Arpita Maitra, Preeti Parashar
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.
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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 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.
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。
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…
The ability to simulate one Hamiltonian with another is an important primitive in quantum information processing. In this paper, a simulation method for arbitrary $σ_z \otimes σ_z$ interaction based on Hadamard matrices (quant-ph/9904100) is generalized for any pairwise interaction. We describe two applications of the generalized framework. First, we obtain a class of protocols for selecting an arbitrary interaction term in an n-qubit Hamiltonian. This class includes the scheme given in quant-ph/0106064v2. Second, we obtain a class of protocols for inverting an arbitrary, possibly unknown n-qu…
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 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 ].
In the shadow of the Hadamard test: Using the garbage state for good and further modifications
本研究将Hadamard测试与经典阴影结合，在辅助量子比特上估计特征值，同时利用工作寄存器获取保真度、能量等额外信息，并探讨反控制酉操作的增强作用。
We present an improved low-overhead implementation of surface code logical H. We describe in full detail logical H applied to a single distance 7 double-defect logical qubit in an otherwise idle scalable array such qubits. Our goal is to provide a clear description of logical H and to emphasize that the surface code possesses low-overhead implementations of the entire Clifford group.
Multipartite entanglement of random states of qubits
研究随机量子比特态的多体纠缠特性，比较海拉典型态与哈达玛态的纯度分布，发现哈达玛态在平均纠缠度上更高，且超图态在寻找最大多体纠缠态中尤为相关。
Approximate Cosine Similarity Estimation via an Angle-Encoding Hadamard Test
本文提出一种角度编码的Hadamard测试方法，用于估计归一化实值向量间的余弦相似性，通过并行执行两量子位Hadamard测试电路实现常数电路深度，尽管需要更大的量子位数和经典后处理。
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…
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。
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.
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…
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…
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…
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…
Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints
本研究提出结合Hadamard检验与近似振幅约束的变分量子Goemans-Williamson算法，仅需n+1个量子比特即可高效求解大规模半定规划，在GSet库的MaxCut问题上性能优于同类经典方法。
We propose a scheme to implement the simplest and best-studied version of quantum random walk, the discrete Hadamard walk, in one dimension using coherent macroscopic sample of ultracold atoms, Bose-Einstein condensate (BEC). Implementation of quantum walk using BEC gives access to the familiar quantum phenomena on a macroscopic scale. This paper uses rf pulse to implement Hadamard operation (rotation) and stimulated Raman transition technique as unitary shift operator. The scheme suggests implementation of Hadamard operation and unitary shift operator while the BEC is trapped in long Rayleigh…
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。
The Isomorphism of 3-Qubit Hadamards and $E_8$
本文揭示了矩阵$\mathbb{U}$与$H_4$和$E_8$同构相关的性质，证明其对于8阶矩阵的意义等同于黄金比例，并展示了其与3-qubit Hadamard矩阵共享回文特征多项式系数。
The solving of linear systems provides a rich area to investigate the use of nearer-term, noisy, intermediate-scale quantum computers. In this work, we discuss hybrid quantum-classical algorithms for skewed linear systems for over-determined and under-determined cases. Our input model is such that the columns or rows of the matrix defining the linear system are given via quantum circuits of poly-logarithmic depth and the number of circuits is much smaller than their Hilbert space dimension. Our algorithms have poly-logarithmic dependence on the dimension and polynomial dependence in other natu…
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…
$N$-Party Hadamard Test for Distributed Quantum Computation
提出一种无需非局域受控操作的N方Hadamard测试协议，利用预共享纠缠和局域操作与经典通信，降低采样开销并增强噪声鲁棒性，推动分布式量子计算应用。
Qubit-count optimization using ZX-calculus
本研究针对量子电路量子比特数优化问题，提出逆转Hadamard门小装置化的方法并证明其NP难度，还给出基于ZX图最小宽度路径分解的通用方法，可适配多种量子计算模型。
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。
The quantum Fourier transform (QFT) is a key ingredient of several quantum algorithms and a qudit-specific implementation of the QFT is hence an important step toward the realization of qudit-based quantum computers. This work develops a circuit decomposition of the QFT for hybrid qudits based on generalized Hadamard and generalized controlled-phase gates, which can be implemented using selective rotations in NMR. We experimentally implement the hybrid qudit QFT on an NMR quantum emulator, which uses four qubits to emulate a single qutrit coupled to two qubits.
Custom Penalty QAOA via Walsh-Hadamard Transform
本文提出WCP-QAOA框架，利用Walsh-Hadamard变换将自定义惩罚函数展开为Pauli-Z字符串加权和，实现无松弛约束编码，并在6量子比特背包问题上验证了其有效性。
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 introduce a trichromatic graphical calculus for quantum computing. The generators represent three complementary observables that are treated on equal footing, hence reflecting the symmetries of the Bloch sphere. We derive the Euler angle decomposition of the Hadamard gate within it as well as the so-called supplementary relationships, which are valid equations for qubits that were not derivable within Z/X-calculus of Coecke and Duncan. More specifically, we have: dichromatic Z/X-calculus + Euler angle decomposition of the Hadamard gate = trichromatic calculus.
