Implementing gate operations between uncoupled qubits in linear nearest neighbor arrays using a learning algorithm

We propose a new scheme to implement gate operations in a one dimensional linear nearest neighbor array, by using dynamic learning algorithm. This is accomplished by training quantum system using a back propagation technique, to find the system parameters that implement gate operations directly. The key feature of our scheme is that, we can reduce the computational overhead of a quantum circuit by finding the parameters to implement the desired gate operation directly, without decomposing them into a sequence of elementary gate operations. We show how the training algorithm can be used as a tool for finding the parameters for implementing controlled-NOT (CNOT) and Toffoli gates between next-to-nearest neighbor qubits in an Ising-coupled linear nearest neighbor system. We then show how the scheme can be used to find parameters for realizing swap gates first, between two adjacent qubits and then, between two next-to-nearest-neighbor qubits, in each case without decomposing it into 3 CNOT gates. Finally, we show how the scheme can be extended to systems with non-diagonal interactions. To demonstrate, we train a quantum system with Heisenberg interactions to find the parameters to realize a swap operation.     

Direct implementation of an N-qubit controlled-unitary gate in a single step

We present a new scheme to implement an N-qubit controlled-unitary operation directly in a single step. The main advantage of our scheme is that we do not use conventional gate decomposition protocols to break an N-qubit controlled-unitary gate into one- and two-qubit gates. This greatly reduces the number of computational steps in implementing quantum algorithms and error-correcting codes, which use multi-control unitary operations. We show how to find analytic solutions to the time evolution of the system, so that system parameters can be found to realize the desired N-qubit controlled-unitary operations.     

量子计算模拟及优化方法综述

在处理某些大规模并行问题时,量子计算因量子位独特的叠加态和纠缠态特性,相比经典计算机在并行处理方面具有更明显的优势。现阶段,物理量子比特计算机受限于可扩展性、相干时间和量子门操作精度,在经典计算机上开展量子计算模拟成为研究量子优越性和量子算法的有效途径。然而,随着量子比特数的增加,模拟所需的计算机资源呈指数增长。因此,研究大规模量子计算模拟在保证计算准确度、精度及效率的情况下减少模拟所需资源具有重要意义。从量子比特、量子门、量子线路、量子操作系统等方面展开,阐述量子计算的基本原理和背景知识。同时总结基于经典计算机的量子计算模拟基本方法,分析不同方法的设计思路和优缺点,列举目前常见的量子计算模拟器。在此基础上,针对量子计算模拟的通信开销问题,从节点拆分和通信优化2个方面出发,讨论基于超级计算机集群的量子计算模拟优化方法。     

三值量子基本门及其对量子Fourier变换的电路实现

理论上可以把量子基本门组合在一起来实现任何量子电路和构建可伸缩的量子计算机。但由于构建量子线路的量子基本门数量庞大,要正确控制这些量子门十分困难。因此,如何减少构建量子线路的基本门数量是一个非常重要和非常有意义的课题。提出采用三值量子态系统构建量子计算机,并给出了一组三值量子基本门的功能定义、算子矩阵和量子线路图。定义的基本门主要包括三值量子非门、三值控制非门、三值Hadamard门、三值量子交换门和三值控制CR_k门等。通过把量子Fourier变换推广到三值量子态,成功运用部分三值量子基本门构建出能实现量子Fourier变换的量子线路。通过定量分析发现,三值量子Fourier变换的线路复杂度比二值情况降低了至少50%,表明三值量子基本门在降低量子计算线路复杂度方面具有巨大优势。     

An Explicit Universal Gate-Set for Exchange-Only Quantum Computation

A single physical interaction might not be universal for quantum computation in general. It has been shown, however, that in some cases it can achieve universal quantum computation over a subspace. For example, by encoding logical qubits into arrays of multiple physical qubits, a single isotropic or anisotropic exchange interaction can generate a universal logical gate-set. Recently, encoded universality for the exchange interaction was explicitly demonstrated on three-qubit arrays, the smallest nontrivial encoding. We now present the exact specification of a discrete universal logical gate-set on four-qubit arrays. We show how to implement the single qubit operations exactly with at most 3 nearest neighbor exchange operations and how to generate the encoded controlled-NOT with 27 parallel nearest neighbor exchange interactions or 50 serial gates, obtained from extensive numerical optimization using genetic algorithms and Nelder–Mead searches. We also give gate-switching times for the three-qubit encoding to much higher accuracy than previously and provide the full speci.cation for exact CNOT for this encoding. Our gate-sequences are immediately applicable to implementations of quantum circuits with the exchange interaction. PACS: 03.67.Lx, 03.65.Ta, 03.65.Fd, 89.70.+c     

