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Considering the emerging applications of quantum technologies, studying energy storage and usage at the quantum level is of great interest. In this context, there is a significant contemporary interest in studying ergotropy, the maximum amount of work that can be extracted unitarily from an energy-storing quantum device. Here, we propose and experimentally demonstrate a feedback-based algorithm (FQErgo) for estimating ergotropy. This method also transforms an arbitrary initial state to its passive state, which allows no further unitary work extraction. FQErgo applies drive fields whose strengths are iteratively adjusted via certain expectation values, conveniently read using a single probe qubit. Thus, FQErgo provides a practical way for unitary energy extraction and for preparing passive states. By numerically analyzing FQErgo on random initial states, we confirm the successful preparation of passive states and estimation of ergotropy, even in the presence of drive errors. Finally, we implement FQErgo on two- and three-qubit NMR registers, prepare their passive states, and accurately estimate their ergotropy.

Classical and quantum machine learning are being increasingly applied to various tasks in quantum information technologies. Here, we present an experimental demonstration of quantum control using a physics-informed neural network (PINN). PINN's salient feature is how it encodes the entire control sequence in terms of its network parameters. This feature enables the control sequence to be later adopted to any hardware with optimal time discretization, which contrasts with conventional methods involving a priory time discretization. Here, we discuss two important quantum information tasks: gate synthesis and state preparation. First, we demonstrate quantum gate synthesis by designing a two-qubit CNOT gate and experimentally implementing it on a heteronuclear two-spin NMR register. Second, we demonstrate quantum state preparation by designing a control sequence to efficiently transfer the thermal state into the long-lived singlet state and experimentally implement it on a homonuclear two-spin NMR register. We present a detailed numerical analysis of the PINN control sequences regarding bandwidth, discretization levels, control field errors, and external noise.

Designing efficient and robust quantum control strategies is vital for developing quantum technologies. One recent strategy is the Quantum Alternating Operator Ansatz (QAOA) sequence that alternatively propagates under two noncommuting Hamiltonians, whose control parameters can be optimized to generate a gate or prepare a state. Here, we describe the design of the QAOA sequence and their variants to prepare long-lived singlet states (LLS) from the thermal state in NMR. With extraordinarily long lifetimes exceeding the spin-lattice relaxation time constant $T_1$, LLS have been of great interest for various applications, from spectroscopy to medical imaging. Accordingly, designing sequences for efficiently preparing LLS in a general spin system is crucial. Using numerical analysis, we study the efficiency and robustness of the QAOA sequences over a wide range of errors in the control parameters. Using a two-qubit NMR register, we conduct an experimental study to benchmark QAOA sequences against other prominent methods of LLS preparation and observe the significantly superior performance of the QAOA sequences., Comment: 9 pages, 8 figures

The quasiprobability distribution of the discrete Wigner function provides a complete description of a quantum state and is, therefore, a useful alternative to the usual density matrix description. Moreover, the experimental quantum state tomography in discrete Wigner phase space can also be implemented. We observe that for a certain class of states, such as harmonic states, the Wigner matrix is far more sparse compared to the density matrix in the computational basis. Additionally, reading only a small part of the Wigner matrix may suffice to infer certain behavior of quantum dynamics. In such cases, selective Wigner phase space tomography (SWPST) can be more efficient than the usual density matrix tomography (DMT). Employing nuclear magnetic resonance methods on a three-qubit nuclear spin register, we experimentally estimate Wigner matrices of various two-qubit quantum states. As a specific example application of SWPST, we study the evolution of spin coherent states under the quantum chaotic kicked top model and extract signatures of quantum-classical correspondence in the Wigner phase space.

Lee-Yang (LY) zeros, points on the complex plane of physical parameters where the partition function goes to zero, have found diverse applications across multiple disciplines like statistical physics, protein folding, percolation, complex networks etc. However, experimental extraction of the complete set of LY zeros for general asymmetrical classical systems remains a crucial challenge to put those applications into practice. Here, we propose a qubit-based method to simulate an asymmetrical classical Ising system, enabling the exploration of LY zeros at arbitrary values of physical parameters like temperature, internal couplings etc. Without assuming system symmetry, the full set of LY zeros forms an algebraic variety in a higher-dimensional complex plane. To determine this variety, we pro ject it into sets representing magnitudes (amoeba ) and phases (coamoeba ) of LY zeros. Our approach uses a probe qubit to initialize the system and to extract LY zeros without assuming any control over the system qubits. This is particularly important as controlling system qubits can get intractable with the increasing complexity of the system. Initializing the system at an amoeba point, coamoeba points are sampled by measuring probe qubit dynamics. Iterative sampling yields the entire algebraic variety. Experimental demonstration of the protocol is achieved through a three-qubit NMR register. This work expands the horizon of quantum simulation to domains where identifying LY zeros in general classical systems is pivotal. Moreover, by extracting abstract mathematical objects like amoeba and coamoeba for a given polynomial, our study integrates pure mathematical concepts into the realm of quantum simulations.

