G. Cattaneo, G. Conte, R. Leporini
We consider the unitary and the anti--unitary operator realizations of two important genuine quantum gates that transform elements of the computational basis of into superpositions: the square root of the identity and the square root of the negation.
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Constructing appropriate unitary matrix operators for new quantum algorithms and finding the minimum cost gate sequences for the implementation of these unitary operators is of fundamental importance in the field of quantum information and quantum computation. Evolution of quantum circuits faces two major challenges: complex and huge search space and the high costs of simulating quantum circuits on classical computers. Here, we use the group leaders optimization algorithm to decompose a given unitary matrix into a proper-minimum cost quantum gate sequence. We test the method on the known decom…
Classification of Schmidt-rank-two multipartite unitary gates by singular number
该研究针对施密特秩为2的真多体幺正门，利用其施密特分解的唯一性提出奇异数概念完成分类，并拓展研究了三量子比特对角幺正门的施密特秩性质，给出了秩为3的充要条件。
We propose a simple formalism to design unitary gates robust against given systematic errors. This formalism generalizes our previous observation [Y. Kondo and M. Bando, J. Phys. Soc. Jpn. 80, 054002 (2011)] that vanishing dynamical phase in some composite gates is essential to suppress amplitude errors. By employing our formalism, we naturally derive a new composite unitary gate which can be seen as a concatenation of two known composite unitary operations. The obtained unitary gate has high fidelity over a wider range of the error strengths compared to existing composite gates.
Constructing three-qubit unitary gates in terms of Schmidt rank and CNOT gates
本文构造了Schmidt秩从一到七的三量子比特酉门，发现Toffoli和Fredkin门秩分别为二和四，并利用CNOT及局域门实现，揭示了CNOT门数量与Schmidt秩的联系。
Decomposition of unitary matrix into quantum gates
本研究提出一种将任意酉矩阵分解为X门及全受控Ry、Rz、R1门序列的算法，可生成对应Q#实现，还探讨了优化并分析了复杂度。
We show that any unitary operator on the $d_A\times d_B$ system ($d_A\ge 2$) can be decomposed into the product of at most $4d_A-5$ controlled unitary operators. The number can be reduced to $2d_A-1$ when $d_A$ is a power of two. We also prove that three controlled unitaries can implement a bipartite complex permutation operator, and discuss the connection to an analogous result on classical reversible circuits. We further show that any $n$-partite unitary on the space $\mathbb{C}^{d_1}\otimes...\otimes\mathbb{C}^{d_n}$ is the product of at most $[2\prod^{n-1}_{j=1}(2d_j-2)-1]$ controlled unit…
Quantum states obey an asymptotic no-cloning theorem, stating that no deterministic machine can reliably replicate generic sequences of identically prepared pure states. In stark contrast, we show that generic sequences of unitary gates can be replicated deterministically at nearly quadratic rates, with an error vanishing on most inputs except for an exponentially small fraction. The result is not in contradiction with the no-cloning theorem, since the impossibility of deterministically transforming pure states into unitary gates prevents the application of the gate replication protocol to sta…
In this paper, we demonstrate n-party controlled unitary gate implementations locally on arbitrary remote state through linear entangled channel where control parties share entanglement with the adjacent control parties and only one of them shares entanglement with the target party. In such a network, we describe the protocol of simultaneous implementation of controlled-Hermitian gate starting from three party scenario. We also explicate the implementation of three party controlled-Unitary gate, a generalized form of Toffoli gate and subsequently generalize the protocol for n-party using minim…
Probabilistic implementation of universal Hadamard and Unitary gates
我们表明，Hadamard和酉门可以实现由一个单一的进化与测量从一组$A = {{| {Ψ_i} > ,| {\bar Ψ_i} >}}({i = 1,2})$选择的任何未知状态，当且仅当$| {Ψ_1} > ,| {Ψ_2} > ,| {\bar Ψ_1} > ,| {\bar Ψ_2} >$是线性独立的。我们还可以获得最佳的转换效率。
