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Estimating many-body Hamiltonians has wide applications in quantum technology. By allowing coherent evolution of quantum systems and entanglement across multiple probes, the precision of estimating a fully connected $k$-body interaction can scale up to $(n^kt)^{-1}$, where $n$ is the number of probes and $t$ is the probing time. However, the optimal scaling may no longer be achievable under quantum noise, and it is important to apply quantum error correction in order to recover this limit. In this work, we study the performance of stabilizer quantum error correcting codes in estimating many-body Hamiltonians under noise. When estimating a fully connected $ZZZ$ interaction under single-qubit noise, we showcase three families of stabilizer codes -- thin surface codes, quantum Reed--Muller codes and Shor codes -- that achieve the scalings of $(nt)^{-1}$, $(n^2t)^{-1}$ and $(n^3t)^{-1}$, respectively, all of which are optimal with $t$. We further discuss the relation between stabilizer structure and the scaling with $n$, and identify several no-go theorems. For instance, we find codes with constant-weight stabilizer generators can at most achieve the $n^{-1}$ scaling, while the optimal $n^{-3}$ scaling is achievable if and only if the code bears a repetition code substructure, like in Shor code.

We propose a class of metrological resource states whose quantum Fisher information scales optimally in both system size and noise rate. In these states, qubits are partitioned into sensing groups with relatively large correlations within a group but small correlations between groups. The states are obtainable from local Hamiltonian evolution, and we design a metrologically optimal and efficient measurement protocol utilizing time-reversed dynamics and single-qubit local measurements. Using quantum domino dynamics, we also present a protocol free of the time-reversal step that has an estimation error roughly twice the best possible value., Comment: 7 + 17 pages, 3 pages

We show that quantum entanglement can provide an exponential advantage in learning properties of a bosonic continuous-variable (CV) system. The task we consider is estimating a probabilistic mixture of displacement operators acting on $n$ bosonic modes, called a random displacement channel. We prove that if the $n$ modes are not entangled with an ancillary quantum memory, then the channel must be sampled a number of times exponential in $n$ in order to estimate its characteristic function to reasonable precision; this lower bound on sample complexity applies even if the channel inputs and measurements performed on channel outputs are chosen adaptively. On the other hand, we present a simple entanglement-assisted scheme that only requires a number of samples independent of $n$, given a sufficient amount of squeezing. This establishes an exponential separation in sample complexity. We then analyze the effect of photon loss and show that the entanglement-assisted scheme is still significantly more efficient than any lossless entanglement-free scheme under mild experimental conditions. Our work illuminates the role of entanglement in learning continuous-variable systems and points toward experimentally feasible demonstrations of provable entanglement-enabled advantage using CV quantum platforms., Comment: 7+26 pages, 3+5 figures

The Heisenberg limit (HL) and the standard quantum limit (SQL) are two quantum metrological limits, which describe the scalings of estimation precision $\Delta \hat\theta$ of an unknown parameter $\theta$ with respect to $n$, the number of one-parameter quantum channels applied. It was known that the HL ($\Delta \hat\theta \propto 1/n$) is achievable using quantum error correction (QEC) strategies when the ``Hamiltonian-not-in-Kraus-span'' (HNKS) condition is satisfied; and when HNKS is violated, the SQL ($\Delta \hat\theta \propto 1/n^{1/2}$) is optimal and can be achieved with $n$ repeated measurements. However, it is unknown whether such limits are still achievable using restricted quantum devices where the required QEC operations are not available -- e.g., finite-size devices where only unitary controls are available or where noiseless ancilla is not available. In this work, we identify various new noisy metrological limits for estimating one-parameter qubit channels in different settings with restricted controls. The HL is proven to be unattainable in these cases, indicating the necessity of QEC in achieving the HL. Furthermore, we find a necessary and sufficient condition for qubit channels to attain the SQL, called the ``rotation-generators-not-in-Kraus-span'' (RGNKS) condition. When RGNKS is satisfied, the SQL is achievable using only unitary controls and a single measurement. When RGNKS is violated, the estimation precision (in most cases) has a constant floor when repeated measurements are not allowed. Demonstration of this separation in metrological powers is within reach of current quantum technologies., Comment: 38 pages, 1 figure, preliminary version, comments welcome

