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Long-term time series forecasting (LTSF) involves predicting a large number of future values of a time series based on the past values and is an essential task in a wide range of domains including weather forecasting, stock market analysis, disease outbreak prediction. Over the decades LTSF algorithms have transitioned from statistical models to deep learning models like transformer models. Despite the complex architecture of transformer based LTSF models `Are Transformers Effective for Time Series Forecasting? (Zeng et al., 2023)' showed that simple linear models can outperform the state-of-the-art transformer based LTSF models. Recently, quantum machine learning (QML) is evolving as a domain to enhance the capabilities of classical machine learning models. In this paper we initiate the application of QML to LTSF problems by proposing QuLTSF, a simple hybrid QML model for multivariate LTSF. Through extensive experiments on a widely used weather dataset we show the advantages of QuLTSF over the state-of-the-art classical linear models, in terms of reduced mean squared error and mean absolute error., Comment: submitted for conference publication

Integrated photonic circuits play a crucial role in implementing quantum information processing in the noisy intermediate-scale quantum (NISQ) era. Variational learning is a promising avenue that leverages classical optimization techniques to enhance quantum advantages on NISQ devices. However, most variational algorithms are circuit-model-based and encounter challenges when implemented on integrated photonic circuits, because they involve explicit decomposition of large quantum circuits into sequences of basic entangled gates, leading to an exponential decay of success probability due to the non-deterministic nature of photonic entangling gates. Here, we present a variational learning approach for designing quantum photonic circuits, which directly incorporates post-selection and elementary photonic elements into the training process. The complicated circuit is treated as a single nonlinear logical operator, and a unified design is discovered for it through variational learning. Engineering an integrated photonic chip with automated control, we adjust and optimize the internal parameters of the chip in real time for task-specific cost functions. We utilize a simple case of designing photonic circuits for a single ancilla CNOT gate with improved success rate to illustrate how our proposed approach works, and then apply the approach in the first demonstration of quantum stochastic simulation using integrated photonics.

The study of undecidability in problems arising from physics has experienced a renewed interest, mainly in connection with quantum information problems. The goal of this review is to survey this recent development. After a historical introduction, we first explain the necessary results about undecidability in mathematics and computer science. Then we briefly review the first results about undecidability in physics which emerged mostly in the 80s and early 90s. Finally we focus on the most recent contributions, which we divide in two main categories: many body systems and quantum information problems., Comment: 77 pages, v1

Dynamical quantum error-correcting codes (QECC) offer wider possibilities in how one can protect logical quantum information from noise and perform fault-tolerant quantum computation compared to static QECCs. A family of dynamical QECCs called the ``Floquet codes'' consists of a periodic sequence of two-body measurements that enables error-correction on many-body systems, relaxing hardware implementation requirements and improving error-correction reliability. Existing results on Floquet codes has been focused on qubits, two-level quantum systems, with very little attention given on higher dimensional quantum systems, or qudits. We bridge this gap by proposing a simple, yet general construction of qudit Floquet codes based on a simple set of conditions on the sequence two-body measurements defining the code. Moreover, this construction applies to a large family of configurations of qudits on the vertices of a three-colorable lattice which connectivity represented by the edges. We show that this construction includes the existing constructions of both qubit and qudit Floquet codes as special cases. In addition, any qudit Floquet code obtained by our construction achieves a rate of encoded logical qudits over physical qudits approaching $\frac{1}{2}$ as the number of physical qudits in total and on the faces of the lattice grows larger, as opposed to vanishing rate in existing qudit Floquet code constructions., Comment: New figures, fixed typos, updated citations; 20 pages, 6 figures

In stochastic modeling, the excess entropy -- the mutual information shared between a processes past and future -- represents the fundamental lower bound of the memory needed to simulate its dynamics. However, this bound cannot be saturated by either classical machines or their enhanced quantum counterparts. Simulating a process fundamentally requires us to store more information in the present than than what is shared between past and future. Here we consider a hypothetical generalization of hidden Markov models beyond classical and quantum models -- n-machines -- that allow for negative quasi-probabilities. We show that under the collision entropy measure of information, the minimal memory of such models can equalize the excess entropy. Our results hint negativity as a necessary resource for memory-advantaged stochastic simulation -- mirroring similar interpretations in various other quantum information tasks., Comment: 22 pages, 13 figures

Decoupling systems into independently evolving components has a long history of simplifying seemingly complex systems. They enable a better understanding of the underlying dynamics and causal structures while providing more efficient means to simulate such processes on a computer. Here we outline a variational decoupling algorithm for decoupling unitary quantum dynamics -- allowing us to decompose a given $n$-qubit unitary gate into multiple independently evolving sub-components. We apply this approach to quantum circuit synthesis - the task of discovering quantum circuit implementations of target unitary dynamics. Our numerical studies illustrate significant benefits, showing that variational decoupling enables us to synthesize general $2$ and $4$-qubit gates to fidelity that conventional variational circuits cannot reach.

