Quantum Physics (quant-ph)

Abstract: In addition to secret splitting, secret reconstruction is another important component of secret sharing. In this paper, the first quantum secret reconstruction protocol based on cluster states is proposed. Before the protocol, a classical secret is divided into multiple shares, which are distributed among shareholders via secret splitting. In the protocol, the dealer utilizes her secret to encrypt a private quantum state, and sends the encrypted state to a combiner chosen by her from the shareholders. With the help of other shareholders, the combiner utilizes the properties of cluster states to recover the privacy quantum state. It is shown that the proposed protocol is secure against several common attacks, including external and internal attacks. Compared with classical secret reconstruction protocols, this protocol not only achieves theoretical security of all shares, but also is more efficient due to reducing the distribution cost and computation cost. To demonstrate the feasibility of the protocol, a corresponding simulation quantum experiment is conducted on the IBM Q platform. Furthermore, in conjunction with quantum fingerprinting, it can be directly applied to achieve the task of multiple secrets sharing, because the classical shares can be reused in the proposed protocol.

Abstract: In recent years, researchers have been exploring the applications of noisy intermediate-scale quantum (NISQ) computation in various fields. One important area in which quantum computation can outperform classical computers is the ground state problem of a many-body system, e.g., the nucleus. However, using a quantum computer in the NISQ era to solve a meaningful-scale system remains a challenge. To calculate the ground energy of nuclear systems, we propose a new algorithm that combines classical shadow and subspace diagonalization techniques. Our subspace is composed of matrices, with the basis of the subspace being the classical shadow of the quantum state. We test our algorithm on nuclei described by Cohen-Kurath shell model and USD shell model. We find that the accuracy of the results improves as the number of shots increases, following the Heisenberg scaling.

Abstract: Multidimensional spectroscopy unveils the interplay of nuclear and electronic dynamics, which characterizes the ultrafast dynamics of various molecular and solid-state systems. In a widely used class of models used for the simulation of such dynamics, field-induced transitions between electronic states result in linear transformations (Duschinsky rotations) between the normal coordinates of the vibrational modes. Here we present an approach for the calculation of the response functions, based on the explicit derivation of the vibrational state. This can be shown to coincide with a multimode squeezed coherent state, whose expression we derive within a quantum-optical formalism, and specifically by the sequential application to the initial state of rotation, displacement and squeeze operators. This approach potentially simplifies the numerical derivation of the response function, avoiding the time integration of the Schr\"odinger equation or the Hamiltonian diagonalization, combined with the sum over infinite vibronic pathways. Besides, it quantitatively substantiates in the considered models the intuitive interpretation of the response function in terms of the vibrational wave packet dynamics.

Abstract: Strong light-matter coupling provides a versatile and novel means to manipulate chemical processes. Here we develop a theoretical framework to investigate the spectroscopy and dynamics of a molecular ensemble embedded in an optical cavity under the collective strong coupling regime. This theory is constructed by a pseudoparticle representation of the molecular Hamiltonians, mapping the polaritonic Hamiltonian into a coupled fermion-boson model under particle number constraints. The mapped model is then analyzed using the non-equilibrium Green function theory with the important self-energy diagrams identified through power counting. Numerical demonstrations are shown for the driven Tavis-Cummings model, which shows an excellent agreement with exact results.

Abstract: We establish a general framework for developing approximation algorithms for a class of counting problems. Our framework is based on the cluster expansion of abstract polymer models formalism of Koteck\'y and Preiss. We apply our framework to obtain efficient algorithms for (1) approximating probability amplitudes of a class of quantum circuits close to the identity, (2) approximating expectation values of a class of quantum circuits with operators close to the identity, (3) approximating partition functions of a class of quantum spin systems at high temperature, and (4) approximating thermal expectation values of a class of quantum spin systems at high temperature with positive-semidefinite operators. Further, we obtain hardness of approximation results for approximating probability amplitudes of quantum circuits and partition functions of quantum spin systems. This establishes a computational complexity transition for these problems and shows that our algorithmic conditions are optimal under complexity-theoretic assumptions. Finally, we show that our algorithmic condition is almost optimal for expectation values and optimal for thermal expectation values in the sense of zero freeness.

Abstract: Entanglement entropy is one of the most prominent measures in quantum physics. We show that it has an interesting ergotropic interpretation in terms of unitarily extracted work. It determines how much energy one can extract from a source of pure unknown states by applying unitary operations when only local measurements can be performed to characterize this source. Additionally, entanglement entropy sets a limit on the minimal temperature to which these partially characterized states can be cooled down, by using only unitary operations.

Abstract: We study a $5$-state Landau-Zener model which cannot be solved by integrability methods. By analyzing analytical constraints on its scattering matrix combined with fitting to results from numerical simulations of the Schr\"{o}dinger equation, we find nearly exact analytical expressions of all its transition probabilities. We further apply this model to study a $5$-site Su-Schrieffer-Heeger chain with couplings changing linearly in time. Our work points out a new possibility to solve multistate Landau-Zener models not necessarily integrable and with insufficient numbers of constraints on their scattering matrices.