Quantum entanglement is a crucial resource for learning properties from nature, but a precise characterization of its advantage can be challenging. In this work, we consider learning algorithms without entanglement to be those that only utilize states, measurements, and operations that are separable between the main system of interest and an ancillary system. Interestingly, we show that these algorithms are equivalent to those that apply quantum circuits on the main system interleaved with mid-circuit measurements and classical feedforward. Within this setting, we prove a tight lower bound for Pauli channel learning without entanglement that closes the gap between the best-known upper and lower bound. In particular, we show that $\Theta(2^n\varepsilon^{-2})$ rounds of measurements are required to estimate each eigenvalue of an $n$-qubit Pauli channel to $\varepsilon$ error with high probability when learning without entanglement. In contrast, a learning algorithm with entanglement only needs $\Theta(\varepsilon^{-2})$ copies of the Pauli channel. The tight lower bound strengthens the foundation for an experimental demonstration of entanglement-enhanced advantages for Pauli noise characterization., Comment: 20 pages, 2 figure; v2: add Fig.2 comparing our lower bound with noisy-entanglement-assisted upper bound, close to accepted version

Quantum error correction (QEC) is theoretically capable of achieving the ultimate estimation limits in noisy quantum metrology. However, existing quantum error-correcting codes designed for noisy quantum metrology generally exploit entanglement between one probe and one noiseless ancilla of the same dimension, and the requirement of noiseless ancillas is one of the major obstacles to implementing the QEC metrological protocol in practice. Here we successfully lift this requirement by explicitly constructing two types of multi-probe quantum error-correcting codes, where the first one utilizes a negligible amount of ancillas and the second one is ancilla-free. Specifically, we consider Hamiltonian estimation under Markovian noise and show that (i)~when the Heisenberg limit (HL) is achievable, our new codes can achieve the HL and its optimal asymptotic coefficient; (ii)~when only the standard quantum limit (SQL) is achievable (even with arbitrary adaptive quantum strategies), the optimal asymptotic coefficient of the SQL is also achievable by our new codes under slight modifications., Comment: 40 pages, 6 figures, new results added, an error fixed, published version

Measurement noise is a major source of noise in quantum metrology. Here, we explore preprocessing protocols that apply quantum controls to the quantum sensor state prior to the final noisy measurement (but after the unknown parameter has been imparted), aiming to maximize the estimation precision. We define the quantum preprocessing-optimized Fisher information, which determines the ultimate precision limit for quantum sensors under measurement noise, and conduct a thorough investigation into optimal preprocessing protocols. First, we formulate the preprocessing optimization problem as a biconvex optimization using the error observable formalism, based on which we prove that unitary controls are optimal for pure states and derive analytical solutions of the optimal controls in several practically relevant cases. Then we prove that for classically mixed states (whose eigenvalues encode the unknown parameter) under commuting-operator measurements, coarse-graining controls are optimal, while unitary controls are suboptimal in certain cases. Finally, we demonstrate that in multi-probe systems where noisy measurements act independently on each probe, the noiseless precision limit can be asymptotically recovered using global controls for a wide range of quantum states and measurements. Applications to noisy Ramsey interferometry and thermometry are presented, as well as explicit circuit constructions of optimal controls., Comment: 41 pages, 3 figures, published version

Mitigating errors in quantum information processing devices is especially important in the absence of fault tolerance. An effective method in suppressing state-preparation errors is using multiple copies to distill the ideal component from a noisy quantum state. Here, we use classical shadows and randomized measurements to circumvent the need for coherent access to multiple copies at an exponential cost. We study the scaling of resources using numerical simulations and find that the overhead is still favorable compared to full state tomography. We optimize measurement resources under realistic experimental constraints and apply our method to an experiment preparing Greenberger-Horne-Zeilinger (GHZ) state with trapped ions. In addition to improving stabilizer measurements, the analysis of the improved results reveals the nature of errors affecting the experiment. Hence, our results provide a directly applicable method for mitigating errors in near-term quantum computers., Comment: 16 pages, 9 figures

Spin systems are an attractive candidate for quantum-enhanced metrology. Here we develop a variational method to generate metrological states in small dipolar-interacting ensembles with limited qubit controls and unknown spin locations. The generated states enable sensing beyond the standard quantum limit (SQL) and approaching the Heisenberg limit (HL). Depending on the circuit depth and the level of readout noise, the resulting states resemble Greenberger-Horne-Zeilinger (GHZ) states or Spin Squeezed States (SSS). Sensing beyond the SQL holds in the presence of finite spin polarization and a non-Markovian noise environment., Comment: 6 pages, 4 figures, 1 table in main text. 21 pages, 8 figures, 3 tables in supplemental material