An unknown unitary gates, which is secretly chosen from several known ones, can always be distinguished perfectly. In this paper, we implement such a task on IBM's quantum processor. More precisely, we experimentally demonstrate the discrimination of two qubit unitary gates, the identity gate and the $\frac{2}{3}π$-phase shift gate, using two discrimination schemes -- the parallel scheme and the sequential scheme. We program these two schemes on the \emph{ibmqx4}, a $5$-qubit superconducting quantum processor via IBM cloud, with the help of the $QSI$ modules [S. Liu et al.,~arXiv:1710.09500, 2…
We study the entangling properties of multipartite unitary gates with respect to the measure of entanglement called one-tangle. Putting special emphasis on the case of three parties, we derive an analytical expression for the entangling power of an $n$-partite gate as an explicit function of the gate, linking the entangling power of gates acting on $n$-partite Hilbert space of dimension $d_1 \ldots d_n$ to the entanglement of pure states in the Hilbert space of dimension $(d_1 \ldots d_n)^2$. Furthermore, we evaluate its mean value averaged over the unitary and orthogonal groups, analyze the m…
We introduce a twisted Heisenberg-type interaction Hamiltonian, a Heisenberg-type spin interaction where the coordinates of the second qubit are twisted according to a unitary gate. We develop parallelizable adiabatic gate teleportation (PAGT) where a sequence of unitary gates is performed in a single step of the adiabatic process. In PAGT, numeric calculations suggest the necessary time for the adiabatic evolution implementing a sequence of $L$ unitary gates increases at most as $O(L^5)$. However, we show that it has the interesting property that it can map the temporal order of gates to the …
On the generation of sequential unitary gates from continuous time Schrodinger equations driven by external fields
在所有关于量子计算机的各种提议中，一个共同的特征是量子线路被期望由作用于量子态的幺正变换的级联组成。提出了一个框架，直接表示这些基本量子门的控制输入进入连续时间强迫薛定谔方程。
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.
Optimal Realizations of Controlled Unitary Gates
受控非门和单量子比特门被认为是量子计算中的基本门。人们自然会问，要实现更复杂的门或电路，需要多少这样的基本门。回想一下，受控U门可以用两个受控非门和四个单量子位门来实现。我们证明了这种实现是最优的，当且仅当矩阵U满足条件tr U！= 0，tr UX！= 0，det U！= 1.我们还得到了最佳的实现在非通用的情况下。
We show that it is possible to reduce the number of two-qubit gates needed for the construction of an arbitrary controlled-unitary transformation by up to two times using a tunable controlled-phase gate. On the platform of linear optics, where two-qubit gates can only be achieved probabilistically, our method significantly reduces the amount of components and increases success probability of a two-qubit gate. The experimental implementation of our technique presented in this paper for a controlled single-qubit unitary gate demonstrates that only one tunable controlled-phase gate is needed inst…
Long time behavior of a unitary quantum gate $U$, acting sequentially on two subsystems of dimension $N$ each, is investigated. We derive an expression describing an arbitrary iteration of a two-qubit gate making use of a link to the dynamics of a free particle in a $3D$ billiard. Due to ergodicity of such a dynamics an average along a trajectory $V^t$ stemming from a generic two-qubit gate $V$ in the canonical form tends for a large $t$ to the average over an ensemble of random unitary gates distributed according to the flat measure in the Weyl chamber - the minimal $3D$ set containing points…
Arbitrary state controlled-unitary gate by adiabatic passage
我们提出了一个鲁棒的方案，涉及原子固定在一个光学腔中，直接实现通用的控制酉门。本技术的基础上绝热通过使用新的暗态非常适合于受控旋转操作。我们表明，这些黑暗的状态允许一个门，这是一个广义的控制酉门的情况下，控制量子位可以被选择为一个任意的状态的强大的实现。这种门具有潜在的应用，以快速实现量子算法，如投影测量算法。这个过程是消相干自由的，因为激发原子态和腔模在动力学过程中没有被填充。
The Controlled-U and Unitary Transformation in Two-Qudit
我们具体地从一些基本门构造了qudit中受控U门的一个推广。我们还利用扩展的受控U门构造了二量子中的幺正变换，并证明了它的普适性。
Quantifying Entangling Power of Controlled Unitary Gates
本文提出可计算量ζ量化受控幺正门的纠缠能力，推导其通用上界并关联其他量子熵量，经对比验证该量可达到对应维度下纠缠能力的已知最优值。
We characterize all maximally entangling bipartite unitary operators, acting on systems $A,B$ of arbitrary finite dimensions $d_A\le d_B$, when use of ancillary systems by both parties is allowed. Several useful and interesting consequences of this characterization are discussed, including an understanding of why the entangling and disentangling capacities of a given (maximally entangling) unitary can differ and a proof that these capacities must be equal when $d_A=d_B$.
