Publications
Eigen-SNAP gate of two photonic qubits coupled via a transmon
Marcus Meschede, and Ludwig Mathey
In the pursuit of robust quantum computing, we put forth a platform based on photonic qubits in a circuit-QED environment. Specifically, we propose a versatile two-qubit gate based on two cavities coupled via a transmon, constituting a selective number-dependent phase gate operating on the in-phase eigenmodes of the two cavities, the Eigen-SNAP gate. This gate natively operates in the dispersive coupling regime of the cavities and the transmon, and operates by driving the transmon externally, to imprint desired phases on the number states. As an example for the utility of the Eigen-SNAP gate, we implement a SWAP−−−−−−√ gate on a system of two logical bosonic qubits encoded in the cavities. Further, we use numerical optimization to determine the optimal implementation of the SWAP−−−−−−√. We find that the fidelities of these optimal protocols are only limited by the coherence times of the system's components. These findings pave the way to continuous variable quantum computing in cavity-transmon systems.
Optimal recoil-free state preparation in an optical atom tweezer
Lia Kley, Nicolas Heimann, Aslam Parvej, Lukas Broers, and Ludwig Mathey
Quantum computing in atom tweezers requires high-fidelity implementations of quantum operations. Here, we demonstrate the optimal implementation of the transition |0⟩→|1⟩ of two levels, serving as a qubit, of an atom in a tweezer potential, driven by a single-photon Rabi pulse. The Rabi pulse generates a photon recoil of the atom, due to the Lamb-Dicke coupling between the internal and motional degree of freedom, driving the system out of the logical subspace. This detrimental effect is strongly suppressed in the protocols that we propose. Using pulse engineering, we generate optimal protocols composed of a Rabi protocol and a force protocol, corresponding to dynamically displacing the tweezer. We generate these for a large parameter space, from small to large values of the Rabi frequency, and a range of pulse lengths. We identify three main regimes for the optimal protocols, and discuss their properties. In all of these regimes, we demonstrate infidelity well below the current technological standard, thus mitigating a universal challenge in atom tweezers and other quantum technology platforms.
Tensor networks enable the calculation of turbulence probability distributions
Nikita Gourianov, Peyman Givi, Dieter Jaksch and Stephen B. Pope
Predicting the dynamics of turbulent fluids has been an elusive goal for centuries. Even with modern computers, anything beyond the simplest turbulent flows is too chaotic and multiscaled to be directly simulatable. An alternative is to treat turbulence probabilistically, viewing flow properties as random variables distributed according to joint probability density functions (PDFs). Such PDFs are neither chaotic nor multiscale, yet remain challenging to simulate due to their high dimensionality. Here, we overcome the dimensionality problem by encoding turbulence PDFs as highly compressed “tensor networks” (TNs). This enables single CPU core simulations that would otherwise be impractical even with supercomputers: for a 5 + 1 dimensional PDF of a chemically reactive turbulent flow, we achieve reductions in memory and computational costs by factors of and , respectively, compared to standard finite-difference algorithms. A future path is opened toward something heretofore thought infeasible: directly simulating high-dimensional PDFs of both turbulent flows and other chaotic systems that can usefully be described probabilistically.
Pulse engineering via projection of response functions
Nico Heimann, Lukas Broers and Ludwig Mathey
We present an iterative optimal control method of quantum systems, aimed at an implementation of a desired operation with optimal fidelity. The update step of the method is based on the linear response of the fidelity to the control operators, and its projection onto the mode functions of the corresponding operator. Our method extends methods such as gradient-ascent pulse engineering (GRAPE) and variational quantum algorithms, by determining the fidelity gradient in a hyperparameter-free manner, and using it for a multiparameter update, capitalizing on the multimode overlap of the perturbation and the mode functions. This directly reduces the number of dynamical trajectories that need to be evaluated in order to update a set of parameters. We demonstrate this approach, and compare it to the standard GRAPE algorithm, for the example of a quantum gate on two qubits, demonstrating a clear improvement in convergence and optimal fidelity of the generated protocol.