The phenomenon of quantum entanglement underlies several important protocols that enable emerging quantum technologies. Entangled states, however, are extremely delicate and often get perturbed by tiny fluctuations in their external environment. Certification of entanglement is therefore immensely crucial for the successful implementation of protocols involving this resource. In this work, we propose a set of entanglement criteria for multi-qubit systems that can be easily verified by measuring certain thermodynamic quantities. In particular, the criteria depend on the difference in optimal global and local works extractable from an isolated quantum system under global and local interactions, respectively. As a proof of principle, we demonstrate the proposed scheme on nuclear spin registers of up to 10 qubits using the Nuclear Magnetic Resonance architecture. We prepare noisy Bell-diagonal state and noisy Greenberger-Horne-Zeilinger class of states in star-topology systems and certify their entanglement through our thermodynamic criteria. Along the same line, we also propose an entanglement certification scheme in many-body systems when only partial or even no knowledge about the state is available., Comment: 4 pages (two-column) + 16.5 pages (one-column) + 12 figures; Accepted in Physical Review A (Letter); Close to accepted version

The quantum adiabatic method, which maintains populations in their instantaneous eigenstates throughout the state evolution, is an established and often a preferred choice for state preparation and manipulation. Though it minimizes the driving cost significantly, its slow speed is a severe limitation in noisy intermediate-scale quantum (NISQ) era technologies. Since adiabatic paths are extensive in many physical processes, it is of broader interest to achieve adiabaticity at a much faster rate. Shortcuts to adiabaticity techniques which overcome the slow adiabatic process by driving the system faster through non-adiabatic paths, have seen increased attention recently. The extraordinarily long lifetime of the long-lived singlet states (LLS) in nuclear magnetic resonance, established over the past decade, has opened several important applications ranging from spectroscopy to biomedical imaging. Various methods, including adiabatic methods, are already being used to prepare LLS. In this article, we report the use of counterdiabatic driving (CD) to speed up LLS preparation with faster drives. Using NMR experiments, we show that CD can give stronger LLS order in shorter durations than conventional adiabatic driving.

Long-lived singlet states (LLS) of nuclear spin pairs have been extensively studied and utilized in the isotropic phase via liquid state NMR. However, there are hardly any reports of LLS in the anisotropic phase that allows contribution from the dipolar coupling in addition to the scalar coupling, thereby opening many exciting possibilities. Here we report observing LLS in a pair of nuclear spins partially oriented in the nematic phase of a liquid crystal solvent. The spins are strongly interacting via the residual dipole-dipole coupling. We observe LLS in the oriented phase living up to three times longer than the usual spin-lattice relaxation time constant ($T_1$). Upon heating, the system undergoes a phase transition from nematic into isotropic phase, wherein the LLS is up to five times longer lived than the corresponding $T_1$. Interestingly, the LLS prepared in the oriented phase can survive the transition from the nematic to the isotropic phase. As an application of LLS in the oriented phase, we utilize its longer life to measure the small translational diffusion coefficient of solute molecules in the liquid crystal solvent. Finally, we propose utilizing the phase transition to lock or unlock access to LLS., Comment: 10 pages, 10 figures

In the usual quantum tunneling, a low-energy quantum particle penetrates across a physical barrier of higher potential energy, by traversing a classically forbidden region, and finally escapes into another region. In an analogous scenario, a classical particle inside a closed regular region in the phase space is dynamically bound from escaping to other regions of the phase space. Here, the physical potential barrier is replaced by dynamical barriers which separate different regions of the phase space. However, in the quantum regime, the system can overcome such dynamical barriers and escape through them, giving rise to dynamical tunneling. In chaotic Hamiltonian systems, dynamical tunneling refers to quantum tunneling between states whose classical limit correspond to symmetry-related regular regions separated by a chaotic zone between which any classical transport is prohibited. Here, an experimental realization of dynamical tunneling in spin systems is reported using nuclear magnetic resonance (NMR) architecture. In particular, dynamical tunneling in quantum kicked tops of spin-1 and spin-3/2 systems using two- and three-qubit NMR registers is investigated. By extracting time-dependent expectation values of the angular momentum operator components, size-dependent tunneling behaviour for various initial states is systematically investigated. Further, by monitoring the adverse effects of dephasing noise on the tunneling oscillations, we assert the importance of quantum coherence in enabling dynamical tunneling.

Quantum controls realize the unitary or nonunitary operations employed in quantum computers, quantum simulators, quantum communications, and other quantum information devices. They implement the desired quantum dynamics with the help of electric, magnetic, or electromagnetic control fields. Quantum optimal control (QOC) deals with designing an optimal control field modulation that most precisely implements a desired quantum operation with minimum energy consumption and maximum robustness against hardware imperfections as well as external noise. Over the last two decades, numerous QOC methods have been proposed. They include asymptotic methods, direct search, gradient methods, variational methods, machine learning methods, etc. In this review, we shall introduce the basic ideas of QOC, discuss practical challenges, and then take an overview of the diverse QOC methods.