The implementation of a high-fidelity two-qubit quantum logic gate remains an outstanding challenge for isolated solid-state qubits such as Nitrogen-Vacancy (NV) centers in diamond. In this work, we show that by driving pairs of NV centers to undergo photon scattering processes that flip their qubit state simultaneously, we can achieve a unitary two-qubit gate conditioned upon a single photon detection event. Further, by exploiting quantum interference between the optical transitions of the NV centers electronic states, we realize the existence of two special drive frequencies: a `magic' point…
Unitary gates are an interesting resource for quantum communication in part because they are always invertible and are intrinsically bidirectional. This paper explores these two symmetries: time-reversal and exchange of Alice and Bob. We will present examples of unitary gates that exhibit dramatic separations between forward and backward capacities (even when the back communication is assisted by free entanglement) and between entanglement-assisted and unassisted capacities, among many others. Along the way, we will give a general time-reversal rule for relating the capacities of a unitary gat…
Entangling characterization of (SWAP)1/m and Controlled unitary gates
我们研究了（SWAP）1/m（m>=1）和受控酉门的纠缠功率和完美纠缠性质。结果表明，（SWAP）1/2是该族中唯一的完美纠缠子。另一方面，确定局部等价于CNOT的CU的子集。结果表明，该子集是一个完美的纠缠，必须具有最大的纠缠功率。
Universal and deterministic manipulation of the quantum state of harmonic oscillators: a route to unitary gates for Fock State qubits
我们提出了一个简单的量子线路，它允许对受限谐振子的量子态进行普适的和确定性的操纵。该方案是基于与辅助三能级系统和一个经典的外部驱动源的参考振荡器的选择性相互作用，并使任何Fock态的幺正操作，二乘二。一个电路等效于Fock态量子位上的单个量子位酉逻辑门。类似协议的序列允许对谐振子量子态进行完全的、确定性的和与状态无关的操纵。
A general scheme is presented to decompose a $d$-by-$d$ unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in $d-1$ classes, where each class is isomorphic to the group of $2\times 2$ unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on $n$-qubit registe…
We study efficient generations of random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis with randomly distributed phases for their eigenvalues. Despite the simple algebraic structure, they cannot be achieved by quantum circuits composed of a few-qubit diagonal gates. We introduce diagonal-unitary $t$-designs and present two quantum circuits that implement diagonal-unitary $2$-designs with the computational basis in $N$-qubit systems. One is composed of single-qubit diagonal gates and controlled-phase gates with randomized phases, which achieves an exact di…
With improved gate calibrations reducing unitary errors, we achieve a benchmarked single-qubit gate fidelity of 99.95% with superconducting qubits in a circuit quantum electrodynamics system. We present a method for distinguishing between unitary and non-unitary errors in quantum gates by interleaving repetitions of a target gate within a randomized benchmarking sequence. The benchmarking fidelity decays quadratically with the number of interleaved gates for unitary errors but linearly for non-unitary, allowing us to separate systematic coherent errors from decoherent effects. With this protoc…
Unitary Gate Synthesis for Continuous Variable Systems
本文研究了连续变量双模幺正门的综合问题，其中两个模A和B由一个固定的二次哈密顿量H耦合。门综合由一系列的演变支配的哈密顿H间隔的局部相移应用到A和B。我们专注于协议，需要最少的必要步骤，我们展示了如何实现分束器和双模挤压器在短短的三个步骤。特别注意的哈密顿量x_AP_B，描述了有效的非共振相互作用的光与集体原子自旋。
Characterization of three-qubit controlled unitary gates of Schmidt rank three
本文刻画了Schmidt秩为三的三量子比特受控酉门，给出了充要条件，并构造了显式例子，分析其纠缠能力，展示了从完全可分到最大GHZ类态的纠缠生成，并提供了达到最小CNOT门数的分解。
SU(4) gate design via unitary process tomography: its application to cross-resonance based superconducting quantum devices
本文提出利用CR哈密顿量传播子与Weyl腔几何结构，在超导量子器件上实现无回波的连续R_{ZZ}(θ)门，并验证了其在SU(4)门设计中的保真度与时间优势。
Operator Spreading in Random Unitary Circuits with Unitary-invariant Gate Distributions
研究酉不变系综门分布下随机酉电路的算子扩散，发现其长期行为由蝴蝶速度和扩散常数控制，但需有限时间达到二元权重形式且具有限畴壁宽度，并以Poisson核分布电路验证。
We perform a functional expansion of the fidelity between two unitary matrices in order to find the necessary conditions for the robust implementation of a target gate. Comparison of these conditions with those obtained from the Magnus expansion and Dyson series shows that they are equivalent in first order. By exploiting techniques from robust design optimization, we account for issues of experimental feasibility by introducing an additional criterion to the search for control pulses. This search is accomplished by exploring the competition between the multiple objectives in the implementatio…
