Final Quantum State Research Articles

Modeling quantum optical phenomena using transition currents

Spontaneous light emission is central to a vast range of physical systems and is a founding pillar for the theory of light–matter interactions. In the presence of complex photonic media, the description of spontaneous light emission usually requires advanced theoretical quantum optics tools such as macroscopic quantum electrodynamics, involving quantized electromagnetic fields. Although rigorous and comprehensive, the complexity of such models can obscure the intuitive understanding of many quantum-optical phenomena. Here, we review a method for calculating spontaneous emission and other quantum-optical processes without making explicit use of quantized electromagnetic fields. Instead, we introduce the concept of transition currents, comprising charges in matter that undergo transitions between initial and final quantum states. We show how predictions that usually demand advanced methods in quantum electrodynamics or quantum optics can be reproduced by feeding these transition currents as sources to the classical Maxwell equations. One then obtains the relevant quantum observables from the resulting classical field amplitudes, without washing out quantum optical effects. We show that this procedure allows for a straightforward description of quantum phenomena, even when going beyond the dipole approximation and single emitters. As illustrative examples, we calculate emission patterns and Purcell-enhanced emission rates in both bound-electron and free-electron systems. For the latter, we derive cathodoluminescence emission and energy-loss probabilities of free electrons interacting with nanostructured samples. In addition, we calculate quantum-beat phenomena in bound-electron systems and wave function-dependent optical coherence in free-electron systems. Remarkably, the transition-current formalism captures more complex phenomena, such as many-body interference effects and super-radiance of both bound- and free-electron systems, second-order processes such as two-photon emission, and quantum recoil corrections to free-electron radiation. We review a variety of light–matter interactions in fields ranging from electron microscopy to nanophotonics and quantum optics, for which the transition-current theoretical formalism facilitates practical simulations and a deeper understanding of novel applications.

A unifying quantum speed limit for time-independent Hamiltonian evolution

Quantum speed limit (QSL) is the study of fundamental limits on the evolution time of quantum systems. For instance, under the action of a time-independent Hamiltonian, the evolution time between an initial and a final quantum state obeys various mutually complementary lower bounds. They include the Mandelstam–Tamm (MT) bound, the Margolus–Levitin (ML) bound, the Luo–Zhang bound, the Lee–Chau (LC) bound together with the dual ML bound introduced by Ness and coworkers. Here we show that the MT bound can be obtained by optimizing the LC bound over a certain parameter. More importantly, we report a QSL that includes all the above bounds as special cases before optimizing over the physically meaningless reference energy level of a quantum system. This unifying bound depends on a certain parameter p. For any fixed p, we find all pairs of time-independent Hamiltonian and initial pure quantum state that saturate this unifying bound. More importantly, these pairs allow us to compute this bound accurately and efficiently using an oracle that returns certain pth moments related to the absolute value of energy of the quantum state. Moreover, this oracle can be simulated by a computationally efficient and accurate algorithm for finite-dimensional quantum systems as well as for certain infinite-dimensional quantum states with bounded and continuous energy spectra. This makes our computational method feasible in a lot of practical situations. We further compare the performance of this bound for the case of a fixed p as well as the case of optimizing over p with existing QSLs. We find that if the dimension of the underlying Hilbert space is ≲2000 , our unifying bound optimized over p can be computed accurately in a few minutes using Mathematica code with just-in-time compilation in a typical desktop. Besides, this optimized unifying QSL is at least as good as all the existing ones combined and can occasionally be a few percent to a few times better.

TRAVOLTA: GPU acceleration and algorithmic improvements for constructing quantum optimal control fields in photo-excited systems

We present an open-source software package, TRAVOLTA (Terrific Refinements to Accelerate, Validate, and Optimize Large Time-dependent Algorithms), for carrying out massively parallelized quantum optimal control calculations on GPUs. The TRAVOLTA software package is a significant overhaul of our previous NIC-CAGE algorithm and also includes algorithmic improvements to the gradient ascent procedure to enable faster convergence. We examine three different variants of GPU parallelization to assess their performance in constructing optimal control fields in a variety of quantum systems. In addition, we provide several examples with extensive benchmarks of our GPU-enhanced TRAVOLTA code to show that it generates the same results as previous CPU-based algorithms but with a speedup that is more than ten times faster. Our GPU enhancements and algorithmic improvements enable large quantum optimal control calculations that can be efficiently and routinely executed on modern multi-core computational hardware. Program summaryProgram Title: TRAVOLTACPC Library link to program files:https://doi.org/10.17632/grwppm37rn.1Licensing provisions: GNU General Public License 3Programming language: C++, openBLAS, and CUDASupplementary material: Brief review of LU decomposition, raw numerical values used to generate Fig. 6 in the main text, and input examples for the TRAVOLTA software package.Nature of problem: The TRAVOLTA software package utilizes GPU accelerated routines and new algorithmic improvements to compute optimized electric fields that can drive a system from a known initial vibrational eigenstate to a specified final quantum state with a large (≈1) transition probability.Solution method: Quantum control, GPU acceleration, analytic gradients, Crank-Nicolson propagation, and gradient ascent optimization.

