Showing papers in "Physical Review A in 2020"

Circuit-centric quantum classifiers

Abstract: A machine learning design is developed to train a quantum circuit specialized in solving a classification problem. In addition to discussing the training method and effect of noise, it is shown that the circuits perform reasonably well on classical benchmarks.

Variational quantum algorithms for nonlinear problems

Abstract: We show that nonlinear problems including nonlinear partial differential equations can be efficiently solved by variational quantum computing. We achieve this by utilizing multiple copies of variational quantum states to treat nonlinearities efficiently and by introducing tensor networks as a programming paradigm. The key concepts of the algorithm are demonstrated for the nonlinear Schr\"odinger equation as a canonical example. We numerically show that the variational quantum ansatz can be exponentially more efficient than matrix product states and present experimental proof-of-principle results obtained on an IBM Q device.

Robust data encodings for quantum classifiers

Abstract: Data representation is crucial for the success of machine-learning models. In the context of quantum machine learning with near-term quantum computers, equally important considerations of how to efficiently input (encode) data and effectively deal with noise arise. In this paper, we study data encodings for binary quantum classification and investigate their properties both with and without noise. For the common classifier we consider, we show that encodings determine the classes of learnable decision boundaries as well as the set of points which retain the same classification in the presence of noise. After defining the notion of a robust data encoding, we prove several results on robustness for different channels, discuss the existence of robust encodings, and prove a lower bound on the number of robust points in terms of fidelities between noisy and noiseless states. Numerical results for several example implementations are provided to reinforce our findings.

Locality and entanglement in table-top testing of the quantum nature of linearized gravity

Abstract: This paper points out the importance of the assumption of locality of physical interactions, and the concomitant necessity of propagation of an entity (in this case, off-shell quanta---virtual gravitons) between two nonrelativistic test masses in unveiling the quantum nature of linearized gravity through a laboratory experiment. At the outset, we will argue that observing the quantum nature of a system is not limited to evidencing $O\left(\ensuremath{\hbar}\right)$ corrections to a classical theory: it instead hinges upon verifying tasks that a classical system cannot accomplish. We explain the background concepts needed from quantum field theory and quantum information theory to fully appreciate the previously proposed table-top experiments, namely forces arising through the exchange of virtual (off-shell) quanta, as well as local operations and classical communication (LOCC) and entanglement witnesses. We clarify the key assumption inherent in our evidencing experiment, namely the locality of physical interactions, which is a generic feature of interacting systems of quantum fields around us, and naturally incorporate microcausality in the description of our experiment. We also present the types of states the matter field must inhabit, putting the experiment on firm relativistic quantum-field-theoretic grounds. At the end, we use a nonlocal theory of gravity to illustrate how our mechanism may still be used to detect the qualitatively quantum nature of a force when the scale of nonlocality is finite. We find that the scale of nonlocality, including the entanglement entropy production in local and nonlocal gravity, may be revealed from the results of our experiment.

Quantum gradient descent for linear systems and least squares

Abstract: Quantum machine learning and optimization are exciting new areas that have been brought forward by the breakthrough quantum algorithm of Harrow, Hassidim, and Lloyd for solving systems of linear equations. The utility of classical linear system solvers extends beyond linear algebra as they can be leveraged to solve optimization problems using iterative methods like gradient descent. In this work, we provide a quantum method for performing gradient descent when the gradient is an affine function. Performing $\ensuremath{\tau}$ steps of the gradient descent requires time $O(\ensuremath{\tau}{C}_{S})$ for weighted least-squares problems, where ${C}_{S}$ is the cost of performing one step of the gradient descent quantumly, which at times can be considerably smaller than the classical cost. We illustrate our method by providing two applications: first, for solving positive semidefinite linear systems, and, second, for performing stochastic gradient descent for the weighted least-squares problem with reduced quantum memory requirements. We also provide a quantum linear system solver in the QRAM data structure model that provides significant savings in cost for large families of matrices.