Quantum error-correcting code (QECC) is the central ingredient in fault-tolerant quantum information processing. An emerging paradigm of dynamical QECC shows that one can robustly encode logical quantum information both temporally and spatially in a more resource-efficient manner than traditional QECCs. Nevertheless, an overarching theory of how dynamical QECCs achieve fault-tolerance is lacking. In this work, we bridge this gap by proposing a unified spatio-temporal QECC framework called the ``strategic code'' built around an ``interrogator'' device which sequentially measures and evolves the spatial QECC in an adaptive manner based on the ``quantum combs'' formalism, a generalization of the channel-state duality. The strategic code covers all existing dynamical and static QECC, as well as all physically plausible QECCs to be discovered in the future, including those that involve adaptivity in its operational dynamics. Within this framework, we show an algebraic and an information-theoretic necessary and sufficient error-correction conditions for a strategic code, which consider spatially and temporally correlated errors. These conditions include the analogous known static QECC conditions as a special case. Lastly, we also propose an optimization-theoretic approach to obtain an approximate strategic code adapting to a correlated error., Comment: 20 pages, 6 figures

In recent years, several experimental groups have claimed demonstrations of ``quantum supremacy'' or computational quantum advantage. A notable first claim by Google Quantum AI revolves around a metric called the Linear Cross Entropy Benchmarking (Linear XEB), which has been used in multiple quantum supremacy experiments since. The complexity-theoretic hardness of spoofing Linear XEB has nevertheless been doubtful due to its dependence on the Cross-Entropy Quantum Threshold (XQUATH) conjecture put forth by Aaronson and Gunn, which has been disproven for sublinear depth circuits. In efforts on demonstrating quantum supremacy by quantum Hamiltonian simulation, a similar benchmarking metric called the System Linear Cross Entropy Score (sXES) holds firm in light of the aforementioned negative result due to its fundamental distinction with Linear XEB. Moreover, the hardness of spoofing sXES complexity-theoretically rests on the System Linear Cross-Entropy Quantum Threshold Assumption (sXQUATH), the formal relationship of which to XQUATH is unclear. Despite the promises that sXES offers for future demonstration of quantum supremacy, in this work we show that it is an unsound benchmarking metric. Particularly, we prove that sXQUATH does not hold for sublinear depth circuits and present a classical algorithm that spoofs sXES for experiments corrupted with noise larger than certain threshold., Comment: 19 pages, 1 figure

Quantum technologies offer a promising route to the efficient sampling and analysis of stochastic processes, with potential applications across the sciences. Such quantum advantages rely on the preparation of a quantum sample state of the stochastic process, which requires a memory system to propagate correlations between the past and future of the process. Here, we introduce a method of lossy quantum dimension reduction that allows this memory to be compressed, not just beyond classical limits, but also beyond current state-of-the-art quantum stochastic sampling approaches. We investigate the trade-off between the saving in memory resources from this compression, and the distortion it introduces. We show that our approach can be highly effective in low distortion compression of both Markovian and strongly non-Markovian processes alike. We further discuss the application of our results to quantum stochastic modelling more broadly.

Efficiently harvesting thermodynamic resources requires a precise understanding of their structure. This becomes explicit through the lens of information engines -- thermodynamic engines that use information as fuel. Maximizing the work harvested using available information is a form of physically-instantiated machine learning that drives information engines to develop complex predictive memory to store an environment's temporal correlations. We show that an information engine's complex predictive memory poses both energetic benefits and risks. While increasing memory facilitates detection of hidden patterns in an environment, it also opens the possibility of thermodynamic overfitting, where the engine dissipates additional energy in testing. To address overfitting, we introduce thermodynamic regularizers that incur a cost to engine complexity in training due to the physical constraints on the information engine. We demonstrate that regularized thermodynamic machine learning generalizes effectively. In particular, the physical constraints from which regularizers are derived improve the performance of learned predictive models. This suggests that the laws of physics jointly create the conditions for emergent complexity and predictive intelligence.

By exploiting the complexity intrinsic to quantum dynamics, quantum technologies promise a whole host of computational advantages. One such advantage lies in the field of stochastic modelling, where it has been shown that quantum stochastic simulators can operate with a lower memory overhead than their best classical counterparts. This advantage is particularly pronounced for continuous-time stochastic processes; however, the corresponding quantum stochastic simulators heretofore prescribed operate only on a quasi-continuous-time basis, and suffer an ever-increasing circuit complexity with increasing temporal resolution. Here, by establishing a correspondence with quantum trajectories -- a method for modelling open quantum systems -- we show how truly continuous-time quantum stochastic simulators can be embedded in such open quantum systems, bridging this gap and obviating previous constraints. We further show how such an embedding can be made for discrete-time stochastic processes, which manifest as jump-only trajectories, and discuss how viewing the correspondence in the reverse direction provides new means of studying structural complexity in quantum systems themselves., Comment: 16 pages, 4 figures

