Abstract:ABSTRACT. The hidden subgroup problem (HSP) plays an important role in quantum computing because many quantum algorithms that are exponentially faster than classical algorithms are special cases of the HSP. In this paper we show that there exists a new efficient quantum algorithm for the HSP on groups Z N Z q s where N is an integer with a special prime factorization, q prime number and s any positive integer.
“…In various quantum algorithms and quantum computations, such as quantum phase estimation algorithm 1–3 , the factoring problem 4–7 , the discrete logarithm problem 2,4,8,9 , and the hidden subgroup problem 10–12 , a discrete quantum Fourier transform (DQFT) 2,13–19 plays a critical role in accomplishing quantum information processing. Thus, for the experimental implementation of DQFT, a variety of physical resources have been used, including those based on linear optical systems 20–22 , nonlinear optical systems 17,23–25 , nuclear magnetic resonance or ion trap systems 26–29 , superconducting circuits 30 , and cavity-QED 15,31–33 .…”
We propose an optical scheme of discrete quantum Fourier transform (DQFT) via ancillary systems using quantum dots (QDs) confined in single-sided cavities (QD-cavity systems). In our DQFT scheme, the main component is a controlled-rotation k (CRk) gate, which utilizes the interactions between photons and QDs, consisting of two QD-cavity systems. Since the proposed CRk gate can be experimentally implemented with high efficiency and reliable performance, the scalability of multi-qubit DQFT scheme can also be realized through the simple composition of the proposed CRk gates via the QD-cavity systems. Subsequently, in order to demonstrate the performance of the CRk gate, we analyze the interaction between a photon and a QD-cavity system, and then indicate the condition to be efficient CRk gate with feasibility under vacuum noise and sideband leakage.
“…In various quantum algorithms and quantum computations, such as quantum phase estimation algorithm 1–3 , the factoring problem 4–7 , the discrete logarithm problem 2,4,8,9 , and the hidden subgroup problem 10–12 , a discrete quantum Fourier transform (DQFT) 2,13–19 plays a critical role in accomplishing quantum information processing. Thus, for the experimental implementation of DQFT, a variety of physical resources have been used, including those based on linear optical systems 20–22 , nonlinear optical systems 17,23–25 , nuclear magnetic resonance or ion trap systems 26–29 , superconducting circuits 30 , and cavity-QED 15,31–33 .…”
We propose an optical scheme of discrete quantum Fourier transform (DQFT) via ancillary systems using quantum dots (QDs) confined in single-sided cavities (QD-cavity systems). In our DQFT scheme, the main component is a controlled-rotation k (CRk) gate, which utilizes the interactions between photons and QDs, consisting of two QD-cavity systems. Since the proposed CRk gate can be experimentally implemented with high efficiency and reliable performance, the scalability of multi-qubit DQFT scheme can also be realized through the simple composition of the proposed CRk gates via the QD-cavity systems. Subsequently, in order to demonstrate the performance of the CRk gate, we analyze the interaction between a photon and a QD-cavity system, and then indicate the condition to be efficient CRk gate with feasibility under vacuum noise and sideband leakage.
In this paper we discuss the Hidden Subgroup Problem (HSP) in relation to post-quantum cryptography. We review the relationship between HSP and other computational problems discuss an optimal solution method, and review the known results about the quantum complexity of HSP. We also overview some platforms for group-based cryptosystems. Notably, efficient algorithms for solving HSP in the proposed infinite group platforms are not yet known.
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