Quantum circuit model is the most popular paradigm for implementing complex quantum computation. Based on Cartan decomposition, we show that $2(N-1)$ generalized controlled-$X$ (GCX) gates, $6$ single-qubit rotations about the $y$- and $z$-axes, and $N+5$ single-partite $y$- and $z$-rotation-types which are defined in this paper are sufficient to simulate a controlled-unitary gate $\mathcal{U}_{cu(2\otimes N)}$ with $A$ controlling on $\mathbb{C}^2\otimes \mathbb{C}^N$. In the scenario of the unitary gate $\mathcal{U}_{cd(M\otimes N)}$ with $M\geq3$ that is locally equivalent to a diagonal uni…
Unitary Synthesis with AlphaZero via Dynamic Circuits
本文提出受 AlphaZero 启发的强化学习智能体，利用离散逻辑门精确编译酉变换，并扩展到含测量与条件门的动态电路，以低推理时间高效综合可复用量子子程序。
The recently demonstrated possibility of entangling opposite values of the orbital angular momentum (OAM) of a photon with its spin enables the realization of nontrivial one-photon spinorbit four-dimensional states for quantum information purposes. Hitherto, however, an optical device able to perform arbitrary unitary transformations on such spinorbit photon states has not been proposed yet. In this work we show how to realize such a ``universal unitary gate'' device, based only on existing optical technology, and describe its operation. Besides the quantum information field, the proposed devi…
Multi-controlled unitary gates have been a subject of interest in quantum computing since its inception, and are widely used in quantum algorithms. The current state-of-the-art approach to implementing n-qubit multi-controlled gates involves the use of a quadratic number of single-qubit and CNOT gates. However, linear solutions are possible for the case where the controlled gate is a special unitary SU(2). The most widely-used decomposition of an n-qubit multi-controlled SU(2) gate requires a circuit with a number of CNOT gates proportional to 28n. In this work, we present a new decomposition …
A universal quantum processor is a device that takes as input a (quantum) program, containing an encoding of an arbitrary unitary gate, and a (quantum) data register, on which the encoded gate is applied. While no perfect universal quantum processor can exist, approximate processors have been proposed in the past two decades. A fundamental open question is how the size of the smallest quantum program scales with the approximation error. Here we answer the question, by proving a bound on the size of the program and designing a concrete protocol that attains the bound in the asymptotic limit. Ou…
Any bipartite nonlocal unitary operation can be carried out by teleporting a quantum state from one party to the other, performing the unitary gate locally, and teleporting a state back again. This paper investigates unitaries which can be carried out using less prior entanglement and classical communication than are needed for teleportation. Large families of such unitaries are constructed using (projective) representations of finite groups. Among the tools employed are: a diagrammatic approach for representing entangled states, a theorem on the necessary absence of information at certain tim…
A universal quantum circuit for two-qubit transformations with three CNOT gates
我们考虑通过CNOT门和单量子比特幺正门实现两量子比特的幺正变换。我们表明，通过一个明确的量子线路，连同本地门三个CNOT门是必要的和足够的，以实现任意幺正变换的两个量子比特。我们还确定了两个量子比特门的子集，可以执行只有两个CNOT门。
We develop a unitary dependence theory to characterize the behaviors of quantum circuits and states in terms of how quantum gates manipulate qubits and determine their measurement probabilities. A qubit has dependence on a 1-qubit unitary gate if its measurement probabilities depend on the parameters of the gate. A 1-qubit unitary creates such a dependence onto the target qubit, and a CNOT gate copies all the dependences from the control qubit to the target qubit. The complete dependence picture of the output state details the connections qubits may have when being manipulated or measured. Com…
Universal unitary gate for single-photon 2-qubit states
在纠缠光子的空间二进制替代与其偏振，可以使用单光子来研究任意的2量子比特状态。通过马赫-曾德尔干涉仪发送光子，配备了一组改变偏振的波片，相当于对2量子比特状态进行幺正变换。我们表明，任何所需的酉门可以实现一个明智的选择参数的设置，并讨论了一些应用。它们包括一个未知的2-qubit状态的诊断，光学Grover搜索，以及Vaidman，Aharonov和Albert发明的思想实验的实现。
