Controlled gates for multi-level quantum computation

Multi-level (ML) quantum logic can potentially reduce the number of inputs/outputs or quantum cells in a quantum circuit which is a limitation in current quantum technology. In this paper we propose theorems about ML-quantum and reversible logic circuits. New efficient implementations for some basic controlled ML-quantum logic gates, such as three-qudit controlled NOT, Cycle, and Self Shift gates are proposed. We also propose lemmas about r-level quantum arrays and the number of required gates for an arbitrary n-qudit ML gate. An equivalent definition of quantum cost (QC) of binary quantum gates for ML-quantum gates is introduced and QC of controlled quantum gates is calculated.     

A model of discrete quantum computation

We present a model of discrete quantum computing focused on a set of discrete quantum states. For this, we choose the set that is the most outstanding in terms of simplicity of the states: the set of Gaussian coordinate states, which includes all the quantum states whose coordinates in the computation base, except for a normalization factor \(\sqrt{2^{-k}}\), belong to the ring of Gaussian integers \(\mathbb {Z}[i]=\{a+bi\ |\ a,b\in \mathbb {Z}\}\). We also introduce a finite set of quantum gates that transforms discrete states into discrete states and generates all discrete quantum states, and the set of discrete quantum gates, as the quantum gates that leave the set of discrete states invariant. We prove that the quantum gates of the model generate the expected discrete states and the discrete quantum gates of 2-qubits and conjecture that they also generate the discrete quantum gates of n-qubits.     

Conceptual aspects of geometric quantum computation

Geometric quantum computation is the idea that geometric phases can be used to implement quantum gates, i.e., the basic elements of the Boolean network that forms a quantum computer. Although originally thought to be limited to adiabatic evolution, controlled by slowly changing parameters, this form of quantum computation can as well be realized at high speed by using nonadiabatic schemes. Recent advances in quantum gate technology have allowed for experimental demonstrations of different types of geometric gates in adiabatic and nonadiabatic evolution. Here, we address some conceptual issues that arise in the realizations of geometric gates. We examine the appearance of dynamical phases in quantum evolution and point out that not all dynamical phases need to be compensated for in geometric quantum computation. We delineate the relation between Abelian and non-Abelian geometric gates and find an explicit physical example where the two types of gates coincide. We identify differences and similarities between adiabatic and nonadiabatic realizations of quantum computation based on non-Abelian geometric phases.     

基于控制K次平方根非门的类Toffoli门构造方法

在量子电路综合算法中,由于非置换量子门比置换量子门具有更复杂的规则,直接使用非置换量子门会大幅度提高综合算法的复杂性,因此可先使用非置换量子门生成相应的置换量子门,然后再用这些置换量子门综合所求量子可逆逻辑电路,从而提高算法性能。本文重点研究如何用非置换量子门构造新的置换量子门,为此吸收了格雷码的思想,提出了一种高效的递归构造方法,实现使用控制非门和控制K次平方根非门(非置换量子门),快速生成最优的类Toffoli门(置换量子门)。     

One-qubit quantum gates associated with topological defects in solids

We show a new proposal for implementing one-qubit quantum gates in a solid associated with the presence of topological defects. We discuss a new way of obtaining quantum holonomies for a spin-half particle, and the implementation of a set of one-qubit quantum gates based on the topological phases provided by the presence of a defect in a crystalline solid.     

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.      

Experimental demonstration of non-local controlled-unitary quantum gates using a five-qubit quantum computer

Local implementation of non-local quantum gates is necessary in a distributed quantum computer. Here, we demonstrate the non-local implementation of controlled-unitary quantum gates proposed by Eisert et al. (Phys Rev A 62:052317, 2000) using the five-qubit IBM quantum computer. We verify the fidelity and accuracy of the implementation through the techniques of quantum state and process tomographies.     

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。     

参数化的广义量子通用相位门

主要研究参数化的广义量子通用相位门,给出了单比特量子门、双比特量子门以及三比特量子门的参数化构造。证明参数化的广义量子门和M.Nielsen给出的广义量子门是等价的。举例说明了参数化的广义量子通用门在量子计算中的作用。     

经典逻辑门与量子逻辑门之比较

本文通过经典逻辑门与量子逻辑门之比较,论述了量子计算的特点、量子算法的巨大威力及量子逻辑门的实现问题。     

A recursive method for synthesizing quantum/reversible quaternary parallel adder/subtractor with look-ahead carry

Multiple-valued quantum logic circuits are a promising choice for future quantum computing technology since they have several advantages over binary quantum logic circuits. Adder/subtractor is the major component of the ALU of a computer and is also used in quantum oracles. In this paper, we propose a recursive method of hand synthesis of reversible quaternary full-adder circuit using macro-level quaternary controlled gates built on the top of ion-trap realizable 1-qudit quantum gates and 2-qudit Muthukrishnan–Stroud quantum gates. Based on this quaternary full-adder circuit we propose a reversible circuit realizing quaternary parallel adder/subtractor with look-ahead carry. We also show the way of adapting the quaternary parallel adder/subtractor circuit to an encoded binary parallel adder/subtractor circuit by grouping two qubits together into quaternary qudit values.     