X Y mixers: Analytical and numerical results for the quantum alternating operator ansatz

Abstract: The quantum alternating operator ansatz (QAOA) is a promising gate-model metaheuristic for combinatorial optimization. Applying the algorithm to problems with constraints presents an implementation challenge for near-term quantum resources. This paper explores strategies for enforcing hard constraints by using $XY$ Hamiltonians as mixing operators (mixers). Despite the complexity of simulating the $XY$ model, we demonstrate that, for an integer variable admitting $\ensuremath{\kappa}$ discrete values represented through one-hot encoding, certain classes of the mixer Hamiltonian can be implemented without Trotter error in depth $O(\ensuremath{\kappa})$. We also specify general strategies for implementing QAOA circuits on all-to-all connected hardware graphs and linearly connected hardware graphs inspired by fermionic simulation techniques. Performance is validated on graph-coloring problems that are known to be challenging for a given classical algorithm. The general strategy of using $XY$ mixers is borne out numerically, demonstrating a significant improvement over the general $X$ mixer, and moreover the generalized $W$ state yields better performance than easier-to-generate classical initial states when $XY$ mixers are used.

Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code

Abstract: The performance of the circuit-based surface-GKP bosonic quantum error correction code under noise due to finite squeezing of the GKP states and photon losses is investigated. The authors show under what conditions fault-tolerant quantum error correction is possible with the surface-GKP code.

Effective Hamiltonian models of the cross-resonance gate

Abstract: Effective Hamiltonian methods are utilized to model the two-qubit cross-resonance gate for both the ideal two-qubit case and when higher levels are included. Analytic expressions are obtained in the qubit case and the higher-level model is solved both perturbatively and numerically, with the solutions agreeing well in the weak-drive limit. The methods are applied to parameters from recent experiments and, accounting for classical cross-talk effects, results in good agreement between theory and experimental results.

Reducing the number of non-Clifford gates in quantum circuits

Abstract: We present a method for reducing the number of non-Clifford quantum gates, in particularly T-gates, in a circuit, an important task for efficiently implementing fault-tolerant quantum computations. This method matches or beats previous approaches to ancillae-free T-count reduction on the majority of our benchmark circuits, in some cases yielding up to 50% improvement. Our method begins by representing the quantum circuit as a ZX-diagram, a tensor networklike structure that can be transformed and simplified according to the rules of the ZX-calculus. We then extend a recent simplification strategy with a different ingredient, phase gadgetization, which we use to propagate non-Clifford phases through a ZX-diagram to find nonlocal cancellations. Our procedure extends unmodified to arbitrary phase angles and to parameter elimination for variational circuits. Finally, our optimization is self-checking, in the sense that the simplification strategy we propose is powerful enough to independently validate equality of the input circuit and the optimized output circuit. We have implemented the routines of this paper in the open-source library pyzx.

Measurement reduction in variational quantum algorithms

Abstract: Variational quantum algorithms are promising applications of noisy intermediate-scale quantum (NISQ) computers. These algorithms consist of a number of separate prepare-and-measure experiments that estimate terms in a Hamiltonian. The number of terms can become overwhelmingly large for problems at the scale of NISQ hardware that may soon be available. We use unitary partitioning (developed independently by Izmaylov et al. [J. Chem. Theory Comput. 16, 190 (2020)]) to define variational quantum eigensolver procedures in which additional unitary operations are appended to the ansatz preparation to reduce the number of terms. This approach may be scaled to use all coherent resources available after ansatz preparation. We also study the use of asymmetric qubitization to implement the additional coherent operations with lower circuit depth. Using this technique, we find a constant factor speedup for lattice and random Pauli Hamiltonians. For electronic structure Hamiltonians, we prove that linear term reduction with respect to the number of orbitals, which has been previously observed in numerical studies, is always achievable. For systems represented on 10--30 qubits, we find that there is a reduction in the number of terms by approximately an order of magnitude. Applied to the plane-wave dual-basis representation of fermionic Hamiltonians, however, unitary partitioning offers only a constant factor reduction. Finally, we show that noncontextual Hamiltonians may be reduced to effective commuting Hamiltonians using unitary partitioning.