Tomographic measurement data of states that never existed
J. Göttsch, S. Grebien, F. Pein, M. Lautzas, D. Abdelkhalek, L. Rebón, B. Hage, J. Fiurašek, R. Schnabel
Microscopic Schr{ö}dinger cat states are generated from quantum correlated fields using a probabilistic heralding photon subtraction event. Subsequent quantum state tomography provides complete information about the state with typical photon numbers of the order of one. Another approach strives for a larger number of quantum-correlated photons by conditioning the measurement analysis on events with exactly this number of photons. Here, we present a new approach to derive measurement data of quantum correlated states with average quantum-correlated photon numbers significantly larger than one. We produce an ensemble of a heralded, photon-subtracted squeezed vacuum state of light. We split the states at a balanced beam splitter and simultaneously measure a pair of orthogonal field quadratures at the outputs using tomographic `Q-function homodyne detection' (QHD). The final act is probabilistic two-copy data post-processing aiming for data from a new state with larger photon number. Evaluating the final tomographic data as that of a grown microscopic Schr{ö}dinger cat state shows that the probabilistic post-processing increased the photon number of |α0|2≈1.2 to |α2|2≈6.8. Our concept for obtaining tomographic measurement data of mesoscopic non-classical states that never existed might be a turning point in measurement-based quantum technology.
Exclusive-or encoded algebraic structure for efficient quantum dynamics
Lukas Broers and Ludwig Mathey
We propose a formalism that captures the algebraic structure of many-body two-level quantum systems, and directly motivates an efficient numerical method. This formalism is based on the binary representation of the enumeration-indices of the elements of the corresponding Lie algebra. The action of arbitrarily large elements of that algebra reduces to a few bit-wise exclusive-or operations. This formalism naturally produces sparse representations of many-body density operators, the size of which we control through a dynamic truncation method. We demonstrate how this formalism applies to real-time evolution, dissipative Lindblad action, imaginary-time evolution, and projective measurement processes. We find that this approach to calculating quantum dynamics scales close to linearly with the number of non-zero components in the density operator. We refer to this exclusive-or represented quantum algebra as ORQA. As a proof of concept, we provide a numerical demonstration of this formalism by simulating quantum annealing processes for the maximum independent set problem for up to 22 two-level systems.
Effect of strong confinement on the order parameter dynamics in fermionic superfluids
Cesar R. Cabrera, René Henke, Lukas Broers, Jim Skulte, Hector P. Ojeda Collado, H. Biss, Ludwig Mathey and Henning Moritz
Fermionic pairing and the superfluid order parameter change dramatically in low-dimensional systems such as high-Tc superconductors. Here we show how the order parameter dynamics, which defines essential collective properties, is modified by strong confinement. Using a model system for strongly correlated superfluidity, an ultracold fermionic gas, we study the response to a weak modulation of the confinement. Surprisingly, we observe a well-defined collective mode throughout the entire crossover from the Bardeen-Cooper-Schrieffer (BCS) state to Bose-Einstein condensation (BEC) of molecules. Starting in the BCS regime, the excitation energy follows twice the pairing gap, then drops below it in the strongly correlated regime, and finally approaches twice the harmonic level spacing imposed by the confinement in the BEC regime. Its spectral weight vanishes when approaching the superfluid critical temperature. The experimental results are in excellent agreement with an effective field theory, providing strong evidence that amplitude oscillations of the order parameter hybridize with and eventually transform into spatial excitations along the confined direction. The strong modification of the excitation spectrum highlights the relevance of confinement to fermionic superfluids and superconductors, and raises questions about its influence on other fundamental quantities.
Directly measured squeeze factors over GHz bandwidth from monolithic ppKTP resonators
Benedict Tohermes, Sophie Verclas and Roman Schnabel
Squeezed vacuum states of light with bandwidths in the gigahertz range are required for ultrafast quantum sensors, for high-bandwidth QKD and for optical quantum computers. Here we present squeeze factors of monolithic periodically poled KTP (ppKTP) resonators measured with two laboratory-built balanced homodyne detectors with gigahertz bandwidth. We realise two complete systems without selection of optical or electronic hardware components to test the reproducibility without rejects. As expected, the systems show clear spectral differences. However, both achieve directly measured squeeze factors in the order of 3 dB over a GHz bandwidth, which is achieved here for the first time. Our direct measurement of quantum correlation is suitable for increasing the key rate of one-sided, device-independent QKD.
