Programmable networks for quantum algorithms

The implementation of a quantum computer requires the realization of a large number of N-qubit unitary operations which represent the possible oracles or which are part of the quantum algorithm. Until now there have been no standard ways to uniformly generate whole classes of N-qubit gates. We develop a method to generate arbitrary controlled phase-shift operations with a single network of one-qubit and two-qubit operations. This kind of network can be adapted to various physical implementations of quantum computing and is suitable to realize the Deutsch-Jozsa algorithm as well as Grover's search algorithm.     

On time-optimal NMR control of states of qutrits represented by quadrupole nuclei with the spin <Emphasis Type="Italic">I</Emphasis> = 1

Elementary logical operators (selective rotation, Fourier transform, controllable phase shift, and SUM gate) are considered for a quantum computer based on three-level systems (qutrits) represented by nuclear spins I = 1 under nuclear magnetic resonance conditions. The computer simulation of the realization of these operators by means of simple and composite selective radiofrequency (RF) pulses and optimized RF pulses is performed. The time dependence of the amplitude of last pulses is found by numerical optimization at different durations. Two variants are proposed for realization of a two-qutrit SUM gate by using one-qutrit or two-qutrit optimized RF pulses. The calculated time dependences of realization errors were used to study the time optimality of different methods for obtaining gates, proposed earlier and in this paper. The advantages and disadvantages of each of the methods are evaluated for different values of physical parameters.     

A quantum search algorithm of two entangled registers to realize quantum discrete Fouriertransform of signal processing

The discrete Fourier transform （DFT） is the base of modern signal processing. 1-dimensional fast Fourier transform （1D FFT） and 2D FFT have time complexity O（N log N） and O（N^2 log N） respectively. Since 1965, there has been no more essential breakthrough for the design of fast DFT algorithm. DFT has two properties. One property is that DFT is energy conservation transform. The other property is that many DFT coefficients are close to zero. The basic idea of this paper is that the generalized Grover＇s iteration can perform the computation of DFT which acts on the entangled states to search the big DFT coefficients until these big coefficients contain nearly all energy. One-dimensional quantum DFT （1D QDFT） and two-dimensional quantum DFT （2D QDFT） are presented in this paper. The quantum algorithm for convolution estimation is also presented in this paper. Compared with FFT, 1D and 2D QDFT have time complexity O（v/N） and O（N） respectively. QDFT and quantum convolution demonstrate that quantum computation to process classical signal is possible.     

Pulse sequences for realizing the quantum Fourier transform on multilevel systems

Sequences of selective pulses of an RF magnetic field for realizing the quantum Fourier transform by NMR methods on systems with four, six, and eight nonequidistant levels are found using the virtual spin formalism. The results can be applied to other quantum systems when laser pulses are used.     

Quantum spin dynamics as a model for quantum computer operation

We study effects of the physical realization of quantum computers on their logical operation. Through simulation of physical models of quantum computer hardware, we analyze the difficulties that are encountered in programming physical realizations of quantum computers. Examples of logically identical implementations of the controlled-NOT operation and Grover's database search algorithm are used to demonstrate that the results of a quantum computation are unstable with respect to the physical realization of the quantum computer. We discuss the origin of these instabilities and discuss possibilities to overcome this, for practical purposes, fundamental limitation of quantum computers. Received 5 November 2001 and Received in final form 8 February 2002     

NMR analogue of the generalized Grover‘s algorithm of multiple marked states and its application

The generalized Grover's algorithm for the case in which there are multiple marked states is demonstrated on a nuclear magnetic resonance (NMR) quantum computer. The Walsh－Hadamard transform and the phase inversion are all replaced. NMR analogues of Einstein－Podolsky－Rosen states (pseudo-EPR states) are synthesized using the above algorithm.     

Ising耦合体系中量子傅里叶变换的优化

量子傅里叶变换是量子计算中一种重要的量子逻辑门. 任意量子位的傅里叶变换可以分解为一系列普适的单比特量子逻辑门和两比特量子逻辑门, 这种分解方式使得傅里叶变换的实验实现简单直观, 但所用的实验时间显然不是最短的. 本文利用优化控制和数值计算方法对Ising耦合体系中多量子位傅里叶变换的实验时间进行优化, 优化后的实现方法明显短于传统方法. 优化方法的核磁共振实验实现验证了其有效性.     

