Preeti Parashar
We establish the non-existence of a universal Hadamard gate for arbitrary unknown qubits, by considering two different principles; namely, no-superluminal signalling and non-increase of entanglement under LOCC. It is also shown that these principles are not violated if and only if the qubit states are of a special form obtained in our recent work [ IJQI (in press), quant-ph/0505068 ].
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Optimal Hadamard gate count for Clifford$+T$ synthesis of Pauli rotations sequences
本文针对Clifford+T综合中Hadamard门减少问题，提出以最少Hadamard门综合π/4 Pauli旋转序列的算法，并进一步最优最小化首末T门之间的Hadamard门数量，从而有效降低T门计数。
We study a reduced quantum circuit computation paradigm in which the only allowable gates either permute the computational basis states or else apply a "global Hadamard operation", i.e. apply a Hadamard operation to every qubit simultaneously. In this model, we discuss complexity bounds (lower-bounding the number of global Hadamard operations) for common quantum algorithms : we illustrate upper bounds for Shor's Algorithm, and prove lower bounds for Grover's Algorithm. We also use our formalism to display a gate that is neither quantum-universal nor classically simulable, on the assumption tha…
Optimal number of parametrized rotations and Hadamard gates in parametrized Clifford circuits with non-repeated parameters
本研究提出一种低复杂度高效算法，可最优缩减参数不重复的参数化Clifford电路中的参数化旋转门数量，同时证明既有Hadamard门优化流程对该类电路的最优性，算法在少内部Hadamard门的电路中效率突出。
We consider one possible implementation of Hadamard gate for optical and ion trap holonomic quantum computers. The expression for its fidelity determining the gate stability with respect to the errors in the single-mode squeezing parameter control is analytically derived. We demonstrate by means of this expression the cancellation of the squeezing control errors up to the fourth order on their magnitude.
We propose a quantum state transfer from an atomic qubit to a cat-like qubit by means of one degenerate Raman interaction and one Hadamard gate operation for coherent states. We show that the coefficients of the atomic qubit can be mapped onto coherent state qubit, with an effective qubit state transfer.
We show that the Hadamard and Unitary gates could be implemented by a unitary evolution together with a measurement for any unknown state chosen from a set $A = {{| {Ψ_i} > ,| {\bar Ψ_i} >}}({i = 1,2})$ if and only if $| {Ψ_1} > ,| {Ψ_2} > ,| {\bar Ψ_1} > ,| {\bar Ψ_2} >$ are linearly independent. We also derive the best transformation efficiencies.
Recently Shi proved that Toffoli and Hadamard are universal for quantum computation. This is perhaps the simplest universal set of gates that one can hope for, conceptually; It shows that one only needs to add the Hadamard gate to make a 'classical' set of gates quantum universal. In this note we give a few lines proof of this fact relying on Kitaev's universal set of gates, and discuss the meaning of the result.
We discuss and experimentally demonstrate a probabilistic Hadamard gate for coherent state qubits. The scheme is based on linear optical components, non-classical resources and the joint projective action of a photon counter and a homodyne detector. We experimentally characterize the gate for the coherent states of the computational basis by full tomographic reconstruction of the transformed output states. Based on the parameters of the experiment we simulate the fidelity for all coherent state qubits on the Bloch sphere.
