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We study the complexity of testing properties of quantum channels. First, we show that testing identity to any channel $\mathcal N: \mathbb C^{d_{\mathrm{in}} \times d_{\mathrm{in}}} \to \mathbb C^{d_{\mathrm{out}} \times d_{\mathrm{out}}}$ in diamond norm distance requires $\Omega(\sqrt{d_{\mathrm{in}}} / \varepsilon)$ queries, even in the strongest algorithmic model that admits ancillae, coherence, and adaptivity. This is due to the worst-case nature of the distance induced by the diamond norm. Motivated by this limitation and other theoretical and practical applications, we introduce an average-case analogue of the diamond norm, which we call the average-case imitation diamond (ACID) norm. In the weakest algorithmic model without ancillae, coherence, or adaptivity, we prove that testing identity to certain types of channels in ACID distance can be done with complexity independent of the dimensions of the channel, while for other types of channels the complexity depends on both the input and output dimensions. Building on previous work, we also show that identity to any fixed channel can be tested with $\tilde O(d_{\mathrm{in}} d_{\mathrm{out}}^{3/2} / \varepsilon^2)$ queries in ACID distance and $\tilde O(d_{\mathrm{in}}^2 d_{\mathrm{out}}^{3/2} / \varepsilon^2)$ queries in diamond distance in this model. Finally, we prove tight bounds on the complexity of channel tomography in ACID distance.

We investigate the precision of estimating the interaction strength between a two-level system (TLS) and a single-photon pulse when the latter is subject to chirping. We consider linear, quadratic, and sinusoidal temporal phases applied to Gaussian and exponential temporal profiles. At the asymptotic time, when the TLS has fully decayed to its ground state, the fundamental precision depends solely on the magnitude of its spectral amplitude. For quadratically phase-modulated Gaussian pulses, this is entirely determined by the spectral bandwidth. We provide expressions for evaluating the fundamental precision for general temporal profiles and phase modulations. Finally, we show that experimentally feasible mode-resolved measurements are optimal, or close to it, for chirped, pulsed single photon spectroscopy.

Quantum vacuum fluctuations of the electromagnetic field result in two signatures on a harmonically trapped charged particle: a shift from the natural trap frequency and generation of quantum coherences. We assess the role of the long-wavelength and rotating-wave approximations in estimating this frequency shift. We estimate the magnitude of the frequency shift using parameters from a single-electron cyclotron experiment and also demonstrate how the dependence of the frequency shift on the magnetic field of the cyclotron is tied to the rotating-wave approximation. We expect the frequency shift to be observable in future experiments. We also suggest a possible route to detecting vacuum-generated quantum coherences. These experiments should settle the debate on the choice of approximations and gauge in capturing the effect of the quantum vacuum fluctuations., Comment: 6 pages, 1 figure

To investigate the transport properties in random alloys, it is important to model the alloy disorder using supercells. Though traditional methods like Virtual Crystal Approximation (VCA) are computationally efficient, the local disorder in the system is not accurately captured as artificial translational symmetry is imposed on the system. However, in the case of supercells, the error introduced by self-image interaction between the impurities is reduced and translational symmetry is explicitly imposed over larger length scales. In this work, we have investigated the Effective Phonon Dispersion (EPD) and transport properties, from first principle calculations using supercells in AlxGa1-xN alloy systems. Using our in-house developed code, the EPD of AlGaN is obtained and the individual modes are identified. Next, we discuss our in-house developed method to calculate low-field transport properties in supercells. First to validate our methods we have solved the Boltzmann Transport Equation using Rode method to compare the phonon limited mobility in the 4 atom GaN primitive cell and 12 atom GaN supercell. Using the same technique, we have investigated the low field transport in random AlxGa1-xN alloy systems. Our calculations show that along with alloy scattering, electron-phonon scattering may also play an important role at room temperature and high-temperature device operation. This technique opens up the path for calculating phonon-limited transport properties in random alloy systems.

In this work, the Effective Phonon Dispersion of $\beta$- (Al$_x$Ga$_{1-x}$)$_2$O$_3$ alloy is investigated using the Phonon unfolding formalism. To capture the true randomness of the system, supercells are designed using the technique of special quasirandom structues. The phonon unfolding technique is then used to obtain the effective phonon dispersion (EPD) for aluminium fractions of 18.75 \% and 37.5 \% with respect to gallium atoms. The impact of disorder on the high frequency and DC dielectric constant is also investigated. The unfolding procedure was also applied to the dipole moment where the projection probability was used to obtain the IR spectra of the disordered system. The calculated properties gives us further insights to calculate the transport properties in these alloy semiconductors while capturing the true randomness of the system.