REALIZATION OF QUANTUM ALGORITHMS ON SIMULATOR AND SPECTROMETER OF NMR

报道了利用NMR谱仪和NMR模拟机实现量子算法.以天然苯为样品,我们分别用500M谱仪和NMR模拟机实现了量子D-J算法,Grover搜寻算法及受控非门(C-NOT).通过比较实验谱和模拟谱发现二者能很好符合.利用NMR模拟机实现量子算法比用NMR谱仪更为方便、清晰.     

Quantum computation through entangling single photons in multipath interferometers

Single-photon interferometry has been used to simulate quantum computations. Its use has been limited to studying few-bit applications due to rapid growth in physical size with numbers of bits. We propose a hybrid approach that employs n photons, each having L degrees of freedom yielding L(n) basis states. The photons are entangled via a quantum nondemolition measurement. This approach introduces the essential element of quantum computing, that is, entanglement into the interferometry. Using these techniques, we demonstrate a controlled-NOT gate and a Grover's search circuit. These ideas are also applicable to the study of nonlocal correlations in many dimensions.     

利用NMR谱仪及模拟机实现量子算法

报道了利用NMR谱仪和NMR模拟机实现量子算法.以天然苯为样品,我们分别用500 M谱仪和NMR模拟机实现了量子D-J算法,Grover搜寻算法及受控非门(C NOT).通过比较实验谱和模拟谱发现二者能很好符合.利用NMR模拟机实现量子算法比用NMR谱仪更为方便、清晰.     

Implementing logic gates and the Deutsch-Jozsa quantum algorithm by two-dimensional NMR using spin- and transition-selective pulses

Quantum logical operations using two-dimensional NMR have recently been described using the scalar coupling evolution technique [J. Chem. Phys. 109, 10603 (1998)]. In the present paper, we describe the implementation of quantum logical operations using two-dimensional NMR, with the help of spin- and transition-selective pulses. A number of logic gates are implemented using two and three qubits with one extra observer spin. Some many-in-one gates (or Portmanteau gates) are also implemented. Toffoli gate (or AND/NAND gate) and OR/NOR gates are implemented on three qubits. The Deutsch-Jozsa quantum algorithm for one and two qubits, using one extra work qubit, has also been implemented using spin- and transition-selective pulses after creating a coherent superposition state in the two-dimensional methodology.     

Magnetic resonance realization of decoherence-free quantum computation

We report the realization, using nuclear magnetic resonance techniques, of the first quantum computer that reliably executes a complete algorithm in the presence of strong decoherence. The computer is based on a quantum error avoidance code that protects against a class of multiple-qubit errors. The code stores two decoherence-free logical qubits in four noisy physical qubits. The computer successfully executes Grover's search algorithm in the presence of arbitrarily strong engineered decoherence. A control computer with no decoherence protection consistently fails under the same conditions.     

A Scheme for Physical Implementation of a Ququadrit Quantum Computation with Cooled-Trapped Ions

The physics realization of a ququadrit quantum computation with cooled trapped ¹³⁸Ba⁺ ions in a Paul trap is investigated. The ground state level 6² S_1/2(m = −1/2) and three metastable levels: 5² D_3/2(m = −1/2), 5² D_5/2(m = −1/2), and 5² D_5/2(m = 1/2), of the fine-structure of the ¹³⁸Ba⁺ ion, are used to store the quantum information of ququadrits. The use of coherent manipulation of populations in single ququadrit, being a four-dimensional Hilbert space, produces a discrete Fourier transform and the manipulation of the first red band transitions with the introduction of an ancillary quantum channel between two ququadrits generates a conditional phase gate. The combination of the both above results in a universal two-ququadrit gate, called XOR⁽⁴⁾ gate corresponding to the controlled-NOT gate operation in qubit systems. The implementation of quantum Fourier transform for n ququadrits is performed by means of the conditional phase-shift gate. The feasibility of physical realization of ququadrit quantum computation with cooled-trapped ¹³⁸Ba⁺ ions is detailed analyzed and described, and the theoretical detection method of logical states is given. Higher entanglement between ququadrits than qutrits or qubits and more security of ququadrit quantum cryptography than qutrit's or qutrit's will lead to more extensive applications ququadrits in quantum information fields. In particular, it is pointed out that this scheme should be the highest dimensional quantum computation in cooled-trapped ions, the entanglement between ququadrits should be the highest dimensional entanglement in it, and the ququadrit quantum cryptography should be the most secure cryptography protocol in it.     

