We describe an implementation of Grover's fixed-point quantum search algorithm on a nuclear magnetic resonance (NMR) quantum computer, searching for either one or two matching items in an unsorted database of four items. In this new algorithm the target state (an equally weighted superposition of the matching states) is a fixed point of the recursive search operator, and so the algorithm always moves towards the desired state. The effects of systematic errors in the implementation are briefly explored.
By designing a proper unitary operator U, we synthesize NMR analogues of Einstein-Podolsky-Rosen states (pseudo-EPR states) using generalized Grover's algorithm on a nuclear magnetic resonance (NMR) quantum computer. Experiments also demonstrate generalized Grover's algorithm for the case in which there are multiple marked states.
We demonstrate the implementation of Grover's quantum search algorithm on a liquid state nuclear magnetic resonance (NMR) quantum computer using essentially pure states. This was achieved using a two qubit device where the initial state is an essentially pure ($ε=1.06\pm0.04$) singlet nuclear spin state of a pair of 1H nuclei arising from a chemical reaction involving para-hydrogen. We have implemented Grover's search to find one of four inputs which satisfies a function.
Grover's quantum search algorithm, involving a large number of qubits, is highly sensitive to errors in the physical implementation of the unitary operators. This poses an intrinsic limitation to the size of the database that can be practically searched. The lack of robustness of Grover's algorithm for a large number of qubits is due to quite stringent "phase-matching" condition. To overcome this limitation, Tulsi suggested a modified search algorithm [PRA 78, 022332] which succeeds as long as the errors are reproducible and reversible while Grover's algorithm fails. Such systematic errors ari…
Generalized Grover's searching algorithm for the case in which there are multiple marked states is demonstrated on a nuclear magnetic resonance (NMR) quantum computer. The entangled basis states (EPR states) are synthesized using the algorithm.
Experimental realization of quantum information processing in the field of nuclear magnetic resonance (NMR) has been well established. Implementation of conditional phase shift gate has been a significant step, which has lead to realization of important algorithms such as Grover's search algorithm and quantum Fourier transform. This gate has so far been implemented in NMR by using coupling evolution method. We demonstrate here the implementation of the conditional phase shift gate using transition selective pulses. As an application of the gate, we demonstrate Grover's search algorithm and qua…
Dense coding has been implemented using the generalized Grover's algorithm and its inverse operation. Exploiting the superpositions of two Einstein-Podolsky-Rosen (EPR) states, messages that are possible to be transmitted increase. Our scheme is demonstrated using nuclear magnetic resonance (NMR). Experimental results show a good agreement between theory and experiment.
The improved quantum scheduling algorithm proposed by Grover has been generalized using the generalized quantum search algorithm, in which a unitary operator replaces the Walsh-Hadamard transform, and $π/2$ phase rotations replace the selective inversions, in order to make the quantum scheduling algorithm suitable for more cases. Our scheme is realized on a nuclear magnetic resonance (NMR) quantum computer. Experimental results show a good agreement between theory and experiment.
A continuous, analog version of the Grover algorithm is realized using NMR. The system studied is 23Na in a liquid crystal medium. The presence of quadrupolar coupling makes the spin I=3/2 nucleus a 2-qubit system. Applying a specially designed pulse sequence, the time evolution of the spin density operator is described in an interaction representation which has no external time-dependent radio-frequency fields. This approach is used to implement one instance of the continuous Grover search for the transform of a uniform state to a target state, and the implementation provides a clear physical…
We describe the experimental implementation of a recently proposed quantum algorithm involving quantum entanglement at the level of two qubits using NMR. The algorithm solves a generalisation of the Deutsch problem and distinguishes between even and odd functions using fewer function calls than is possible classically. The manipulation of entangled states of the two qubits is essential here, unlike the Deutsch-Jozsa algorithm and the Grover's search algorithm for two bits.
Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. In some cases, such as the adiabatic versions of Grover's search algorithm and Deutsch-Jozsa algorithm, applying the global adiabatic evolution yields a complexity similar to their classical algorithms. However, using the local adiabatic evolution, the algorithms given by J. Roland and N. J. Cerf for Grover's search [ Phys. Rev. A. {\bf 65} 0423…
Searching a marked item or several marked items from an unsorted database is a very difficult mathematical problem. Using classical computer, it requires $O(N=2^n)$ steps to find the target. Using a quantum computer, Grover's algorithm uses $O(\sqrt{N=2^n})$ steps. In NMR ensemble computing, Brushweiler's algorithm uses $\log N$ steps. In this Letter, we propose an algorithm that fetches marked items in an unsorted database directly. It requires only a single query. It can find a single marked item or multiple number of items.
