Quantum Physics (quant-ph)

Abstract: It is well known that artificial neural networks initialized from independent and identically distributed priors converge to Gaussian processes in the limit of large number of neurons per hidden layer. In this work we prove an analogous result for Quantum Neural Networks (QNNs). Namely, we show that the outputs of certain models based on Haar random unitary or orthogonal deep QNNs converge to Gaussian processes in the limit of large Hilbert space dimension $d$. The derivation of this result is more nuanced than in the classical case due the role played by the input states, the measurement observable, and the fact that the entries of unitary matrices are not independent. An important consequence of our analysis is that the ensuing Gaussian processes cannot be used to efficiently predict the outputs of the QNN via Bayesian statistics. Furthermore, our theorems imply that the concentration of measure phenomenon in Haar random QNNs is much worse than previously thought, as we prove that expectation values and gradients concentrate as $\mathcal{O}\left(\frac{1}{e^d \sqrt{d}}\right)$ -- exponentially in the Hilbert space dimension. Finally, we discuss how our results improve our understanding of concentration in $t$-designs.

Abstract: We investigate, both analytically and numerically, the scattering of one-dimensional quantum droplets by a P\"{o}schl-Teller reflectionless potential well, confirming that there is a sharp transition between full reflection and full transmission at a certain critical incident speed for both small droplets and large flat-top droplets. We observe sharp differences between small quantum droplet scattering and large quantum droplet scattering. The scattering of small quantum droplets is similar to that of solitons, where a spatially symmetric trapped mode is formed at the critical speed, whereas for large quantum droplets a spatially asymmetric trapped mode is formed. Additionally, a nonmonotonous dependence of the critical speed on the atom number is identified$:$ on the small-droplet side, the critical speed increases with the atom number, while in the flat-top regime, the critical speed decreases with increasing the atom number. Strikingly, the scattering excites internal modes below the particle-emission threshold, preventing the quantum droplets from emitting radiation upon interaction with the potential. Analysis of the small-amplitude excitation spectrum shows that as the number of particles increases, it becomes increasingly difficult to emit particles outside the droplet during scattering, while radiation from solitons cannot be completely avoided. Finally, we study the collision of two quantum droplets at the reflectionless potential, revealing the role of the $\pi$-phase difference ``generator'' played by the reflectionless potential.

Abstract: We establish a coding theorem for rate-limited quantum-classical optimal transport systems with limited classical common randomness. This theorem characterizes the rate region of measurement protocols on a product source state for faithful construction of a given destination state while maintaining the source-destination distortion below a prescribed threshold with respect to a general distortion observable. It also provides a solution to the problem of rate-limited optimal transport, which aims to find the optimal cost of transforming a source quantum state to a destination state via an entanglement-breaking channel with a limited communication rate. The coding theorem is further extended to cover Bosonic continuous-variable quantum systems. The analytical evaluation is performed for the case of a qubit measurement system with unlimited common randomness.

Abstract: In the same silicon photonic integrated circuit, we compare two types of integrated degenerate photon-pair sources (microring resonators or waveguides) by means of Hong-Ou-Mandel (HOM) interference experiments. Two nominally identical microring resonators are coupled to two nominally identical waveguides which form the arms of a Mach-Zehnder interferometer. This is pumped by two lasers at two different wavelengths to generate by spontaneous four-wave mixing degenerate photon pairs. In particular, the microring resonators can be thermally tuned in or out of resonance with the pump wavelengths, thus choosing either the microring resonators or the waveguides as photon-pair sources, respectively. In this way, an on-chip HOM visibility of 94% with microring resonators and 99% with straight waveguides is measured. We compare our experimental results with theoretical simulations of the joint spectral intensity and the purity of the degenerate photon pairs. We verify that the visibility is connected to the sources' indistinguishability, which can be quantified by the overlap between the joint spectral amplitudes (JSA) of the photon pairs generated by the two sources. We estimate a JSA overlap of 98% with waveguides and 89% with microring resonators.