Quantum control optimization algorithms are routinely used to generate optimal quantum gates or efficient quantum state transfers. However, there are two main challenges in designing efficient optimization algorithms, namely overcoming the sensitivity to local optima and improving the computational speed. The former challenge can be dealt with by designing hybrid algorithms, such as a combination of gradient and simulated annealing methods. Here, we propose and demonstrate the use of a machine learning method, specifically the recommender system (RS), to deal with the latter challenge of enhancing computational efficiency. We first describe ways to set up a rating matrix involving gradients or gate fidelities. We then establish that RS can rapidly and accurately predict elements of a sparse rating matrix. Using this approach, we expedite a gradient ascent based quantum control optimization, namely GRAPE and demonstrate the faster performance for up to 8 qubits. Finally, we describe and implement the enhancement of the computational speed of a hybrid algorithm, namely SAGRAPE., Comment: 7 pages, 4 figures, 2 tables

Theoretical explorations have revealed that quantum batteries can exploit quantum correlation to achieve faster charging, thus promising exciting applications in future technologies. Using NMR architecture, here we experimentally investigate various aspects of quantum battery with the help of nuclear spin systems in star-topology configuration. We first carry out numerical analysis to study how charging a quantum battery depends on the relative purity factors of charger and battery spins. By experimentally measuring the polarization of the battery spin undergoing charging, we estimate the battery energy and establish the theoretically predicted quantum advantage. We propose using the quantum advantage, which depends on the entanglement among chargers and battery, as a measure for estimating the size of the entangled cluster. We develop a simple iterative method to realize asymptotic charging avoiding oscillatory behaviour of charging and discharging. Finally, we introduce a load spin and realize a charger-battery-load circuit and experimentally demonstrate battery energy consumption after varying durations of battery storage, for up to two minutes.

We report an experimental study of phase-synchronization in a pair of interacting nuclear spins subjected to an external drive in nuclear magnetic resonance architecture. A weak transition-selective radio-frequency field applied on one of the spins is observed to cause phase-localization, which is experimentally established by measuring the Husimi distribution function under various drive conditions. To this end, we have developed a general interferometric technique to directly extract values of the Husimi function via the transverse magnetization of the undriven nuclear spin. We further verify the robustness of synchronization to detuning in the system by studying the Arnold tongue behaviour.

The development of efficient algorithms that generate robust quantum controls is crucial for the realization of quantum technologies. The commonly used gradient-based optimization algorithms are limited by their sensitivity to the initial guess, which affects their performance. Here we propose combining the gradient method with the simulated annealing technique to formulate a hybrid algorithm. Our numerical analysis confirms its superior convergence rate. Using the hybrid algorithm, we generate spin-selective $\pi$ pulses and employ them for experimental measurement of local noise-spectra in a three-qubit NMR system. Moreover, here we describe a general method to construct noise-resilient quantum controls by incorporating noisy fields within the optimization routine of the hybrid algorithm. On experimental comparison with similar sequences obtained from standard algorithms, we find remarkable robustness of the hybrid sequences against dephasing errors., Comment: 8 pages, 7 figures

In recent years, the performance of different entanglement indicators obtained directly from tomograms has been assessed in continuous-variable and hybrid quantum systems. In this paper, we carry out this task in the case of spin systems. We compute the entanglement indicators from actual experimental data obtained from three liquid-state NMR experiments, and compare them with standard entanglement measures calculated from the corresponding density matrices, both experimentally reconstructed and numerically computed. The gross features of entanglement dynamics and spin squeezing properties are found to be reproduced by these entanglement indicators. However, the extent to which these indicators and spin squeezing track the entanglement during time evolution of the multipartite systems in the NMR experiments is very sensitive to the precise nature and strength of interactions as well as the manner in which the full system is partitioned into subsystems. We also use the IBM quantum computer to implement equivalent circuits that capture the dynamics of the multipartite system in one of the NMR experiments. We compute and compare the entanglement indicators obtained from the tomograms corresponding to the experimental execution and simulation of these equivalent circuits. This exercise shows that these indicators can estimate the degree of entanglement without necessitating detailed state reconstruction procedures, establishing the advantage of the tomographic approach., Comment: 26 pages, 17 figures

With the advent of quantum technology, nitrogen vacancy ($NV$) centers in diamond turn out to be a frontier which provides an efficient platform for quantum computation, communication and sensing applications. Due to the coupled spin-charge dynamics of the $NV$ system, knowledge about $NV$ charge state dynamics can help to formulate efficient spin control sequences strategically. Through this paper we report two spectroscopy-based deconvolution methods to create charge state mapping images of ensembles of $NV$ centers in diamond. First, relying on the fact that an off axis external magnetic field mixes the electronic spins and selectively modifies the photoluminescence (PL) of $NV^-$, we perform decomposition of the optical spectrum for an ensemble of $NV$s and extract the spectra for $NV^-$ and $NV^0$ states. Next, we introduce an optical filter based decomposition protocol and perform PL imaging for $NV^-$ and $NV^0$. Earlier obtained spectra for $NV^-$ and $NV^0$ states are used to calculate their transmissivities through a long pass optical filter. These results help us to determine the spatial distribution of the $NV$ charge states in a diamond sample., Comment: 7 pages, 6 figures