Verifying multipartite entangled Greenberger-Horne-Zeilinger states via multiple quantum coherences

Abstract: An 18-qubit GHZ state with multipartite entanglement is prepared and measured on a 20-qubit device. The detection technique used, which is based on measuring multiple quantum coherences, is robust to noise and only requires measuring the population in the ground state.

Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes

Abstract: Encoding a qubit in the continuous degrees of freedom of an oscillator is a promising path to error-corrected quantum computation. One advantageous way to achieve this is through Gottesman-Kitaev-Preskill (GKP) grid states, whose symmetries allow for the correction of any small continuous error on the oscillator. Unfortunately, ideal grid states have infinite energy, so it is important to find finite-energy approximations that are realistic, practical, and useful for applications. In the first half of this work we investigate the impact of imperfect GKP states on computational circuits independently of the physical architecture. To this end, we analyze the behaviour of the physical and logical content of normalizable GKP states through several figures of merit, employing a recently-developed modular subsystem decomposition. By tracking the errors that enter into the computational circuit due to imperfections in the GKP states, we are able to gauge the utility of these states for NISQ (Noisy Intermediate-Scale Quantum) devices. In the second half, we focus on a state preparation approach in the photonic domain wherein photon-number-resolving measurements on some modes of Gaussian states produce non-Gaussian states in others. We produce detailed numerical results for the preparation of GKP states alongside estimating the resource requirements in practical settings and probing the quality of the resulting states with the tools we develop. Our numerical experiments indicate that we can generate any state in the GKP Bloch sphere with nearly equal resources, which has implications for magic state preparation overheads.

Quantum Gravity Witness via Entanglement of Masses: Casimir Screening

Abstract: A recently proposed experimental protocol for quantum gravity induced entanglement of masses (QGEM) requires in principle realizable, but still very ambitious, set of parameters in matter-wave interferometry. Motivated by easing the experimental realization, in this paper, we consider the parameter space allowed by a slightly modified experimental design, which mitigates the Casimir potential between two spherical neutral test masses by separating the two macroscopic interferometers by a thin conducting plate. Although this setup will reintroduce a Casimir potential between the conducting plate and the masses, there are several advantages of this design. First, the quantum gravity induced entanglement between the two superposed masses will have no Casimir background. Secondly, the matter-wave interferometry itself will be greatly facilitated by allowing both the mass ${10}^{\ensuremath{-}16}--{10}^{\ensuremath{-}15}\phantom{\rule{0.28em}{0ex}}\mathrm{kg}$ and the superposition size $\mathrm{\ensuremath{\Delta}}x\ensuremath{\sim}20\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{m}$ to be a one-two order of magnitude smaller than those proposed earlier, and thereby also two orders of magnitude smaller magnetic field gradient of ${10}^{4}\phantom{\rule{0.28em}{0ex}}{\mathrm{Tm}}^{\ensuremath{-}1}$ to create that superposition through the Stern-Gerlach effect. In this context, we will further investigate the collisional decoherences and decoherence due to vibrational modes of the conducting plate.

Nonreciprocal ground-state cooling of multiple mechanical resonators

Abstract: The simultaneous ground-state cooling of multiple degenerate or near-degenerate mechanical modes coupled to a common cavity-field mode has become an outstanding challenge in cavity optomechanics. This is because the dark modes formed by these mechanical modes decouple from the cavity mode and prevent extracting energy from the dark modes through the cooling channel of the cavity mode. Here we propose a universal and reliable dark-mode-breaking method to realize the simultaneous ground-state cooling of two degenerate or nondegenerate mechanical modes by introducing a phase-dependent phonon-exchange interaction, which is used to form a loop-coupled configuration. We find an asymmetrical cooling performance for the two mechanical modes and expound this phenomenon based on the nonreciprocal energy transfer mechanism, which leads to the directional flow of phonons between the two mechanical modes. We also generalize this method to cool multiple mechanical modes. The physical mechanism in this cooling scheme has general validity and this method can be extended to break other dark-mode and dark-state effects in physics.