We systematically study the fundamental competition between quantum error correction (QEC) and continuous symmetries, two key notions in quantum information and physics, in a quantitative manner. Three meaningful measures of approximate symmetries in quantum channels and in particular QEC codes, respectively based on the violation of covariance conditions over the entire symmetry group or at a local point, and the violation of charge conservation, are introduced and studied. Each measure induces a corresponding characterization of approximately covariant codes. We explicate a host of different ideas and techniques that enable us to derive various forms of trade-off relations between the QEC inaccuracy and all symmetry violation measures. More specifically, we introduce two frameworks for understanding and establishing the trade-offs respectively based on the notions of charge fluctuation and gate implementation error, and employ methods including the Knill--Laflamme conditions as well as quantum metrology and quantum resource theory for the derivation. From the perspective of fault-tolerant quantum computing, our bounds on symmetry violation indicate limitations on the precision or density of transversally implementable logical gates for general QEC codes, refining the Eastin--Knill theorem. To exemplify nontrivial approximately covariant codes and understand the achievability of the above fundamental limits, we analyze the behaviors of two explicit types of codes: a parametrized extension of the thermodynamic code (which gives a construction of a code family that continuously interpolates between exact QEC and exact symmetry), and the quantum Reed--Muller codes. We show that both codes can saturate the scaling of the bounds for group-global covariance and charge conservation asymptotically, indicating the near-optimality of these bounds and codes., Comment: 46 pages, 4 figures, long version of arXiv:2111.06355 published as supplementary information

Quantum error correction (QEC) is a key concept in quantum computation as well as many areas of physics. There are fundamental tensions between continuous symmetries and QEC. One vital situation is unfolded by the Eastin--Knill theorem, which forbids the existence of QEC codes that admit transversal continuous symmetry actions (transformations). Here, we systematically study the competition between continuous symmetries and QEC in a quantitative manner. We first define a series of meaningful measures of approximate symmetries motivated from different perspectives, and then establish a series of trade-off bounds between them and QEC accuracy utilizing multiple different methods. Remarkably, the results allow us to derive general quantitative limitations of transversally implementable logical gates, an important topic in fault-tolerant quantum computation. As concrete examples, we showcase two explicit types of quantum codes, obtained from quantum Reed--Muller codes and thermodynamic codes, respectively, that nearly saturate our bounds. Finally, we discuss several potential applications of our results in physics., Comment: 19 pages, 2 figures, published version, concise version of arXiv:2111.06360

We show that entangled measurements provide an exponential advantage in sample complexity for Pauli channel estimation, which is both a fundamental problem and a practically important subroutine for benchmarking near-term quantum devices. The specific task we consider is to simultaneously learn all the eigenvalues of an $n$-qubit Pauli channel to $\pm\varepsilon$ precision. We give an estimation protocol with an $n$-qubit ancilla that succeeds with high probability using only $O(n/\varepsilon^{2})$ copies of the Pauli channel, while prove that any ancilla-free protocol (possibly with adaptive control and channel concatenation) would need at least $\Omega(2^{n/3})$ rounds of measurement. We further study the advantages provided by a small number of ancillas. For the case that a $k$-qubit ancilla ($k\le n$) is available, we obtain a sample complexity lower bound of $\Omega(2^{(n-k)/3})$ for any non-concatenating protocol, and a stronger lower bound of $\Omega(n2^{n-k})$ for any non-adaptive, non-concatenating protocol, which is shown to be tight. We also show how to apply the ancilla-assisted estimation protocol to a practical quantum benchmarking task in a noise-resilient and sample-efficient manner, given reasonable noise assumptions. Our results provide a practically-interesting example for quantum advantages in learning and also bring new insight for quantum benchmarking., Comment: 21 pages, 5 figures. Introduction rewritten, additional references added, typo corrected

Quantum sensing is an important application of emerging quantum technologies. We explore whether a hybrid system of quantum sensors and quantum circuits can surpass the classical limit of sensing. In particular, we use optimization techniques to search for encoder and decoder circuits that scalably improve sensitivity under given application and noise characteristics. Our approach uses a variational algorithm that can learn a quantum sensing circuit based on platform-specific control capacity, noise, and signal distribution. The quantum circuit is composed of an encoder which prepares the optimal sensing state and a decoder which gives an output distribution containing information of the signal. We optimize the full circuit to maximize the Signal-to-Noise Ratio (SNR). Furthermore, this learning algorithm can be run on real hardware scalably by using the "parameter-shift" rule which enables gradient evaluation on noisy quantum circuits, avoiding the exponential cost of quantum system simulation. We demonstrate up to 13.12x SNR improvement over existing fixed protocol (GHZ), and 3.19x Classical Fisher Information (CFI) improvement over the classical limit on 15 qubits using IBM quantum computer. More notably, our algorithm overcomes the decreasing performance of existing entanglement-based protocols with increased system sizes., Comment: 12 pages, 11 figures