Yang–Baxterizations,Universal Quantum Gates and Hamiltonians

The unitary braiding operators describing topological entanglements can be viewed as universal quantum gates for quantum computation. With the help of the Brylinski’s theorem, the unitary solutions of the quantum Yang–Baxter equation can be also related to universal quantum gates. This paper derives the unitary solutions of the quantum Yang–Baxter equation via Yang–Baxterization from the solutions of the braid relation. We study Yang–Baxterizations of the non-standard and standard representations of the six-vertex model and the complete solutions of the non-vanishing eight-vertex model. We construct Hamiltonians responsible for the time-evolution of the unitary braiding operators which lead to the Schrödinger equations.     

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.     

Towards Scalable Linear-Optical Quantum Computers

Scalable quantum computation with linear optics was considered to be impossible due to the lack of efficient two-qubit logic gates, despite the ease of implementation of one-qubit gates. Two-qubit gates necessarily need a non-linear interaction between the two photons, and the efficiency of this non-linear interaction is typically very small in bulk materials. However, it has recently been shown that this barrier can be circumvented with effective non-linearities produced by projective measurements, and with this work linear-optical quantum computing becomes a new avenue towards scalable quantum computation. We review several issues concerning the principles and requirements of this scheme. PACS: 03.67.Lx, 03.67.Pp, 42.50.Dv, 42.65.Lm     

Algebraic Characterization of Reversible Logic Gates

Reversible logic plays an important role in quantum computing. This paper investigates the universality and composition power of various known and new reversible gates. We present the algebraic characterization of selected new families of Boolean reversible gates. Some theoretical results on the relation between reversible w*w gates and the corresponding symmetric group are derived. Different combinations of reversible gate classes are proven to generate the entire class of reversible w*w gates.     

Coupling of quantum angular momenta: an insight into analogic/discrete and local/global models of computation

In the past few years there has been a tumultuous activity aimed at introducing novel conceptual schemes for quantum computing. The approach proposed in (Marzuoli and Rasetti, 2002, 2005a) relies on the (re)coupling theory of SU(2) angular momenta and can be viewed as a generalization to arbitrary values of the spin variables of the usual quantum-circuit model based on ‘qubits’ and Boolean gates. Computational states belong to finite-dimensional Hilbert spaces labelled by both discrete and continuous parameters, and unitary gates may depend on quantum numbers ranging over finite sets of values as well as continuous (angular) variables. Such a framework is an ideal playground to discuss discrete (digital) and analogic computational processes, together with their relationships occurring when a consistent semiclassical limit takes place on discrete quantum gates. When working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum finite states-machines which in turn represent discrete versions of topological quantum computation models. We argue that our model embodies a sort of unifying paradigm for computing inspired by Nature and, even more ambitiously, a universal setting in which suitably encoded quantum symbolic manipulations of combinatorial, topological and algebraic problems might find their ‘natural’ computational reference model.     

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.      

Compact quantum gates for hybrid photon–atom systems assisted by Faraday rotation

We present some compact circuits for a deterministic quantum computing on the hybrid photon–atom systems, including the Fredkin gate and SWAP gate. These gates are constructed by exploiting the optical Faraday rotation induced by an atom trapped in a single-sided optical microcavity. The control qubit of our gates is encoded on the polarization states of the single photon, and the target qubit is encoded on the ground states of an atom confined in an optical microcavity. Since the decoherence of the flying qubit with atmosphere for a long distance is negligible and the stationary qubits are trapped inside single-sided microcavities, our gates are robust. Moreover, ancillary single photon is not needed and only some linear-optical devices are adopted, which makes our protocols efficient and practical. Our schemes need not meet the condition that the transmission for the uncoupled cavity is balanceable with the reflectance for the coupled cavity, which is different from the quantum computation with a double-sided optical microcavity. Our calculations show that the fidelities of the two hybrid quantum gates are high with the available experimental technology.     

Quantum computation in the decoherence-free subspaces with cavity QED

We present a scheme to implement quantum computation in decoherence-free subspaces (DFSs) with four atoms in a single-mode cavity. A four-dimensional DFS is constituted to protect quantum information when the full symmetry of interaction between system and environment is broken in a specific way, and entangling two-qubit logic gates and noncommuting single-qubit gates are implemented in such DFS. The gate fidelity is numerically calculated, and the feasibility of the approximations taken in this work is verified based on the numerical calculations.     

Exact canonical decomposition of two-qubit operators in terms of CNOT

The canonical decomposition for two-qubit operators has proven very useful for applications in quantum computing. This decomposition generates equivalence classes up to local quantum gates. We provide a variety of complete, explicit decompositions of given two-qubit operators in terms of single, double, and triple controlled-NOT (CNOT) gates. By analytically addressing the needed pre- and post-tensor product factors, we demonstrate that exact results are possible, even when a parameter is included. The examples given are of interest to superconducting qubit, spin-based, dipolar molecule, and other quantum information processing systems.