Force and acceleration sensing with optically levitated nanogram masses at microkelvin temperatures

Abstract: An apparatus that uses optically levitated objects, together with active feedback cooling, is used to provide an order-of-magnitude improvement on the sensitivity of state-of-the-art accelerometers. The results pave the way to using such sensors as a part of large-scale particle detectors, for assessing the neutrality of matter and dark matter, and for studying short-ranged Yukawa forces.

Demonstration of a parametrically activated entangling gate protected from flux noise

Abstract: In state-of-the-art quantum computing platforms, including superconducting qubits and trapped ions, imperfections in the two-qubit entangling gates are the dominant contributions of error to systemwide performance. Recently, a novel two-qubit parametric gate was proposed and demonstrated with superconducting transmon qubits. This gate is activated through rf modulation of the transmon frequency and can be operated at an amplitude where the performance is first-order insensitive to flux noise. In this work we experimentally validate the existence of this ac sweet spot and demonstrate its dependence on white-noise power from room-temperature electronics. With these factors in place, we observe entangling-gate fidelity with coherence-limited performance. An ensemble of repeated observations has a median fidelity of 98.8%, with roughly 10% of observations above 99%.

Quantum simulation of hyperbolic space with circuit quantum electrodynamics: From graphs to geometry.

Abstract: We show how quantum many-body systems on hyperbolic lattices with nearest-neighbor hopping and local interactions can be mapped onto quantum field theories in continuous negatively curved space. The underlying lattices have recently been realized experimentally with superconducting resonators and therefore allow for a table-top quantum simulation of quantum physics in curved background. Our mapping provides a computational tool to determine observables of the discrete system even for large lattices, where exact diagonalization fails. As an application and proof of principle we quantitatively reproduce the ground state energy, spectral gap, and correlation functions of the noninteracting lattice system by means of analytic formulas on the Poincare disk, and show how conformal symmetry emerges for large lattices. This sets the stage for studying interactions and disorder on hyperbolic graphs in the future. Importantly, our analysis reveals that even relatively small discrete hyperbolic lattices emulate the continuous geometry of negatively curved space, and thus can be used to experimentally resolve fundamental open problems at the interface of interacting many-body systems, quantum field theory in curved space, and quantum gravity.

First-principles analysis of cross-resonance gate operation

Abstract: We present a comprehensive theoretical study of the cross-resonance gate operation covering estimates for gate parameters and gate error as well as analyzing spectator qubits and multiqubit frequency collisions. Starting from the Josephson nonlinearity and by accounting for the eigenstates renormalization, due to counterrotating terms, we derive a starting model for the cross-resonance gate with modified qubit-qubit interaction and drive matrix elements. Employing time-dependent Schrieffer-Wolff perturbation theory, we derive an effective Hamiltonian for the cross-resonance gate with estimates for the gate parameters calculated up to the fourth order in drive amplitude. The model with renormalized eigenstates lead to 10%--15% relative correction of the effective gate parameters compared to Kerr theory. We find that gate operation is strongly dependent on the ratio of qubit-qubit detuning and anharmonicity. In particular, we characterize five distinct regions of operation, and propose candidate parameter choices for achieving high gate speed and low coherent gate error when the cross-resonance tone is equipped with an echo pulse sequence. Furthermore, we generalize our method to include a third spectator qubit and characterize possible detrimental multiqubit frequency collisions.