Quantum control of large spin registers is crucial for many applications ranging from spectroscopy to quantum information. A key factor that determines the efficiency of a register for implementing a given information processing task is its network topology. One particular type, called star-topology, involves a central qubit uniformly interacting with a set of ancillary qubits. A particular advantage of the star-topology quantum registers is in the efficient preparation of large entangled states, called NOON states, and their generalized variants. Thanks to the robust generation of such correlated states, spectral simplicity, ease of polarization transfer from ancillary qubits to the central qubit, as well as the availability of large spin-clusters, the star-topology registers have been utilized for several interesting applications over the last few years. Here we review some recent progress with the star-topology registers, particularly via nuclear magnetic resonance methods., Comment: 15 pages, 10 figures

We demonstrate characterizing quantum evolutions via matrix factorization algorithm, a particular type of the recommender system (RS). A system undergoing a quantum evolution can be characterized in several ways. Here we choose (i) quantum correlations quantified by measures such as entropy, negativity, or discord, and (ii) state-fidelity. Using quantum registers with up to 10 qubits, we demonstrate that an RS can efficiently characterize both unitary and nonunitary evolutions. After carrying out a detailed performance analysis of the RS in two qubits, we show that it can be used to distinguish a clean database of quantum correlations from a noisy or a fake one. Moreover, we find that the RS brings about a significant computational advantage for building a large database of quantum discord, for which no simple closed-form expression exists. Also, RS can efficiently characterize systems undergoing nonunitary evolutions in terms of quantum discord reduction as well as state-fidelity. Finally, we utilize RS for the construction of discord phase space in a nonlinear quantum system., Comment: Accepted in Quantum

We experimentally emulate interaction induced blockade and local spin freezing in two and three qubit Nuclear Magnetic Resonance (NMR) architecture. These phenomena are identical to the Rydberg blockade and Rydberg biased freezing. In Rydberg blockade, the simultaneous excitation of two or more atoms is blocked due to the level shift induced by the strong Van der Waal's interaction. In such a strong interaction regime, one can also observe Rydberg biased freezing, wherein the dynamics is confined to a subspace, with the help of multiple drives with unequal amplitudes. Here we drive NMR qubits with specific transition-selective radio waves, while intermittently characterizing the quantum states via quantum state tomography. This not only allows us to track the population dynamics, but also helps to probe quantum correlations, by means of quantum discord, evolving under blockade and freezing phenomena. While, our work constitutes the first experimental simulations of these phenomena in the NMR platform, it is also the first experimental demonstration of Rydberg biased freezing. Moreover, these studies open up interesting quantum control perspectives in exploiting the above phenomena for entanglement generation as well as subspace manipulations., Comment: 9 pages, 6 figures

A cost-precision trade-off relationship, the so-called thermodynamic uncertainty relation (TUR), has been recently discovered in stochastic thermodynamics. It bounds certain thermodynamic observables in terms of the associated entropy production. In this work, we experimentally study the TUR in a two-qubit system using an NMR setup. Each qubit is prepared in an equilibrium state, but at different temperatures. The qubits are then coupled, allowing energy exchange (in the form of heat). Using the quantum state tomography technique we obtain the moments of heat exchange within a certain time interval and analyze the relative uncertainty of the energy exchange process. We find that generalized versions of the TUR, which are based on the fluctuation relation, are obeyed. However, the specialized TUR, a tighter bound that is valid under specific dynamics, is violated in certain regimes of operation, in excellent agreement with analytic results. Altogether, this experiment-theory study provides a deep understanding of heat exchange in quantum systems, revealing favorable noise-dissipation regimes of operation., Comment: 9 pages, 6 figures

Quantum entanglement is a form of correlation between quantum particles that cannot be increased via local operations and classical communication. It has therefore been proposed that an increment of quantum entanglement between probes that are interacting solely via a mediator implies non-classicality of the mediator. Indeed, under certain assumptions regarding the initial state, entanglement gain between the probes indicates quantum coherence in the mediator. Going beyond such assumptions, there exist other initial states which produce entanglement between the probes via only local interactions with a classical mediator. In this process the initial entanglement between any probe and the rest of the system "flows through" the classical mediator and gets localised between the probes. Here we theoretically characterise maximal entanglement gain via classical mediator and experimentally demonstrate, using liquid-state NMR spectroscopy, the optimal growth of quantum correlations between two nuclear spin qubits interacting through a mediator qubit in a classical state. We additionally monitor, i.e., dephase, the mediator in order to emphasise its classical character. Our results indicate the necessity of verifying features of the initial state if entanglement gain between the probes is used as a figure of merit for witnessing non-classical mediator. Such methods were proposed to have exemplary applications in quantum optomechanics, quantum biology and quantum gravity., Comment: 9 pages, 2 figures

Quantum optimal control involves setting up an objective function that evaluates the quality of an operator representing the realized process w.r.t. the target process. Here we propose a stronger objective function which incorporates not only the target operator but also a set of its orthogonal operators. We find significantly superior convergence of optimization routines with the combined influences of all the operators. We refer to this method as the $\textit{push-pull}$ optimization. In particular, we describe adopting the push-pull optimization to a gradient based approach and a variational-principle based approach. We carry out extensive numerical simulations of the push-pull optimization of quantum controls on a pair of Ising coupled qubits. Finally, we demonstrate its experimental application by preparing a long-lived singlet-order in a two-qubit system using NMR techniques.