Machine Learning based Noise Characterization and Correction on Neutral Atoms NISQ Devices

AbstractNeutral atoms devices represent a promising technology using optical tweezers to geometrically arrange atoms and modulated laser pulses to control their quantum states. They are exploited as noisy intermediate‐scale quantum (NISQ) processors. Indeed, like all real quantum devices, they are affected by noise introducing errors in the computation. Therefore, it is important to understand and characterize the noise sources and possibly to correct them. Here, two machine‐learning based approaches are proposed respectively to estimate the noise parameters and to mitigate their effects using only measurements of the final quantum state. Our analysis is then tested on a real neutral atom platform, comparing our predictions with a priori estimated parameters. It turns out that increasing the number of atoms is less effective than using more measurements on a smaller scale. The agreement is not always good but this may be due to the limited amount of real data that are obtained from a still under development device. Finally, reinforcement learning is employed to design a pulse that mitigates the noise effects. Our machine learning‐based approach is espected to be very useful for the noise benchmarking of NISQ processors and, more in general, of real quantum technologies.

High-dimensional multi-fidelity Bayesian optimization for quantum control

We present the first multi-fidelity Bayesian optimization (BO) approach for solving inverse problems in the quantum control of prototypical quantum systems. Our approach automatically constructs time-dependent control fields that enable transitions between initial and desired final quantum states. Most importantly, our BO approach gives impressive performance in constructing time-dependent control fields, even for cases that are difficult to converge with existing gradient-based approaches. We provide detailed descriptions of our machine learning methods as well as performance metrics for a variety of machine learning algorithms. Taken together, our results demonstrate that BO is a promising approach to efficiently and autonomously design control fields in general quantum dynamical systems.

High-Fidelity Multiplexed Single-Shot Readout for Transmon Qubits With High-Power Measurement

Single-shot readout of a quantum state is an essential component in building a large-scale quantum processor, andreduc-ing the readout error is the key challenge. To achieve a high signal-to-noiseratio,quantum-limitedlow-noiseamplifierssuchasJoseph-son parametric amplifiers are typically used. However, a high-fidel-ity single-shot readout method without a parametric amplifier would be advantageous, as it would reduce system complexity. In this paper, we present a high-fidelity result of frequency-multi-plexed, high-power single-shot readout for superconducting qubits without using a parametric amplifier. We achieved a high readout fidelity of 98.3 % with a 400 ns pulse and confirmed that the high-power readout can be usedintomography of anentangledstate. We havedemonstratedthepotentialusefulnessofthistechniqueinchar-acterizing the final quantum state at the end of quantum opera-tions in a superconducting quantum processor.

A Quantum States Preparation Method Based on Difference-Driven Reinforcement Learning

Due to the large state space of the two-qubit system, and the adoption of ladder reward function in the existing quantum state preparation methods, the convergence speed is slow and it is difficult to prepare the desired target quantum state with high fidelity under limited conditions. To solve the above problems, a difference-driven reinforcement learning (RL) algorithm for quantum state preparation of two-qubit system is proposed by improving the reward function and action selection strategy. Firstly, a model is constructed for the problem of preparing quantum states of a two-qubit system, with restrictions on the type of quantum gates and the time for quantum state evolution. In the preparation process, a weighted differential dynamic reward function is designed to assist the algorithm quickly obtain the maximum expected cumulative reward. Then, an adaptive [Formula: see text]-greedy action selection strategy is adopted to achieve a balance between exploration and utilization to a certain extent, thereby improving the fidelity of the final quantum state. The simulation results show that the proposed algorithm can prepare quantum state with high fidelity under limited conditions. Compared with other algorithms, it has different degrees of improvement in convergence speed and fidelity of the final quantum state.

