Quantum Physics (quant-ph)

Abstract: As one of the most elegant theories in physics, Yang-Mills theory not only incorporates Maxwell's equations unifying the classical electromagnetic phenomena, but also underpins the standard model explaining the electroweak and strong interactions in a succinct way. As an Abelian $U(1)$ case, the electromagnetic field is the simplest classical solution of Yang-Mills equation. Notwithstanding, there is a paucity of studies about the simplest quantum situation, namely the consideration of the ``magnetic'' and ``electric'' fields in Maxwell's equations with non-Abelian potentials, which is exactly the staple of our present work. Akin to the electromagnetic waves predicted by Maxwell's equations, the quantum solution of the simplest Yang-Mills equation may predict the SU(2) angular-momentum waves. Such angular-momentum waves can be possibly realized in the experiments with oscillations of the spin angular momentum (such as the ``spin Zitterbewegung'' of Dirac's electron).

Abstract: Quantum protocols commonly require a certain number of quantum resource states to be available simultaneously. An important class of examples is quantum network protocols that require a certain number of entangled pairs. Here, we consider a setting in which a process generates a quantum resource state with some probability $p$ in each time step, and stores it in a quantum memory that is subject to time-dependent noise. To maintain sufficient quality for an application, each resource state is discarded from the memory after $w$ time steps. Let $s$ be the number of desired resource states required by a protocol. We characterise the probability distribution $X_{(w,s)}$ of the ages of the quantum resource states, once $s$ states have been generated in a window $w$. Combined with a time-dependent noise model, the knowledge of this distribution allows for the calculation of fidelity statistics of the $s$ quantum resources. We also give exact solutions for the first and second moments of the waiting time $\tau_{(w,s)}$ until $s$ resources are produced within a window $w$, which provides information about the rate of the protocol. Since it is difficult to obtain general closed-form expressions for statistical quantities describing the expected waiting time $\mathbb{E}(\tau_{(w,s)})$ and the distribution $X_{(w,s)}$, we present two novel results that aid their computation in certain parameter regimes. The methods presented in this work can be used to analyse and optimise the execution of quantum protocols. Specifically, with an example of a Blind Quantum Computing (BQC) protocol, we illustrate how they may be used to infer $w$ and $p$ to optimise the rate of successful protocol execution.

Abstract: Since its discovery, the quantum entanglement becomes a promising resource in quantum communication and computation. However, the entanglement is fragile due to the presence of noise in quantum channels. Entanglement purification is a powerful tool to distill high quality entangled states from the low quality entangled states. In this review, we present an overview of entanglement purification, including the basic entanglement purification theory, the entanglement purification protocols (EPPs) with linear optics, EPPs with cross-Kerr nonlinearities, hyperentanglement EPPs, deterministic EPPs, and measurement-based EPPs. We also review experimental progresses of EPPs in linear optics. Finally, we give the discussion on potential outlook for the future development of EPPs. This review may pave the way for practical implementations in future long-distance quantum communication and quantum network.

Abstract: Suppose two separated parties, Alice and Bob, share a bipartite quantum state or a classical correlation called a seed, and they try to generate a target classical correlation by performing local quantum or classical operations on the seed, i.e., any communications are not allowed. We consider the following fundamental problem about this setting: whether Alice and Bob can use a given seed to generate a target classical correlation. We show that this problem has rich mathematical structures. Firstly, we prove that even if the seed is a pure bipartite state, the above decision problem is already NP-hard and a similar conclusion can also be drawn when the seed is also a classical correlation, implying that this problem is hard to solve generally. Furthermore, we prove that when the seed is a pure quantum state, solving the problem is equivalent to finding out whether the target classical correlation has some canonical form of positive semi-definite factorizations that matches the seed pure state, revealing an interesting connection between the current problem and optimization theory. Based on this observation and other insights, we give several necessary conditions where the seed pure state has to satisfy to generate the target classical correlation, and it turns out that these conditions can also be generalized to the case that the seed is a mixed quantum state. Lastly, since canonical forms of positive semi-definite factorizations play a crucial role in solving the problem, we develop an algorithm that can compute them for an arbitrary classical correlation, which has decent performance on the cases we test.