Quantum information processing in self-assembled crystals of cold polar molecules

We discuss the implementation of quantum gate operations in a self-assembled dipolar crystal of polar molecules. Here qubits are encoded in long-lived spin states of the molecular ground state and stabilized against collisions by repulsive dipole–dipole interactions. To overcome the single site addressability problem in this high density crystalline phase, we describe a new approach for implementing controlled single and two-qubit operations based on resonantly enhanced spin–spin interactions mediated by a localized phonon mode. This local mode is created at a specified lattice position with the help of an additional marker molecule such that individual qubits can be manipulated by using otherwise global static and microwave fields only. We present a general strategy for generating state and time dependent dipole moments to implement a universal set of gate operations for molecular qubits and we analyze the resulting gate fidelities under realistic conditions. Our analysis demonstrates the experimental feasibility of this approach for scalable quantum computing or digital quantum simulation schemes with polar molecules.     

Improved quantum circuit modelling based on Heisenberg representation

Heisenberg model allows a more compact representation of certain quantum states and enables efficient modelling of stabilizer gates operation and single-qubit measurement in computational basis on classical computers. Since generic quantum circuit modelling appears intractable on classical computers, the Heisenberg representation that makes the modelling process at least practical for certain circuits is crucial. This paper proposes efficient algorithms to facilitate accurate global phase maintenance for both stabilizer and non-stabilizer gates application that play a vital role in the stabilizer frames data structure, which is based on the Heisenberg representation. The proposed algorithms are critical as maintaining global phase involves compute-intensive operations that are necessary for the modelling of each quantum gate. In addition, the proposed work overcomes the limitations of prior work where the phase factors due to non-stabilizer gates application was not taken into consideration. The verification of the proposed algorithms is made against the golden reference model that is constructed based on the conventional state vector approach.     

An atom�Cmolecule platform for quantum computing

We propose a combined atom–molecule system for quantum information processing in individual traps, such as provided by optical lattices. In this platform, different species of atoms—one atom carrying a qubit and the other enabling the interaction—are used to store and process quantum information via intermediate molecular states. We show how gates, initialization, and readout operations could be implemented using this approach. In particular, we describe in some detail the implementation of a two-qubit phase gate in which a pair of atoms is transferred into the ground rovibrational state of a polar molecule with a large dipole moment, thus allowing atoms transferred into molecules to interact via their dipole-dipole interaction. We also discuss how the reverse process could be used as a non-destructive readout tool of molecular qubit states. Finally, we generalize these ideas to use a decoherence-free subspace for qubit encoding to minimize the decoherence due to magnetic field fluctuations. In this case, qubits will be encoded into field-insensitive states of two identical atoms, while a third atom of a different species will be used to realize a phase gate.     

On synthesis of 3 × 3 reversible logic functions

Reversible logic plays an important role in quantum computing. This article presents some novel results on synthesis of 3?×?3 reversible Boolean gates. We derive the relationship between reversible 3?×?3 gates and corresponding symmetric groups. By introducing a set of universal libraries, we show how to use group theory to synthesize any 3?×?3 reversible gate.     

Quantum computing implementations with neutral particles

We review quantum information processing with cold neutral particles, that is, atoms or polar molecules. First, we analyze the best suited degrees of freedom of these particles for storing quantum information, and then we discuss both single- and two-qubit gate implementations. We focus our discussion mainly on collisional quantum gates, which are best suited for atom-chip-like devices, as well as on gate proposals conceived for optical lattices. Additionally, we analyze schemes both for cold atoms confined in optical cavities and hybrid approaches to entanglement generation, and we show how optimal control theory might be a powerful tool to enhance the speed up of the gate operations as well as to achieve high fidelities required for fault tolerant quantum computation.     