We propose the implementation of a deterministic Hadamard gate for logical photonic qubits encoded in superpositions of coherent states of a harmonic oscillator. The proposed scheme builds on a recently introduced set of conditional operations in the strong dispersive regime of circuit QED [Z. Leghtas et al. Phys. Rev. A. {\bf 87} (2013)]. We further propose an architecture for coupling two such logical qubits and provide a universal set of deterministic quantum gates. Based on parameter values taken from the current state of the art, we give estimates for the achievable gate fidelities accoun…
We have implemented a Walsh-Hadamard gate, which performs a quantum Fourier transform, in a superconducting qutrit. The qutrit is encoded in the lowest three energy levels of a capacitively shunted flux device, operated at the optimal flux-symmetry point. We use an efficient decomposition of the Walsh-Hadamard gate into two unitaries, generated by off-diagonal and diagonal Hamiltonians respectively. The gate implementation utilizes simultaneous driving of all three transitions between the three pairs of energy levels of the qutrit, one of which is implemented with a two-photon process. The gat…
We consider protecting a Hadamard operation from independent dephasing, bit flipping, and dissipation. These environment-induced errors are represented by three uncorrelated reservoirs of thermalized bosons and we show that the protection is achievable through continuous dynamical decoupling. We find that, to decouple the Hadamard evolution from the environmental influence, we need a control field of higher frequency if the boson spectral density is super-ohmic than if it is ohmic. We also study the relevance of bit flipping and dissipation to the gate fidelity when it is protected from dephas…
The Hadamard gate cannot be replaced by a resource state in universal quantum computation
研究涉及固定资源态的量子计算模型，证明仅靠非相干酉操作、经典控制、测量及任意资源态无法以非零概率精确实现相干酉操作（如Hadamard门），并扩展至近似情况与有限门实现更多门的no-go结果。
Theoretical realization of a two qubit quantum controlled-not logic gate and a single qubit Hadamard logic gate in the anti-Jaynes-Cummings model
本研究在反-Jaynes-Cummings模型中理论实现了Hadamard门和量子受控非门，通过精确选择相互作用时间，在非正交基态子空间中获得单位成功概率，为量子计算提供了新方案。
Topological quantum computation (TQC) is one of the most striking architectures that can realize fault-tolerant quantum computers. In TQC, the logical space and the quantum gates are topologically protected, i.e., robust against local disturbances. The topological protection, however, requires rather complicated lattice models and hard-to-manipulate dynamics; even the simplest system that can realize universal TQC--the Fibonacci anyon system--lacks a physical realization, let alone braiding the non-Abelian anyons. Here, we propose a disk model that can realize the Fibonacci anyon system, and c…
Error-corrected Hadamard gate simulated at the circuit level
本文首次在电路级噪声下模拟表面码的逻辑Hadamard门，通过补丁变形的两种畴壁方案实现逻辑X和Z弦交换，优化物理电路并证明其逻辑失败概率与量子存储实验相当，还将SWAP纠错轮编译为仅四层双量子比特门。
A Sound and Complete Equational Theory for 3-Qubit Toffoli-Hadamard Circuits
本文为3量子比特Toffoli-Hadamard电路建立了可靠且完备的等式理论，通过先研究生成E8格自同构有限群的Toffoli-K电路，再借助Li等人的工作扩展得到完整等式体系。
We obtain the most general ensemble of qubits, for which it is possible to design a universal Hadamard gate. These states when geometrically represented on the Bloch sphere, give a new trajectory. We further consider some Hadamard `type' of operations and find ensembles of states for which such transformations hold. Unequal superposition of a qubit and its orthogonal complement is also investigated.
We cut the volume of surface code S gates by 25% by omitting a Hadamard gate.
A Quantum Approximate Optimization Method For Finding Hadamard Matrices
本文提出一种基于QAOA的门量子计算方法，通过避免辅助量子比特将寻找M阶Hadamard矩阵的量子比特需求降至O(M)，并在模拟器和真实量子计算机上验证了有效性。
What additional gates are needed for a set of classical universal gates to do universal quantum computation? We answer this question by proving that any single-qubit real gate suffices, except those that preserve the computational basis. The result of Gottesman and Knill[quant-ph/9807006] implies that any quantum circuit involving only the Controlled-NOT and Hadamard gates can be efficiently simulated by a classical circuit. In contrast, we prove that Controlled-NOT plus any single-qubit real gate that does not preserve the computational basis and is not Hadamard (or its alike) are universal f…
Quantum computing in terms of geometric phases, i.e. Berry or Aharonov-Anandan phases, is fault-tolerant to a certain degree. We examine its implementation based on Zeeman coupling with a rotating field and isotropic Heisenberg interaction, which describe NMR and can also be realized in quantum dots and cold atoms. Using a novel physical representation of the qubit basis states, we construct $π/8$ and Hadamard gates based on Berry and Aharonov-Anandan phases. For two interacting qubits in a rotating field, we find that it is always impossible to construct a two-qubit gate based on Berry phases…
Consider the problem: Alice wishes to send the same key to $n-1$ users (Bob, Carol,. . . , Nathan), while preventing eavesdropper Eve from acquiring information without being detected. The problem has no solution in the classical cryptography but in quantum telecommunication there are some codes to solve the problem. In the paper \cite{zengetall}, Guo-Jyun Zeng, Kuan-Hung Chen, Zhe-Hua Chang, Yu-Shan Yang, and Yao-Hsin Chou from one side and Cabello in \cite{cabello} from other side, used Hadamard gates, Pauli gates in providing the quantum communication code for two-partity telecommunication …
In this paper, we develop a new classical simulation of quantum bit (qubit) by use of analog components in order to be able to simulate the quantum properties such as the superposition of states. As part of this new approach, we have also implemented many logical gates (Pauli-X, Hadamard, and Conditional phase-shift gates). Simulation of Hadamard and Conditional phase-shift gates using Proteus Design Suite is also presented and discussed in this work.