We compare pseudopure state ensemble implementations, quantified by their initial polarization and ensemble size, of Grover's search algorithm to probabilistic classical sequential search algorithms in terms of their success and failure probabilities. We propose a criterion for quantifying the resources used by the ensemble implementation via the aggregate number of oracle invocations across the entire ensemble and use this as a basis for comparison with classical search algorithms. We determine bounds for a critical polarization such that the ensemble algorithm succeeds with a greater probabi…
In general, a quantum circuit is constructed with elementary gates, such as one-qubit gates and CNOT gates. It is possible, however, to speed up the execution time of a given circuit by merging those elementary gates together into larger modules, such that the desired unitary matrix expressing the algorithm is directly implemented. We demonstrate this by taking the two-qubit Grover's algorithm implemented in NMR quantum computation, whose pseudopure state is generated by cyclic permutations of the state populations. This is the first exact time-optimal solution, to our knowledge, obtained for …
A generalized quantum search algorithm, where phase inversions for the marked state and the prepared state are replaced by $π/2$ phase rotations, is realized in a 2-qubit NMR heteronuclear system. The quantum algorithm searches a marked state with a smaller step compared to standard Grover algorithm. Phase matching requirement in quantum searching is demonstrated by comparing it with another generalized algorithm where the two phase rotations are $π/2$ and $3π/2$ respectively. Pulse sequences which include non 90 degree pulses are given.
As the matching condition in Grover search algorithm is transgressed due to inevitable errors in phase inversions, it gives a reduction in maximum probability of success. With a given degree of maximum success, we have derive the generalized and imroved criterion for tolerated error and corresponding size of quantum database under the inevitable gate imperfections. The vanished inaccurancy to this condition has also been shown. Besides, a concise formula for evaluating minimum number of iterations is also presented in this work.
Grover's algorithm has achieved great success. But quantum search algorithms still are not complete algorithms because of Grover's Oracle. We concerned on this problem and present a new quantum search algorithm in adiabatic model without Oracle. We analyze the general difficulties in quantum search algorithms and show how to solve them in the present algorithm. As well this algorithm could deal with both single-solution and multi-solution searches without modification. We also implement this algorithm on NMR quantum computer. It is the first experiment which perform a real quantum database sea…
Geometric phase (GP) independent of energy and time rely only on the geometry of state space. It has been argued to have potential fault tolerance and plays an important role in quantum information and quantum computation. We present the first experiment for producing and measuring an Abelian geometric phase shift in a three-level system by using NMR interferometry. In contrast to existing experiments, based on the geometry of $S^2$, our experiment concerns the geometric phase with the geometry of $SU(3)/U(2)$. Two interacting qubits have been used to provide such a three-dimensional Hilbert s…
This thesis addresses the problems of initialization and separability in liquid state NMR based quantum information processors. We prepare pure quantum states lying above the entanglement threshold. Our pure state quantum computer derives its purity from the highly polarized nuclear spin states in the para-hydrogen molecule. The thesis begins with a critique of conventional NMR based quantum information processing outlining the major strengths and weaknesses of the technology. We describe the enhanced magnetic ordering of the nuclear spin states in para-hydrogen and an initialization experimen…
In this work, we present an implementation of quantum logic gates and algorithms in a three effective qubits system, represented by a (I = 7/2) NMR quadrupolar nuclei. To implement these protocols we have used the strong modulating pulses (SMP). The various stages of each implementation were verified by quantum state tomography (QST). It is presented here the results for the computational base states, Toffolli logic gates, and Deutsch-Jozsa and Grover algorithms. Also, we discuss the difficulties and advantages of implementing such protocols using the SMP technique in quadrupolar systems.
The phenomenon of quantum entanglement is fundamental to the implementation of quantum computation, and requires at least two qubits for its demonstration. However, both Deutsch algorithm and Grover's search algorithm for two bits do not use entanglement. We develop a Deutsch-like problem, where we consider all possible binary functions for two bit inputs and distinguish their even or odd nature. The quantum algorithm to solve this problem requires entanglement at the level of two qubits. The final solution suggests that an NMR implementation of the problem would lead to interesting results.