We analyze a discrimination problem of a single-qubit unitary gate with two candidates, where the candidates are not provided with their classical description, but their quantum sample is. More precisely, there are three unitary quantum gates -- one target and one sample for each of the two candidates -- whose classical description is unknown except for their dimension. The target gate is chosen equally among the candidates. We obtain the optimal protocol that maximizes the expected success probability, assuming the Haar distribution for the candidates. This problem is originally introduced in…
Non-local properties of ensembles of quantum gates induced by the Haar measure on the unitary group are investigated. We analyze the entropy of entanglement of a unitary matrix U equal to the Shannon entropy of the vector of singular values of the reshuffled matrix. Averaging the entropy over the Haar measure on U(N^2) we find its asymptotic behaviour. For two--qubit quantum gates we derive the induced probability distribution of the interaction content and show that the relative volume of the set of perfect entanglers reads 8/3 π\approx 0.85. We establish explicit conditions under which a giv…
Using a braid group representation based on the Temperley-Lieb algebra, we construct braid quantum gates that could generate entangled $n$-partite $D$-level qudit states. $D$ different sets of $D^n\times D^n$ unitary representation of the braid group generators are presented. With these generators the desired braid quantum gates are obtained. We show that the generalized GHZ states, which are maximally entangled states, can be obtained directly from these braid quantum gates without resorting to further local unitary transformations. We also point out an interesting observation, namely for a g…
We consider the effects of plane-wave states scattering off finite graphs, as an approach to implementing single-qubit unitary operations within the continuous-time quantum walk framework of universal quantum computation. Four semi-infinite tails are attached at arbitrary points of a given graph, representing the input and output registers of a single qubit. For a range of momentum eigenstates, we enumerate all of the graphs with up to $n=9$ vertices for which the scattering implements a single-qubit gate. As $n$ increases, the number of new unitary operations increases exponentially, and for …
A memory and gate efficient algorithm for unitary mixed Schur sampling
该研究形式化并推广酉Schur采样任务至酉混合Schur采样，提出流式算法实现内存指数级、门复杂度多项式级降低，有限秩输入下可进一步优化，推广改进了已有结果。
Practical implementation of arbitrary nonlocal controlled-unitary gate via indefinite causal order
本文提出了一种基于不定因果顺序的实用协议，实现任意受控单元arity门，通过调整单量子比特操作减少电路复杂度并提高实验灵活性。
The processing of quantum information always has a cost in terms of physical resources such as energy or time. Determining the resource requirements is not only an indispensable step in the design of practical devices - the resources need to be actually provided - but may also yield fundamental constraints on the class of processes that are physically possible. Here we study how much energy is required to implement a desired unitary gate on a quantum system with a non-trivial energy spectrum. We derive a general lower bound on the energy requirement, extending the main result of Ref. [1] from …
By using an ancilla qubit as a mediator, two distant qubits can undergo a non-local entangling unitary operation. This is desirable for when attempting to scale up or distribute quantum computation by combining fixed static local sets of qubits with ballistic mediators. Using a model driven by measurements on the ancilla, it is possible to generate a maximally entangling CZ gate while only having access to a less entangling gate between the pair qubits and the ancilla. However this results in a stochastic process of generating control phase rotation gates where the expected time for success do…
On the capacities of bipartite Hamiltonians and unitary gates
我们认为相互作用是双向通道。我们研究了相互作用哈密顿量和非局域幺正门产生纠缠和传递经典信息的能力。我们给出了相互作用的纠缠产生容量和纠缠辅助单向经典通信容量的解析表达式，并证明了这些量是可加的，使得渐近容量等于相应的单次容量.我们给其他能力的一般界限，讨论了一些例子，并得出结论与一些开放的问题。
Fernando Galve \emph{et al.} $[Phys. Rev. Lett. \textbf{110}, 010501 (2013)]$ introduced discording power for a two-qubit unitary gate to evaluate its capability to produce quantum discord, and found that a $π/8$ gate has maximal discording power. This work analyzes the entangling power of a two-qubit unitary gate, which reflects its ability to generate quantum entanglement in another way. Based on the renowned Cartan decomposition of two-qubit unitary gates, we show that the magic power of the $π/8$ gate produces maximal entanglement for a general value of purities for two-qubit states.
