Classical randomness has emerged as an important tool to address the challenge of designing quantum algorithms and protocols. Current methods for calibrating and evaluating quantum gates, such as randomized benchmarking, rely heavily on classical randomness. Many researchers are exploring ways to incorporate classical randomness to reduce the requirements of traditional quantum algorithms due to progress toward quantum advantage and the development of early fault-tolerant quantum hardware. However, these techniques, especially randomized benchmarking, have been limited to specific areas such as Trotterized Hamiltonian simulation and phase estimation, leaving a gap for other quantum algorithms.
Existing methods discussed in this paper include a quantum computing model using a constant-size control register, strongly coupled to many qubits with local connectivity. While this setup supports controlled time evolution using the Trotter approximation, it struggles to implement Hamiltonian simulation with quantum signal processing (QSP) due to the small size of the control register. Other efforts have aimed at optimizing the implementation of QSP, particularly when dealing with block-encoded unitary operators using controlled U operations. While there are ways to remove parity constraints for real polynomials, these methods often introduce an unwanted factor of 1/2.
Researchers at the Massachusetts Institute of technology (MIT) Center for Theoretical Physics and IBM Quantum, MIT-IBM Watson ai Lab, have proposed an approach called Stochastic QSP to address the limitations of randomized quantum algorithms. This method aims to reduce the error in QSP polynomial approximations of objective functions with the help of randomized compilation. Furthermore, Stochastic QSP can achieve query complexity scaling with an error ϵ of O(log(1/ϵ)) for almost all QSP-based algorithms. This leads to an asymptotic halving of the cost of QSP-based algorithms compared to their deterministic versions, effectively combining the strengths of QSP and randomization.
The Stochastic QSP architecture is designed to apply random compilation techniques to common polynomials used in quantum algorithms. This method is evaluated on four specific polynomials:
- Jacobi-Anger expansions of cosine
- The Jacobi-Anger expansion of an exponential decay
- A smooth approximation of 1/x in a domain far from the origin, where x ∈ (−1, 1).
- An approximation of erf(kx) obtained from the integration of the Jacobi-Anger expansion of a Gaussian, where k is a parameter.
Each polynomial includes a cost parameter, which determines the degree of truncation required for an accurate approximation.
The results of applying Stochastic QSP to the selected polynomials demonstrate its effectiveness in reducing query complexity. As the degree d increases, the cost reduction ratio davg/d approaches 1/2, with a discrepancy scale of O(1/d). This confirms the ability of the method to halve the query complexity of QSP-based algorithms in practical applications. For some cost parameter values and functions, davg/d approaches 1/2 from below, indicating even better performance for smaller d values. This advantage is due to the optimization of the C constants and q values in the implementation process. Furthermore, a pattern is observed in the cost reduction ratio, linked to the ceiling function used when setting the cut-off degree d*.
In this paper, researchers introduced the stochastic quantum algorithm QSP to overcome the limitations of randomized quantum algorithms. This marks a major step in the optimization of quantum algorithms by combining the quantum algorithm QSP with random compilation. It can reduce circuit complexity by a factor of 2 in various quantum algorithms, including real-time/imaginary-time evolution, matrix inversion, phase estimation, and ground state preparation. The results highlight the importance of classical randomness as a resource in quantum computing, bringing quantum algorithms closer to practical use. Future research includes exploring the stochastic quantum algorithm QSP with noisy gates, which may further improve practical applications.
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Sajjad Ansari is a final year student from IIT Kharagpur. As a technology enthusiast, he delves into practical applications of ai, focusing on understanding the impact of ai technologies and their real-world implications. He aims to articulate complex ai concepts in a clear and accessible manner.
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