Abstract: Quantum many-body systems are commonly considered as quantum chaotic if their spectral statistics, such as the level spacing distribution, agree with those of random matrix theory. Using the example of the kicked Ising chain we demonstrate that even if both level spacing distribution and eigenvector statistics agree well with random matrix predictions, the entanglement entropy deviates from the expected Page curve. To explain this observation we propose a new measure of the effective spin interactions and obtain the corresponding random matrix result. By this the deviations of the entanglement entropy can be attributed to significantly different behavior of the $k$-spin interactions compared to RMT.

Abstract: We introduce a numerical method to simulate nonlinear open quantum dynamics of a particle in situations where its state undergoes significant expansion in phase space while generating small quantum features at the phase-space Planck scale. Our approach involves simulating the Wigner function in a time-dependent frame that leverages information from the classical trajectory to efficiently represent the quantum state in phase space. To demonstrate the capabilities of our method, we examine the open quantum dynamics of a particle evolving in a one-dimensional weak quartic potential after initially being ground-state cooled in a tight harmonic potential. This numerical approach is particularly relevant to ongoing efforts to design, optimize, and understand experiments targeting the preparation of macroscopic quantum superposition states of massive particles through nonlinear quantum dynamics.

Abstract: Much recent research has focused on establishing the thermodynamic cost of quantum operations acting on single input states. However, information processing, quantum or classical, relies on channels transforming multiple input states to different corresponding outputs. In Ref. [1] the existence of a bound on the work extraction for multiple inputs was proven. However, no specifics were provided for how optimal multipurpose operations may be constructed, and no upper limit on the dissipated work was given. For the insightful case of qubits, we here give explicit protocols to implement work extraction on multiple states. We first prove conditions on the feasibility of carrying out such transformations at all. Furthermore, we quantify the achievable work extraction, and find that there is a dramatic penalty for multipurpose operations. Our results will be relevant for the growing field of quantum technologies in the thermodynamic assessment of all quantum information processing tasks.

Abstract: These are pedagogical notes on Shor's factoring algorithm, which is a quantum algorithm for factoring very large numbers (of order of hundreds to thousands of bits) in polynomial time. In contrast, all known classical algorithms for the factoring problem take an exponential time to factor large numbers. In these notes, we assume no prior knowledge of Shor's algorithm beyond a basic familiarity with the circuit model of quantum computing. The literature is thick with derivations and expositions of Shor's algorithm, but most of them seem to be lacking in essential details, and none of them provide a pedagogical presentation. We develop the theory of modular exponentiation (ME) operators in some detail, one of the fundamental components of Shor's algorithm, and the place where most of the quantum resources are deployed. We also discuss the post-quantum processing and the method of continued fractions, which is used to extract the exact period of the modular exponential function from the approximately measured phase angles of the ME operator. The manuscript then moves on to a series of examples. We first verify the formalism by factoring N=15, the smallest number accessible to Shor's algorithm. We then proceed to factor larger numbers, developing a systematic procedure that will find the ME operators for any semi-prime $N = p \times q$ (where $q$ and~$p$ are prime). Finally, we factor the numbers N=21, 33, 35, 143, 247 using the Qiskit simulator. It is observed that the ME operators are somewhat forgiving, and truncated approximate forms are able to extract factors just as well as the exact operators. This is because the method of continued fractions only requires an approximate phase value for its input, which suggests that implementing Shor's algorithm might not be as difficult as first suspected.

Abstract: Dynamical decoupling has been shown to be effective in reducing gate errors in most quantum computation platforms and is therefore projected to play an essential role in future fault-tolerant constructions. In superconducting circuits, however, it has proven difficult to utilize the benefits of dynamical decoupling. In this work, we present a theoretical proposal that incorporates a continuous version of dynamical decoupling, namely spin locking, with a coupler-based CZ gate for transmons and provide analytical and numerical results that demonstrate its effectiveness.

Abstract: The Quantum Approximate Optimization Algorithm (QAOA) is a highly promising variational quantum algorithm that aims to solve combinatorial optimization problems that are classically intractable. This comprehensive review offers an overview of the current state of QAOA, encompassing its performance analysis in diverse scenarios, its applicability across various problem instances, and considerations of hardware-specific challenges such as error susceptibility and noise resilience. Additionally, we conduct a comparative study of selected QAOA extensions and variants, while exploring future prospects and directions for the algorithm. We aim to provide insights into key questions about the algorithm, such as whether it can outperform classical algorithms and under what circumstances it should be used. Keywords: QAOA, Variational Quantum Algorithms (VQAs), Quantum Optimization, Combinatorial Optimization Problems, NISQ Algorithms