In this work we investigate that whether one can construct single and two qubit gates for arbitrary quantum states from the principle of no signalling. We considered the problem for Pauli gates, Hadamard gate, C-Not gate.
Hadamard-Pi: Equational Quantum Programming
Hadamard-Pi通过引入Hadamard门的量子编程语言，提供了对量子计算行为的等式理论刻画，并通过有限呈现和合成算法证明了其完备性。
Controlled-NOT (CNOT) gates are commonly included in the standard gate set of quantum processors and provide an important way to entangle qubits. For fixed-frequency qubits using the cross-resonance entangling technique, using the higher-frequency qubit to control the lower-frequency qubit enables much shorter entangling times than using the lower-frequency qubit as the control. Consequently, when implementing a CNOT gate where logical control by the lower-frequency qubit is needed, compilers may implement this functionality by using an equivalent circuit such as placing Hadamard gates on both…
We study decoherence induced by stochastic squeezing control errors considering the particular implementation of Hadamard gate on optical and ion trap holonomic quantum computers. We analytically obtain both the purity of the final state and the fidelity for Hadamard gate when the control noise is modeled by Ornstein-Uhlenbeck stochastic process. We demonstrate the purity and the fidelity oscillations depending on the choice of the initial superimposed state. We derive a linear formulae connecting the gate fidelity and the purity of the final state.
The basic one-bit gates (X, Y, Z, Hadamard, phase, pi/8) as well as the controlled cnot and Toffoli gates are reformulated in the language of geometric-algebra quantum-like computation. Thus, all the quantum algorithms can be reformulated in purely geometric terms without any need of tensor products.
We study decoherence induced by stochastic squeezing control errors considering the particular implementation of Hadamard gate on optical and ion trap holonomic quantum computers. We find the fidelity for Hadamard gate and compute the purity of the final state when the control noise is modeled by Ornstein-Uhlenbeck stochastic process. We demonstrate that in contradiction to the case of the systematic control errors the stochastic ones lead to decoherence of the final state. In the small errors limit we derive a simple formulae connecting the gate fidelity and the purity of the final state.
A novel universal and fault-tolerant basis (set of gates) for quantum computation is described. Such a set is necessary to perform quantum computation in a realistic noisy environment. The new basis consists of two single-qubit gates (Hadamard and ${σ_z}^{1/4}$), and one double-qubit gate (Controlled-NOT). Since the set consisting of Controlled-NOT and Hadamard gates is not universal, the new basis achieves universality by including only one additional elementary (in the sense that it does not include angles that are irrational multiples of $π$) single-qubit gate, and hence, is potentially the…
An implementation is proposed of single qubit gates, e.g., phase, NOT, \sqrt{NOT} and Hadamard, operating on polarized photons and based on light storage. Instead of processing photons themselves, qubit transformations are performed on atomic excitations due to photon storage in a medium of atoms in the tripod configuration.
Improved Synthesis of Toffoli-Hadamard Circuits
本文研究了 Toffoli-Hadamard 门集的精确电路综合问题，通过引入两种改进方法，将综合电路的复杂度从 $O(2^n \log(n)k)$ 显著降低至 $O(n^2 \log(n)k)$，并将结果扩展至二进分数正交矩阵。
We consider how the Hamiltonian Quantum Computing scheme introduced in [arXiv:1509.01278] can be implemented using a 2D array of superconducting transmon qubits. We show how the scheme requires the engineering of strong attractive cross-Kerr and weak flip-flop or hopping interactions and we detail how this can be achieved. Our proposal uses a new electric circuit for obtaining the attractive cross-Kerr coupling between transmons via a dipole-like element. We discuss and numerically analyze the forward motion and execution of the computation and its dependence on coupling strengths and their va…
An Improved Quantum Private Set Intersection Protocol Based on Hadamard Gates
针对 Liu-Yin 量子保密集合交集协议存在内部信息泄露的问题，提出基于 Hadamard 门和额外编码量子比特的改进协议，并证明其正确且能抵抗内外部攻击。
This is an exposition of some basic mathematical aspects of quantum logic gates. At first we established some general formulas for the case of arbitrary quantum gate A with unique restriction A^2=I. The explicit form of the generators and roots of matrix A have been found . Then we apply general results to the particular cases of one-qubit and multi-qubit quantum gates. Some interesting properties of square roots of basic Pauli and Hadamard gates are demonstrated.