It depends. For a single molecule interacting with one mode of a biphoton probe, we show that the spectroscopic information has three contributions, only one of which is a genuine two-photon contribution. When all the scattered light can be measured, solely this contribution exists and can be fully extracted using unentangled measurements. Furthermore, this two-photon contribution can, in principle, be matched by an optimised but unentangled single-photon probe. When the matter system spontaneously emits into inaccessible modes, an advantage due to entanglement can not be ruled out. In practice, time-frequency entanglement does enhance spectroscopic performance of the oft-studied weakly-pumped spontaneous parametric down conversion (PDC) probes. For two-level systems and coupled dimers, more entangled PDC probes yield more spectroscopic information, even in the presence of emission into inaccessible modes. Moreover, simple, unentangled measurements can capture between 60% - 90% of the spectroscopic information. We thus establish that biphoton spectroscopy using source-engineered PDC probes and unentangled measurements can provide tangible quantum enhancement. Our work underscores the intricate role of entanglement in single-molecule spectroscopy using quantum light., Comment: 18+14 pages, 12+14 figures, 1+0 tables

We present a formalism to discern the effects of fluctuations of the spacetime metric on electromagnetic radiation. The formalism works via the measurement of electromagnetic field correlations, while allowing a clear assessment of the assumptions involved. As an application of the formalism, we present a model of spacetime fluctuations that appear as random fluctuations of the refractive index of the vacuum in single, and two co-located Michelson interferometers. We compare an interferometric signal predicted using this model to experimental data from the Holometer and aLIGO. We show that if the signal manifests at a frequency at which the interferometers are sensitive, the strength and scale of possible spacetime fluctuations can be constrained. The bounds, thus obtained, on the strength and scale of the spacetime fluctuations, are also shown to be more stringent than the bounds obtained previously using astronomical observation at optical frequencies. The formalism enables us to evaluate proposed experiments such as QUEST for constraining quantum spacetime fluctuations and to design new ones., Comment: 21 pages, 3 figures, 2 tables

We present an accreditation protocol for analogue, i.e., continuous-time, quantum simulators. For a given simulation task, it provides an upper bound on the variation distance between the probability distributions at the output of an erroneous and error-free analogue quantum simulator. As its overheads are independent of the size and nature of the simulation, the protocol is ready for immediate usage and practical for the long term. It builds on the recent theoretical advances of strongly universal Hamiltonians and quantum accreditation as well as experimental progress towards the realisation of programmable hybrid analogue-digital quantum simulators., Comment: 16 pages, 4 figures

In this article, we investigate the Single Events Effects (SEE) leading to Single Event Burnout (SEB) in $\beta$-Ga$_2$O$_3$ MOSFETs. Using Silvaco TCAD, 2D simulations were performed to understand the mechanism behind the SEB mechanism in lateral Ga$_2$O$_3$ MOSFETS. The high electric fields in the channel played a critical role leading to high impact generation rates and eventual SEB. To reduce the electric field in the channel, radiation hardened designs are then proposed with rounded gates and the use of a combination of high permittivity (k) dielectric with SiO$_2$. With HfO$_2$-SiO$_2$ dielectric combination, the SEB threshold of 550V at LET=10 MeV/mg/cm$^2$ is seen. However, to operate under extreme radiation conditions, a combination of very high-k dielectric material BaTiO$_3$ with SiO$_2$ is proposed. Using the radiation hardened design, SEB thresholds up to 1000 V for LET=75 MeV/mg/cm$^2$ could be achieved which is higher than the state-of-the-art technology. The energy dissipated during the ion strike event is also calculated and it is observed that it is lower than that of SiC MOSFETs.

We study the fundamental limits of the precision of estimating parameters of a quantum matter system when it is probed by a travelling pulse of quantum light. In particular, we focus on the estimation of the interaction strength between the pulse and a two-level atom, equivalent to the estimation of the dipole moment. Our analysis of single-photon pulses highlights the interplay between the information gained from the absorption of the photon by the atom as measured in absorption spectroscopy, and the perturbation to the temporal mode of the photon due to spontaneous emission. Beyond the single-photon regime, we introduce an approximate model to study more general states of light in the limit of short pulses, where spontaneous emission can be neglected. We also show that for a vast class of entangled biphoton states, quantum entanglement between the signal mode interacting with the atom and the idler mode provides no fundamental advantage and the same precision can be obtained with a separable state. We conclude by studying the estimation of the electric dipole moment of a sodium atom using quantum light. Our work initiates a quantum information theoretic methodology for developing the theory and practice of quantum light spectroscopy., Comment: v2: 30 pages, 10 figures, Fig. 7 fixes an error in Fig. 8 of v1, appendix E replaces appendix B of v1