Analog Quantum Variational Embedding Classifier

Quantum machine learning has the potential to provide powerful algorithms for artificial intelligence. The pursuit of quantum advantage in quantum machine learning is an active area of research. For current noisy, intermediate-scale quantum (NISQ) computers, various quantum-classical hybrid algorithms have been proposed. One such previously proposed hybrid algorithm is a gate-based variational embedding classifier, which is composed of a classical neural network and a parameterized gate-based quantum circuit. We propose a quantum variational embedding classifier based on an analog quantum computer, where control signals vary continuously in time. In our algorithm, the classical data is transformed into the parameters of the time-varying Hamiltonian of the analog quantum computer by a linear transformation. The nonlinearity needed for a nonlinear classification problem is purely provided by the analog quantum computer through the nonlinear dependence of the final quantum state on the control parameters of the Hamiltonian. We performed numerical simulations that demonstrate the effectiveness of our algorithm for performing binary and multi-class classification on linearly inseparable datasets such as concentric circles and MNIST digits. Our classifier can reach accuracy comparable with the best classical classifiers. We find the performance of our classifier can be increased by increasing the number of qubits until the performance saturates and fluctuates. Moreover, the number of optimization parameters of our classifier scales linearly with the number of qubits. The increase of number of training parameters when the size increases is therefore not as fast as that of neural network. Our algorithm presents the possibility of using current quantum annealers for solving practical machine-learning problems, and it could also be useful to explore quantum advantage in quantum machine learning.

Quantum Optimal Control for Pauli Operators Based on Spin-1/2 System

Quantum control is an important field for quantum computing and quantum simulation. The key of quantum control is to realize quantum logic operators with high fidelity. In this work, based on spin-1/2 system, the optimal simulation of three Pauli logic operators are carried out by using quantum optimal control theory. Under Pauli z spin-presentation, the results show that under the given quantum initial state, the Pauli operators achieve the expected target quantum state with fidelity (0.9999). When the control pulse is applied on x-axis, the number of iterations required to optimize Pauli x operator to achieve the target state is the least, and the number of iterations required to optimize Pauli z operator is the most. In addition, the comparison shows that when the fidelity reach 0.9999, the population of the quantum final state can reach the ideal theoretical expectation. Besides, the optimization fidelity of Hadamard gate can also reach 0.9999 based on spin-1/2 system. Finally, the study on the phase evolution of quantum states shows that the phase difference between the initial and final quantum states of optimized Pauli x and Pauli z logic operator are π/2 respectively, and there is no phase difference between the final quantum state and the initial quantum state after the evolution of optimized Pauli y operator.

Quantum brachistochrone.

Quantum brachistochrone (QB) is a quantum analogue of classical brachistochrone (shortest path). It is a solution to the following problem: How can we perform a desired quantum operation (or obtain a desired final quantum state) most quickly, by a time-dependent Hamiltonian subject to given constraints? Finding QB is a fundamental problem in quantum mechanics in its own right. Moreover, it will be useful in the study of quantum information and quantum engineering, such as quantum speed limits and implementations of quantum computers. A general framework for finding QBs, called QB formalism, has been developed. It is based on Pontryagin's maximum principle. We review the basics of the QB formalism, give simple examples, and briefly discuss some related studies. This article is part of the theme issue 'Shortcuts to adiabaticity: theoretical, experimental and interdisciplinary perspectives'.

Quantum nondemolition measurements of photon number in monolithic microcavities

We revisit the idea of quantum nondemolition measurement (QND) of optical quanta via a resonantly enhanced Kerr nonlinearity taking into account quantum backaction. We show that the monolithic microcavities enable QND measurement of the number of quanta in a weak signal field using a classical probe field spatially overlapping with the signal. The phase of the probe field acquires information about the signal number of quanta without altering it due to the cross-phase modulation effect. We find the exact solution to the Heisenberg equations of motion of this system and calculate the measurement error, accounting for the optical losses in the measurement path. We identify a realistic approximation to obtain the explicit form of the final conditional quantum state of the signal field, accounting for the undesirable self-phase modulation effect and designing the optimal homodyne measurement of the probe beam to evade this effect. We show that the best modern monolithic microcavities allow achieving the measurement imprecision several times better than the standard quantum limit.