Solving lattice gauge theories using the quantum Krylov algorithm and qubitization
Lewis W. Anderson, Martin Kiffner, Tom O'Leary, Jason Crain, Dieter Jaksch
Computing vacuum states of lattice gauge theories (LGTs) containing fermionic degrees of freedom can present significant challenges for classical computation using Monte-Carlo methods. Quantum algorithms may offer a pathway towards more scalable computation of groundstate properties of LGTs. However, a comprehensive understanding of the quantum computational resources required for such a problem is thus far lacking. In this work, we investigate using the quantum subspace expansion (QSE) algorithm to compute the groundstate of the Schwinger model, an archetypal LGT describing quantum electrodynamics in one spatial dimension. We perform numerical simulations, including the effect of measurement noise, to extrapolate the resources required for the QSE algorithm to achieve a desired accuracy for a range of system sizes. Using this, we present a full analysis of the resources required to compute LGT vacuum states using a quantum algorithm using qubitization within a fault tolerant framework. We develop of a novel method for performing qubitization of a LGT Hamiltonian based on a 'linear combination of unitaries' (LCU) approach. The cost of the corresponding block encoding operation scales as O~(N) with system size N. Including the corresponding prefactors, our method reduces the gate cost by multiple orders of magnitude when compared to previous LCU methods for the QSE algorithm, which scales as O~(N2) when applied to the Schwinger model. While the qubit and single circuit T-gate cost resulting from our resource analysis is appealing to early fault-tolerant implementation, we find that the number of shots required to avoid numerical instability within the QSE procedure must be significantly reduced in order to improve the feasibility of the methodology we consider and discuss how this might be achieved.
A Quantum information perspective on many-body dispersive forces
Christopher Willby, Martin Kiffner, Joseph Tindall, Jason Crain, Dieter Jaksch
Despite its ubiquity, many-body dispersion remains poorly understood. Here we investigate the distribution of entanglement in quantum Drude oscillator assemblies, minimal models for dispersion bound systems. We analytically determine a relation between entanglement and energy, showing how the entanglement distribution governs dispersive bonding. This suggests that the monogamy of entanglement explains deviations of multipartite dispersive binding energies compared to the commonly used pairwise prediction. We illustrate our findings using examples of a trimer and extended crystal lattices.
Floquet Schrieffer-Wolff transform based on Sylvester equations
Xiao Wang, Fabio Pablo Miguel Mendez-Cordoba, Dieter Jaksch and Frank Schlawin
We present a Floquet Schrieffer-Wolff transform (FSWT) to obtain effective Floquet Hamiltonians and micromotion operators of periodically driven many-body systems for any nonresonant driving frequency. The FSWT perturbatively eliminates the oscillatory components in the driven Hamiltonian by solving operator-valued Sylvester equations with systematic approximations. It goes beyond various high-frequency expansion methods commonly used in Floquet theory, as we demonstrate with the example of the driven Fermi-Hubbard model. In the limit of high driving frequencies, the FSWT Hamiltonian reduces to the widely used Floquet-Magnus result. We anticipate this method will be useful for designing Rydberg multiqubit gates, controlling correlated hopping in quantum simulations in optical lattices, and describing multiorbital and long-range interacting systems driven in-gap.
Contactless transition for wideband double-ridge waveguides
Omar, Jabi, Nico Weiss, Georg Frederik Riemschneider, Ralf Riedinger, Alexander Kölpin
As quantum computers require temperatures near absolute zero, cryostats are utilized to deliver such conditions. This necessitates thermal decoupling in the signal feedlines. This paper introduces contactless waveguide transitions as an alternative to the currently used coaxial cables. Specifically, we present a contactless transition for WRD650 waveguides. The structure was optimised for transmission within the frequency range of 10 GHz to 18 GHz and validated by measurement. Additionally, the transition exhibits a relatively low susceptibility to misalignments and a decreased dispersion in contrast to rectangular waveguides.