We investigate the scrambling of information in a hierarchical star-topology system using out-of-time-ordered correlation (OTOC) functions. The system consists of a central qubit directly interacting with a set of satellite qubits, which in turn interact with a second layer of satellite qubits. This particular topology not only allows convenient preparation and filtering of multiple quantum coherences between the central qubit and the first layer but also to engineer scrambling in a controlled manner. Hence, it provides us with an opportunity to experimentally study scrambling of information localized in multi-spin correlations via the construction of relevant OTOCs. Since the measurement of OTOC requires a time evolution, the non-scrambling processes such as decoherence and certain experimental errors create an ambiguity. Therefore, the unambiguous quantification of information scrambling requires suppressing contributions from decoherence to the OTOC dynamics. To this end, we propose and experimentally demonstrate a constant time protocol which is able to filter contribution exclusively from information scrambling., Comment: 10 pages, 7 figures, comments are welcome

We experimentally verify the Jarzynski and W\"ojcik quantum heat exchange fluctuation relation by implementing the interferometric technique in liquid-state Nuclear Magnetic Resonance setup and study the exchange heat statistics between two weakly coupled spin-1/2 quantum systems. In presence of uncorrelated initial state with individual spins prepared in local Gibbs thermal states at different temperatures, the exchange fluctuation symmetry is verified for arbitrary transient time. In contrast, when the initial preparation includes correlation, the fluctuation symmetry breaks down and further leads to an apparent spontaneous flow of heat from cold to hot. Our experimental approach is general and can be systematically extended to study heat statistics for more complex out-of-equilibrium many-body quantum systems.

Quantum chaotic kicked top model is implemented experimentally in a two qubit system comprising of a pair of spin-1/2 nuclei using Nuclear Magnetic Resonance techniques. The essential nonlinear interaction was realized using indirect spin-spin coupling, while the linear kicks were realized using RF pulses. After a variable number of kicks, quantum state tomography was employed to reconstruct the single-qubit density matrices using which we could extract various measures such as von Neumann entropies, Husimi distributions, and Lyapunov exponents. These measures enabled the study of correspondence with classical phase space as well as to probe distinct features of quantum chaos, such as symmetries and temporal periodicity in the two-qubit kicked top., Comment: 8 pages, 8 figures

We experimentally emulate, in a controlled fashion, the non-Markovian dynamics of a pure dephasing spin-boson model at zero temperature. Specifically, we use a randomized set of external radio-frequency fields to engineer a desired noise power-spectrum to effectively realize a non-Markovian environment for a single NMR qubit. The information backflow, characteristic to the non-Markovianity, is captured in the nonmonotonicity of the decoherence function and von Neumann entropy of the system. Using such emulated non-Markovian environments, we experimentally study the efficiency of the Carr-Purcell-Meiboom-Gill dynamical decoupling (DD) sequence to inhibit the loss of coherence. Using the filter function formalism, we design optimized DD sequences that maximize coherence protection for non-Markovian environments and study their efficiencies experimentally. Finally, we discuss DD-assisted tuning of the effective non-Markovianity., Comment: 8 Pages, 6 Figures, close to published version

A quantum memory interacts with its environment and loses information via decoherence as well as incoherence. A robust quantum control that prepares, preserves, and manipulates nonclassical correlations even in the presence of environmental influence is of paramount importance in quantum information processing. A well-known technique to suppress decoherence, namely Dynamical Decoupling (DD), consists of a sequence of rapid flips applied to the system in order to refocus the system-environment interactions. In this work, we integrate DD with quantum gates using optimal control techniques to realize robust quantum gates which offer protection against decoherence. To investigate the protection of non-classical correlation, we study the evolution of quantum discord in Grover's search algorithm implemented with dynamically protected gates. Using a two-qubit NMR system, we experimentally demonstrate a significant protection against decoherence and incoherence. We find better performances by phase alternating DD sequences with suitable spacings between the DD pulses. Interestingly, we also find that DD sequences based on $\pi/2$ pulses perform as well as or even better than those with $\pi$ pulses in protecting the non-classical correlation. We also support the experimental results by analyzing the robustness of various DD schemes.

Given a quantum state with an unknown parameter, the Quantum Fisher Information (QFI) is a measure of the amount of information that an observable can extract about the parameter. QFI also quantifies the maximum achievable precision in estimating the parameter with a given amount of resource via an inequality known as quantum Cramer-Rao bound. In this work, we describe a protocol to amplify QFI of a single target qubit precorrelated with a set of ancillary qubits. A single quadrature measurement of only ancillary qubits suffices to perform the complete quantum state tomography (QST) of the target qubit. We experimentally demonstrate this protocol using an NMR system consisting of a $^{13}$C nuclear spin as the target qubit and three $^1$H nuclear spins as ancillary qubits. We prepare the target qubit in various initial states, perform QST, and estimate the amplification of QFI in each case. We also show that the QFI-amplification scales linearly with the number of ancillary qubits and quadratically with their purity., Comment: 8 pages, 2 figures, missing factor of 1/2 in quadrature QFI included, minor correction in Fig. 1