Abstract: With the emergence of quantum computing, a growing number of quantum devices is accessible via cloud offerings. However, due to the rapid development of the field, these quantum-specific service offerings vary significantly in capabilities and requirements they impose on software developers. This is particularly challenging for practitioners from outside the quantum computing domain who are interested in using these offerings as parts of their applications. In this paper, we compare several devices based on different hardware technologies and provided through different offerings, by conducting the same experiment on each of them. By documenting the lessons learned from our experiments, we aim to simplify the usage of quantum-specific offerings and illustrate the differences between predominant quantum hardware technologies.

Abstract: Spectroscopy of gases under high-pressure conditions is of interest in various fields such as plasma physics or astrophysics. Recently, it has also been proposed to utilize a high-pressure noble gas environment as a thermalization medium to extend the wavelength range of photon Bose-Einstein condensates to the vacuum-ultraviolet, from the presently accessible visible and near-infrared spectral regimes. In this work, we report on experimental results of two-photon spectroscopy of gaseous and supercritical xenon for pressures as high as $95 \; \text{bar}$, probing the transitions from the $5p^6$ electronic ground state to the $5p^56p$ and $5p^56p^\prime$ excited state configurations. Aiming at the exploration of possible pumping schemes for future vacuum-ultraviolet photon condensates, we have recorded degenerate two-photon excitation spectra of such dense xenon samples. In further measurements, we have investigated whether irradiation of an auxiliary light field can enhance the reabsorption of the emission on the second excimer continuum of xenon, which is subject to a large Stokes shift. To this end, absorption measurements have been conducted, driving the $5p^6 \rightarrow 5p^56p$ two-photon transitions non-degenerately.

Abstract: Multipartite high-dimensional entanglement presents different physics from multipartite two-dimensional entanglement. However, how to prepare multipartite high-dimensional entanglement is still a challenge with linear optics. In this paper, a multiphoton GHZ state with arbitrary dimensions preparation protocol is proposed in optical systems. In this protocol, we use auxiliary entanglements to realize a high-dimensional entanglement gate, so that high-dimensional entangled pairs can be connected into a multipartite high-dimensional GHZ state. Specifically, we give an example of using photons' path degree of freedom to prepare a 4-particle 3-dimensional GHZ state. Our method can be extended to other degrees of freedom and can generate arbitrary GHZ entanglement in any dimension.

Abstract: Quantum fingerprinting is a technique that maps classical input word to a quantum state. The resulting quantum state is much shorter than original word, and its processing requires less resources, making it useful in quantum algorithms, communication and cryptography. One of the examples of quantum fingerprinting is quantum automaton for $MOD_{p}=\{a^{i\cdot p} \mid i \geq 0\}$ language, where $p$ is a prime number. However, implementing this automata in current quantum hardware is not efficient. Quantum fingeprinting maps a word $x \in \{0,1\}^{n}$ of length $n$ to a state $|\psi(x)\rangle$ of $O(\log \log n)$ qubits, and requires $O(\log n)$ unitary operations. Computing quantum fingerprint using all memory of the current quantum computers is currently infeasible due to the large number of quantum operations necessary. In order to make quantum fingerprinting practical, we must optimize the circuit for depth instead of width as previous works did. We propose explicit methods of quantum fingerprinting based on tools from additive combinatorics, such as generalized arithmetic progressions (GAPs), and prove that these methods provide circuit depth comparable to probabilistic method. We also compare our method to prior work on explicit quantum fingerprinting methods.

Abstract: This paper explores the phenomenon of avoided level crossings in quantum annealing, a promising framework for quantum computing that may provide a quantum advantage for certain tasks. Quantum annealing involves letting a quantum system evolve according to the Schr\"odinger equation, with the goal of obtaining the optimal solution to an optimization problem through measurements of the final state. However, the continuous nature of quantum annealing makes analytical analysis challenging, particularly with regard to the instantaneous eigenenergies. The adiabatic theorem provides a theoretical result for the annealing time required to obtain the optimal solution with high probability, which is inversely proportional to the square of the minimum spectral gap. Avoided level crossings can create exponentially closing gaps, which can lead to exponentially long running times for optimization problems. In this paper, we use a perturbative expansion to derive a condition for the occurrence of an avoided level crossing during the annealing process. We then apply this condition to the MaxCut problem on bipartite graphs. We show that no exponentially small gaps arise for regular bipartite graphs, implying that QA can efficiently solve MaxCut in that case. On the other hand, we show that irregularities in the vertex degrees can lead to the satisfaction of the avoided level crossing occurrence condition. We provide numerical evidence to support this theoretical development, and discuss the relation between the presence of exponentially closing gaps and the failure of quantum annealing.