Compact quantum kernel-based binary classifier

Quantum computing opens exciting opportunities for kernel-based machine learning methods, which have broad applications in data analysis. Recent works show that quantum computers can efficiently construct a model of a classifier by engineering the quantum interference effect to carry out the kernel evaluation in parallel. For practical applications of these quantum machine learning methods, an important issue is to minimize the size of quantum circuits. We present the simplest quantum circuit for constructing a kernel-based binary classifier. This is achieved by generalizing the interference circuit to encode data labels in the relative phases of the quantum state and by introducing compact amplitude encoding, which encodes two training data vectors into one quantum register. When compared to the simplest known quantum binary classifier, the number of qubits is reduced by two and the number of steps is reduced linearly with respect to the number of training data. The two-qubit measurement with post-selection required in the previous method is simplified to single-qubit measurement. Furthermore, the final quantum state has a smaller amount of entanglement than that of the previous method, which advocates the cost-effectiveness of our method. Our design also provides a straightforward way to handle an imbalanced data set, which is often encountered in many machine learning problems.

Relativistic quantum field theory of stochastic dynamics in the Hilbert space

We develop an action formulation of stochastic dynamics in the Hilbert space. By generalizing the Wiener process into 1+3-dimensional spacetime, we define a Lorentz-invariant random field. By coupling the random to quantum fields, we obtain a random-number action which has the statistical spacetime translation and Lorentz symmetries. The canonical quantization of the theory results in a Lorentz-invariant equation of motion for the state vector or density matrix. We derive the path integral formula of $S$-matrix and the diagrammatic rules for both the stochastic free field theory and stochastic $\phi^4$-theory. The Lorentz invariance of the random $S$-matrix is strictly proved. We then develop a diagrammatic technique for calculating the density matrix. Without interaction, we obtain the exact $S$-matrix and density matrix. With interaction, we prove a simple relation between the density matrices of stochastic and conventional $\phi^4$-theory. Our formalism leads to an ultraviolet divergence which has the similar origin as that in QFT. The divergence is canceled by renormalizing the coupling strength to random field. We prove that the stochastic QFT is renormalizable even in the presence of interaction. In the models with a linear coupling between random and quantum fields, the random field excites particles out of the vacuum, driving the universe towards an infinite-temperature state. The number of excited particles follows the Poisson distribution. The collision between particles is not affected by the random field. But the signals of colliding particles are gradually covered by the background excitations caused by random field.

Reverse quantum speed limit and minimum Hilbert space norm

The reverse quantum speed limit (RQSL) gives an upper limit to the time for evolution between initial and final quantum states. We show that, in conjunction with the existence of a minimum time scale, the RQSL implies a lower limit to the norm of the change in a quantum state, and confirm that this limit is satisfied in two-state and ideal-measurement models. Such a lower limit is of relevance for interpretational issues in probability and for understanding the meaning of probability in Everett quantum theory.

Applying the quantum approximate optimization algorithm to the minimum vertex cover problem

The minimum vertex cover problem belongs to a NP- complete problem, which is difficult to obtain the near-optimal solution in the polynomial time range using classical algorithms. In this paper, a quantum circuit solution scheme based on the quantum approximate optimization algorithm is presented for the minimum vertex cover problem. Firstly, the quantum Ising model and Hamiltonian of the problem are obtained based on the Ising model corresponding to the problem, which is quantized by the rotation operator and Pauli operator. Secondly, the parametric unitary transformation with the initial Hamiltonian and the problem Hamiltonian as the generator is obtained respectively. Through the alternating evolution of two parametric unitary transformations, the final quantum state and the problem Hamiltonian expectation are derived. In the process of evolution, the parameters in the parametric unitary transformations which are optimized by the classical processor can adjust the problem Hamiltonian expectation, so as to improve the probability of the problem solution. Then, the initial state of the algorithm and the quantum logic gate corresponding to the parametric unitary transformation are derived to generate the quantum circuit which can be implemented on the quantum computer. Simulation results show that the scheme can obtain the problem solution with high probability in polynomial time, realizes exponential acceleration, and has certain feasibility, effectiveness and innovation.

Quantum Search Algorithm for Weighted Solutions

This paper presents a quantum search algorithm (QSA) for weighted solutions. In a QSA, the final quantum state has an equal probability of solutions. However, solutions should have different probabilities in specific problems, such as dynamic spectrum management (DSM). We propose a method to make differences with probability in accordance with rewards for each solution, and present an application to DSM. The proposed algorithm has some error terms for the final quantum state, and we provide the error terms in accordance with the number of iterations. In addition, we present the simulation results of the proposed algorithm compared with a classical exact algorithm and QSA. The proposed algorithm has a lower complexity and relative difference than the classical exact algorithm. Furthermore, the proposed algorithm has a higher performance than the QSA in specific cases.