We present two algorithms, one quantum and one classical, for estimating partition functions of quantum spin Hamiltonians. The former is a DQC1 (Deterministic quantum computation with one clean qubit) algorithm, and the first such for complex temperatures. The latter, for real temperatures, achieves performance comparable to a state-of-the-art DQC1 algorithm [Chowdhury et al. Phys. Rev. A 103, 032422 (2021)]. Both our algorithms take as input the Hamiltonian decomposed as a linear combination Pauli operators. We show this decomposition to be DQC1-hard for a given Hamiltonian, providing new insight into the hardness of estimating partition functions.

The objective of the proposed MAQRO mission is to harness space for achieving long free-fall times, extreme vacuum, nano-gravity, and cryogenic temperatures to test the foundations of physics in macroscopic quantum experiments. This will result in the development of novel quantum sensors and a means to probe the foundations of quantum physics at the interface with gravity. Earlier studies showed that the proposal is feasible but that several critical challenges remain, and key technologies need to be developed. These new technologies will open up the potential for achieving additional science objectives. The proposed research campaign aims to advance the state of the art and to perform the first macroscopic quantum experiments in space. Experiments on the ground, in micro-gravity, and in space will drive the proposed research campaign during the current decade to enable the implementation of MAQRO within the subsequent decade., Comment: 9 pages, 2 figures, submitted as a Research Campaign Whitepaper for the BPS2023 Decadal Survey

Quantum states of light have been shown to enhance precision in absorption estimation over classical strategies. By exploiting interference and resonant enhancement effects, we show that coherent-state probes in all-pass ring resonators can outperform any quantum probe single-pass strategy even when normalized by the mean input photon number. We also find that under optimal conditions coherent-state probes equal the performance of arbitrarily bright pure single-mode squeezed probes in all-pass ring resonators., Comment: 7 pages, 4 figures, typos corrected, expanded section 3 and conclusion

We show that a signature of the quantum nature of gravity is the quantum mechanical squeezing of the differential motion of two identical masses with respect to their common mode. This is because the gravitational interaction depends solely on the relative position of the two masses. In principle, this squeezing is equivalent to quantum entanglement between the masses. In practice, detecting the squeezing is more feasible than detecting the entanglement. To that end, we propose an optical interferometric scheme to falsify hypothetical models of gravity., Comment: Published version. Added analysis of gravity-gradient and seismic noise. 6 pages, 1 figure. Comments welcome!

We provide and experimentally demonstrate an accreditation protocol that upper-bounds the variation distance between noisy and noiseless probability distributions of the outputs of arbitrary quantum computations. We accredit the outputs of twenty-four quantum circuits executed on programmable superconducting hardware, ranging from depth nine circuits on ten qubits to depth twenty-one circuits on four qubits. Our protocol requires implementing the "target" quantum circuit along with a number of random Clifford circuits and subsequently post-processing the outputs of these Clifford circuits. Importantly, the number of Clifford circuits is chosen to obtain the bound with the desired confidence and accuracy, and is independent of the size and nature of the target circuit. We thus demonstrate a practical and scalable method of ascertaining the correctness of the outputs of arbitrary-sized noisy quantum computers--the ultimate arbiter of the utility of the computer itself., Comment: Accepted version

We present a model-independent measure of dynamical complexity based on simulating complex quantum dynamics using stroboscopic Markovian dynamics. Tools from classical signal processing enable us to infer the Hilbert space dimension of a complex quantum system evolving under a time-independent Hamiltonian via pulsed interrogation. We evaluate our model-independent simulation complexity (MISC) for the spin-boson model and simulated third-order pump-probe spectroscopy data for exciton transport in coupled dimers with vibrational levels. The former provides insights into coherence and population dynamics in the two-level system while the latter reveals the dimension of the singly-excited manifold of the dimer. Finally, we probe the complexity of excitonic transport in light harvesting 2 (LH2) and Fenna-Matthews-Olson (FMO) complexes using data from two recent nonlinear ultrafast optical spectroscopy experiments. For the latter we make some model-independent inferences that are commensurate with model-specific ones. This includes estimating the fewest number of parameters needed to fit the experimental data and identifying the spatial extent, i.e., delocalization size, of quantum states occurring in this complex quantum dynamics., Comment: 21 pages, inculding 6 appendices and 10 figures