Measuring the similarity of graphs with a Gaussian boson sampler

Abstract: Gaussian boson samplers (GBSs) have initially been proposed as a near-term demonstration of classically intractable quantum computation. We show here that they have a potential practical application: Samples from these devices can be used to construct a feature vector that embeds a graph in Euclidean space, where similarity measures between graphs---so-called graph kernels---can be naturally defined. This is crucial for machine learning with graph-structured data, and we show that the GBS-induced kernel performs remarkably well in classification benchmark tasks. We provide a theoretical motivation for this success, linking the extracted features to the number of $r$ matchings in subgraphs. Our results contribute to a new way of thinking about kernels as a quantum hardware-efficient feature mapping, and lead to a promising application for near-term quantum computing.

Zero-noise extrapolation for quantum-gate error mitigation with identity insertions

Abstract: Quantum-gate errors are a significant challenge for achieving precision measurements on noisy intermediate-scale quantum (NISQ) computers. This paper focuses on zero-noise extrapolation (ZNE), a technique that can be implemented on existing hardware, studying it in detail and proposing modifications to existing approaches. In particular, we consider identity insertion methods for amplifying noise because they are hardware agnostic. We build a mathematical formalism for studying existing ZNE techniques and show how higher order polynomial extrapolations can be used to systematically reduce depolarizing errors. Furthermore, we introduce a method for amplifying noise that uses far fewer gates than traditional methods. This approach is compared with existing methods for simulated quantum circuits. Comparable or smaller errors are possible with fewer gates, which illustrates the potential for empowering an entirely new class of moderate-depth circuits on near term hardware.

Collective excitations of a one-dimensional quantum droplet

Abstract: We calculate the excitation spectrum of a one-dimensional self-bound quantum droplet in a two-component bosonic mixture described by the Gross-Pitaevskii equation (GPE) with cubic and quadratic nonlinearities. The cubic term originates from the mean-field energy of the mixture proportional to the effective coupling constant $\delta g$, whereas the quadratic nonlinearity corresponds to the attractive beyond-mean-field contribution. The droplet properties are governed by a control parameter $\gamma\propto \delta g N^{2/3}$, where $N$ is the particle number. For large $\gamma>0$ the droplet features the flat-top shape with the discrete part of its spectrum consisting of plane-wave Bogoliubov phonons propagating through the flat-density bulk and reflected by edges of the droplet. With decreasing $\gamma$ these modes cross into the continuum, sequentially crossing the particle-emission threshold at specific critical values. A notable exception is the breathing mode which we find to be always bound. The balance point $\gamma = 0$ provides implementation of a system governed by the GPE with an unusual quadratic nonlinearity. This case is characterized by the ratio of the breathing-mode frequency to the particle-emission threshold equal to 0.8904. As $\gamma$ tends to $-\infty$ this ratio tends to 1 and the droplet transforms into the soliton solution of the integrable cubic GPE.

Measurement-induced transitions of the entanglement scaling law in ultracold gases with controllable dissipation

Abstract: Recent studies of quantum circuit models have theoretically shown that frequent measurements induce a transition in a quantum many-body system, which is characterized by a change in the scaling law of the entanglement entropy from a volume law to an area law. In order to propose a way to experimentally observe this measurement-induced transition, we present numerical analyses using matrix-product states on the quench dynamics of a dissipative Bose-Hubbard model with controllable two-body losses, which has been realized in recent experiments with ultracold atoms. We find that when the strength of dissipation increases, there occurs a measurement-induced transition from volume-law scaling to area-law scaling with a logarithmic correction in a region of relatively small dissipation. We also find that the strong dissipation leads to a revival of the volume-law scaling due to a continuous quantum Zeno effect. We show that dynamics starting with the area-law states exhibits strong suppression of particle transport stemming from ergodicity breaking, which can be used in experiments to distinguish them from the volume-law states.