Abstract: We propose quantum tree networks for hierarchical multi-flow entanglement routing. The end nodes on the leaves of the tree communicate through the routers at internal nodes. In a $k$-ary tree network, each node is connected to $k$ nodes in the lower layer, and the channel length connecting two nodes grows with rate $a_k$ as we move from the leaf to the root node. This architecture enables the qubit-per-node overhead for congestion-free and error-corrected operations to be sublinear in the number of end nodes, $N$. The overhead scaling for $k$-ary tree is $O(N^{\log_k a_k} \cdot \log_k N)$. Specifically, the square-lattice distributed end nodes with the quaternary tree routing leads to an overhead $\sim O(\sqrt{N}\cdot\log_4 N)$. For a minimal surface-covering tree, the overhead $\sim O(N^{0.25}\cdot\log_4 N)$ for $k=4$ and is sublinear for all $k$. We performed network simulations of quantum tree networks that exhibits size-independent threshold behavior. The routing in tree network does not require time-consuming multi-path finding algorithms. These properties satisfy the essential requirements for scalable quantum networks.

Abstract: The acceleration theorem for wavepacket propagation in periodic potentials disentangles the kspace dynamics and real-space dynamics. This is well known and understood for Bloch oscillations and super Bloch oscillations in the presence of position-independent forces. Here, we analyze the dynamics of a model system in which the k-space dynamics and the real-space dynamics are inextricably intertwined due to a position-dependent force which is provided by a parabolic trap. We demonstrate that this coupling gives rise to significantly modified and rich dynamics when the lattice is shaken by a modulated parabolic potential. The dynamics range from chirped Bloch-Harmonic oscillations to the asymmetric spreading oscillations. We analyze these findings by tracing the spatio-temporal dynamics in real space and by visualizing the relative phase in the k-space dynamics which leads to an accurate explanation of the obtained phenomena. We also compare our numerical results to a local acceleration model and obtain very good agreement for the case of coherent oscillations, however, deviations for oscillations with spreading dynamics which altogether supports the interpretations of our findings.

Abstract: We introduce systematic protocols to perform stabilizer testing for quantum states and gates. These protocols are based on quantum convolutions and swap-tests, realized by quantum circuits that implement the quantum convolution for both qubit and qudit systems. We also introduce ''magic entropy'' to quantify magic in quantum states and gates, in a way which may be measurable experimentally.

Abstract: The JPEG algorithm compresses a digital image by filtering its high spatial-frequency components. Similarly, we introduce a quantum protocol that uses the quantum Fourier transform to discard the high-frequency qubits of the image. This allows to capture, compress and send images even with limited quantum resources for storage and communication, at the cost of reducing the resolution of the output. We show under which conditions this protocol is advantageous with respect to its classical counterpart.

Abstract: Instantaneous two-party quantum computation is a computation process in which there are initial shared entanglement, and the nonlocal interactions are limited to simultaneous classical communication in both directions. It is almost equivalent to the problem of instantaneous measurements, and is related to some topics in quantum foundations and position-based quantum cryptography. In this work we show an efficient protocol for instantaneous two-party quantum computation (or measurement). Its entanglement cost is proportional to the T-gate count when the quantum circuit is decomposed into Clifford gates and T gates. It makes use of a garden-hose gadget from the study of quantum homomorphic encryption. This protocol makes a class of quantum position verification schemes insecure. Independent from the main result, we show that any unitary circuit consisting of layers of Clifford gates and T gates can be implemented using a circuit with measurements (or a unitary circuit) of depth proportional to the T-depth of the original circuit. This matches a corresponding result in measurement-based quantum computation. This is of limited use since interesting quantum algorithms often require a high ratio of T gates, but still we discuss some extensions and applications of the second result.

Abstract: Understanding universal aspects of quantum dynamics is an unresolved problem in statistical mechanics. In particular, the spin dynamics of the 1D Heisenberg model were conjectured to belong to the Kardar-Parisi-Zhang (KPZ) universality class based on the scaling of the infinite-temperature spin-spin correlation function. In a chain of 46 superconducting qubits, we study the probability distribution, $P(\mathcal{M})$, of the magnetization transferred across the chain's center. The first two moments of $P(\mathcal{M})$ show superdiffusive behavior, a hallmark of KPZ universality. However, the third and fourth moments rule out the KPZ conjecture and allow for evaluating other theories. Our results highlight the importance of studying higher moments in determining dynamic universality classes and provide key insights into universal behavior in quantum systems.

Abstract: From correlations in measurement outcomes alone, can two otherwise isolated parties establish whether such correlations are atemporal? That is, can they rule out that they have been given the same system at two different times? Classical statistics says no, yet quantum theory disagrees. Here, we introduce the necessary and sufficient conditions by which such quantum correlations can be identified as atemporal. We demonstrate the asymmetry of atemporality under time reversal, and reveal it to be a measure of spatial quantum correlation distinct from entanglement. Our results indicate that certain quantum correlations possess an intrinsic arrow of time, and enable classification of general quantum correlations across space-time based on their (in)compatibility with various underlying causal structures.