We study quantum-limited 3D magnetometry using two qubits. Two qubits form the smallest multi-qubit system for 3D magnetometry, the simultaneous estimation of three phases, as it is impossible with a single qubit. We provide an analytical expression for the Holevo-Cram\'er-Rao bound (HCRB),the fundamental attainable quantum bound of multiparameter estimation, for 3D magnetometry using two-qubit pure states and show its attainability by rank-1 projective measurements. We also examine the attainability of the HCRB in the presence of dephasing noise using numerical methods. While attaining the HCRB may require collective measurements over infinitely many copies, we find that for high noise the HCRB is practically saturated by two copies only. In the low noise regime, up to three copies are unable to attain the HCRB. More generally, we introduce new multiparameter channel bounds to compare quantum-classical and classical-quantum strategies where multiple independent copies of the state are entangled before or after recording the parameters respectively. We find that their relative performance depends on the noise strength, with theclassical-quantum strategy performing better for high noise. We end with shallow quantum circuits that approach the fundamental quantum limit set by the HCRB for two-qubit 3D magnetometry using up to three copies., Comment: 19 pages, 9 figures

Weak value amplification (WVA) is a metrological protocol that amplifies ultra-small physical effects. However, the amplified outcomes necessarily occur with highly suppressed probabilities, leading to the extensive debate on whether the overall measurement precision is improved in comparison to that of conventional measurement (CM). Here, we experimentally demonstrate the unambiguous advantages of WVA that overcome practical limitations including noise and saturation of photo-detection and maintain a shot-noise-scaling precision for a large range of input light intensity well beyond the dynamic range of the photodetector. The precision achieved by WVA is six times higher than that of CM in our setup. Our results clear the way for the widespread use of WVA in applications involving the measurement of small signals including precision metrology and commercial sensors., Comment: 6pages, 4 figures

Nonlinear optical microscopy techniques have emerged as a set of successful tools for biological imaging. Stimulated emission microscopy belongs to a small subset of pump-probe techniques which can image non-fluorescent samples without requiring fluorescent labelling. However, its sensitivity has been shown to be ultimately limited by the quantum fluctuations in the probe beam. We propose and experimentally implement sub-shot-noise limited stimulated emission microscopy by preparing the probe pulse in an intensity-squeezed state. This technique paves the way for imaging delicate biological samples that have no detectable fluorescence with sensitivity beyond standard quantum fluctuations., Comment: The following article has been submitted to Applied Physics Letters

We formulate multiparameter quantum estimation in the parametric and semiparametric setting. While the Holevo Cram\'er-Rao bound (CRB) requires no substantial modifications in moving from the former to the latter, we generalize the Helstrom CRB appropriately. We show that the Holevo CRB cannot be greater than twice the generalized Helstrom CRB. We also present a tighter, intermediate, bound. Finally, we show that for parameters encoded in the first moments of a Gaussian state there always exists a Gaussian measurement that gives a classical Fisher information matrix that is one-half of the quantum Fisher information matrix., Comment: v2 New coauthor, new and more general results, streamlined presentation. 10 pages, no figures. v3 (same manuscript) Main results have been merged with arXiv:1906.09871 and published on PRX (DOI:10.1103/PhysRevX.10.031023)

We show that localisation microscopy of multiple weak, incoherent point sources with possibly different intensities in one spatial dimension is equivalent to estimating the amplitudes of a classical mixture of coherent states of a simple harmonic oscillator. This enables us to bound the multi-parameter covariance matrix for an unbiased estimator for the locations in terms of the quantum Fisher information matrix, which we obtained analytically. In the regime of arbitrarily small separations we find it to be no more than rank two -- implying that no more than two independent parameters can be estimated irrespective of the number of point sources. We use the eigenvalues of the classical and quantum Fisher information matrices to compare the performance of spatial-mode demultiplexing and direct imaging in localisation microscopy with respect to the quantum limits., Comment: Published version

In the study of quantum limits to parameter estimation, the high dimensionality of the density operator and that of the unknown parameters have long been two of the most difficult challenges. Here we propose a theory of quantum semiparametric estimation that can circumvent both challenges and produce simple analytic bounds for a class of problems in which the dimensions are arbitrarily high, few prior assumptions about the density operator are made, but only a finite number of the unknown parameters are of interest. We also relate our bounds to Holevo's version of the quantum Cram\'er-Rao bound, so that they can inherit the asymptotic attainability of the latter in many cases of interest. The theory is especially relevant to the estimation of a parameter that can be expressed as a function of the density operator, such as the expectation value of an observable, the fidelity to a pure state, the purity, or the von Neumann entropy. Potential applications include quantum state characterization for many-body systems, optical imaging, and interferometry, where full tomography of the quantum state is often infeasible and only a few select properties of the system are of interest., Comment: 26 pages, 12 figures. v6: merged with arXiv:1911.11036. v7: added Appendix M: post-publication notes

