International Journal of Quantum Information

This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.

We propose a dynamic quantum secret sharing (DQSS) protocol based on traveling single-qubits where any number of players can join or leave the protocol according to the dealer’s and players’ will. The protocol can resist collusion attacks of dishonest players by using an honesty test based on some unambiguous unitary operations. Specifically, the proposed protocol is secure against active attacks from dishonest players, which is different from existing protocols that consider only passive attacks based on the assumption that players are honest but curious. By using the asymmetric strategy, the efficiency of our DQSS protocol can be made asymptotically close to 100%. Compared with previous DQSS protocols, our protocol is securer, more efficient and more practical.

In recent years, there has been an increasing tendency towards designing circuits based on reversible logic, and has received much attention because of preventing internal power dissipation. In digital computing systems, multiplier circuits are one of the most fundamental and practical circuits used in the development of a wide range of hardware such as arithmetic circuits and Arithmetic Logic Unit (ALU). Vedic multiplier, which is based on Urdhva Tiryakbhayam (UT) algorithm, has many applications in circuit designing because of its high speed in performing multiplication compared to other multipliers. In Vedic multipliers, partial products are obtained through vertical and cross multiplication. In this paper, we propose four $2\times 2$ reversible Vedic multiplier blocks and use each one of them in its right place. Then, we propose a $4\times 4$ reversible Vedic multiplier using the four aforementioned multipliers. We prove that our design leads to better results in terms of quantum cost, number of constant inputs and number of garbage outputs, compared to the previous ones. We also expand our proposed design to $n\times n$ multipliers which enable us to develop our proposed design in every dimension. Moreover, we propose a formula in order to calculate the quantum cost of our proposed $n\times n$ reversible Vedic multiplier, which allows us to calculate the quantum cost even before designing the multiplier.

In this paper, we present some properties through two two-level atoms interacting with a two-mode quantized cavity field. We study this system in the presence of detuning parameter, Kerr nonlinearity, Stark shift, relative phase and intensity-dependent atoms-field coupling. Also, the coupling parameter is modulated to be time dependent. The exact solution of this model is given by using the Schrődinger equation when the atoms and the field are initially prepared in superposition states and coherent states, respectively. We employed the results to calculate some aspects such as linear entropy, total atomic inversion and cross-correlation function.

Let us denote by ${{𝒞}}_n$ the Clifford group (the circuit or operations generated by Hadamard, $\pi ∕4$ phase and the controlled-NOT gates) and by ${{𝒬}{𝒞}}_n$ the set of qubit states that can be prepared by circuits from the Clifford group. In other words, a state $|\varphi 〉\in {{𝒬}{𝒞}}_n$ if $|\varphi 〉=U|0,\dots ,0〉$ where $U\in {{𝒞}}_n$. We will refer to states in ${{𝒬}{𝒞}}_n$ as Clifford states. This paper studies the set of all three-qubit Clifford states. We prove that ${{𝒞}{𝒬}}_3$ has 8640 states and if we define two states $|{\varphi }_1〉$ and $|{\varphi }_2〉$ in ${{𝒞}{𝒬}}_3$ to be equivalent if $|{\varphi }_1〉=U|{\varphi }_2〉$, with $U$ a local transformation in ${{𝒞}}_3$, then the resulting quotient space has five orbits. More exactly, $\frac{{{𝒞}{𝒬}}_3}{\sim }=\{{{𝒪}}_{1,0},{{𝒪}}_{1,2∕3},{{𝒪}}_{2,2∕3},{{𝒪}}_{3,2∕3},{{𝒪}}_{1,1}\}$ where the orbit ${{𝒪}}_{i,x}$ is made up of states with entanglement entropy $S=x$. For example, the first orbit ${{𝒪}}_{1,0}$ contains the state $|000〉$ and corresponds to the unentangled Clifford states. We say that $|\varphi 〉={\alpha }_1|0,\dots ,0〉+{\alpha }_{2^n}|1,\dots ,1〉$ is a real state if all its amplitudes ${\alpha }_i$ are real numbers. We also say that an operator is real if all the entries of its matrix representation with respect to the computational basis are real numbers. In this paper, we also study the set of real Clifford 3 qubits and the way this set splits when we identify two real Clifford states $|{\varphi }_1〉$ and $|{\varphi }_2〉$ to be equivalent if $|{\varphi }_1〉=U_{ℝ}|{\varphi }_2〉$ where $U_{ℝ}$ is a local real Clifford operator. An interesting aspect that follows from this study of Clifford states is the existence of two real Clifford states $|{\xi }_1〉$ and $|{\xi }_2〉$ that can be connected with a Clifford local transformation but they cannot be connected with a real Clifford local transformation. This is, the equation $|{\xi }_2〉=U|{\xi }_1〉$ for $U$, does have a solution in the set of local transformations from ${{𝒞}}_3$ but it does not have a solution among the local transformations from ${{𝒞}}_3$ that are real. We go a little deeper and show that the equation $|{\xi }_2〉=U|{\xi }_1〉$ does not have a solution for any local operation (not necessarily Clifford) whose entries are real numbers. Finally, we show how the CNOT gates act on the set of Clifford states and also in the set of real Clifford states.