A frequency beam splitter (FBS) with the split ratio of 0.5 or 1 can be used as the frequency-mode Hadamard gate (FHG) for frequency-encoded photonic qubits or as the quantum frequency converter (QFC) for frequency up or down conversion of photons. Previous works revealed that all kinds of the FHG or QFC operating at the single-photon level had overall efficiency or output-to-input ratio around 50% or less. In this work, our FHG and QFC are made with the four-wave mixing process based on the dual-$Λ$ electromagnetically induced transparency scheme. We achieved an overall efficiency of 90$\pm$4…
We prove a general limitation in quantum information that unifies the impossibility principles such as no-cloning and no-anticloning. Further, we show that for an unknown qubit one cannot design a universal Hadamard gate for creating equal superposition of the original and its complement state. Surprisingly, we find that Hadamard transformations exist for an unknown qubit chosen either from the polar or equatorial great circles. Also, we show that for an unknown qubit one cannot design a universal unitary gate for creating unequal superpositions of the original and its complement state. We dis…
The Qudit ZH-Calculus: Generalised Toffoli+Hadamard and Universality
本文引入qudit ZH-演算，将无相位规则推广至qudit，证明素数维度下其普适性，并推广Toffoli+Hadamard连接，构造经典可逆逻辑电路，证明近似普适性。
It is shown that the two qubit CNOT (controlled NOT) gate can also be realised using q-deformed angular momentum states via the Jordan-Schwinger mechanism.Thus all the three gates necessary for universality i.e. Hadamard, Phase Shift and the two qubit CNOT gate are realisable with q-deformed oscillators.
We consider stimulated Raman adiabatic passage (STIRAP) processes in tripod systems and show how to generate purely geometric phase changes of the quantum states involved. The geometric phases are controlled by three laser fields where pulse shapes, relative field strength and phases can be controlled. We present a robust set of universal gates for quantum computing based on these geometric phases: a one-qubit phase gate, a Hadamard gate and a two-qubit phase gate.
We show that the quantum logic gates, {\it viz.} the single qubit Hadamard and Phase Shift gates, can also be realised using q-deformed angular momentum states constructed via the Jordan-Schwinger mechanism with two q-deformed oscillators. {\it Keywords :} quantum logic gates ; q-deformed oscillators ; quantum computation {\it PACS:} 03.67.Lx ; 02.20.Uw
We theoretically study operations with a four-level superconducting circuit as a two-qubit system. Using a mapping on a two-qubit system, we show how to implement iSWAP gates and Hadamard gates through pulses on transitions between particular pairs of energy levels. Our approach allows one to prepare pure two-qubit entangled states with desired form of reduced density matrices of the same purity and, in particular, arbitrary identical reduced states of qubits. We propose using schemes for the Hadamard gate and two-qubit entangled states with identical reduced density matrices in order to verif…
The quantum switch is a quantum computational primitive that provides computational advantage by applying operations in a superposition of orders. In particular, it can reduce the number of gate queries required for solving promise problems where the goal is to discriminate between a set of properties of a given set of unitary gates. In this work, we use Complex Hadamard matrices to introduce more general promise problems, which reduce to the known Fourier and Hadamard promise problems as limiting cases. Our generalization loosens the restrictions on the size of the matrices, number of gates a…
Optimisation of Pulse Waveforms for Qubit Gates using Deep Learning
提出利用DNN学习脉冲幅度与门保真度关系并以两阶段优化脉冲参数的方法，在Qiskit模拟中实现了高保真度单量子比特门。
We report the characterization of a universal set of logic gates for one-way quantum computing using a four-photon `star' cluster state generated by fusing photons from two independent photonic crystal fibre sources. We obtain a fidelity for the cluster state of 0.66 +/- 0.01 with respect to the ideal case. We perform quantum process tomography to completely characterize a controlled-NOT, Hadamard and T gate all on the same compact entangled resource. Together, these operations make up a universal set of gates such that arbitrary quantum logic can be efficiently constructed from combinations o…
We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximately (real-valued) universal family of Hadamard+CCZ circuits. The diagrammatic language is generated by two kinds of nodes: the so-called 'spider' associated with the computational basis, as well as a new arity-N generalisation of the Hadamard gate, which satisfies a variation of the spider fusion law. Unlike previous graphical calculi, this admits compact encodings of non-linear classical functions. For exampl…
Composite quantum gates simultaneously compensated for multiple errors
本文提出一种复合脉冲序列，实现X和Hadamard门的同时补偿幅度、频移和持续时间误差，通过导数消去误差项和最小化门保真度，提高门保真度。
Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits
本文将 Clifford-分圆电路精确综合的代数表征从量子比特推广到量子三态，定义了阶为 3^k 的多 qutrit Clifford-分圆门集，并证明一个 3^n×3^n 酉矩阵可由该门集电路表示当且仅当其元素属于环 Z[1/3,ω_k]。