Hybridization of the amplitude mode in a confined fermionic superfluid
C.R. Cabrera, R. Henke, L. Broers, J. Skulte, H.P Ojeda Collado, H. Biss, L. Mathey and H. Moritz
In phase transitions, spontaneous symmetry breaking results in a nonzero order parameter and two collective excitations: the Goldstone and the amplitude mode. These modes, which define key properties of superconductors and fermionic superfluids, are well understood in homogeneous systems. However, their behavior under strong confinement remains largely unexplored, particularly when their excitation energy becomes comparable to the imposed discrete level spacing. In this scenario, hybridization between different collective modes is expected to take place. Here, we show how the amplitude mode hybridizes with a spatial mode in a confined fermionic superfluid. Using lattice modulation spectroscopy, we observe the evolution of the mode throughout the entire crossover from the Bardeen-Cooper-Schrieffer (BCS) state to Bose-Einstein condensation (BEC) of molecules. In the BCS regime, the excitation energy is located at twice the pairing gap, then gradually becomes an in-gap excitation in the strongly correlated regime. Further to the BEC limit, the excitation energy approaches twice the level spacing. The spectral weight of this mode vanishes when approaching the superfluid critical temperature. Our experimental results are in excellent agreement with an effective field theory, providing strong evidence that amplitude oscillations hybridize with and eventually transform into breathing oscillations of the order parameter. The strong modification of the excitation spectrum reveals how confinement and finite-size effects impact fundamental modes of symmetry-broken states.
Quantum-inspired tensor-network fractional-step method for incompressible flow in curvilinear coordinates
N.-L. van Hülst, P. Siegl, P. Over, S. Bengoechea, T. Hashizume, M. G. Cecile, T. Rung and D. Jaksch
We introduce an algorithmic framework based on tensor networks for computing fluid flows around immersed objects in curvilinear coordinates. We show that the tensor network simulations can be carried out solely using highly compressed tensor representations of the flow fields and the differential operators and discuss the numerical implementation of the tensor operations required for computing fluid flows in detail. The applicability of our method is demonstrated by applying it to the paradigm example of steady and transient flows around stationary and rotating cylinders. We find excellent quantitative agreement in comparison to finite difference simulations for Strouhal numbers, forces and velocity fields. The properties of our approach are discussed in terms of reduced order models. We estimate the memory saving and potential runtime advantages in comparison to standard finite difference simulations. We find accurate results with errors of less than 0.3% for flow-field compressions by a factor of up to 20 and differential operators compressed by factors of up to 1000 compared to sparse matrix representations. We provide strong numerical evidence that the runtime scaling advantages of the tensor network approach with system size will provide substantial resource savings when simulating larger systems. Finally, we note that, like other tensor network-based fluid flow simulations, our algorithmic framework is directly portable to a quantum computer leading to further scaling advantages.
Dynamical quantum phase transitions on random networks
T. Hashizume, F. Herbort, J. Tindall and D. Jaksch
We investigate two types of dynamical quantum phase transitions (DQPTs) in the transverse-field Ising model on ensembles of Erdős–Rényi networks of size N. These networks consist of vertices connected randomly with probability p (0 < p ⩽ 1). Using analytical derivations and numerical techniques, we compare the characteristics of the transitions for p < 1 against the fully connected network (p = 1). We analytically show that the overlap between the wave function after a quench and the wave function of the fully connected network after the same quench deviates by at most O(N−1/2). For a DQPT defined by an order parameter, the critical point remains unchanged for all p. For a DQPT defined by the rate function of the Loschmidt echo, we find that the rate function deviates from the p = 1 limit near vanishing points of the overlap with the initial state, while the critical point remains independent for all p. Our analysis suggests that this divergence arises from
persistent non-trivial global many-body correlations absent in the p = 1 limit.