Abstract: The Margolus-Levitin (ML) bound says that for any time-independent Hamiltonian, the time needed to evolve from one quantum state to another is at least $\pi \alpha(\epsilon) / (2 \langle E-E_0 \rangle)$, where $\langle E-E_0 \rangle$ is the expected energy of the system relative to the ground state of the Hamiltonian and $\alpha(\epsilon)$ is a function of the fidelity $\epsilon$ between the two state. Nonetheless, only a upper bound $\alpha_{>}(\epsilon)$ and lower bound $\alpha_{<}(\epsilon)$ are known to date although they agree up to at least seven significant figures. By giving a new proof of the ML bound, I show that $\alpha_{>}(\epsilon)$ is indeed equal to $\alpha_{<}(\epsilon)$ and explain why this is the case, thereby filling in this longstanding gap. I also point out a numerical stability issue in computing $\alpha_{>}(\epsilon)$ and report a simple way to evaluate it efficiently and accurately.

Abstract: In this article, we generalize a proof technique by Alicki, Fannes and Winter and introduce a method to prove continuity bounds for entropic quantities derived from different quantum relative entropies. For the Umegaki relative entropy, we mostly recover known almost optimal bounds, whereas, for the Belavkin-Staszewski relative entropy, our bounds are new. Finally, we use these continuity bounds to derive a new entropic uncertainty relation.

Abstract: We investigate a graph-theoretic approach to the problem of distinguishing quantum product states in the fundamental quantum communication framework called local operations and classical communication (LOCC). We identify chordality as the key graph structure that drives distinguishability in one-way LOCC, and we derive a one-way LOCC characterization for chordal graphs that establishes a connection with the theory of matrix completions. We also derive minimality conditions on graph parameters that allow for the determination of indistinguishability of states. We present a number of applications and examples built on these results.

Abstract: In this conceptual paper, we discuss quantum formalisms which do not use the famous Axiom of Choice. We also consider the fundamental problem which addresses the (in)correctness of having the complex numbers as the base field for Hilbert spaces in the K{\o}benhavn interpretation of quantum theory, and propose a new approach to this problem (based on the Lefschetz principle). Rather than a Theorem--Proof--paper, this paper describes two new research programs on the foundational level, and focuses on fundamental open questions in these programs which come along the way.

Abstract: The notorious `measurement problem' has been roving around quantum mechanics for nearly a century since its inception, and has given rise to a variety of `interpretations' of quantum mechanics, which are meant to evade it. We argue that no less than six problems need to be distinguished, and that several of them classify as different types of problems. One of them is what traditionally is called `the measurement problem' (here: the Reality Problem of Measurement Outcomes). Another of them has nothing to do with measurements but is a profound metaphysical problem. We also analyse critically Maudlin's (1995) well-known statement of `three measurements problems', and the clash of the views of Brown (1986) and Stein (1997) on one of the six measurement problems, concerning so-called Insolubility Theorems. Finally, we summarise a solution to one measurement problem which has been largely ignored but tacitly if not explicitly acknowledged.

Abstract: We study the competing effects of collective generalized measurements and interaction-induced scrambling in the dynamics of an ensemble of spin-1/2 particles at the level of quantum trajectories. This setup can be considered as analogous to the one leading to measurement-induced transitions in quantum circuits. We show that the interplay between collective unitary dynamics and measurements leads to three regimes of the average Quantum Fisher Information (QFI), which is a witness of multipartite entanglement, as a function of the monitoring strength. While both weak and strong measurements lead to extensive QFI density (i.e., individual quantum trajectories yield states displaying Heisenberg scaling), an intermediate regime of classical-like states emerges for all system sizes where the measurement effectively competes with the scrambling dynamics and precludes the development of quantum correlations, leading to sub-Heisenberg-limited states. We characterize these regimes and the transitions between them using numerical and analytical tools, and discuss the connections between our findings, entanglement phases in monitored many-body systems, and the quantum-to-classical transition.

Abstract: We present algorithms and a C code to decide quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al (J. Phys. A: Math. Theor. 55 475301, 2022), but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from two to seven. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension two and higher, (ii) non-existence of negative subspaces of dimension three and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for ranks four, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree.