On figures of merit in reversible and quantum logic designs

Five figures of merit including number of gates, quantum cost, number of constant inputs, number of garbage outputs, and delay are used casually in the literature to compare the performance of different reversible or quantum logic circuits. In this paper we propose new definitions and enhancements, and identify similarities between these figures of merit. We evaluate these measures to show their strength and weakness. Instead of the number of gates, we introduce the weighted number of gates, where a weighting factor is assigned to each quantum or reversible gate, based on its type, size and technology. We compare the quantum cost with weighted number of gates of a circuit and show three major differences between these measures. It is proved that it is not possible to define a universal reversible logic gate without adding constant inputs. We prove that there is an optimum value for number of constant inputs to obtain a circuit with minimum quantum cost. Some reversible logic benchmarks have been synthesized using Toffoli and Fredkin gates to obtain their optimum values of number of constant inputs. We show that the garbage outputs can also be used to decrease the quantum cost of the circuit. A new definition of delay in quantum and reversible logic circuits is proposed for music line style representation. We also propose a procedure to calculate the delay of a circuit, based on the quantum cost and the depth of the circuit. The results of this research show that to achieve a fair comparison among designs, figures of merit should be considered more thoroughly.      

Function synthesis algorithm based on RTD-based three-variable universal logic gates

Compared with complementary metal–oxide semiconductor (CMOS), the resonant tunneling device (RTD) has better performances; it is the most promising candidate for next-generation integrated circuit devices. The universal logic gate is an important unit circuit because of its powerful logic function, but there are few function synthesis algorithms that can implement an n-variable logical function by RTD-based universal logic gates. In this paper, we propose a new concept, i.e., the truth value matrix. With it a novel disjunctive decomposition algorithm can be used to decompose an arbitrary n-variable logical function into three-variable subset functions. On this basis, a novel function synthesis algorithm is proposed, which can implement arbitrary n-variable logical functions by RTD-based universal threshold logic gates (UTLGs), RTD-based three-variable XOR gates (XOR3s), and RTD-based three-variable universal logic gate (ULG3s). When this proposed function synthesis algorithm is used to implement an n-variable logical function, if the function is a directly disjunctive decomposition one, the circuit structure will be very simple, and if the function is a non-directly disjunctive decomposition one, the circuit structure will be simpler than when using only UTLGs or ULG3s. The proposed function synthesis algorithm is straightforward to program, and with this algorithm it is convenient to implement an arbitrary n-variable logical function by RTD-based universal logic gates.     

Mapping from multiple-control Toffoli circuits to linear nearest neighbor quantum circuits

In recent years, quantum computing research has been attracting more and more attention, but few studies on the limited interaction distance between quantum bits (qubit) are deeply carried out. This paper presents a mapping method for transforming multiple-control Toffoli (MCT) circuits into linear nearest neighbor (LNN) quantum circuits instead of traditional decomposition-based methods. In order to reduce the number of inserted SWAP gates, a novel type of gate with the optimal LNN quantum realization was constructed, namely NNTS gate. The MCT gate with multiple control bits could be better cascaded by the NNTS gates, in which the arrangement of the input lines was LNN arrangement of the MCT gate. Then, the communication overhead measurement model on inserted SWAP gate count from the original arrangement to the new arrangement was put forward, and we selected one of the LNN arrangements with the minimum SWAP gate count. Moreover, the LNN arrangement-based mapping algorithm was given, and it dealt with the MCT gates in turn and mapped each MCT gate into its LNN form by inserting the minimum number of SWAP gates. Finally, some simplification rules were used, which can further reduce the final quantum cost of the LNN quantum circuit. Experiments on some benchmark MCT circuits indicate that the direct mapping algorithm results in fewer additional SWAP gates in about 50%, while the average improvement rate in quantum cost is 16.95% compared to the decomposition-based method. In addition, it has been verified that the proposed method has greater superiority for reversible circuits cascaded by MCT gates with more control bits.     