Experimental realization of one-way quantum computing with two-photon four-qubit cluster states

We report an experimental realization of one-way quantum computing on a two-photon four-qubit cluster state. This is accomplished by developing a two-photon cluster state source entangled both in polarization and spatial modes. With this special source, we implemented a highly efficient Grover's search algorithm and high-fidelity two-qubit quantum gates. Our experiment demonstrates that such cluster states could serve as an ideal source and a building block for rapid and precise optical quantum computation.     

Arbitrary Phase Rotation of the Marked State Cannot Be Used for Grover's Quantum Search Algorithm

A misunderstanding that an arbitrary phase rotation of the marked state together with the inversion about average operation can be used to construct a (less efficient) quantum search algorithm is cleared. The π rotation of the phase of the marked state is not only the choice for efficiency, but also vital in Grover's quantum search algorithm. The results also show that Grover's quantum search algorithm is robust.     

二维光子晶体中Grover搜索算法的实现

最近,Angelakis等人将光子晶体引入量子计算。本文主要讨论在二维光子晶体中两比特Grover搜索算法的实现。沿用由Angelakis等所提出的量子比特与可控相位门,简单有效的实现了Grover算法。具体的实现方案文中将详细讨论。     

Novel strategy for database searching in spin liouville space by NMR ensemble computing

Quantum computing by nuclear magnetic resonance using pseudopure spin states is bound by the maximal speed of quantum computing algorithms operating on pure states. In contrast to these quantum computing algorithms, a novel algorithm for searching an unsorted database is presented here that operates on truly mixed states in spin Liouville space. It provides an exponential speedup over Grover's quantum search algorithm with the sensitivity scaling exponentially with the number of spins, as for pseudopure state implementations. The minimal decoherence time required is exponentially shorter than that for Grover's algorithm.     

A comparative analysis of two methods of realizing elementary logic operators for a quantum computer on qutrits

In the present work, elementary logic operators (including selective rotation, Fourier transform, controlled phase shift, and controlled NOT operators) for a quantum computer on tristable systems (qutrits) are examined. Computer modeling of realization of these operators based on a system of two nuclear spins I = 1 is carried out by the methods of nuclear magnetic resonance. Two different methods of realizing the controlled NOT operator are presented: with the help of weak pulses of radio-frequency magnetic field selective in spin-spin interaction and with the help of strong pulses selective in quadrupole interaction. A dependence of realization errors on the interaction parameters, variable field amplitude, and pulse duration is calculated. Advantages and disadvantages of each method in the process of realization are indicated. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 41–47, June, 2007.     

Use of non-adiabatic geometric phase for quantum computing by NMR

Geometric phases have stimulated researchers for its potential applications in many areas of science. One of them is fault-tolerant quantum computation. A preliminary requisite of quantum computation is the implementation of controlled dynamics of qubits. In controlled dynamics, one qubit undergoes coherent evolution and acquires appropriate phase, depending on the state of other qubits. If the evolution is geometric, then the phase acquired depend only on the geometry of the path executed, and is robust against certain types of error. This phenomenon leads to an inherently fault-tolerant quantum computation. Here we suggest a technique of using non-adiabatic geometric phase for quantum computation, using selective excitation. In a two-qubit system, we selectively evolve a suitable subsystem where the control qubit is in state |1, through a closed circuit. By this evolution, the target qubit gains a phase controlled by the state of the control qubit. Using the non-adiabatic geometric phase we demonstrate implementation of Deutsch-Jozsa algorithm and Grover's search algorithm in a two-qubit system.     

Implementation of the quantum Fourier transform

A quantum Fourier transform (QFT) has been implemented on a three qubit nuclear magnetic resonance (NMR) quantum computer to extract the periodicity of an input state. Implementation of a QFT provides a first step towards the realization of Shor's factoring and other quantum algorithms. The experimental implementation of the QFT on a periodic state is presented along with a quantitative measure of its efficiency measured through state tomography. Experimentally realizing the QFT is a clear demonstration of the ability of NMR to control quantum systems.