Only with the simultaneous estimation of multiple parameters are the quantum aspects of metrology fully revealed. This is due to the incompatibility of observables. The fundamental bound for multi-parameter quantum estimation is the Holevo Cram\'er-Rao bound (HCRB) whose evaluation has so far remained elusive. For finite-dimensional systems we recast its evaluation as a semidefinite program, with reduced size for rank-deficient states. We use this result to study phase and loss estimation in optical interferometry and three-dimensional magnetometry with noisy multi-qubit systems. For the former, we show that, in some regimes, it is possible to attain the HCRB with the optimal (single-copy) measurement for phase estimation. For the latter, we show a non-trivial interplay between the HCRB and incompatibility, and provide numerical evidence that projective single-copy measurements attain the HCRB in the noiseless two-qubit case., Comment: 6+9 pages, 2+2 figures. Comments are welcome

The role of quantum effects in excitonic energy transport (EET) has been scrutinised intensely and with increasingly sophisticated experimental techniques. This increased complexity requires invoking correspondingly elaborate models to fit spectroscopic data before molecular parameters can be extracted. Possible quantum effects in EET can then be studied, but the conclusions are strongly contingent on the efficacy of the fitting and the accuracy of the model. To circumvent this challenge, we propose a witness for quantum coherence in EET that can be extracted directly from two-pulse pump-probe spectroscopy experimental data. We provide simulations to judge the feasibility of our approach. Somewhat counterintuitively, our protocol does not probe quantum coherence directly, but only indirectly through its implicit deletion. It allows for classical models with no quantum coherence to be decisively ruled out., Comment: 13 pages, 9 figures

We propose to explore the quantum nature of gravity using the correlation of light between two optomechanical cavities, and the quantumness of the correlation is witnessed by squeezing. As long as the gravity between the end mirrors of two cavities is quantum in the Newtonian limit, we show that the squeezing is always nonzero and monotonically increases as the mechanical property of the mirrors is improved. The proposed scheme provides a new pathway for testing the quantum nature of gravity systematically with tabletop experiments., Comment: Use squeezing to show the quantumness of the correlation

Momentum diffusion is a possible mechanism for driving macroscopic quantum systems towards classical behaviour. Experimental tests of this hypothesis rely on a precise estimation of the strength of this diffusion. We show that quantum-mechanical squeezing offers significant improvements, including when measuring position. For instance, with 10dB of mechanical squeezing, experiments would require a tenth of proposed free-fall times. Momentum measurement is better by an additional factor of three, while another quadrature is close to optimal. These have particular implications for the space-based MAQRO proposal -- where it could rule out the spontaneous collapse theory due to Ghirardi, Rimini, and Weber -- as well as terrestrial optomechanical sensing.

We propose a scheme for covert active sensing using floodlight illumination from a THz-bandwidth amplified spontaneous emission (ASE) source and heterodyne detection. We evaluate the quantum-estimation-theoretic performance limit of covert sensing, wherein a transmitter's attempt to sense a target phase is kept undetectable to a quantum-equipped passive adversary, by hiding the signal photons under the thermal noise floor. Despite the quantum state of each mode of the ASE source being mixed (thermal), and hence inferior compared to the pure coherent state of a laser mode, the thousand-times higher optical bandwidth of the ASE source results in achieving a substantially superior performance compared to a narrowband laser source by allowing the probe light to be spread over many more orthogonal temporal modes within a given integration time. Even though our analysis is restricted to single-mode phase sensing, this system could be applicable extendible for various practical optical sensing applications., Comment: We present new results and discuss some results found in arXiv:1701.06206. Comments are welcome

We present an accreditation protocol for the outputs of noisy intermediate-scale quantum devices. By testing entire circuits rather than individual gates, our accreditation protocol can provide an upper-bound on the variation distance between noisy and noiseless probability distribution of the outputs of the target circuit of interest. Our accreditation protocol requires implementation of quantum circuits no larger than the target circuit, therefore it is practical in the near term and scalable in the long term. Inspired by trap-based protocols for the verification of quantum computations, our accreditation protocol assumes that noise in single-qubit gates is bounded (but potentially gate-dependent) in diamond norm. We allow for arbitrary spatial and temporal correlations in the noise affecting state preparation, measurements and two-qubit gates. We describe how to implement our protocol on real-world devices, and we also present a novel cryptographic protocol (which we call `mesothetic' protocol) inspired by our accreditation protocol., Comment: Accepted version