We investigate quantum synchronization and correlations of two qutrits in one non-Markovian environment using the hierarchy equation method. There is no direct interaction between two qutrits and each qutrit interacts with the same non-Markovian environment. The influence of the temperature of the bath, correlation time and coupling strength between qutrits and bath on the quantum synchronization and correlations of two qutrits are studied without the Markovian, Born and rotating wave approximations. We also discuss the influence of dissipation and dephasing on the synchronization of two qutrits. In the presence of dissipation, the phase locking between two qutrits without any direct interaction can be achieved when each qutrit interacts with the common bath. Two qutrits within one common bath cannot be syncrhonized in the purely dephasing case. In addition, the Arnold tongue can be significantly broadened by decreasing the correlation time of two qutrits and bath. Markovian baths are more suitable for synchronizing qutrits than non-Markovian baths.

The Gaussian kernel is a very popular kernel function used in many machine learning algorithms, especially in support vector machines (SVMs). It is more often used than polynomial kernels when learning from nonlinear datasets and is usually employed in formulating the classical SVM for nonlinear problems. Rebentrost et al. discussed an elegant quantum version of a least square support vector machine using quantum polynomial kernels, which is exponentially faster than the classical counterpart. This paper demonstrates a quantum version of the Gaussian kernel and analyzes its runtime complexity using the quantum random access memory (QRAM) in the context of quantum SVM. Our analysis shows that the runtime computational complexity of the quantum Gaussian kernel is approximated to $O[{ϵ}^{-1}d\mathrm{\text{log}}N],\mathrm{\text{with}}\mathrm{\text{error}}{R}_d(X_{\mathrm{\text{qtm}}})$ and even $\approx O({ϵ}^{-1}\mathrm{\text{log}}N)$ when $d$ and the error $R_d(X_{\mathrm{\text{qtm}}})$ are small enough to be ignored, where $N$ is the dimension of the training instances, $ϵ$ is the accuracy, $X_{\mathrm{\text{qtm}}}$ is the dot product of the two quantum states, and $R_d(X_{\mathrm{\text{qtm}}})$ is the Taylor remainder error term. Therefore, the run time complexity of the quantum version of the Gaussian kernel seems to be significantly faster when compared with its classical version.

We combine the long-distance quantum state transfer and simple operations with the elements of the transferred (nor perfectly) density matrix. These operations are turning some matrix elements to zero, rearranging the matrix elements and preparing their linear combinations with required coefficients. The basic tool performing these operations is the unitary transformation on the extended receiver. A system of linear algebraic equations can be solved in this way as well. Such operations are numerically simulated on the basis of 42-node spin-1/2 chain with the two-qubit sender and receiver.

In order to solve the high complexity of classical image processing, a quantum scheme for image sharpness estimation based on the Laplacian operator is proposed. The mean of grayscale gradients of all pixels is regarded as the sharpness estimation metric. A new quantum image representation model is presented by extending the Novel Enhanced Quantum Representation (NEQR) model, which is greatly useful for quantum image convolution. In quantum platforms, it has been shown that the mean calculation of numbers is rather difficult because the numbers are stored in a quantum superposition state. In order to solve this problem, we put forward an algorithm which essential idea is cyclically shifting the superposition state and iteratively calculating the mean of the new one and the original state. The mean can be obtained from the superposition state by only one quantum measurement. By analyzing the space complexity and time complexity, the scheme is far superior to classical ones in terms of resource consumption and execution speed. In addition, the results of simulation experiments show that for noiseless images, the performance of the scheme is consistent with subjective visual perception of images sharpness.