We experimentally study the response of star-shaped clusters of initially unentangled $N=4$, 10 and 37 nuclear spin-$\frac{1}{2}$ moments to an inexact $\pi$-pulse sequence, and show that an Ising coupling between the centre and the satellite spins results in robust period-two magnetization oscillations. The period is stable against bath-effects but the amplitude decays with a time scale that depends on the inexactness of the pulse. Simulations reveal a semiclassical picture where the rigidity of the period is due to a randomizing effect of the Larmor precession under the magnetization of surrounding spins. The time scales with stable periodicity increase with net initial magnetization even in the presence of perturbations, indicating a robust temporal ordered phase for large systems with finite magnetization per spin.

Matrix exponentiation (ME) is widely used in various fields of science and engineering. For example, the unitary dynamics of quantum systems is described by exponentiation of Hamiltonian operators. However, despite a significant attention, the numerical evaluation of ME remains computationally expensive, particularly for large dimensions. Often this process becomes a bottleneck in algorithms requiring iterative evaluation of ME. Here we propose a method for approximating ME of a single operator with a bounded coefficient into a product of certain discrete operators. This approach, which we refer to as Rapid Exponentiation using Discrete Operators (REDO), is particularly efficient for iterating ME over large numbers. We describe REDO in the context of a quantum system with a constant as well as a time-dependent Hamiltonian, although in principle, it can be adapted in a more general setting. As concrete examples, we choose two applications. First, we incorporate REDO in optimal quantum control algorithms and report a speed-up of several folds over a wide range of system size. Secondly, we propose REDO for numerical simulations of quantum dynamics. In particular, we study exotic quantum freezing with noisy drive fields.

Using a Bang-Bang optimal control (BB) technique, we transfer polarization from abundant high-$\gamma$ nuclei directly to singlet order. This approach is analogous to algorithmic cooling (AC) procedure used in quantum state purification. Specifically, we apply this method for enhancing the singlet order in a natural abundant $^{13}$C-$^{13}$C spin pair using a set of nine equivalent protons of an 11-spin system. Compared to the standard method not involving polarization transfer, we find an enhancement of singlet order by about three times. In addition, since the singlet magnetization is contributed by the faster relaxing protons, the recycle delay is halved. Thus effectively we observe a sensitivity enhancement by 4.2 times or a reduction in the overall experimental time by a factor of 18. We also discuss a possible extension of AC, known as heat-bath algorithmic cooling (HBAC)., Comment: 7 pages, 7 figures

Cooling the qubit into a pure initial state is crucial for realizing fault-tolerant quantum information processing. Here we envisage a star-topology arrangement of reset and computation qubits for this purpose. The reset qubits cool or purify the computation qubit by transferring its entropy to a heat-bath with the help of a heat-bath algorithmic cooling procedure. By combining standard NMR methods with powerful quantum control techniques, we cool central qubits of two large star topology systems, with 13 and 37 spins respectively. We obtain polarization enhancements by a factor of over 24, and an associated reduction in the spin temperature from 298 K down to 12 K. Exploiting the enhanced polarization of computation qubit, we prepare combination-coherences of orders up to 15. By benchmarking the decay of these coherences we investigate the underlying noise process. Further, we also cool a pair of computation qubits and subsequently prepare them in an effective pure-state., Comment: 9 pages, 10 figures; close to published version

Quantum processors are potentially superior to their classical counterparts for many computational tasks including factorization. Circuit methods as well as adiabatic methods have already been proposed and implemented for finding the factors of a given composite number. The main challenge in scaling it to larger numbers is the unavailability of large number of qubits. Here we propose a hybrid scheme that involves both classical and quantum computation, which reduces the number of qubits required for factorization. The classical part involves setting up and partially simplifying a set of bit-wise factoring equations and the quantum part involves solving these coupled equations using a quantum adiabatic process. We demonstrate the hybrid scheme by factoring 551 using a three qubit NMR quantum register.

Characterizing and understanding noise affecting quantum states has immense benefits in spectroscopy as well as in realizing quantum devices. Transverse relaxation times under a set of dynamical decoupling (DD) sequences with varying interpulse delays were earlier used for obtaining the noise spectral densities of single-qubit coherences. In this work, using a pair of homonuclear spins and NMR techniques, we experimentally characterize noise in certain decoherence-free subspaces. We also explore the noise of similar states in a heteronuclear spin pair. Further, using a 10-qubit system, we investigate noise profiles of various multiqubit coherences and study the scaling of noise with respect to the coherence order. Finally, using the experimentally obtained noise spectrum of the 10-qubit NOON state, we predict the performance of a Uhrig DD sequence and verify it experimentally., Comment: 8 pages, 10 figures

In conventional NMR experiments, the Zeeman energy gaps of the nuclear spin ensembles are much lower than their thermal energies, and accordingly exhibit tiny polarizations. Generally such low-purity quantum states are devoid of quantum entanglement. However, there exist certain nonclassical correlations which can be observed even in such systems. In this chapter, we discuss three such quantum correlations, namely, quantum contextuality, Leggett-Garg temporal correlations, and quantum discord. In each case, we provide a brief theoretical background and then describe some results from NMR experiments., Comment: 21 pages, 7 figures