We develop the general framework of virtual resource distillation -- an alternative distillation strategy proposed in [Phys. Rev. Lett. 132, 050203 (2024)], which extends conventional quantum resource distillation by integrating the power of classical postprocessing. The framework presented here is applicable not only to quantum states, but also dynamical quantum objects such as quantum channels and higher-order processes. We provide a general characterization and benchmarks for the performance of virtual resource distillation in the form of computable semidefinite programs as well as several operationally motivated quantities. We apply our general framework to various concrete settings of interest, including standard resource theories such as entanglement, coherence, and magic, as well as settings involving dynamical resources such as quantum memory, quantum communication, and non-Markovian dynamics. The framework of probabilistic distillation is also discussed., Comment: 22 pages, 5 figures. Published version. Companion paper to arXiv:2303.00955, where the framework is introduced and some results on virtual resource distillation of quantum states are presented. Some of the content of this submission was originally a part of arXiv:2303.00955v1

The ability of quantum states to be in superposition is one of the key features that sets them apart from the classical world. This `coherence' is rigorously quantified by resource theories, which aim to understand how such properties may be exploited in quantum technologies. There has been much research on what the resource theory of coherence can reveal about quantum metrology, almost all of which has been from the viewpoint of Fisher information. We prove however that the relative entropy of coherence, and its recent generalisation to POVMs, naturally quantify the performance of Bayesian metrology. In particular, we show how a coherence measure can be applied to an ensemble of states. We then prove that during parameter estimation, the ensemble relative entropy of coherence is equal to the difference between the information gained, and the optimal Holevo information. We call this relation the CXI equality. The ensemble coherence lets us visualise how much information is locked away in superposition inaccessible with a given measurement scheme, and quantify the advantage that would be gained by using a joint measurement on multiple states. Our results hold regardless of how the parameter is encoded in the state, encompassing unitary, dissipative, and discrete settings. We consider both projective measurements, and general POVMs. This work suggests new directions for research in quantum resource theories, provides a novel operational interpretation for the relative entropy of coherence and its POVM generalisation, and introduces a new tool to study the role of quantum features in metrology., Comment: 10 pages main + 4 pages supplementary

In covert target detection, Alice attempts to send optical or microwave probes to determine the presence or absence of a weakly-reflecting target embedded in thermal background radiation within a target region, while striving to remain undetected by an adversary, Willie, who is co-located with the target and collects all light that does not return to Alice. We formulate this problem in a realistic setting and derive quantum-mechanical limits on Alice's error probability performance in entanglement-assisted target detection for any fixed level of her detectability by Willie. We demonstrate how Alice can approach this performance limit using two-mode squeezed vacuum probes in the regime of small to moderate background brightness, and how such protocols can outperform any conventional approach using Gaussian-distributed coherent states. In addition, we derive a universal performance bound for non-adversarial quantum illumination without requiring the passive-signature assumption., Comment: 20 pages, 5 figures

Why is it that a ticking clock typically becomes less accurate when subject to outside noise but rarely the reverse? Here, we formalize this phenomenon by introducing process causal asymmetry - a fundamental difference in the amount of past information an autonomous agent must track to transform one stochastic process to another over an agent that transforms in the opposite direction. We then illustrate that this asymmetry can paradoxically be reversed when agents possess a quantum memory. Thus, the spontaneous direction in which processes get 'simpler' may be different, depending on whether quantum information processing is allowed or not., Comment: 12 pages, 4 figures

We obtain universal (i.e., probe and measurement-independent) performance bounds on ancilla-assisted quantum sensing of multiple parameters of phase-covariant optical channels under energy and mode-number constraints. We first show that for any such constrained problem, an optimal ancilla-entangled probe can always be found whose reduced state on the modes probing the channel is diagonal in the photon-number basis. For parameters that are encoded in single-mode Gaussian channels, we derive a universal upper bound on the quantum Fisher information matrix that delineates the roles played by the energy and mode constraints. We illustrate our results for sensing of the transmittance of a thermal loss channel under both the no-passive-signature and passive-signature paradigms, and in the problem of sensing the noise variance of an additive-noise channel. In both cases, we show that two-mode squeezed vacuum probes are near-optimal under the constraints in the regime of low signal brightness, i.e., per-mode average photon number. More generally, our work sets down a uniform framework for readily evaluating universal limits for any sensing problem involving Gaussian channels., Comment: 18 pages, 6 figures

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., Comment: 29 pages

Heisenberg's uncertainty principle implies fundamental constraints on what properties of a quantum system can we simultaneously learn. However, it typically assumes that we probe these properties via measurements at a single point in time. In contrast, inferring causal dependencies in complex processes often requires interactive experimentation - multiple rounds of interventions where we adaptively probe the process with different inputs to observe how they affect outputs. Here we demonstrate universal uncertainty principles for general interactive measurements involving arbitrary rounds of interventions. As a case study, we show that they imply an uncertainty trade-off between measurements compatible with different causal dependencies., Comment: 7 pages (main text) + 47 pages (supplemental material), 5 figures (main text) + 17 figures (supplemental material). Accepted for publication in Physical Review Letters

