
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Identifying the Logarithmic Noise Boundary
Characterizing the impacts of partial-distinguishability noise, a significant obstacle to Quantum Computational Advantage QCA in optical systems and in more recent bosonic platforms such as atomic arrays and ion traps currently being employed for boson sampling, is the main contribution of the Go, Oh, and Jeong. The task’s quantum computing difficulty is directly related to photons’ indistinguishable nature.
The researchers found that Boson Sampling retains the same level of complexity as the ideal boson-sampling situation even when there are, on average, distinct photons out of input photons.
For traditional hardness arguments, this result indicates a notable improvement in the computed noise robustness. Prior research on photon loss in particular had only shown that complexity equivalence was preserved if a maximum of a predetermined number of input photons were lost. In contrast to these previous findings, the current research demonstrates that the allowed number of acceptable noisy photons scales logarithmically with the system size. The implementation of Quantum Computational Advantage QCA with noisy boson samplers is made possible by this enhanced threshold.
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Complexity-Theoretical Equivalence via Reduction
The computational challenge of predicting the ideal output probability within specific error boundaries is the foundation of the traditional hardness of ideal Boson Sampling. It is hypothesized that this problem is #P-hard. It is quite improbable that the polynomial hierarchy will collapse if this estimation problem can be resolved by an effective classical algorithm.
Similar to earlier research on photon loss, the methodology employed in this study was to demonstrate that the average-case hardness of ideal boson sampling could be lowered to the average-case hardness of noisy boson sampling. The similar problem for the noisy system, which entails calculating the noisy output probability given an indistinguishability rate, was formalized by the authors. As long as the noise rate is below the logarithmic threshold, the decrease demonstrates that solving the noisy problem is at least as challenging as solving the ideal problem.
The employment of a low-degree polynomial approximation in conjunction with polynomial interpolation is the primary mechanism that makes this finding possible. In the indistinguishability rate, the noisy output probability is inherently a polynomial.
However, there would be an exponential imprecision blowup if the complete degree polynomial were interpolated in order to determine the optimal output probability. The researchers were successful in creating a logarithmically scaled degree polynomial approximation. As long as the average number of identifiable photons is maintained, this low-degree approximation permits the inference of the ideal output probability (which corresponds to through interpolation) without experiencing an exponentially huge imprecision blowup.
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Combined Noise and Future Challenges
To handle the real-world situation where photon loss and partial distinguishability happen at the same time, the approach was further generalized. The results verify that when the system experiences an average number of identifiable photons and photon loss, the classical intractability of ideal Boson Sampling is maintained.
In addition to providing a baseline for the amount of physical noise required in near-term experiments to successfully show Quantum Computational Advantage QCA, these results are meant to aid academics in understanding the regimes of classical intractability for noisy boson sampling.
But there are still a lot of unresolved issues. Only when the average number of noisy photons scales logarithmically does the current finding hold true. Indistinguishability or a constant fraction of noisy photons are common problems with experimental implementations. Since the existing low-degree polynomial approximation strategy fails, it is necessary to develop completely new advanced techniques in order to extend the hardness proof to this constant noise rate domain.
The classical simulation hardness of Gaussian Boson Sampling (GBS), a variation that frequently fits in better with existing experimental setups using Gaussian state inputs, also needs more theoretical work. Another intriguing future issue is including non-uniform noise models in place of idealized uniform distinguishability noise represented by a single parameter.
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