Measurement of an observable on a quantum system involves a probabilistic collapse of the quantum state and a corresponding measurement outcome. L\"uders and von Neumann state update rules attempt to describe the above phenomenological observations. These rules are identical for a nondegenerate observable, but differ for a degenerate observable. While L\"uders rule preserves superpositions within a degenerate subspace under a measurement of the corresponding degenerate observable, the von Neumann rule does not. Recently Hegerfeldt and Mayato [Phys. Rev. A, 85, 032116 (2012)] had formulated a protocol to discriminate between the two types of measuring devices. Here we have reformulated this protocol for quantum registers comprising of system and ancilla qubits. We then experimentally investigated this protocol using nulear spin systems with the help of NMR techniques, and found that L\"uders rule is favoured., Comment: 6 pages,6 figures, updated with a few more calculations

A key task in quantum computation is the application of a sequence of gates implementing a specific unitary operation. However, the decomposition of an arbitrary unitary operation into simpler quantum gates is a nontrivial problem. Here we propose a general and robust protocol to decompose any target unitary into a sequence of Pauli rotations. The procedure involves identifying a commuting subset of Pauli operators having a high trace overlap with the target unitary, followed by a numerical optimization of their corresponding rotation angles. The protocol is demonstrated by decomposing several standard quantum operations. The applications of the protocol for quantum state preparation and quantum simulations are also described. Finally, we describe an NMR experiment implementing a three-body quantum simulation, wherein the above decomposition technique is used for the efficient realization of propagators.

A robust control over quantum dynamics is of paramount importance for quantum technologies. Many of the existing control techniques are based on smooth Hamiltonian modulations involving repeated calculations of basic unitaries resulting in time complexities scaling rapidly with the length of the control sequence. On the other hand, the bang-bang controls need one-time calculation of basic unitaries and hence scale much more efficiently. By employing a global optimization routine such as the genetic algorithm, it is possible to synthesize not only highly intricate unitaries, but also certain nonunitary operations. Here we demonstrate the unitary control through the first implementation of the optimal fixed-point quantum search algorithm in a three-qubit NMR system. More over, by combining the bang-bang pulses with the twirling process, we also demonstrate a nonunitary transformation of the thermal equilibrium state into an effective pure state in a five-qubit NMR system.

Quantum ensembles form easily accessible architectures for studying various phenomena in quantum physics, quantum information science, and spectroscopy. Here we review some recent protocols for measurements in quantum ensembles by utilizing ancillary systems. We also illustrate these protocols experimentally via nuclear magnetic resonance techniques. In particular, we shall review noninvasive measurements, extracting expectation values of various operators, characterizations of quantum states, and quantum processes, and finally quantum noise engineering., Comment: 10 pages, 11 figures

Quantum simulators based on nuclear spin-systems controlled by NMR techniques have been used for studying various quantum phenomena. In this work, using a four-qubit NMR quantum simulator, we investigate the recently postulated quantum pigeon-hole effect. In mathematics, the pigeonhole effect is described by a set of three objects being allocated with only two containers. Classically, one would expect at least one container to accommodate more than one object. However, recently it was predicted that there exist quantum scenarios wherein three quantum particles appear to reside in two containers in such a way that no two particles can be simultaneously assigned with a single container. In our experiments, quantum pigeons are emulated by three nuclear qubits whose states are probed jointly and noninvasively by an ancillary spin. The qubit-states $\{\ket{0}$, $\ket{1}\}$ emulate the two containers available for each of the qubits. The experimental results are in good agreement with quantum theoretical predictions., Comment: 5 pages, 5 figures

Physical potentials are routinely approximated to harmonic potentials so as to analytically solve the system dynamics. Often it is important to know when a quantum harmonic oscillator (QHO) behaves quantum mechanically and when classically. Recently Su et. al. [Phys. Rev. A {\bf 85}, 052126 (2012)] have theoretically shown that QHO exhibits quantum contextuality (QC) for a certain set of pseudospin observables. In this work, we encode the four eigenstates of a QHO onto four Zeeman product states of a pair of spin-1/2 nuclei. Using the techniques of NMR quantum information processing, we then demonstrate the violation of a state-dependent inequality arising from the noncontextual hidden variable model, under specific experimental arrangements. We also experimentally demonstrate the violation of a state-independent inequality by thermal equilibrium states of nuclear spins, thereby assessing their quantumness., Comment: 5 Pages, 3 Figures, context dependency illustrated, error below eq. 5 corrected

Benford's law is a statistical inference to predict the frequency of significant digits in naturally occurring numerical databases. In such databases this law predicts a higher occurrence of the digit 1 in the most significant place and decreasing occurrences to other larger digits. Although counter-intuitive at first sight, Benford's law has seen applications in a wide variety of fields like physics, earth-science, biology, finance etc. In this work, we have explored the use of Benford's law for various spectroscopic applications. Although, we use NMR signals as our databases, the methods described here may also be extended to other spectroscopic techniques. In particular, with the help of Benford analysis, we demonstrate the detection of weak NMR signals and spectral corrections. We also explore a potential application of Benford analysis in the image-processing of MRI data., Comment: 6 pages, 6 figures