Searching large databases is an important problem with broad applications. The Grover search algorithm provides a powerful method for quantum computers to perform searches with a quadratic speedup in the number of required database queries over classical computers. It is an optimal search algorithm for a quantum computer, and has further applications as a subroutine for other quantum algorithms. Searches with two qubits have been demonstrated on a variety of platforms and proposed for others, but larger search spaces have only been demonstrated on a non-scalable NMR system. Here, we report res…
Magic-angle spinning (MAS) solid state nuclear magnetic resonance (NMR) spectroscopy is shown to be a promising technique for implementing quantum computing. The theory underlying the principles of quantum computing with nuclear spin systems undergoing MAS is formulated in the framework of formalized quantum Floquet theory. The procedures for realizing state labeling, state transformation and coherence selection in Floquet space are given. It suggests that by this method, the largest number of qubits can easily surpass that achievable with other techniques. Unlike other modalities proposed for…
Fault-tolerant quantum computers which can solve hard problems rely on quantum error correction. One of the most promising error correction codes is the surface code, which requires universal gate fidelities exceeding the error correction threshold of 99 per cent. Among many qubit platforms, only superconducting circuits, trapped ions, and nitrogen-vacancy centers in diamond have delivered those requirements. Electron spin qubits in silicon are particularly promising for a large-scale quantum computer due to their nanofabrication capability, but the two-qubit gate fidelity has been limited to …
Recently it has been shown that time-optimal quantum computation is attained by using the Cartan decomposition of a unitary matrix. We extend this approach by noting that the unitary group is compact. This allows us to reduce the execution time of a quantum algorithm $U_{\rm alg}$ further by adding an extra gate $W$ to it. This gate $W$ sends $U_{\rm alg}$ to another algorithm $WU_{\rm alg}$ which is executable in a shorter time than $U_{\rm alg}$. We call this technique warp-drive. Here we show both theoretically and experimentally that the execution time of Grover's algorithm is reduced in t…
Compared with classical search algorithms, Grover quantum algorithm [ Phys. Rev. Lett., 79, 325(1997)] achieves quadratic speedup and Bruschweiler hybrid quantum algorithm [Phys. Rev. Lett., 85, 4815(2000)] achieves an exponential speedup. In this paper, we report the experimental realization of the Bruschweiler$ algorithm in a 3-qubit NMR ensemble system. The pulse sequences are used for the algorithms and the measurement method used here is improved on that used by Bruschweiler, namely, instead of quantitatively measuring the spin projection of the ancilla bit, we utilize the shape of the an…
Quantum superposition and entanglement of physical states can be harnessed to solve some problems which are intractable on a classical computer implementing binary logic. Several algorithms have been proposed to utilize the quantum nature of physical states and solve important problems. For example, Shor's quantum algorithm is extremely important in the field of cryptography since it factors large numbers exponentially faster than any known classical algorithm. Another celebrated example is the Grovers quantum algorithm. These algorithms can only be implemented on a quantum computer which oper…
Risk-aware neural pulse compilation of continuously parameterized single-qubit gates with optimal-control benchmarking and NMR hardware validation
Koç University（科奇大学）；TÜBİTAK Research Institute for Fundamental Sciences (TBAE)（土耳其科学技术研究委员会基础科学研究院（TBAE））；Middle East Technical University（中东理工大学）；Shenzhen SpinQ Technology Co., Ltd.（深圳本源量子技术有限公司）
Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best classical algorithms and taking into account realistic hardware parameters and overheads for fault-tolerance. All known examples of such speedups correspond to problems related to simulation of quantum systems and cryptography. Here we apply general-purpose quantum algorithms for solving constraint satisfaction problems to two families of prototypical NP-com…
Any physical quantum device for quantum information processing is subject to errors in implementation. In order to be reliable and efficient, quantum computers will need error correcting or error avoiding methods. Fault-tolerance achieved through quantum error correction will be an integral part of quantum computers. Of the many methods that have been discovered to implement it, a highly successful approach has been to use transversal gates and specific initial states. A critical element for its implementation is the availability of high-fidelity initial states such as |0> and the Magic State.…
A quantum memory interacts with its environment and loses information via decoherence as well as incoherence. A robust quantum control that prepares, preserves, and manipulates nonclassical correlations even in the presence of environmental influence is of paramount importance in quantum information processing. A well-known technique to suppress decoherence, namely Dynamical Decoupling (DD), consists of a sequence of rapid flips applied to the system in order to refocus the system-environment interactions. In this work, we integrate DD with quantum gates using optimal control techniques to rea…
ROM-based quantum computation (QC) is an alternative to oracle-based QC. It has the advantages of being less ``magical'', and being more suited to implementing space-efficient computation (i.e. computation using the minimum number of writable qubits). Here we consider a number of small (one and two-qubit) quantum algorithms illustrating different aspects of ROM-based QC. They are: (a) a one-qubit algorithm to solve the Deutsch problem; (b) a one-qubit binary multiplication algorithm; (c) a two-qubit controlled binary multiplication algorithm; and (d) a two-qubit ROM-based version of the Deutsc…
University of Technology, Sydney（悉尼科技大学）；Australian National University（澳大利亚国立大学）
In this paper, we inaugurate the field of quantum fair machine learning. We undertake a comparative analysis of differences and similarities between classical and quantum fair machine learning algorithms, specifying how the unique features of quantum computation alter measures, metrics and remediation strategies when quantum algorithms are subject to fairness constraints. We present the first results in quantum fair machine learning by demonstrating the use of Grover's search algorithm to satisfy statistical parity constraints imposed on quantum algorithms. We provide lower-bounds on iteration…