We investigate and ascertain the ideal inputs to any finite-time thermodynamic process. We demonstrate that the expectation values of entropy flow, heat, and work can all be determined via Hermitian observables of the initial state. These Hermitian operators encapsulate the breadth of behavior and the ideal inputs for common thermodynamic objectives. We show how to construct these Hermitian operators from measurements of thermodynamic output from a finite number of effectively arbitrary inputs. Behavior of a small number of test inputs thus determines the full range of thermodynamic behavior from all inputs. For any process, entropy flow, heat, and work can all be extremized by pure input states -- eigenstates of the respective operators. In contrast, the input states that minimize entropy production or maximize the change in free energy are non-pure mixed states obtained from the operators as the solution of a convex optimization problem. To attain these, we provide an easily implementable gradient descent method on the manifold of density matrices, where an analytic solution yields a valid direction of descent at each iterative step. Ideal inputs within a limited domain, and their associated thermodynamic operators, are obtained with less effort. This allows analysis of ideal thermodynamic inputs within quantum subspaces of infinite-dimensional quantum systems; it also allows analysis of ideal inputs in the classical limit. Our examples illustrate the diversity of 'ideal' inputs: Distinct initial states minimize entropy production, extremize the change in free energy, and maximize work extraction.

The amount of information that a noisy channel can transmit has been one of the primary subjects of interest in information theory. In this work we consider a practically-motivated family of optical quantum channels that can be implemented without an external energy source. We optimize the Holevo information over procedures that encode information in attenuations and phase-shifts applied by these channels on a resource state of finite energy. It is shown that for any given input state and environment temperature, the maximum Holevo information can be achieved by an encoding procedure that uniformly distributes the channel's phase-shift parameter. Moreover for large families of input states, any maximizing encoding scheme has a finite number of channel attenuation values, simplifying the codewords to a finite number of rings around the origin in the output phase space. The above results and numerical evidence suggests that this property holds for all resource states. Our results are directly applicable to the quantum reading of an optical memory in the presence of environmental thermal noise., Comment: 32 pages, 11 figures; Accepted for publication in Quantum

Distillation, or purification, is central to the practical use of quantum resources in noisy settings often encountered in quantum communication and computation. Conventionally, distillation requires using some restricted 'free' operations to convert a noisy state into one that approximates a desired pure state. Here, we propose to relax this setting by only requiring the approximation of the measurement statistics of a target pure state, which allows for additional classical postprocessing of the measurement outcomes. We show that this extended scenario, which we call virtual resource distillation, provides considerable advantages over standard notions of distillation, allowing for the purification of noisy states from which no resources can be distilled conventionally. We show that general states can be virtually distilled with a cost (measurement overhead) that is inversely proportional to the amount of existing resource, and we develop methods to efficiently estimate such cost via convex and semidefinite programming, giving several computable bounds. We consider applications to coherence, entanglement, and magic distillation, and an explicit example in quantum teleportation (distributed quantum computing). This work opens a new avenue for investigating generalized ways to manipulate quantum resources., Comment: 6 pages, 2 figures; See also the companion paper arXiv:2404.13048 for the broader framework and complete technical proofs

Complex systems are embedded in our everyday experience. Stochastic modelling enables us to understand and predict the behaviour of such systems, cementing its utility across the quantitative sciences. Accurate models of highly non-Markovian processes -- where the future behaviour depends on events that happened far in the past -- must track copious amounts of information about past observations, requiring high-dimensional memories. Quantum technologies can ameliorate this cost, allowing models of the same processes with lower memory dimension than corresponding classical models. Here we implement such memory-efficient quantum models for a family of non-Markovian processes using a photonic setup. We show that with a single qubit of memory our implemented quantum models can attain higher precision than possible with any classical model of the same memory dimension. This heralds a key step towards applying quantum technologies in complex systems modelling., Comment: 11+2 pages, 5 figures

Numerous quantum error-mitigation protocols have been proposed, motivated by the critical need to suppress noise effects on intermediate-scale quantum devices. Yet, their general potential and limitations remain elusive. In particular, to understand the ultimate feasibility of quantum error mitigation, it is crucial to characterize the fundamental sampling cost -- how many times an arbitrary mitigation protocol must run a noisy quantum device. Here, we establish universal lower bounds on the sampling cost for quantum error mitigation to achieve the desired accuracy with high probability. Our bounds apply to general mitigation protocols, including the ones involving nonlinear postprocessing and those yet-to-be-discovered. The results imply that the sampling cost required for a wide class of protocols to mitigate errors must grow exponentially with the circuit depth for various noise models, revealing the fundamental obstacles in the scalability of useful noisy near-term quantum devices., Comment: 5+15 pages, 4 figures, published version; See also a related work by Tsubouchi et al., which appeared in the same arXiv posting as arXiv:2208.09385

Quantum information-processing techniques enable work extraction from a system's inherently quantum features, in addition to the classical free energy it contains. Meanwhile, the science of computational mechanics affords tools for the predictive modeling of non-Markovian classical and quantum stochastic processes. We combine tools from these two sciences to develop a technique for predictive work extraction from non-Markovian stochastic processes with quantum outputs. We demonstrate that this technique can extract more work than non-predictive quantum work extraction protocols, on one hand, and predictive work extraction without quantum information processing, on the other. We discover a phase transition in the efficacy of memory for work extraction from quantum processes, which is without classical precedent. Our work opens up the prospect of machines that harness environmental free energy in an essentially quantum, essentially time-varying form.