Strong-field control by reverse engineering

Based on the idea of reverse engineering, we design an optimal laser pulse to control strong-field multiphoton atomic transitions. Starting from the time-dependent Schr\"odinger equation of the full system, we adiabatically eliminate the nonessential states and apply the rotating-wave approximation to arrive at an effective two-state representation that involves dynamic Stark shifts and multiphoton coupling. Solving this equation inversely for the field, we obtain an analytical laser pulse shape that is expected to induce the full system's evolution according to user-defined quantum pathways. In our procedure, the amplitude and phase of the laser pulse are engineered such that the dynamically shifted electronic states are resonantly coupled during the action of the pulse at each moment of time. As a result, the driven system evolves from an arbitrary initial population distribution to any desired final quantum state superposition at a predefined rate. The proposed scheme is demonstrated using the example of the $3s\ensuremath{\rightarrow}4s$ two-photon transition of atomic sodium. By solving the time-dependent Schr\"odinger equation of the single-active electron with two different methods, either propagating time-dependent coefficients of many field-free states or directly propagating the three-dimensional electronic wave packet on a grid, we demonstrate the robustness as well as the limitations of the presented reverse engineering scheme.

Quantifying the hybrid entanglement of the Stern-Gerlach experiment using discrete reductions

Hybrid entanglement occurs between discrete and continuous variables, resulting in a considerable difficulty for its quantification. However, some advancements have been made, one of them is the discretization of the continuous variable as a tool to quantify it. Here, in this paper, the applicability of discrete entanglement measures to the final quantum state of the Stern-Gerlach Experiment is discussed in order to quantify its hybrid entanglement. Additionally, the possibility of using an equivalent discrete state is compared to the employment of the partial trace in order to consider the discrete component only, we show that the entanglement entropy distinctively represents the discrete part. Hence, the known entanglement capabilities of the SGE allow us to support the use of both techniques. Therefore, these measures give a detailed description of the entanglement creation in this famous experiment.

NIC-CAGE: An open-source software package for predicting optimal control fields in photo-excited chemical systems

We present an open-source software package, NIC-CAGE (Novel Implementation of Constrained Calculations for Automated Generation of Excitations), for predicting quantum optimal control fields in photo-excited chemical systems. Our approach utilizes newly derived analytic gradients for maximizing the transition probability (based on a norm-conserving Crank–Nicolson propagation scheme) for driving a system from a known initial quantum state to another desired state. The NIC-CAGE code is written in the MATLAB and Python programming environments to aid in its readability and general accessibility to both users and practitioners. Throughout this work, we provide several examples and outputs on a variety of different potentials, propagation times, and user-defined parameters to demonstrate the robustness of the NIC-CAGE software package. As such, the use of this predictive tool by both experimentalists and theorists could lead to further advances in both understanding and controlling the dynamics of photo-excited systems. Program summaryProgram Title: NIC-CAGECPC Library link to program files:http://dx.doi.org/10.17632/82jcpk5svt.1Licensing provisions: GNU General Public License 3Programming language: MATLAB or PythonSupplementary material: Comparisons of propagated wavefunctions obtained from analytical π pulses vs wavefunctions resulting from numerically optimized electric fields predicted by the NIC-CAGE programNature of problem: The NIC-CAGE software package utilizes analytic Crank–Nicolson gradients to compute optimized (and constrained) electric fields that can drive a system from a known initial vibrational eigenstate to a specified final quantum state with a large (≈1) transition probability.Solution method: Analytic gradients, Crank–Nicolson propagation, and gradient ascent optimization

Quantum State and Doppler-Resolved Scattering of Thermal/Hyperthermal DCl at the Gas–Liquid Interface: Support for a Simple “Lever Arm” Model of the Energy-Transfer Dynamics

Supersonically cooled deuterium chloride (10 K, DCl(J ≈ 0, 1)) has been scattered from the gas–liquid interface under thermal (Einc = 1.5(1) kcal/mol) and hyperthermal (Einc = 11.5(5) kcal/mol) conditions for a series of prototypical liquids (perfluoropolyether (PFPE), squalane, and glycerol), with the final DCl quantum states detected with narrow-band IR laser absorption methods to achieve rotational (J) and transverse Doppler (vy) resolutions. First of all, we see direct evidence for both trapping desorption (TD) and impulsive scattering (IS) components in the DCl distributions, with both TD rotational and transverse velocity components fully equilibrated with the liquid (TS ≈ Trot ≈ TDopp). Second, high-resolution laser dopplermetry on the IS component reveals quite efficient transfer from Einc into out-of-plane scattering of the DCl, in contrast with the notably inefficient channeling of the collision energy into end-over-end rotation. Third, though the rotational excitation efficiency is low, the imp...