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Register a new userScaling laws for Shors algorithm with a banded quantum Fourier transform
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We investigate the performance of a streamlined version of Shors algorithm in which the quantum Fourier transform is replaced by a banded version that for each qubit retains only coupling to its $b$ nearest neighbors. Defining the performance $P(n,b)$ of the $n$-qubit algorithm for bandwidth $b$ as the ratio of the success rates of Shors algorithm equipped with the banded and the full bandwidth ($b=n-1
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Quantum computers will allow calculations beyond existing classical computers. However, current technology is still too noisy and imperfect to construct a universal digital quantum computer with quantum error correction. Inspired by the evolution of classical computation, an alternative paradigm merging the flexibility of digital quantum computation with the robustness of analog quantum simulation has emerged. This universal paradigm is known as digital-analog quantum computing. Here, we introduce an efficient digital-analog quantum algorithm to compute the quantum Fourier transform, a subroutine widely employed in several relevant quantum algorithms. We show that, under reasonable assumptions about noise models, the fidelity of the quantum Fourier transformation improves considerably using this approach when the number of qubits involved grows. This suggests that, in the Noisy Intermediate-Scale Quantum (NISQ) era, hybrid protocols combining digital and analog quantum computing could be a sensible approach to reach useful quantum supremacy.
Shors powerful quantum algorithm for factoring represents a major challenge in quantum computation and its full realization will have a large impact on modern cryptography. Here we implement a compiled version of Shors algorithm in a photonic system using single photons and employing the non-linearity induced by measurement. For the first time we demonstrate the core processes, coherent control, and resultant entangled states that are required in a full-scale implementation of Shors algorithm. Demonstration of these processes is a necessary step on the path towards a full implementation of Shors algorithm and scalable quantum computing. Our results highlight that the performance of a quantum algorithm is not the same as performance of the underlying quantum circuit, and stress the importance of developing techniques for characterising quantum algorithms.
The Quantum Fourier Transformation ($QFT$) is a key building block for a whole wealth of quantum algorithms. Despite its proven efficiency, only a few proof-of-principle demonstrations have been reported. Here we utilize $QFT$ to enhance the performance of a quantum sensor. We implement the $QFT$ algorithm in a hybrid quantum register consisting of a nitrogen-vacancy (NV) center electron spin and three nuclear spins. The $QFT$ runs on the nuclear spins and serves to process the sensor - NV electron spin signal. We demonstrate $QFT$ for quantum (spins) and classical signals (radio frequency (RF) ) with near Heisenberg limited precision scaling. We further show the application of $QFT$ for demultiplexing the nuclear magnetic resonance (NMR) signal of two distinct target nuclear spins. Our results mark the application of a complex quantum algorithm in sensing which is of particular interest for high dynamic range quantum sensing and nanoscale NMR spectroscopy experiments.
Quantum Fourier Transform (QFT) plays a principal role in the development of efficient quantum algorithms. Since the number of quantum bits that can currently built is limited, while many quantum technologies are inherently three- (or more) valued, we consider extending the reach of the realistic quantum systems by building a QFT over ternary quantum digits. Compared to traditional binary QFT, the q-valued transform improves approximation properties and increases the state space by a factor of (q/2)n. Further, we use non-binary QFT derivation to generalize and improve the approximation bounds for QFT.
The quantum multicomputer consists of a large number of small nodes and a qubus interconnect for creating entangled state between the nodes. The primary metric chosen is the performance of such a system on Shors algorithm for factoring large numbers: specifically, the quantum modular exponentiation step that is the computational bottleneck. This dissertation introduces a number of optimizations for the modular exponentiation. My algorithms reduce the latency, or circuit depth, to complete the modular exponentiation of an n-bit number from O(n^3) to O(n log^2 n) or O(n^2 log n), depending on architecture. Calculations show that these algorithms are one million times and thirteen thousand times faster, when factoring a 6,000-bit number, depending on architecture. Extending to the quantum multicomputer, five different qubus interconnect topologies are considered, and two forms of carry-ripple adder are found to be the fastest for a wide range of performance parameters. The links in the quantum multicomputer are serial; parallel links would provide only very modest improvements in system reliability and performance. Two levels of the Steane [[23,1,7]] error correction code will adequately protect our data for factoring a 1,024-bit number even when the qubit teleportation failure rate is one percent.
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