Users of quantum computers must be able to confirm they are indeed functioning as intended, even when the devices are remotely accessed. In particular, if the Hilbert space dimension of the components are not as advertised -- for instance if the qubits suffer leakage -- errors can ensue and protocols may be rendered insecure. We refine the method of delayed vectors, adapted from classical chaos theory to quantum systems, and apply it remotely on the IBMQ platform -- a quantum computer composed of transmon qubits. The method witnesses, in a model-independent fashion, dynamical signatures of higher-dimensional processes. We present evidence, under mild assumptions, that the IBMQ transmons suffer state leakage, with a $p$ value no larger than $5{\times}10^{-4}$ under a single qubit operation. We also estimate the number of shots necessary for revealing leakage in a two-qubit system., Comment: 11 pages, 5 figures

Identifying non-classicality unambiguously and inexpensively is a long-standing open challenge in physics. The No-Signalling-In-Time protocol was developed as an experimental test for macroscopic realism, and serves as a witness of quantum coherence in isolated quantum systems by comparing the quantum state to its completely dephased counterpart. We show that it provides a lower bound on a certain resource-theoretic coherence monotone. We go on to generalise the protocol to the case where the system of interest is coupled to an environment. Depending on the manner of the generalisation, the resulting witness either reports on system coherence alone, or on a disjunction of system coherence with either (i) the existence of non-classical system-environment correlations or (ii) non-negligible dynamics in the environment. These are distinct failure modes of the Born approximation in non-isolated systems., Comment: 16pp, 2 figs, 5 thms. v2: typos corrected, references added and small change to title to reflect that of published version

We introduce the notion of fault-tolerant quantum metrology to overcome noise beyond our control -- associated with sensing the parameter, by reducing the noise in operations under our control -- associated with preparing and measuring probes and ancillae. To that end, we introduce noise thresholds to quantify the noise resilience of parameter estimation schemes. We demonstrate improved noise thresholds over the non-fault-tolerant schemes. We use quantum Reed-Muller codes to retrieve more information about a single phase parameter being estimated in the presence of full-rank Pauli noise. Using only error detection, as opposed to error correction, allows us to retrieve higher thresholds. We show that better devices, which can be engineered, can enable us to counter larger noise in the field beyond our control. Further improvements in fault-tolerant quantum metrology could be achieved by optimising in tandem parameter-specific estimation schemes and transversal quantum error correcting codes., Comment: 16 pages, 12 figures

Optomechanical sensors involving multiple optical carriers can experience mechanically mediated interactions causing multi-mode correlations across the optical fields. One instance is laser-interferometric gravitational wave detectors which introduce multiple carrier frequencies for classical sensing and control purposes. An outstanding question is whether such multi-carrier optomechanical sensors outperform their single-carrier counterpart in terms of quantum-limited sensitivity. We show that the best precision is achieved by a single-carrier instance of the sensor. For the current LIGO detection system this precision is already reachable.

We present two verification protocols where the correctness of a "target" computation is checked by means of "trap" computations that can be efficiently simulated on a classical computer. Our protocols rely on a minimal set of noise-free operations (preparation of eight single-qubit states or measurement of four observables, both on a single plane of the Bloch sphere) and achieve linear overhead. To the best of our knowledge, our protocols are the least demanding techniques able to achieve linear overhead. They represent a step towards further reducing the quantum requirements for verification., Comment: Accepted version

The quantum enhanced classical sensor network consists of $K$ clusters of $N_e$ entangled quantum states that have been trialled $r$ times, each feeding into a classical estimation process. Previous literature has shown that each cluster can {ideally} achieve an estimation variance of $1/N_e^2r$ for sufficient $r$. We begin by deriving the optimal values for the minimum mean squared error of this quantum enhanced classical system. We then show that if noise is \emph{absent} in the classical estimation process, the mean estimation error will decay like $\Omega(1/KN_e^2r)$. However, when noise is \emph{present} we find that the mean estimation error will decay like $\Omega(1/K)$, so that \emph{all} the sensing gains obtained from the individual quantum clusters will be lost.