Abstract: Bell's inequality plays an important role with respect to the Einsteinian question about the physical reality of quantum theory. While Bell's inequality is usually viewed within the geometric framework of a Hilbert space quantum model, the present note extends the theory of Heisenberg measurements to quantum systems with representations in general orthogonal geometric spaces and, in particular, the Minkowski spaces of relativity theory. A Feynmanian numerical example exhibits two measurements that admit a joint probabilistic interpretation in Minkowski space while they are not jointly observable in Hilbert space. The analysis shows that probabilistic interpretations of quantum measurements may depend not only on the measuring instruments and the system states but also on the geometric space in which the measurements are conducted. In particular, an explicit numerical example is given of a Heisenberg measurement with a complete set of common observables that violates Bell's inequality in Minkowski space but, mutatatis mutandis, satisfies it in Hilbert space.

Abstract: We present tools and methods to generalize parity compilation to digital quantum computing devices with arbitrary connectivity graphs and construct circuit implementations for the constraint Hamiltonian of higher-order constrained binary optimization problems. In particular, we show how even non-local constraints can be efficiently implemented without expensive SWAP gates. We show how the presented tools can be used to optimize the total circuit depth and CNOT count of the quantum approximate optimization algorithm in the parity architecture and highlight the advantages of the flexible compilation using various examples. We derive the relation between the developed gate sequences and the traditional approach that uses SWAP gates. The result can be applied to improve the implementation of many other non-local operators.

Abstract: Optical mirrors determine cavity properties by means of light reflection. Imperfect reflection gives rise to open cavities with photon loss. We study an open cavity made of atom-dimer mirrors with a tunable reflection spectrum. We find that the atomic cavity shows anti-$\mathcal{PT}$ symmetry. The anti-$\mathcal{PT}$ phase transition controlled by atomic couplings in mirrors indicates the emergence of two degenerate cavity supermodes. Interestingly, a threshold of mirror reflection is identified for realizing strong coherent cavity-atom coupling. This reflection threshold reveals the criterion of atomic mirrors to produce a good cavity. Moreover, cavity quantum electrodynamics with a probe atom shows mirror-tuned properties, including reflection-dependent polaritons formed by the cavity and probe atom. Our work presents a non-Hermitian theory of an anti-$\mathcal{PT}$ atomic cavity, which may have applications in quantum optics and quantum computation.

Abstract: The eigenvalue of a non-Hermitian Hamiltonian often forms a self-intersecting Riemann surface, leading to a unique mode conversion phenomenon when the Hamiltonian evolves along certain loop paths around an exceptional point (EP). However, two fundamental problems exist with the conventional scheme of EP encircling: the speed of mode conversion is restricted by the adiabatic requirement, and the chirality cannot be freely controlled. We introduce a method for dynamically engineering adiabaticity in the evolution of non-Hermitian Hamiltonians that allows for both chiral and non-chiral mode conversion on the same path. Our method is based on quantifying and controlling the instantaneous adiabaticity, allowing for non-uniform evolution throughout the entire path. By optimizing the evolution based on the distributed nature of adiabaticity, we achieve the same quality as conventional quasi-adiabatic evolution in only one-third of the time. Our approach provides a comprehensive and universal solution to address the speed and chirality challenges associated with EP encircling. It also facilitates the dynamic manipulation and regulation of non-adiabatic processes, thereby accelerating the operation and allowing for a selection among various mode conversion patterns.