Recent investigations show that conservation laws limit the accuracy of gate operations in quantum computing. The inevitable error under the angular momentum conservation law has been evaluated so far for the CNOT, Hadamard, and NOT gates for spin 1/2 qubits, while the SWAP gate has no constraint. Here, we extend the above results to general single-qubit gates. We obtain an upper bound of the gate fidelity of arbitrary single-qubit gates implemented under arbitrary conservation laws, determined by the geometry of the conservation law and the gate operation on the Bloch sphere as well as the si…
Lower T-count with faster algorithms
本研究提出多种高效低耗时的T-count优化算法，大幅改进TODD算法复杂度并减少T门数量，同时证明了在无Hadamard门及Clifford+T电路中T门数量的更优理论上界。
We numerically study the implementation of a universal two-qubit gate set, composed of CNOT, Hadamard, phase and $π/8$ gates, for transmon-based systems. The control signals to implement such gates are obtained using the Chopped Random Basis optimal control technique, with a target gate infidelity of $10^{-2}$. During the optimization processes we account for the leakage toward non-computational states, an important non-ideality affecting transmon qubits. We also test and benchmark the optimal control solutions against the introduction of Gaussian white noise and spectral distortion, two key n…
Among the many proposals for the realization of a quantum computer, holonomic quantum computation (HQC) is distinguished from the rest in that it is geometrical in nature and thus expected to be robust against decoherence. Here we analyze the realization of various quantum gates by solving the inverse problem: Given a unitary matrix, we develop a formalism by which we find loops in the parameter space generating this matrix as a holonomy. We demonstrate for the first time that such a one-qubit gate as the Hadamard gate and such two-qubit gates as the CNOT gate, the SWAP gate and the discrete F…
We propose a Lyapunov control design to achieve specific (or a family of) unitary time-evolution operators, i.e., quantum gates in the Schrödinger picture by tracking control. Two examples are presented. In the first, we illustrate how to realize the Hadamard gate in a single-qubit system, while in the second, the controlled-NOT (CNOT) gate is implemented in two-qubit systems with the Ising and Heisenberg interactions. Furthermore, we demonstrate that the control can drive the time-evolution operator into the local equivalence class of the CNOT gate and the operator keeps in this class forever…
We present a universal set of quantum gate operations based on exchange-only spin qubits in a double quantum dot, where each qubit is obtained by three electrons in the (2,1) filling. Gate operations are addressed by modulating electrostatically the tunneling barrier and the energy offset between the two dots, singly and doubly occupied respectively. We propose explicit gate sequences of single qubit operations for Hadamard gate and $π$/8 gate, and the two-qubit controlled NOT (CNOT) gate, to complete the universal set. The unswitchable interaction between the two electrons of the doubly occup…
We show that an n-th root of the Walsh-Hadamard transform (obtained from the Hadamard gate and a cyclic permutation of the qubits), together with two diagonal matrices, namely a local qubit-flip (for a fixed but arbitrary qubit) and a non-local phase-flip (for a fixed but arbitrary coefficient), can do universal quantum computation on n qubits. A quantum computation, making use of n qubits and based on these operations, is then a word of variable length, but whose letters are always taken from an alphabet of cardinality three. Therefore, in contrast with other universal sets, no choice of qubi…
Universal quantum computation via scalable measurement-free error correction
本文提出Bacon-Shor码无测量形变协议与一次性Toffoli小装置，实现无中间测量、无前馈的容错通用量子计算，测得约0.12%的电路级退极化噪声容错阈值，适配中性原子平台。
Depth-Optimal Quantum Compilation
本文提出深度最优的量子编译方法，实现任意单量子比特门的常数深度合成，并证明有界宽度门下的对数对数深度下界，同时揭示浅层量子电路与Forrelation电路的新联系。
Transversal implementations of encoded unitary gates are highly desirable for fault-tolerant quantum computation. Though transversal gates alone cannot be computationally universal, they can be combined with specially distilled resource states in order to achieve universality. We show that "triorthogonal" stabilizer codes, introduced for state distillation by Bravyi and Haah [Phys. Rev. A 86 052329 (2012)], admit transversal implementation of the controlled-controlled-Z gate. We then construct a universal set of fault-tolerant gates without state distillation by using only transversal controll…
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。
In this work, we show that ma-QAOA is equivalent to a restriction of continuous-time quantum walks on dynamic graphs. We then show it is universal for computation by finding the appropriate $B$ and $C$ operators and angles that implement the universal gate set consisting of the Hadamard, $π/8$ and Controlled-Not gates in the ma-QAOA framework. This result begins to bridge the gap between the continuous-time quantum walk model and gate model of quantum computation.
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