We show that the von Neumann entropy (from herein referred to as the von Neumann index) of a graph's trace normalized combinatorial Laplacian provides structural information about the level of centralization across a graph. This is done by considering the Theil index, which is an established statistical measure used to determine levels of inequality across a system of `agents', e.g., income levels across a population. Here, we establish a Theil index for graphs, which provides us with a macroscopic measure of graph centralization. Concretely, we show that the von Neumann index can be used to bound the graph's Theil index, and thus we provide a direct characterization of graph centralization via the von Neumann index. Because of the algebraic similarities between the bound and the Theil index, we call the bound the von Neumann Theil index. %From an information theoretic perspective, the von Neumann Theil index describes the mutual information between the quantum state corresponding to a graph and the maximally mixed state (which corresponds to the complete graph). We elucidate our ideas by providing examples and a discussion of different $n=7$ vertex graphs. We also discuss how the von Neumann Theil index provides a more comprehensive measure of centralization when compared to traditional centralization measures, and when compared to the graph's classical Theil index. This is because it more accurately accounts for macro-structural changes that occur from micro-structural changes in the graph (e.g., the removal of a vertex). Finally, we provide future direction, showing that the von Neumann Theil index can be generalized by considering the R\'enyi entropy. We then show that this generalization can be used to bound the negative logarithm of the graph's Jain fairness index.

A quantum theory of multiphase estimation is crucial for quantum-enhanced sensing and imaging and may link quantum metrology to more complex quantum computation and communication protocols. In this letter we tackle one of the key difficulties of multiphase estimation: obtaining a measurement which saturates the fundamental sensitivity bounds. We derive necessary and sufficient conditions for projective measurements acting on pure states to saturate the maximal theoretical bound on precision given by the quantum Fisher information matrix. We apply our theory to the specific example of interferometric phase estimation using photon number measurements, a convenient choice in the laboratory. Our results thus introduce concepts and methods relevant to the future theoretical and experimental development of multiparameter estimation., Comment: 4 pages + appendix, 2 figures

We study a large number of physically-plausible arrangements of chromophores, generated via a computational method involving stochastic real-space transformations of a naturally occurring `reference' structure, illustrating our methodology using the well-studied Fenna-Matthews-Olson complex (FMO). To explore the idea that the natural structure has been tuned for efficient energy transport we use an atomic transition charge method to calculate the excitonic couplings of each generated structure and a Lindblad master equation to study the quantum transport of an exciton from a `source' to a `drain' chromophore. We find significant correlations between structure and transport efficiency: High-performing structures tend to be more compact and, among those, the best structures display a certain orientation of the chromophores, particularly the chromophore closest to the source-to-drain vector. We conclude that, subject to reasonable, physically-motivated constraints, the FMO complex is highly attuned to the purpose of energy transport, partly by exploiting these structural motifs., Comment: 7+6pp, 3+10 figures. Accepted version. Data available at http://doi.org/10.6084/m9.figshare.c.3784874.v1

Quantum samplers are believed capable of sampling efficiently from distributions that are classically hard to sample from. We consider a sampler inspired by the classical Ising model. It is nonadaptive and therefore experimentally amenable. Under a plausible conjecture, classical sampling upto additive errors from this model is known to be hard. We present a trap-based verification scheme for quantum supremacy that only requires the verifier to prepare single-qubit states. The verification is done on the same model as the original sampler, a square lattice, with only a constant overhead. We next revamp our verification scheme in two distinct ways using fault tolerance that preserves the nonadaptivity. The first has a lower overhead based on error correction with the same threshold as universal quantum computation. The second has a higher overhead but an improved threshold (1.97\%) based on error detection. We show that classically sampling upto additive errors is likely hard in both these schemes. Our results are applicable to other sampling problems such as the Instantaneous Quantum Polynomial-time (IQP) computation model. They should also assist near-term attempts at experimentally demonstrating quantum supremacy and guide long-term ones., Comment: 33 pages, 11 figures, 3 theorems, 2 conjectures

We show that the (normalized) symmetric Laplacian of a simple graph can be obtained from the partial trace over a pure bipartite quantum state that resides in a bipartite Hilbert space (one part corresponding to the vertices, the other corresponding to the edges). This suggests an interpretation of the symmetric Laplacian's Von Neumann entropy as a measure of bipartite entanglement present between the two parts of the state. We then study extreme values for a connected graph's generalized R\'enyi-$p$ entropy. Specifically, we show that (1) the complete graph achieves maximum entropy, (2) the $2$-regular graph: a) achieves minimum R\'enyi-$2$ entropy among all $k$-regular graphs, b) is within $\log 4/3$ of the minimum R\'enyi-$2$ entropy and $\log4\sqrt{2}/3$ of the minimum Von Neumann entropy among all connected graphs, c) achieves a Von Neumann entropy less than the star graph. Point $(2)$ contrasts sharply with similar work applied to (normalized) combinatorial Laplacians, where it has been shown that the star graph almost always achieves minimum Von Neumann entropy. In this work we find that the star graph achieves maximum entropy in the limit as the number of vertices grows without bound. Keywords: Symmetric; Laplacian; Quantum; Entropy; Bounds; R\'enyi.