Digital-analog quantum computation

Abstract: Digital quantum computing paradigm offers highly desirable features such as universality, scalability, and quantum error correction. However, physical resource requirements to implement useful error-corrected quantum algorithms are prohibitive in the current era of NISQ devices. As an alternative path to performing universal quantum computation, within the NISQ era limitations, we propose to merge digital single-qubit operations with analog multiqubit entangling blocks in an approach we call digital-analog quantum computing (DAQC). Along these lines, although the techniques may be extended to any resource, we propose to use unitaries generated by the ubiquitous Ising Hamiltonian for the analog entangling block and we prove its universal character. We construct explicit DAQC protocols for efficient simulations of arbitrary inhomogeneous Ising, two-body, and $M$-body spin Hamiltonian dynamics by means of single-qubit gates and a fixed homogeneous Ising Hamiltonian. Additionally, we compare a sequential approach where the interactions are switched on and off (stepwise DAQC) with an always-on multiqubit interaction interspersed by fast single-qubit pulses (banged DAQC). Finally, we perform numerical tests comparing purely digital schemes with DAQC protocols, showing a remarkably better performance of the latter. The proposed DAQC approach combines the robustness of analog quantum computing with the flexibility of digital methods.

Hybrid-Liouvillian formalism connecting exceptional points of non-Hermitian Hamiltonians and Liouvillians via postselection of quantum trajectories

Abstract: Exceptional points (EPs) are degeneracies of classical and quantum open systems, which are studied in many areas of physics including optics, optoelectronics, plasmonics, and condensed matter physics. In the semiclassical regime, open systems can be described by phenomenological effective non-Hermitian Hamiltonians (NHHs) capturing the effects of gain and loss in terms of imaginary fields. The EPs that characterize the spectra of such Hamiltonians (HEPs) describe the time evolution of a system without quantum jumps. It is well known that a full quantum treatment describing more generic dynamics must crucially take into account such quantum jumps. In a recent paper [F. Minganti et al., Phys. Rev. A 100, 062131 (2019)], we generalized the notion of EPs to the spectra of Liouvillian superoperators governing open system dynamics described by Lindblad master equations. Intriguingly, we found that in situations where a classical-to-quantum correspondence exists, the two types of dynamics can yield different EPs. In a recent experimental work [M. Naghiloo et al., Nat. Phys. 15, 1232 (2019)], it was shown that one can engineer a non-Hermitian Hamiltonian in the quantum limit by postselecting on certain quantum jump trajectories. This raises an interesting question concerning the relation between Hamiltonian and Lindbladian EPs, and quantum trajectories. We discuss these connections by introducing a hybrid-Liouvillian superoperator, capable of describing the passage from an NHH (when one postselects only those trajectories without quantum jumps) to a true Liouvillian including quantum jumps (without postselection). Beyond its fundamental interest, our approach allows to intuitively relate the effects of postselection and finite-efficiency detectors.

Bridging entanglement dynamics and chaos in semiclassical systems

Abstract: It is widely recognized that entanglement generation and dynamical chaos are intimately related in semiclassical models via the process of decoherence. In this paper, we propose a unifying framework which directly connects the bipartite and multipartite entanglement growth to the quantifiers of classical and quantum chaos. In the semiclassical regime, the dynamics of the von Neumann entanglement entropy, the spin squeezing, the quantum Fisher information, and the out-of-time-order square commutator are governed by the divergence of nearby phase-space trajectories via the local Lyapunov spectrum, as suggested by previous conjectures in the literature. General analytical predictions are confirmed by detailed numerical calculations for two paradigmatic models, relevant in atomic and optical experiments, which exhibit a regular-to-chaotic transition: the quantum kicked top and the Dicke model.

Witnessing the nonclassical nature of gravity in the presence of unknown interactions

Abstract: General relativity is a classical field theory and therefore does not predict gravitationally induced entanglement. As such, recent proposals have focused on seeking an empirical demonstration of this feature. We introduce improvements to a spin witness protocol that reduce the highly challenging experimental requirements. After rigorously assessing approximations from the original proposal [S. Bose et al., Phys. Rev. Lett. 119, 240401 (2017)], we focus on entanglement witnessing. We propose a witness which greatly reduces the required interaction time, thereby making the experiment feasible for higher decoherence rates, and we show how statistical analysis can separate the gravitational contribution from other possibly dominant and ill-known interactions. We point out a potential loophole and show how it can be closed using state tomography.