Quantum computation is a continuously growing research area which is based on nature and resources of quantum mechanics, as superposition and entanglement. In its quantum circuits version, the use of convenient and appropriate gates is essential. But while those gates adopt convenient forms for computational algorithms, their design depends on specific quantum systems and stu? being used. These gates need manage quantum systems based on physical interactions ruled by quantum Hamiltonians. With this, gates design is restricted to properties and limitations of interactions and physical elements …
Optimized synthesis of circuits for diagonal unitary matrices with reflection symmetry
针对NISQ时代量子电路优化需求，本文研究具有反射对称性的对角酉矩阵电路综合，提出优化纠缠门数的构造性算法，相比通用方法将门数和电路深度均减少近一半。
We perform optimal-control-theory calculations to determine the minimum number of two-qubit CNOT gates needed to perform quantum state preparation and unitary operator synthesis for few-qubit systems. By considering all possible gate configurations, we determine the maximum achievable fidelity as a function of quantum circuit size. This information allows us to identify the minimum circuit size needed for a specific target operation and enumerate the different gate configurations that allow a perfect implementation of the operation. We find that there are a large number of configurations that …
We developed a general framework for synthesizing target gates by using a finite set of basic gates, which is a crucial step in quantum compilation. When approximating a gate in SU($n$), a naive brute-force search requires a computational complexity of $O(1/\varepsilon^{(n^2 - 1)})$ to achieve an approximation with error $\varepsilon$. In contrast, by using our method, the complexity can be reduced to $O(-n^2 \log\varepsilon/\varepsilon^{((n^2 - 1)/2)})$. This method requires almost no assumptions and can be applied to a variety of gate sets, including Clifford+$T$ and Clifford+$V$. Further, w…
Operator spreading in random unitary circuits with orthogonal/symplectic-invariant gate distributions
研究从正交不变或辛不变系综中抽取门组成的随机量子电路中的算子扩散，揭示了与酉不变情况的关键区别，包括权重三元结构、有限宽度畴壁以及蝴蝶速度的二分性。
Transitions of the Lyapunov spectrum and entanglement entropy in monitored quantum dynamics with homogeneous unitary gates
本文研究了受测量和空间均匀酉门演化的量子系统中的 Lyapunov 谱与纠缠熵，发现 Lyapunov 指数通常收敛于与测量结果无关的值，且谱转变与纠缠转变在相同阈值处发生，并基于 CPTP 动力学解释了该收敛机制。
We introduce magic measures to quantify the nonstabilizerness of multiqubit quantum gates and establish lower bounds on the $T$ count for fault-tolerant quantum computation. First, we introduce the stabilizer nullity of multi-qubit unitary, which is based on the subgroup of the quotient Pauli group associated with the unitary. This unitary stabilizer nullity extends the state-stabilizer nullity by Beverland et al. to a dynamic version. In particular, we show this nonstabilizerness measure has desirable properties such as subadditivity under composition and additivity under tensor product. Seco…
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 using non-adiabatic holonomy in decoherence-free subspaces and the required resource for the d…
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