Phase-insensitive optical amplifiers uniformly amplify each quadrature of an input field and are of both fundamental and technological importance. We find the quantum limit on the precision of estimating the gain of a quantum-limited phase-insensitive optical amplifier using a multimode probe that may also be entangled with an ancilla system. In stark contrast to the sensing of loss parameters, the average photon number $N$ and number of input modes $M$ of the probe are found to be equivalent and interchangeable resources for optimal gain sensing. All pure-state probes whose reduced state on the input modes to the amplifier is diagonal in the multimode number basis are proven to be quantum-optimal under the same gain-independent measurement. We compare the best precision achievable using classical probes to the performance of an explicit photon-counting-based estimator on quantum probes and show that an advantage exists even for single-photon probes and inefficient photodetection. A closed-form expression for the energy-constrained Bures distance between two product amplifier channels is also derived., Comment: Text, figures, and appendices re-organized and updated. To appear in Physical Review Letters

The inevitable accumulation of errors in near-future quantum devices represents a key obstacle in delivering practical quantum advantages, motivating the development of various quantum error-mitigation methods. Here, we derive fundamental bounds concerning how error-mitigation algorithms can reduce the computation error as a function of their sampling overhead. Our bounds place universal performance limits on a general error-mitigation protocol class. We use them to show (1) that the sampling overhead that ensures a certain computational accuracy for mitigating local depolarizing noise in layered circuits scales exponentially with the circuit depth for general error-mitigation protocols and (2) the optimality of probabilistic error cancellation among a wide class of strategies in mitigating the local dephasing noise on an arbitrary number of qubits. Our results provide a means to identify when a given quantum error-mitigation strategy is optimal and when there is potential room for improvement., Comment: 12+10 pages, 10 figures. v5: extended the results on protocol benchmarking; published version

Central to the success of adaptive systems is their ability to interpret signals from their environment and respond accordingly -- they act as agents interacting with their surroundings. Such agents typically perform better when able to execute increasingly complex strategies. This comes with a cost: the more information the agent must recall from its past experiences, the more memory it will need. Here we investigate the power of agents capable of quantum information processing. We uncover the most general form a quantum agent need adopt to maximise memory compression advantages, and provide a systematic means of encoding their memory states. We show these encodings can exhibit extremely favourable scaling advantages relative to memory-minimal classical agents, particularly when information must be retained about events increasingly far into the past., Comment: 16 pages, 4 figures

Given a quantum system on many qubits split into a few different parties, how many total correlations are there between these parties? Such a quantity, aimed to measure the deviation of the global quantum state from an uncorrelated state with the same local statistics, plays an important role in understanding multipartite correlations within complex networks of quantum states. Yet, the experimental access of this quantity remains challenging as it tends to be non-linear, and hence often requires tomography which becomes quickly intractable as dimensions of relevant quantum systems scale. Here, we introduce a much more experimentally accessible quantifier of total correlations, which can be estimated using only single-qubit measurements. It requires far fewer measurements than state tomography and obviates the need to coherently interfere multiple copies of a given state. Thus, we provide a tool for proving multipartite correlations that can be applied to near-term quantum devices., Comment: 28 pages, 8 figures. Comments are welcome

In modelling complex processes, the potential past data that influence future expectations are immense. Models that track all this data are not only computationally wasteful but also shed little light on what past data most influence the future. There is thus enormous interest in dimensional reduction-finding automated means to reduce the memory dimension of our models while minimizing its impact on its predictive accuracy. Here we construct dimensionally reduced quantum models by machine learning methods that can achieve greater accuracy than provably optimal classical counterparts. We demonstrate this advantage on present-day quantum computing hardware. Our algorithm works directly off classical time-series data and can thus be deployed in real-world settings. These techniques illustrate the immediate relevance of quantum technologies to time-series analysis and offer a rare instance where the resulting quantum advantage can be provably established., Comment: 16 Pages, 17 Figures

Error mitigation has been one of the recently sought after methods to reduce the effects of noise when computation is performed on a noisy near-term quantum computer. Interest in simulating stochastic processes with quantum models gained popularity after being proven to require less memory than their classical counterparts. With previous work on quantum models focusing primarily on further compressing memory, this work branches out into the experimental scene; we aim to bridge the gap between theoretical quantum models and practical use with the inclusion of error mitigation methods. It is observed that error mitigation is successful in improving the resultant expectation values. While our results indicate that error mitigation work, we show that its methodology is ultimately constrained by hardware limitations in these quantum computers., Comment: 20 pages, 8 figures