Abstract: We consider a rational agent who at time $0$ enters into a financial contract for which the payout is determined by a quantum measurement at some time $T>0$. The state of the quantum system is given by a known density matrix $\hat p$. How much will the agent be willing to pay at time $0$ to enter into such a contract? In the case of a finite dimensional Hilbert space, each such claim is represented by an observable $\hat X_T$ where the eigenvalues of $\hat X_T$ determine the amount paid if the corresponding outcome is obtained in the measurement. We prove, under reasonable axioms, that there exists a pricing state $\hat q$ which is equivalent to the physical state $\hat p$ on null spaces such that the pricing function $\Pi_{0T}$ takes the form $\Pi_{0T}(\hat X_T) = P_{0T}\,{\rm tr} ( \hat q \hat X_T) $ for any claim $\hat X_T$, where $P_{0T}$ is the one-period discount factor. By "equivalent" we mean that $\hat p$ and $\hat q$ share the same null space: thus, for any $|\xi \rangle \in \mathcal H$ one has $\langle \bar \xi | \hat p | \xi \rangle = 0$ if and only if $\langle \bar \xi | \hat q | \xi \rangle = 0$. We introduce a class of optimization problems and solve for the optimal contract payout structure for a claim based on a given measurement. Then we consider the implications of the Kochen-Specker theorem in such a setting and we look at the problem of forming portfolios of such contracts.

Abstract: Nonlinear mechanical resonators display rich and complex dynamics and are important in many areas of fundamental and applied sciences. In this letter, we show that a particle confined in a funnel-shaped potential features a Duffing-type nonlinearity due to the coupling between its radial and axial motion. Employing an ion trap platform, we study the nonlinear oscillation, bifurcation and hysteresis of a single calcium ion driven by radiation pressure. Harnessing the bistability of this atomic oscillator, we demonstrate a 20-fold enhancement of the signal from a zeptonewton-magnitude harmonic force through the effect of vibrational resonance. Our findings open up a range of possibilities for controlling and exploiting nonlinear phenomena of mechanical oscillators close to the quantum regime.

Abstract: Multimode bright squeezed vacuum is a non-classical state of light hosting a macroscopic photon number while offering promising capacity for encoding quantum information in its spectral degree of freedom. Here, we employ an accurate model for parametric downconversion in the high-gain regime and use nonlinear holography to design quantum correlations of bright squeezed vacuum in the frequency domain. We propose the design of quantum correlations over two-dimensional lattice geometries that are all-optically controlled, paving the way toward continuous-variable cluster state generation on an ultrafast timescale. Specifically, we investigate the generation of a square cluster state in the frequency domain and calculate its covariance matrix and the quantum nullifier uncertainties, that exhibit squeezing below the vacuum noise level.

Abstract: The approximate stabilizer rank of a quantum state is the minimum number of terms in any approximate decomposition of that state into stabilizer states. Bravyi and Gosset showed that the approximate stabilizer rank of a so-called "magic" state like $|T\rangle^{\otimes n}$, up to polynomial factors, is an upper bound on the number of classical operations required to simulate an arbitrary quantum circuit with Clifford gates and $n$ number of $T$ gates. As a result, an exponential lower bound on this quantity seems inevitable. Despite this intuition, several attempts using various techniques could not lead to a better than a linear lower bound on the "exact" rank of $|T\rangle^{\otimes n}$, meaning the minimal size of a decomposition that exactly produces the state. However, an "approximate" rank is more realistically related to the cost of simulating quantum circuits because exact rank is not robust to errors; there are quantum states with exponentially large exact ranks but constant approximate ranks even with arbitrarily small approximation parameters. No lower bound better than $\tilde \Omega(\sqrt n)$ has been known for the approximate rank. In this paper, we improve this lower bound to $\tilde \Omega (n)$ for a wide range of the approximation parameters. Our approach is based on a strong lower bound on the approximate rank of a quantum state sampled from the Haar measure and a step-by-step analysis of the approximate rank of a magic-state teleportation protocol to sample from the Haar measure.