Benford's Law is an empirical law which predicts the frequency of significant digits in databases corresponding to various phenomena, natural or artificial. Although counter intuitive at the first sight, it predicts a higher occurrence of digit 1, and decreasing occurrences to other larger digits. Here we report the Benford analysis of various NMR databases and draw several interesting inferences. We observe that, in general, NMR signals follow Benford distribution in time-domain as well as in frequency domain. Our survey included NMR signals of various nuclear species in a wide variety of molecules in different phases, namely liquid, liquid-crystalline, and solid. We also studied the dependence of Benford distribution on NMR parameters such as signal to noise ratio, number of scans, pulse angles, and apodization. In this process we also find that, under certain circumstances, the Benford analysis can distinguish a genuine spectrum from a visually identical simulated spectrum. Further we find that chemical-shift databases and amplitudes of certain radio frequency pulses generated using optimal control techniques also satisfy Benford's law to a good extent., Comment: 6 pages, 7 figures

Interaction of molecules with light may lead to electronic transitions and simultaneous vibrational excitations. Franck-Condon factors (FCFs) play an important role in quantifying the intensities of such vibronic transitions occurring during molecular photo-excitations. In this article, we describe a general method for estimating FCFs using a quantum information processor. The method involves the application of a translation operator followed by the measurement of certain projections. We also illustrate the method by experimentally estimating FCFs with the help of a three-qubit NMR quantum information processor. We describe two methods for the measurement of projections - (i) using diagonal tomography and (ii) using Moussa protocol. The experimental results agree fairly well with the theory., Comment: 6 pages, 7 figures

The standard procedure for quantum process tomography (QPT) involves applying the quantum process on a system initialized in each of a complete set of orthonormal states. The corresponding outputs are then characterized by quantum state tomography (QST), which itself requires the measurement of non-commuting observables realized by independent experiments on identically prepared system states. Thus QPT procedure demands a number of independent measurements, and moreover, this number increases rapidly with the size of the system. However, the total number of independent measurements can be greatly reduced with the availability of ancilla qubits. Ancilla assisted process tomography (AAPT) has earlier been shown to require a single QST of system-ancilla space. Ancilla assisted quantum state tomography (AAQST) has also been shown to perform QST in a single measurement. Here we combine AAPT with AAQST to realize a `single-shot QPT' (SSPT), a procedure to characterize a general quantum process in a single collective measurement of a set of commuting observables. We demonstrate experimental SSPT by characterizing several single-qubit processes using a three-qubit NMR quantum register. Furthermore, using the SSPT procedure we experimentally characterize the twirling process and compare the results with theory., Comment: 6 pages, 6 figures

Due to omnipresent environmental interferences, quantum coherences inevitably undergo irreversible transformations over certain time-scales, thus leading to the loss of encoded information. This process, known as decoherence, has been a major obstacle in realizing efficient quantum information processors. Understanding the mechanism of decoherence is crucial in developing tools to inhibit it. Here we utilize a method proposed by Cory and co-workers [Phys. Rev. A 67, 062316 (2003)] to engineer artificial decoherence in the system qubits by randomly perturbing their surrounding ancilla qubits. Using a two qubit nuclear magnetic resonance quantum register, we characterize the artificial decoherence by noise spectroscopy and quantum process tomography. Further, we study the efficacy of dynamical decoupling sequences in suppressing the artificial decoherence. Here we describe the experimental results and their comparisons with theoretical simulations.

We experimentally demonstrate the phenomenon of dynamical many-body freezing in a periodically driven Ising chain. Theoretically [Phys. Rev. B 82, 172402 (2010)], for certain values of the drive parameters all fundamental degrees of freedom contributing to the response dynamics freeze for all time and for arbitrary initial states. Also, since the condition of freezing involves only the drive parameters and not on the quantization of the momentum (i.e., the system-size), our simulation with a small (3-spin) chain captures all salient features of the freezing phenomenon predicted for the infinite chain. Using optimal control techniques, we realize high-fidelity cosine modulated drive, and observe non-monotonic freezing of magnetization at specific frequencies of modulation. Time-evolution of the excitations in momentum space has been tracked directly through magnetization measurements.

We report an experimental quantum simulation of unitary dynamics of an XY spin chain with pre-engineered couplings. Using this simulation, we demonstrate the mirror inversion of quantum states, proposed by Albanese et al. [Phys. Rev. Lett. 93, 230502 (2004)]. The experiment is performed with a 5-qubit dipolar coupled spin system using nuclear magnetic resonance techniques. To perform quantum simulation we make use of the recently proposed unitary operator decomposition algorithm of Ajoy et al. [Phys. Rev. A 85, 030303 (2012)] along with numerical pulse optimization techniques. Further, using mirror inversion, we demonstrate that entangled states can be transferred from one end of the chain to the other end. The simulations are implemented with high experimental fidelity, which implies that these kind of simulations may be possible in larger systems., Comment: 9 pages, 6 figures