While Landauer's Principle sets a lower bound for the work required for a computation, that work is recoverable for efficient computations. However, practical physical computers, such as modern digital computers or biochemical systems, are subject to constraints that make them inefficient -- irreversibly dissipating significant energy. Recent results show that the dissipation in such systems is bounded by the nonreciprocity of the embedded computation. We investigate the consequences of this bound for different types of memory, showing that different memory devices are better suited for different computations. This correspondence comes from the time-reversal symmetries of the memory, which depend on whether information is stored positionally or magnetically. This establishes that the time symmetries of the memory device play an essential roll in determining energetics. The energetic consequences of time symmetries are particularly pronounced in nearly deterministic computations, where the cost of computing diverges as minus log of the error rate. We identify the coefficient of that divergence as the dissipation divergence. We find that the dissipation divergence may be zero for a computation when implemented in one type of memory while it's maximal when implemented with another. Given flexibility in the type of memory, the dissipation divergence is bounded below by the average state compression of the computation. Moreover, we show how to explicitly construct the memory to achieve this minimal dissipation. As a result, we find that logically reversible computations are indeed thermodynamically efficient, but logical irreversibility comes at a much higher cost than previously anticipated.

Elementary cellular automata (ECA) present iconic examples of complex systems. Though described only by one-dimensional strings of binary cells evolving according to nearest-neighbour update rules, certain ECA rules manifest complex dynamics capable of universal computation. Yet, the classification of precisely which rules exhibit complex behaviour remains a significant challenge. Here we approach this question using tools from quantum stochastic modelling, where quantum statistical memory -- the memory required to model a stochastic process using a class of quantum machines -- can be used to quantify the structure of a stochastic process. By viewing ECA rules as transformations of stochastic patterns, we ask: Does an ECA generate structure as quantified by the quantum statistical memory, and if so, how quickly? We illustrate how the growth of this measure over time correctly distinguishes simple ECA from complex counterparts. Moreover, it provides a more refined means for quantitatively identifying complex ECAs -- providing a spectrum on which we can rank the complexity of ECA by the rate in which they generate structure., Comment: 22 pages, 9 figures

The thermodynamic cost of resetting an arbitrary initial state to a particular desired state is lower bounded by Landauer's bound. However, here we demonstrate that this lower bound is necessarily unachievable for nearly every initial state, for any reliable reset mechanism. Since local heating threatens rapid decoherence, this issue is of substantial importance beyond mere energy efficiency. For the case of qubit reset, we find the minimally dissipative state analytically for any reliable reset protocol, in terms of the entropy-flow vector introduced here. This allows us to verify a recent theorem about initial-state dependence of entropy production for any finite-time transformation, as it pertains to quantum state preparation., Comment: 9 pages plus 3 pages of appendices, 3 figures

Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels -- that is, not completely positive -- can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation., Comment: 25 pages. v3: published version

Quantum illumination is the task of determining the presence of an object in a noisy environment. We determine the optimal continuous variable states for quantum illumination in the limit of zero object reflectivity. We prove that the optimal single mode state is a coherent state, while the optimal two mode state is the two-mode squeezed-vacuum state. We find that these probes are not optimal at non-zero reflectivity, but remain near optimal. This demonstrates the viability of the continuous variable platform for an experimentally accessible, near optimal quantum illumination implementation., Comment: 7 pages, 3 figures

The measurement of circular dichroism (CD) has widely been exploited to distinguish the different enantiomers of chiral structures. It has been applied to natural materials (e.g. molecules) as well as to artificial materials (e.g. nanophotonic structures). However, especially for chiral molecules the signal level is very low and increasing the signal-to-noise ratio is of paramount importance to either shorten the necessary measurement time or to lower the minimum detectable molecule concentration. As one solution to this problem, we propose here to use quantum states of light in CD sensing to reduce the noise below the shot noise limit that is encountered when using coherent states of light. Through a multi-parameter estimation approach, we identify the ultimate quantum limit to precision of CD sensing, allowing for general schemes including additional ancillary modes. We show that the ultimate quantum limit can be achieved by various optimal schemes. It includes not only Fock state input in direct sensing configuration but also twin-beam input in ancilla-assisted sensing configuration, for both of which photon number resolving detection needs to be performed as the optimal measurement setting. These optimal schemes offer a significant quantum enhancement even in the presence of additional system loss. The optimality of a practical scheme using a twin-beam state in direct sensing configuration is also investigated in details as a nearly optimal scheme for CD sensing when the actual CD signal is very small. Alternative schemes involving single-photon sources and detectors are also proposed. This work paves the way for further investigations of quantum metrological techniques in chirality sensing., Comment: 9 pages, 5 figures