Phase diffusion invariably accompanies all phase estimation strategies -- quantum or classical. A precise estimation of the former can often provide valuable understanding of the physics of the phase generating phenomena itself. We theoretically examine the performance of fixed-particle number probe states in the simultaneous estimation of phase and collective phase diffusion. We derive analytical quantum limits associated with the simultaneous local estimation of phase and phase diffusion within the quantum Cramer-Rao bound framework in the regimes of large and small phase diffusive noise. The former is for a general fixed-particle number state and the latter for Holland Burnett states, for which we show quantum-enhanced estimation of phase as well as phase diffusion. We next investigate the simultaneous attainability of these quantum limits using projective measurements acting on a single copy of the state in terms of a trade-off relation. In particular, we are interested how this trade-off varies as a function of the dimension of the state. We derive an analytical bound for this trade-off in the large phase diffusion regime for a particular form of the measurement, and show that the maximum of 2, set by the quantum Cramer-Rao bound, is attainable. Further, we show numerical evidence that as diffusion approaches zero, the optimal trade-off relation approaches 1 for Holland-Burnett states. These numerical results are valid in the small particle number regime and suggest that the trade-off for estimating one parameter with quantum-limited precision leads to a complete lack of precision for the other parameter as the diffusion strength approaches zero. Finally, we provide numerical results showing behaviour of the trade-off for a general value of phase diffusion when using Holland-Burnett probe states.

We present a square root law for active sensing of phase $\theta$ of a single pixel using optical probes that pass through a single-mode lossy thermal-noise bosonic channel. Specifically, we show that, when the sensor uses an $n$-mode covert optical probe, the mean squared error (MSE) of the resulting estimator $\hat{\theta}_n$ scales as $\langle (\theta-\hat{\theta}_n)^2\rangle=\mathcal{O}(1/\sqrt{n})$; improving the scaling necessarily leads to detection by the adversary with high probability. We fully characterize this limit and show that it is achievable using laser light illumination and a heterodyne receiver, even when the adversary captures every photon that does not return to the sensor and performs arbitrarily complex measurement as permitted by the laws of quantum mechanics., Comment: 13 pages, 1 figure, submitted to ISIT 2017

We consider the problem of estimating an unknown but constant carrier phase modulation $\theta$ using a general -- possibly entangled -- $n$-mode optical probe through $n$ independent and identical uses of a lossy bosonic channel with additive thermal noise. We find an upper bound to the quantum Fisher information (QFI) of estimating $\theta$ as a function of $n$, the mean and variance of the total number of photons $N_{\rm S}$ in the $n$-mode probe, the transmissivity $\eta$ and mean thermal photon number per mode ${\bar n}_{\rm B}$ of the bosonic channel. Since the inverse of QFI provides a lower bound to the mean-squared error (MSE) of an unbiased estimator $\tilde{\theta}$ of $\theta$, our upper bound to the QFI provides a lower bound to the MSE. It already has found use in proving fundamental limits of covert sensing, and could find other applications requiring bounding the fundamental limits of sensing an unknown parameter embedded in a correlated field., Comment: No major changes to previous version. Change in the title and abstract, change in the presentation and structure, an example of the bound is now included, and some references were added. Comments are welcome

For a fixed average energy, the simultaneous estimation of multiple phases can provide a better total precision than estimating them individually. We show this for a multimode interferometer with a phase in each mode, using Gaussian inputs and passive elements, by calculating the covariance matrix. The quantum Cram\'{e}r-Rao bound provides a lower bound to the covariance matrix via the quantum Fisher information matrix, whose elements we derive to be the covariances of the photon numbers across the modes. We prove that this bound can be saturated. In spite of the Gaussian nature of the problem, the calculation of non-Gaussian integrals is required, which we accomplish analytically. We find our simultaneous strategy to yield no more than a factor-of-2 improvement in total precision, possibly because of a fundamental performance limitation of Gaussian states. Our work shows that no modal entanglement is necessary for simultaneous quantum-enhanced estimation of multiple phases.

The simultaneous quantum estimation of multiple parameters can provide a better precision than estimating them individually. This is an effect that is impossible classically. We review the rich background of multi-parameter quantum metrology, some of the main results in the field and its recent advances. We close by highlighting future challenges and open questions.

We present a framework for the quantum enhanced estimation of multiple parameters corresponding to non-commuting unitary generators. Our formalism provides a recipe for the simultaneous estimation of all three components of a magnetic field. We propose a probe state that surpasses the precision of estimating the three components individually and discuss measurements that come close to attaining the quantum limit. Our study also reveals that too much quantum entanglement may be detrimental to attaining the Heisenberg scaling in quantum metrology., Comment: 9 pages, 1 figure