Abstract: We explore potential quantum speedups for the fundamental problem of testing properties of distributions. In particular, we focus on two different problems: the first one is to test whether two unknown classical distributions are close or far enough, and the second one is to test whether a given distribution over $\{0, 1\}^n$ is $k$-wise uniform or far from any $k$-wise uniform distribution. For the first problem, we propose the currently best quantum algorithm under the metrics of $l^1$-distance and $l^2$-distance. Compared with the latest result given in \cite{gilyen2019distributional} which relied on the technique of quantum singular value transformation (QSVT), our algorithm is not only more concise, but also more efficient. For the latter problem, we propose the first quantum algorithm achieving a quadratic speedup over the state-of-the-art classical algorithm. It is worthy noting that the analysis of our quantum algorithm is much more intuitive and concise than that of the classical one.

Abstract: Gaussian process regression is a well-established Bayesian machine learning method. We propose a new approach to Gaussian process regression using quantum kernels based on parameterized quantum circuits. By employing a hardware-efficient feature map and careful regularization of the Gram matrix, we demonstrate that the variance information of the resulting quantum Gaussian process can be preserved. We also show that quantum Gaussian processes can be used as a surrogate model for Bayesian optimization, a task that critically relies on the variance of the surrogate model. To demonstrate the performance of this quantum Bayesian optimization algorithm, we apply it to the hyperparameter optimization of a machine learning model which performs regression on a real-world dataset. We benchmark the quantum Bayesian optimization against its classical counterpart and show that quantum version can match its performance.

Abstract: Temporal-to-spatial demultiplexing routes non-simultaneous events of the same spatial mode to distinct output trajectories. This technique has now been widely adopted because it gives access to higher-number multi-photon states when exploiting solid-state quantum emitters. However, implementations so far have required an always-increasing number of active elements, rapidly facing resource constraints. Here, we propose and demonstrate a demultiplexing approach that utilizes only a single active element for routing to, in principle, an arbitrary number of outputs. We employ our device in combination with a high-efficiency quantum dot based single-photon source, and measure up to eight demultiplexed highly indistinguishable single photons. We discuss the practical limitations of our approach, and describe in which conditions it can be used to demultiplex, e.g., tens of outputs. Our results thus provides a path for the preparation of resource-efficient larger-scale multi-photon sources.

Abstract: Repeated projective measurements in unitary circuits can lead to an entanglement phase transition as the measurement rate is tuned. In this work, we consider a different setting in which the projective measurements are replaced by dynamically chosen unitary gates that minimize the entanglement. This can be seen as a one-dimensional unitary circuit game in which two players get to place unitary gates on randomly assigned bonds at different rates: The "entangler" applies a random local unitary gate with the aim of generating extensive (volume law) entanglement. The "disentangler", based on limited knowledge about the state, chooses a unitary gate to reduce the entanglement entropy on the assigned bond with the goal of limiting to only finite (area law) entanglement. In order to elucidate the resulting entanglement dynamics, we consider three different scenarios: (i) a classical discrete height model, (ii) a Clifford circuit, and (iii) a general $U(4)$ unitary circuit. We find that both the classical and Clifford circuit models exhibit phase transitions as a function of the rate that the disentangler places a gate, which have similar properties that can be understood through a connection to the stochastic Fredkin chain. In contrast, the ``entangler'' always wins when using Haar random unitary gates and we observe extensive, volume law entanglement for all non-zero rates of entangling.

Abstract: A set of classical or quantum states is equivalent to another one if there exists a pair of classical or quantum channels mapping either set to the other one. For dichotomies (pairs of states) this is closely connected to (classical or quantum) R\'enyi divergences (RD) and the data-processing inequality: If a RD remains unchanged when a channel is applied to the dichotomy, then there is a recovery channel mapping the image back to the initial dichotomy. Here, we prove for classical dichotomies that equality of the RDs alone is already sufficient for the existence of a channel in any of the two directions and discuss some applications. We conjecture that equality of the minimal quantum RDs is sufficient in the quantum case and prove it for special cases. We also show that neither the Petz quantum nor the maximal quantum RDs are sufficient. As a side-result of our techniques we obtain an infinite list of inequalities fulfilled by the classical, the Petz quantum, and the maximal quantum RDs. These inequalities are not true for the minimal quantum RDs.