Realizing a long coherence time quantum memory is a major challenge of current quantum technology. Here, we report a single \Yb ion-qubit memory with over one hour coherence time, an order of improvement compared to the state-of-the-art record. The long coherence time memory is realized by addressing various technical challenges such as ambient magnetic-field noise, phase noise and leakage of the microwave oscillator. Moreover, systematically study the decoherence process of our quantum memory by quantum process tomography, which enables to apply the strict criteria of quantum coherence, relative entropy of coherence. We also benchmark our quantum memory by its ability in preserving quantum information, i.e., the robustness of quantum memory, which clearly shows that over 6000 s, our quantum memory preserves non-classical quantum information. Our results verify the stability of the quantum memory in hours level and indicate its versatile applicability in various scenarios., Comment: 7 pages, 7 figures

Experimentally achieving the precision that standard quantum metrology schemes promise is always challenging. Recently, additional controls were applied to design feasible quantum metrology schemes. However, these approaches generally does not consider ease of implementation, raising technological barriers impeding its realization. In this paper, we circumvent this problem by applying closed-loop learning control to propose a practical controlled sequential scheme for quantum metrology. Purity loss of the probe state, which relates to quantum Fisher information, is measured efficiently as the fitness to guide the learning loop. We confirm its feasibility and certain superiorities over standard quantum metrology schemes by numerical analysis and proof-of-principle experiments in a nuclear magnetic resonance (NMR) system.

A universal squeezing gate capable of squeezing arbitrary input states is essential for continuous-variable quantum computation~\cite{PRA79062318,PRL112120504}. However, in present state-of-the-art techniques~\cite{PRA90060302,PRL106240504}, the fidelity of such gates is ultimately limited by the need to create squeezed vacuum modes of unbounded energy. Here we circumvent this fundamental limitation by using a heralded squeezing gate. We propose and experimentally demonstrate a squeezing gate that can achieve near unit fidelity for coherent input states. In particular, for a target squeezing of \SI{2.3}{\dB}, we report a fidelity of \SI{98.5}{\%}. This result cannot be reproduced by conventional schemes even if the currently best available squeezing of \SI{15}{\dB}~\cite{PRL117110801} is utilised when benchmarked on identical detection inefficiencies. Our technique can be applied to non-Gaussian states and provides a promising pathway towards high-fidelity gate operations and fault-tolerant quantum computation.

Traditional machine learning applications, such as optical character recognition, arose from the inability to explicitly program a computer to perform a routine task. In this context, learning algorithms usually derive a model exclusively from the evidence present in a massive dataset. Yet in some scientific disciplines, obtaining an abundance of data is an impractical luxury, however; there is an explicit model of the domain based upon previous scientific discoveries. Here we introduce a new approach to machine learning that is able to leverage prior scientific discoveries in order to improve generalizability over a scientific model. We show its efficacy in predicting the entire energy spectrum of a Hamiltonian on a superconducting quantum device, a key task in present quantum computer calibration. Our accuracy surpasses the current state-of-the-art by over $20\%.$ Our approach thus demonstrates how artificial intelligence can be further enhanced by "standing on the shoulders of giants."

We quantify the maximum amount of entanglement of formation (EoF) that can be achieved by continuous-variable states under passive operations, which we refer to as EoF-potential. Focusing, in particular, on two-mode Gaussian states we derive analytical expressions for the EoF-potential for specific classes of states. For more general states, we demonstrate that this quantity can be upper-bounded by the minimum amount of squeezing needed to synthesize the Gaussian modes, a quantity called squeezing of formation. Our work, thus, provides a new link between non-classicality of quantum states and the non-classicality of correlations., Comment: Revised version

Quadrature correlations between subsystems of a Gaussian quantum state are fully characterised by its covariance matrix. For example, the covariance matrix determines the amount of entanglement or decoherence of the state. Here, we establish when it is possible to remove correlations between conjugate quadratures using only passive operations. Such correlations are usually undesired and arise due to experimental cross-quadrature contamination. Using the Autonne--Takagi factorisation, we present necessary and sufficient conditions to determine when such removal is possible. Our proof is constructive, and whenever it is possible we obtain an explicit expression for the required passive operation.

In Quantum Illumination (QI), a signal beam initially entangled with an idler beam held at the receiver interrogates a target region bathed in thermal background light. The returned beam is measured jointly with the idler in order to determine whether a weakly reflecting target is present. Using tools from quantum information theory, we derive lower bounds on the average error probability of detecting both specular and fading targets and on the mean squared error of estimating the reflectance of a detected target, which are obeyed by any QI transmitter satisfying a signal energy constraint. For bright thermal backgrounds, we show that the QI system using multiple copies of low-brightness two-mode squeezed vacuum states is nearly optimal. More generally, our results place limits on the best possible performance achievable using QI systems at all wavelengths, and at all signal and background noise levels., Comment: Approximates published version