Efficient experimental design of high-fidelity three-qubit quantum gates via genetic programming

We have designed efficient quantum circuits for the three-qubit Toffoli (controlled–controlled-NOT) and the Fredkin (controlled-SWAP) gate, optimized via genetic programming methods. The gates thus obtained were experimentally implemented on a three-qubit NMR quantum information processor, with a high fidelity. Toffoli and Fredkin gates in conjunction with the single-qubit Hadamard gates form a universal gate set for quantum computing and are an essential component of several quantum algorithms. Genetic algorithms are stochastic search algorithms based on the logic of natural selection and biological genetics and have been widely used for quantum information processing applications. We devised a new selection mechanism within the genetic algorithm framework to select individuals from a population. We call this mechanism the “Luck-Choose” mechanism and were able to achieve faster convergence to a solution using this mechanism, as compared to existing selection mechanisms. The optimization was performed under the constraint that the experimentally implemented pulses are of short duration and can be implemented with high fidelity. We demonstrate the advantage of our pulse sequences by comparing our results with existing experimental schemes and other numerical optimization methods.     

Efficient quantum computing between remote qubits in linear nearest neighbor architectures

We propose a new scheme for implementing gate operations between remote qubits in linear nearest neighbor (LNN) architectures, one that does not require qubits to be adjacent to each other in order to perform a gate operation between them. The key feature of our scheme is a new two-control, one-target controlled-unitary gate operation, which we refer to as the C²(?I) gate. The gate operation can be implemented easily in a single step, requiring only a single control parameter of the system Hamiltonian. Using the C²(?I) gate, we show how to implement CNOT gate operations between remote qubits that do not have any direct coupling between them, along an LNN array. Since this is achieved without requiring swap operations or additional ancilla qubits in the circuit, the quantum cost of our circuit can be more than 50 % lower than those using conventional swap methods. All CNOT gate operations between remote qubits can be achieved with fidelity greater than 99.5 %.     

Controllability and Universal Three-qubit Quantum Computation with Trapped Electron States

We show how to control and perform universal three-qubit quantum computation with trapped electron quantum states. The three qubits are the electron spin, and the first two quantum states of the cyclotron and axial harmonic oscillators. We explicitly show how universal three-qubit gates can be performed. As an example of a quantum algorithm, we outline the implementation of the three-qubit Deutsch-Jozsa algorithm in this system.      

Multi-qubit non-adiabatic holonomic controlled quantum gates in decoherence-free subspaces

Non-adiabatic holonomic quantum gate in decoherence-free subspaces is of greatly practical importance due to its built-in fault tolerance, coherence stabilization virtues, and short run-time. Here, we propose some compact schemes to implement two- and three-qubit controlled unitary quantum gates and Fredkin gate. For the controlled unitary quantum gates, the unitary operator acting on the target qubit is an arbitrary single-qubit gate operation. The controlled quantum gates can be directly implemented by utilizing non-adiabatic holonomy in decoherence-free subspaces and the required resource for the decoherence-free subspace encoding is minimal by using only two neighboring physical qubits undergoing collective dephasing to encode a logical qubit.     

Implementing a Quantum Algorithm with Exchange-Coupled Quantum Dots: A Feasibility Study

We present Monte Carlo wavefunction simulations for quantum computations employing an exchange-coupled array of quantum dots. Employing a combination of experimentally and theoretically available parameters, we find that gate fidelities greater than 98% may be obtained with current experimental and technological capabilities. Application to an encoded 3 qubit (nine physical qubits) Deutsch-Josza computation indicates that the algorithmic fidelity is more a question of the total time to implement the gates than of the physical complexity of those gates. PACS: 81.07.Ta, 02.70.Ss, 03.67.Lx, 03.65.Yz     

Synthesis of reversible circuits with minimal costs

We present fast algorithms to synthesize exact minimal reversible circuits for various types of gate and cost. By reducing reversible logic synthesis problems to permutation group problems, we use the powerful algebraic software GAP to solve such problems. Our approach can minimize for arbitrary cost functions of gates. In addition, we show that Peres gates are a better choice than the standard Toffoli gates in libraries of universal reversible gates. This work was supported by the NNSF of China under Grant 60773205 and the Fund of Cultivating Leading Scholars in UESTC.