Partioned quantum subspace expansion
T. O. Leary, L. W. Anderson, D. Jaksch and M. Kiffner
We present an iterative generalisation of the quantum subspace expansion algorithm used with a Krylov basis. The iterative construction connects a sequence of subspaces via their lowest energy states. Diagonalising a Hamiltonian in a given Krylov subspace requires the same quantum resources in both the single step and sequential cases. We propose a variance-based criterion for determining a good iterative sequence and provide numerical evidence that these good sequences display improved numerical stability over a single step in the presence of finite sampling noise. Implementing the generalisation requires additional classical processing with a polynomial overhead in the subspace dimension. By exchanging quantum circuit depth for additional measurements the quantum subspace expansion algorithm appears to be an approach suited to near term or early error-corrected quantum hardware. Our work suggests that the numerical instability limiting the accuracy of this approach can be substantially alleviated in a parameter-free way.
Tensor-programmable quantum circuits for solving differential equations
P. Siegl, G. S. Reese, H. Hashizume, N.-L. van Hülst and D. Jaksch
We present a quantum solver for partial differential equations based on a flexible matrix product operator representation. Utilizing mid-circuit measurements and a state-dependent norm correction, this scheme overcomes the restriction of unitary operators. Hence, it allows for the direct implementation of a broad class of differential equations governing the dynamics of classical and quantum systems. The capabilities of the framework are demonstrated for an example system governed by Euler equations with absorbing boundaries.
Enhancing measurement precision of non-degenerate two-photon absorption
G. Shukla, S. Panahiyan, D. K. Mishra and F. Schlawin
Recent theoretical and experimental studies have shown that squeezed states of light can be engineered to enhance the resolution of nonlinear optical measurements. Here, we analyze non-degenerate two-photon absorption signals obtained from transmission measurements using two-mode squeezed light and compare different measurement strategies. In particular, we investigate how correlations between the light modes may be used to improve the achievable precision. We find that intensity correlation measurements offer the best performance compared to normalized intensity correlation and noise reduction factor approaches. Under experimental imperfections modeled as linear photon losses, the enhancements from intensity and noise reduction measurements are reduced. In contrast, the normalized intensity correlation remains robust to loss, though this comes at the cost of losing the enhancement from non-classical light fields. This establishes a trade-off between robustness to loss and the achievable quantum advantage.
Enhanced multiphoton ionization driven by quantum light
V. P. Kosheleva, S. Panahiyan, A. Rubio and F. Schlawin
Recent advances in nanoscale and microscale photon-pair sources have enabled quantum-light-matter interactions in regimes where standard approximations, such as the paraxial limit, break down. Identifying quantum advantages in these domains requires a unified, first-principles framework that treats arbitrary quantum states of light and realistic material systems without restrictive assumptions. Here, we present a fully relativistic, beyond-paraxial theory of multiphoton ionization driven by quantum light, incorporating the full spatial structure and correlations of the field alongside material properties. As a case study, we apply the formalism to resonance-enhanced multiphoton ionization (REMPI) of neutral sodium atoms by momentum-entangled photons. We predict cross-section enhancements of several orders of magnitude compared to coherent light, arising entirely from non-paraxial effects and vanishing in the paraxial limit. The enhancement is strongly channel-dependent, with odd multipole transitions benefiting most from favorable parity and interference conditions. The predicted enhancement is accessible with current state-of-the-art photonic and atomic technologies, highlighting a promising avenue for exploiting quantum light in high-precision ionization and spectroscopy experiments.
Lamb-Dicke dynamics of interacting Rydberg atoms coupled to the motion of an optical tweezer array
A. Parvej and L. Mathey
Neutral Rydberg atoms trapped in optical tweezer arrays provide a platform for quantum simulation and computation. In this work, we investigate the Lamb-Dicke dynamics of coupled Rydberg atoms for different trapping frequencies. We model the atomic motion by both internal and motional degrees of freedom, in which the motional states arise due to the oscillation of each atom in optical tweezer traps due to the light-atom interaction. In this setup, the internal states are coupled to a laser light with a Rabi frequency, while each internal state of each atom is also harmonically trapped with a trap frequency that depends on the internal state. The impact of the coherent motion of the optical tweezers on the collective dynamics of the many-body Rydberg atoms is explored for varying Lamb-Dicke parameters and with different trap frequencies. We see the occurrence of dynamical phases e.g., Rabi oscillations in the decoupled limit, the limit torus phase for magic trapping, and the limit cycle phase as the trap frequency is further increased.