Abstract: Recently, measurement-based quantum thermal machines draw more attention in the field of quantum thermodynamics. However, the previous results on quantum Otto heat engines were either limited to special unital and non-unital channels in the bath stages, or a specific driving protocol at the work strokes and assuming the cycle being time-reversal symmetric i.e. $V^{\dagger}=U$ (or $V=U$). In this paper, we consider a single spin-1/2 quantum Otto heat engine, by first replacing one of the heat baths by an arbitrary unital channel and then we give the exact analytical expression of the characteristic function from which all the cumulants of heat and work emerge. We prove that under the effect of monitoring, $\nu_{2}>\nu_{1}$ is a necessary condition for positive work, either for a symmetric or asymmetric-driven Otto cycle. We trace this back to the negative role of projective measurement. We found that considering an arbitrary unital map would enhance the efficiency and the extracted work. Then we prove the system can never work as a refrigerator. This is forbidden by the second law of thermodynamics. Furthermore, going beyond the average we show that the ratio of the fluctuations of work and heat is lower and upper-bounded when the system is working as a heat engine. However, differently from the previous results in the literature we consider and analyze, skewness and kurtosis as well. We show that in the adiabatic regime, the skewness can be arbitrary and that kurtosis can not be below -2. Finally, we consider applying a specific unital map that plays the role of a heat bath in a coherently superposed manner and we show the role of the initial coherence of the control qubit on efficiency and the first four cumulants of work. In the non-adiabatic regime,...

Abstract: Quantum random-access memory (QRAM) is a mechanism to access data (quantum or classical) based on addresses which are themselves a quantum state. QRAM has a long and controversial history, and here we survey and expand arguments and constructions for and against. We use two primary categories of QRAM from the literature: (1) active, which requires external intervention and control for each QRAM query (e.g. the error-corrected circuit model), and (2) passive, which requires no external input or energy once the query is initiated. In the active model, there is a powerful opportunity cost argument: in many applications, one could repurpose the control hardware for the qubits in the QRAM (or the qubits themselves) to run an extremely parallel classical algorithm to achieve the same results just as fast. Escaping these constraints requires ballistic computation with passive memory, which creates an array of dubious physical assumptions, which we examine in detail. Considering these details, in everything we could find, all non-circuit QRAM proposals fall short in one aspect or another. We apply these arguments in detail to quantum linear algebra and prove that most asymptotic quantum advantage disappears with active QRAM systems, with some nuance related to the architectural assumptions. In summary, we conclude that cheap, asymptotically scalable passive QRAM is unlikely with existing proposals, due to fundamental limitations that we highlight. We hope that our results will help guide research into QRAM technologies that attempt to circumvent or mitigate these limitations. Finally, circuit-based QRAM still helps in many applications, and so we additionally provide a survey of state-of-the-art techniques as a resource for algorithm designers using QRAM.

Abstract: We investigate the one-way zero-error classical and quantum communication complexities for a class of relations induced by a distributed clique labelling problem. We consider two variants: 1) the receiver outputs an answer satisfying the relation - the traditional communication complexity of relations (CCR) and 2) the receiver has non-zero probabilities of outputting every valid answer satisfying the relation (equivalently, the relation can be fully reconstructed), that we denote the strong communication complexity of the relation (S-CCR). We prove that for the specific class of relations considered here when the players do not share any resources, there is no quantum advantage in the CCR task for any graph. On the other hand, we show that there exist, classes of graphs for which the separation between one-way classical and quantum communication in the S-CCR task grow with the order of the graph, specifically, the quantum complexity is $O(1)$ while the classical complexity is $\Omega(\log m)$. Secondly, we prove a lower bound (that is linear in the number of cliques) on the amount of shared randomness necessary to overcome the separation in the scenario of fixed restricted communication and connect this to the existence of Orthogonal Arrays. Finally, we highlight some applications of this task to semi-device-independent dimension witnessing as well as to the detection of Mutually Unbiased Bases.

Abstract: Quantum metrology aims at achieving enhanced performance in measuring unknown parameters by utilizing quantum resources. Thus, quantum metrology is an important application of quantum technologies. Photonic systems can implement these metrological tasks with simpler experimental techniques. We present a scheme for improved parameter estimation by introducing entangled generalized coherent states (EGCS) for photonic quantum metrology. These states show enhanced sensitivity beyond the classical and Heisenberg limits and prove to be advantageous as compared to the entangled coherent and NOON states. Further, we also propose a scheme for experimentally generating certain entangled generalized coherent states with current technology.