Characterizing the memory properties of the environment has become critical for the high-fidelity control of qubits and other advanced quantum systems. However, current non-Markovian tomography techniques are either limited to discrete superoperators, or they employ machine learning methods, neither of which provide physical insight into the dynamics of the quantum system. To circumvent this limitation, we design learning architectures that explicitly encode physical constraints like the properties of completely-positive trace-preserving maps in a differential form. This method preserves the versatility of the machine learning approach without sacrificing the efficiency and fidelity of traditional parameter estimation methods. Our approach provides the physical interpretability that machine learning and opaque superoperators lack. Moreover, it is aware of the underlying continuous dynamics typically disregarded by superoperator-based tomography. This paradigm paves the way to noise-aware optimal quantum control and opens a path to exploiting the bath as a control and error mitigation resource.

We analyze the ultimate quantum limit of resolving two identical sources in a noisy environment. We prove that in the presence of noise causing false excitation, such as thermal noise, the quantum Fisher information of arbitrary quantum states for the separation of the objects, which quantifies the resolution, always converges to zero as the separation goes to zero. Noisy cases contrast with a noiseless case where it has been shown to be nonzero for a small distance in various circumstances, revealing the superresolution. In addition, we show that false excitation on an arbitrary measurement, such as dark counts, also makes the classical Fisher information of the measurement approach to zero as the separation goes to zero. Finally, a practically relevant situation resolving two identical thermal sources, is quantitatively investigated by using the quantum and classical Fisher information of finite spatial mode multiplexing, showing that the amount of noise poses a limit on the resolution in a noisy system., Comment: 12 pages, 2 figures

Covariant codes are quantum codes such that a symmetry transformation on the logical system could be realized by a symmetry transformation on the physical system, usually with limited capability of performing quantum error correction (an important case being the Eastin--Knill theorem). The need for understanding the limits of covariant quantum error correction arises in various realms of physics including fault-tolerant quantum computation, condensed matter physics and quantum gravity. Here, we explore covariant quantum error correction with respect to continuous symmetries from the perspectives of quantum metrology and quantum resource theory, establishing solid connections between these formerly disparate fields. We prove new and powerful lower bounds on the infidelity of covariant quantum error correction, which not only extend the scope of previous no-go results but also provide a substantial improvement over existing bounds. Explicit lower bounds are derived for both erasure and depolarizing noises. We also present a type of covariant codes which nearly saturates these lower bounds., Comment: 25 pages, 2 figures

The quantum Fisher information (QFI), as a function of quantum states, measures the amount of information that a quantum state carries about an unknown parameter. The (entanglement-assisted) QFI of a quantum channel is defined to be the maximum QFI of the output state assuming an entangled input state over a single probe and an ancilla. In quantum metrology, people are interested in calculating the QFI of $N$ identical copies of a quantum channel when $N \rightarrow \infty$, which is called the asymptotic QFI. Over the years, researchers found various types of upper bounds of the asymptotic QFI, but they were proven achievable only in several specific situations. It was known that the asymptotic QFI of an arbitrary quantum channel grows either linearly or quadratically with $N$. Here we show that a simple criterion can determine whether the scaling is linear or quadratic. In both cases, the asymptotic QFI and a quantum error correction protocol to achieve it are computable via a semidefinite program. When the scaling is quadratic, the Heisenberg limit, a feature of noiseless quantum channels, is recovered. When the scaling is linear, we show the asymptotic QFI is still in general larger than $N$ times the single-channel QFI and furthermore, sequential estimation strategies provide no advantage over parallel ones., Comment: 23 pages, 4 figures, published version

The quantum information and the Bures metric are equivalent to each other, except at points where the rank of the density matrix changes. Here we show that by slightly modifying the definition of the Bures metric, the quantum information will be fully equivalent to the Bures metric without exception., Comment: 5 pages

For a generic set of Markovian noise models, the estimation precision of a parameter associated with the Hamiltonian is limited by the $1/\sqrt{t}$ scaling where $t$ is the total probing time, in which case the maximal possible quantum improvement in the asymptotic limit of large $t$ is restricted to a constant factor. However, situations arise where the constant factor improvement could be significant, yet no effective quantum strategies are known. Here we propose an optimal approximate quantum error correction (AQEC) strategy asymptotically saturating the precision lower bound in the most general adaptive parameter estimation scheme where arbitrary and frequent quantum controls are allowed. We also provide an efficient numerical algorithm finding the optimal code. Finally, we consider highly-biased noise and show that using the optimal AQEC strategy, strong noises are fully corrected, while the estimation precision depends only on the strength of weak noises in the limiting case., Comment: 6+7 pages, 1 figure