A hybrid classical-quantum approach for evaluation of multi-dimensional Walsh-Hadamard transforms and its applications to quantum image processing are proposed. In this approach, multidimensional Walsh-Hadamard transforms are obtained using quantum Hadamard gates (along with state-preparation, shifting, scaling and measurement operations). The proposed approach for evaluation of multidimensional Walsh-Hadamard transform has a considerably lower computational complexity (involving $O(N^d)$ operations) in contrast to classical Fast Walsh-Hadamard transform (involving $O(N^d~\log_2 N^d)$ operatio…
Large-scale universal quantum computing requires the implementation of quantum error correction (QEC). While the implementation of QEC has already been demonstrated for quantum memories, reliable quantum computing requires also the application of nontrivial logical gate operations to the encoded qubits. Here, we present examples of such operations by implementing, in addition to the identity operation, the NOT and the Hadamard gate to a logical qubit encoded in a five qubit system that allows correction of arbitrary single qubit errors. We perform quantum process tomography of the encoded gate…
In this paper, different from the other existing methods that only the special and limited numbers of qubits transmit between Alice and Bob, a general method is proposed to implement Bidirectional Quantum Controlled/uncontrolled Teleportation (BQCT/BQT) that Alice and Bob can transmit each type of arbitrary n and m-qubits to each other, simultaneously. This protocol is based on Controlled-NOT (CNOT) operation, appropriate single-qubit unitary operations and single-qubit measurement in the Z-basis and X-basis. Also, the quantum channel can be created by using Hadamard and CNOT operations. One o…
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…
We study theoretically the quantum dynamics of an electron in the singly-ionized double-donor structure in the semiconductor host under the influence of two strongly detuned laser pulses. This structure can be used as a charge qubit where the logical states are defined by the lowest two energy states of the remaining valence electron localized around one or another donor. The quantum operations are performed via Raman-like transitions between the localized (qubit) states and the manyfold of states delocalized over the structure. The possibility of realization of arbitrary single-qubit rotation…
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…
Scalable noisy quantum circuits for biased-noise qubits
针对偏置噪声量子比特，设计含噪声哈达玛测试并发现其可被经典高效模拟，据此提出可检验大规模电路噪声偏置性、对串扰等错误敏感且适用于任意偏置的基准。
Hadamard Random Forest: Reconstructing real-valued quantum states with exponential reduction in measurement settings
针对仅需实值振幅的量子计算场景，提出基于Hadamard随机森林的实值量子态读出方法，将测量设置降至O(N_q)，经10量子比特IBM处理器验证，在特性提取、态重叠估计等方面表现优异，可用于量子线性求解器读出。
Necessary conditions for separability are most easily expressed in the computational basis, while sufficient conditions are most conveniently expressed in the spin basis. We use the Hadamard matrix to define the relationship between these two bases and to emphasize its interpretation as a Fourier transform. We then prove a general sufficient condition for complete separability in terms of the spin coefficients and give necessary and sufficient conditions for the complete separability of a class of generalized Werner densities. As a further application of the theory, we give necessary and suffi…
Phase-Sensitive Quantum Measurement without Controlled Operations
针对量子振幅相位测量中受控操作开销大的问题，提出基于复分析的算法，仅需实时演化与浅层虚时演化电路，在电路深度上优于Hadamard测试，并适用于含噪量子计算机。
Very recently the most general ensemble of qubits are identified using the notion of linearity; any of these qubits gets accepted by a Hadamard gate to generate the equal superposition of the qubit and its orthogonal. Towards more generalization, we investigate the possibility and impossibility results related to Discrete Fourier Transform (DFT) type of operations for a more general set up of qutrits.