Robust Mølmer-Sørensen gate for neutral atoms using rapid adiabatic Rydberg dressing

Abstract: The Rydberg blockade mechanism is now routinely considered for entangling qubits encoded in clock states of neutral atoms. Challenges towards implementing entangling gates with high fidelity include errors due to thermal motion of atoms, laser amplitude inhomogeneities, and imperfect Rydberg blockade. We show that adiabatic rapid passage by Rydberg dressing provides a mechanism for implementing two-qubit entangling gates by accumulating phases that are robust to these imperfections. We find that the typical error in implementing a two-qubit gate, such as the controlled phase gate, is dominated by errors in the single-atom light shift, and that this can be easily corrected using adiabatic dressing interleaved with a simple spin echo sequence. This results in a two-qubit M\o{}lmer-S\o{}rensen gate. A gate fidelity $\ensuremath{\sim}0.995$ is achievable with modest experimental parameters and a path to higher fidelities is possible for Rydberg states in atoms with a stronger blockade, longer lifetimes, and larger Rabi frequencies.

Sending-or-not-sending twin-field quantum key distribution: Breaking the direct transmission key rate

Abstract: We present improved results of sending-or-not-sending twin-field quantum key distribution by using error rejection through two-way classical communications. Our error rejection method, especially our method of actively odd-parity pairing (AOPP) can drastically improve the performance of sending-or-not-sending twin-field protocol in both secure distance and key rate. Taking a typical experimental parameter setting, our method here improves the secure distance by 70 km to more than 100 km in comparison with the prior art results. Comparative study also shows advantageous in key rates at regime of long distance and large misalignment error rate for our method here. The numerical results show that our method here can significantly exceed the absolute limit of direct transmission key rate, and also have an advantageous key rates higher than various prior art results by 10 to 20 times.

Symmetry-adapted variational quantum eigensolver

Abstract: We propose a scheme to restore spatial symmetry of Hamiltonian in the variational-quantum-eigensolver (VQE) algorithm for which the quantum circuit structures used usually break the Hamiltonian symmetry. The symmetry-adapted VQE scheme introduced here simply applies the projection operator, which is Hermitian but not unitary, to restore the spatial symmetry in a desired irreducible representation of the spatial group. The entanglement of a quantum state is still represented in a quantum circuit but the nonunitarity of the projection operator is treated classically as postprocessing in the VQE framework. By numerical simulations for a spin-$1/2$ Heisenberg model on a one-dimensional ring, we demonstrate that the symmetry-adapted VQE scheme with a shallower quantum circuit can achieve significant improvement in terms of the fidelity of the ground state and has a great advantage in terms of the ground-state energy with decent accuracy, as compared to the non-symmetry-adapted VQE scheme. We also demonstrate that the present scheme can approximate low-lying excited states that can be specified by symmetry sectors, using the same circuit structure for the ground-state calculation.

Improved tripartite uncertainty relation with quantum memory

Abstract: The uncertainty principle is a striking and fundamental feature in quantum mechanics, distinguishing it from classical mechanics. It offers an important lower bound to predict outcomes of two arbitrary incompatible observables measured on a particle. In quantum information theory, this uncertainty principle is popularly formulized in terms of entropy. Here, we present an improvement of the tripartite quantum-memory-assisted entropic uncertainty relation. The uncertainty's lower bound is derived by considering mutual information and the Holevo quantity. It shows that the bound derived by this method will be tighter than the lower bound of Renes and Boileau [Phys. Rev. Lett. 103, 020402 (2009)]. Furthermore, regarding a pair of mutual unbiased bases as the incompatibility, our bound will become extremely tight for the three-qubit $X$-state system, completely coinciding with the entropy-based uncertainty, and can restore Renes and Boileau's bound with respect to arbitrary tripartite pure states. In addition, by applying our lower bound one can attain the tighter bound of the quantum secret key rate, which is of basic importance to enhance the security of quantum key distribution protocols.