New exact results about the nonequilibrium thermodynamics of open quantum systems at arbitrary timescales are obtained by considering all possible variations of initial conditions of a system, its environment, and correlations between them. First we obtain a new quantum-information theoretic equality for entropy production, valid for an arbitrary initial joint state of system and environment. For any finite-time process with a fixed initial environment, we then show that the contraction of the system's distinction -- relative to the minimally dissipative state -- exactly quantifies its thermodynamic dissipation. The quantum component of this dissipation is the change in coherence relative to the minimally dissipative state. Implications for quantum state preparation and local control are explored. For nonunitary processes -- like the preparation of any particular quantum state -- we find that mismatched expectations lead to divergent dissipation as the actual initial state becomes orthogonal to the anticipated one., Comment: 6 pages plus 14 pages of appendices, 1 figure

A growing body of work has established the modelling of stochastic processes as a promising area of application for quantum techologies; it has been shown that quantum models are able to replicate the future statistics of a stochastic process whilst retaining less information about the past than any classical model must -- even for a purely classical process. Such memory-efficient models open a potential future route to study complex systems in greater detail than ever before, and suggest profound consequences for our notions of structure in their dynamics. Yet, to date methods for constructing these quantum models are based on having a prior knowledge of the optimal classical model. Here, we introduce a protocol for blind inference of the memory structure of quantum models -- tailored to take advantage of quantum features -- direct from time-series data, in the process highlighting the robustness of their structure to noise. This in turn provides a way to construct memory-efficient quantum models of stochastic processes whilst circumventing certain drawbacks that manifest solely as a result of classical information processing in classical inference protocols., Comment: 10 pages, 6 figures

Quantifying how distinguishable two stochastic processes are lies at the heart of many fields, such as machine learning and quantitative finance. While several measures have been proposed for this task, none have universal applicability and ease of use. In this Letter, we suggest a set of requirements for a well-behaved measure of process distinguishability. Moreover, we propose a family of measures, called divergence rates, that satisfy all of these requirements. Focussing on a particular member of this family -- the co-emission divergence rate -- we show that it can be computed efficiently, behaves qualitatively similar to other commonly-used measures in their regimes of applicability, and remains well-behaved in scenarios where other measures break down.

Thermodynamic resources, beyond their well-known usefulness in work extraction and other thermodynamic tasks, are often important also in tasks that are not evidently thermodynamic. Here we develop a framework for identifying such resources in diverse applications of bosonic continuous-variable systems. Introducing the class of bosonic linear thermal operations to model operationally-feasible processes, we apply this model to identify uniquely quantum properties of bosonic states that refine classical notions of thermodynamic resourcefulness. Among these are (1) a suite of temperature-like quantities generalizing the equilibrium temperature to quantum, non-equilibrium scenarios; (2) signal-to-noise ratios quantifying a system's capacity to carry information in phase-space displacement; and (3) well-established non-classicality measures quantifying the resolution in sensing and parameter estimation tasks., Comment: 9 pages (6 figures), 4-page supplemental material

Effective and efficient forecasting relies on identification of the relevant information contained in past observations -- the predictive features -- and isolating it from the rest. When the future of a process bears a strong dependence on its behaviour far into the past, there are many such features to store, necessitating complex models with extensive memories. Here, we highlight a family of stochastic processes whose minimal classical models must devote unboundedly many bits to tracking the past. For this family, we identify quantum models of equal accuracy that can store all relevant information within a single two-dimensional quantum system (qubit). This represents the ultimate limit of quantum compression and highlights an immense practical advantage of quantum technologies for the forecasting and simulation of complex systems., Comment: 6+3 pages, 3+1 figures

Modern computation relies crucially on modular architectures, breaking a complex algorithm into self-contained subroutines. A client can then call upon a remote server to implement parts of the computation independently via an application programming interface (API). Present APIs relay only classical information. Here we implement a quantum API that enables a client to estimate the absolute value of the trace of a server-provided unitary $U$. We demonstrate that the algorithm functions correctly irrespective of what unitary $U$ the server implements or how the server specifically realizes $U$. Our experiment involves pioneering techniques to coherently swap qubits encoded within the motional states of a trapped \Yb ion, controlled on its hyperfine state. This constitutes the first demonstration of modular computation in the quantum regime, providing a step towards scalable, parallelization of quantum computation., Comment: 6 pages, 4 figures (7 pages, 4 figures, and 6 tables for supplementary information)

We characterize the operational capabilities of quantum channels which can neither create nor detect quantum coherence vis-\`a-vis efficiently manipulating coherence as a resource. We study the class of dephasing-covariant operations (DIO), unable to detect the coherence of any input state, as well as introduce an operationally-motivated class of channels $\rho$-DIO which is tailored to a specific input state. We first show that pure-state transformations under DIO are completely governed by majorization, establishing necessary and sufficient conditions for such transformations and adding to the list of operational paradigms where majorization plays a central role. We then show that $\rho$-DIO are strictly more powerful: although they cannot detect the coherence of the input state $\rho$, the operations $\rho$-DIO can distill more coherence than DIO. However, the advantage disappears in the task of coherence dilution as well as generally in the asymptotic limit, where both sets of operations achieve the same rates in all transformations., Comment: 6+4 pages; v2: minor additions and clarifications