We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cram\'er-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace., Comment: 15+8 pages, 3 figures

Estimating the parameters governing the dynamics of a system is a prerequisite for its optimal control. We present a simple but powerful method that we call STEADY, for STochastic Estimation algorithm for DYnamical variables, to estimate the Hamiltonian (or Lindbladian) governing a quantum system of a few qubits. STEADY makes efficient use of all measurements and its performance scales as the information-theoretic limits for such an estimator. Importantly, it is inherently robust to state preparation and measurement errors. It is not limited to evaluating only a fixed set of possible gates, rather it estimates the complete Hamiltonian of the system. The estimator is applicable to any Hamiltonian that can be written as a piecewise-differentiable function and it can easily include estimators for the non-unitary parameters as well. At the heart of our approach is a stochastic gradient descent over the difference between experimental measurement and model prediction.

Quantum error correction has recently emerged as a tool to enhance quantum sensing under Markovian noise. It works by correcting errors in a sensor while letting a signal imprint on the logical state. This approach typically requires a specialized error-correcting code, as most existing codes correct away both the dominant errors and the signal. To date, however, few such specialized codes are known, among which most require noiseless, controllable ancillas. We show here that such ancillas are not needed when the signal Hamiltonian and the error operators commute; a common limiting type of decoherence in quantum sensors. We give a semidefinite program for finding optimal ancilla-free sensing codes in general, as well as closed-form codes for two common sensing scenarios: qubits undergoing dephasing, and a lossy bosonic mode. Finally, we analyze the sensitivity enhancement offered by the qubit code under arbitrary spatial noise correlations, beyond the ideal limit of orthogonal signal and noise operators., Comment: 18 pages, 2 figures. The first two authors contributed equally to this paper

The quantum Cram\'er-Rao bound (QCRB) provides an ultimate precision limit allowed by quantum mechanics in parameter estimation. Given any quantum state dependent on a single parameter, there is always a positive-operator valued measurement (POVM) saturating the QCRB. However, the QCRB-saturating POVM cannot always be implemented efficiently, especially in multipartite systems. In this paper, we show that the POVM based on local operations and classical communication (LOCC) is QCRB-saturating for arbitrary pure states or rank-two mixed states with varying probability distributions over fixed eigenbasis. Local measurements without classical communication, however, is not QCRB-saturating in general., Comment: 13 pages, 2 figures, published version

Rayleigh's criterion states that it becomes essentially difficult to resolve two incoherent optical point sources separated by a distance below the width of point spread functions (PSF), namely in the subdiffraction limit. Recently, researchers have achieved superresolution for two incoherent point sources with equal strengths using a new type of measurement technique, surpassing Rayleigh's criterion. However, situations where more than two point sources needed to be resolved have not been fully investigated. Here we prove that for any incoherent sources with arbitrary strengths, a one- or two-dimensional image can be precisely resolved up to its second moment in the subdiffraction limit, i.e. the Fisher information (FI) is non-zero. But the FI with respect to higher order moments always tends to zero polynomially as the size of the image decreases, for any type of non-adaptive measurement. We call this phenomenon a modern description of Rayleigh's criterion. For PSFs under certain constraints, the optimal measurement basis estimating all moments in the subdiffraction limit for 1D weak-source imaging is constructed. Such basis also generates the optimal-scaling FI with respect to the size of the image for 2D or strong-source imaging, which achieves an overall quadratic improvement compared to direct imaging., Comment: 28 pages, 2 figures; summary of results added, further justification of the order-of-magnitude argument added

Quantum metrology has many important applications in science and technology, ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on measurement precision, called the Heisenberg limit, which can be achieved for noiseless quantum systems, but is not achievable in general for systems subject to noise. Here we study how measurement precision can be enhanced through quantum error correction, a general method for protecting a quantum system from the damaging effects of noise. We find a necessary and sufficient condition for achieving the Heisenberg limit using quantum probes subject to Markovian noise, assuming that noiseless ancilla systems are available, and that fast, accurate quantum processing can be performed. When the sufficient condition is satisfied, a quantum error-correcting code can be constructed which suppresses the noise without obscuring the signal; the optimal code, achieving the best possible precision, can be found by solving a semidefinite program., Comment: 16 pages, 2 figures, see also arXiv:1704.06280