Universal quantum computing relies on high-fidelity entangling operations. Here we demonstrate that four coupled qubits can operate as a quantum gate, where two qubits control the operation on two target qubits (a four-qubit gate). This configuration can implement four different controlled two-qubit gates: two different entangling swap and phase operations, a phase operation distinguishing states of different parity, and the identity operation (idle quantum gate), where the choice of gate is set by the state of the control qubits. The device exploits quantum interference to control the operati…
Quantum Circuit Implementation of Two Matrix Product Operations and Elementary Column Transformations
本文针对Hadamard积、Kronecker积和初等列变换设计了基于酉变换和辅助量子测量的高效量子电路方案，显著降低了计算深度、量子门使用量及能耗。
The structure of all completely positive quantum operations is investigated which transform pure two-qubit input states of a given degree of entanglement in a covariant way. Special cases thereof are quantum NOT operations which transform entangled pure two-qubit input states of a given degree of entanglement into orthogonal states in an optimal way. Based on our general analysis all covariant optimal two-qubit quantum NOT operations are determined. In particular, it is demonstrated that only in the case of maximally entangled input states these quantum NOT operations can be performed perfectl…
Superconducting quantum circuit is a promising system for building quantum computer. With this system we demonstrate the universal quantum computations, including the preparing of initial states, the single-qubit operations, the two-qubit universal gate operations between arbitrary qubits, the multiple pairs of two-qubit gate operations in parallel, the coupling operations on a group of qubits in parallel, the coupling operations on multiple groups of qubits in parallel, the coupling operations on multiple pairs and multiple groups of qubits in parallel. Within available technology, a universa…
A six-qubit quantum network consisting of conditional unitary gates is presented which is capable of implementing a large class of covariant two-qubit quantum operations. Optimal covariant NOT operations for one and two-qubit systems are special cases contained in this class. The design of this quantum network exploits basic algebraic properties which also shed new light onto these covariant quantum processes.
We investigate the use of continuously-applied external fields to maximize the fidelity of quantum logic operations performed on a decohering qubit. Assuming a known error operator and an environment represented by a scalar boson field at a finite temperature, we show how decoherence during logical operations can be efficiently reduced by applying a superposition of two external vector fields: one rotating orthogonally to the direction of the other, which remains static. The required field directions, frequency of rotation and amplitudes to decouple noise dynamically are determined by the coup…
Anticoherent $k$-planes and coding techniques for a 3-qubit scheme of universal quantum computing
该研究利用反相干$k$-平面与编码技术，在3量子比特系统中实现Hadamard门和广义Toffoli门，提出抗任意路径形变的通用量子计算方案。
A scheme of universal quantum computation on a chain of qubits is described that does not require local control. All the required operations, an Ising-type interaction and spatially uniform simultaneous one-qubit gates, are translation-invariant.
Two-Unitary Complex Hadamard Matrices of Order $36$
本文构造阶 $36$ 的二酉复 Hadamard 矩阵族，利用等振幅与仅含六个六次单位根相位的结构，给出量子 Euler 问题更简洁且便于实验实现的新解。
We show that the entanglement of a $2\times 2$ bipartite state can be improved and maximized probabilistically through single-qubit operations only. An experiment is proposed and it is numerically simulated.
L. Lamata et al. have presented an entanglement classification of four qubits under stochastic local operation and classical communication (SLOCC) based on inductive approach [Phys. Rev. A \textbf{75}, 022318 (2007)]. While it works well for three qubits, we show that some of the families for four qubits have overlaps and hence this approach is not valid for entanglement classification of four or more qubits under SLOCC.
We present a fast quantum entangling operation on superconducting qubits assisted by a resonator in the quasi-dispersive regime with a new effect --- the selective resonance coming from the amplified qubit-state-dependent resonator transition frequency and the tunable period relation between a wanted quantum Rabi oscillation and an unwanted one. This operation does not require any kind of drive fields and the interaction between qubits. More interesting, the non-computational third excitation states of the charge qubits can play an important role in shortening largely the operation time of the…
Optimal construction of quantum operations is a fundamental problem in the realization of quantum computation. We here introduce a newly discovered quantum gate, B, that can implement any arbitrary two-qubit quantum operation with minimal number of both two- and single-qubit gates. We show this by giving an analytic circuit that implements a generic nonlocal two-qubit operation from just two applications of the B gate. We also demonstrate that for the highly scalable Josephson junction charge qubits, the B gate is also more easily and quickly generated than the CNOT gate for physically feasibl…
Bipartite operations underpin both classical communication and entanglement generation. Using a superposition of classical messages, we show that the capacity of a two-qubit operation for error-free entanglement-assisted bidirectional classical communication can not exceed twice the entanglement capability. In addition we show that any bipartite two-qubit operation can increase the communication that may be performed using an ensemble by twice the entanglement capability.
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。
In a recent paper [Phys. Rev. A 76, 032304(2007)], Li et al. proposed the definition of the residual entanglement for n qubits by means of the Stochastic local operations and classical communication. Here we argue that their definition is not suitable for the case of odd-n qubits.
In a superconducting quantum processor with nearest neighbor coupling, the dispersive interaction between adjacent qubits can result in an effective next-nearest-neighbor coupling whose strength depends on the state of the intermediary qubit. Here, we theoretically explore the possibility of engineering this next-nearest-neighbor coupling to implement controlled two-qubit operations where the intermediary qubit controls the operation on the next-nearest neighbor pair of qubits. In particular, in a system comprising two types of superconducting qubits with anharmonicities of opposite-sign arran…
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