Hardware Efficient Quantum Simulation of Non-Abelian Gauge Theories with Qudits on Rydberg Platforms

Unconditional advantage of noisy qudit quantum circuits over biased threshold circuits in constant depth

The rapid evolution of quantum devices fuels concerted efforts to experimentally establish quantum advantage over classical computing. Many demonstrations of quantum advantage, however, rely on computational assumptions and face verification challenges. Furthermore, steady advances in classical algorithms and machine learning make the issue of provable, practically demonstrable quantum advantage a moving target. In this work, we unconditionally demonstrate that parallel quantum computation can exhibit greater computational power than previously recognized. We prove that polynomial-size biased threshold circuits of constant depth-which model neural networks with tunable expressivity-fail to solve certain problems solvable by small constant-depth quantum circuits with local gates, for values of the bias that allow quantifiably large computational power. Additionally, we identify a family of problems that are solvable in constant depth by a universal quantum computer over prime-dimensional qudits with bounded connectivity, but remain hard for polynomial-size biased threshold circuits. We thereby bridge the foundational theory of non-local games in higher dimensions with computational advantage on emerging devices operating on a wide range of physical platforms. Finally, we show that these quantum advantages are robust to noise across all prime qudit dimensions with all-to-all connectivity, enhancing their practical appeal.

Simulating two-dimensional lattice gauge theories on a qudit quantum computer

Particle physics describes the interplay of matter and forces through gauge theories. Yet, the intrinsic quantum nature of gauge theories makes important problems notoriously difficult for classical computational techniques. Quantum computers offer a promising way to overcome these roadblocks. We demonstrate two essential requirements on this path: first, we perform a quantum computation of the properties of the basic building block of two-dimensional lattice quantum electrodynamics, involving both gauge fields and matter. Second, we show how to refine the gauge-field discretization beyond its minimal representation, using a trapped-ion qudit quantum processor, where quantum information is encoded in several states per ion. Such qudits are ideally suited for describing gauge fields, which are naturally high dimensional, leading to reduced register size and circuit complexity. We prepare the ground state of the model using a variational quantum eigensolver and observe the effect of dynamical matter on quantized magnetic fields. By controlling the qudit dimension, we also show how to seamlessly observe the effect of different gauge-field truncations. Finally, we experimentally study the dynamics of pair creation and magnetic energy. Our results open the door for hardware-efficient quantum simulations of gauge theories with qudits in near-term quantum devices.

Deterministic Photonic Entanglement Arising from Non-Abelian Quantum Holonomy

Realizing deterministic, high-fidelity entangling interactions-of the kind that can be utilized for efficient quantum information processing-between photons remains an elusive goal. Here, we address this long-standing issue by devising a protocol for creating and manipulating highly entangled superpositions of well-controlled states of light by using an on-chip photonic system that has recently been shown to implement three-dimensional, non-Abelian quantum holonomy. Our calculations indicate that a subset of such entangled superpositions are maximally entangled, "volume-law" states, and that the underlying entanglement can be distilled and purified for applications in quantum science. Crucially, we generalize this approach to demonstrate the potentiality of deterministically entangling two arbitrarily high, N-dimensional quantum systems, by formally establishing a deep connection between the matrix representations of the unitary quantum holonomy-within energy-degenerate subspaces in which the total excitation number is conserved-and the (2j+1)-dimensional irreducible representations of the rotation operator, where j=(N-1)/2 and N≥2. Specifically, our protocol deterministically entangles spatially localized modes that are not only distinguishable but are also individually accessible and amenable to state preparation and measurement, and therefore, we envisage that this entangling mechanism could be utilized for deterministic quantum information processing with light.

Photonic Simulation of Majorana-Based Jones Polynomials

By braiding non-Abelian anyons it is possible to realize fault-tolerant quantum algorithms through the computation of Jones polynomials. So far, this has been an experimentally formidable task. In this Letter, a photonic quantum system employing two-photon correlations and nondissipative imaginary-time evolution is utilized to simulate two inequivalent braiding operations of Majorana zero modes. The resulting amplitudes are shown to be mathematically equivalent to Jones polynomials. The high fidelity of our optical platform allows us to distinguish between a wide range of links, such as Hopf links, Solomon links, Trefoil knots, Figure Eight knots and Borromean rings, through determining their corresponding Jones polynomials. Our photonic quantum simulator represents a significant step towards executing fault-tolerant quantum algorithms based on topological quantum encoding and manipulation.

Magnetically Tunable Electric Dipolar Interactions of Ultracold Polar Molecules in the Quantum Ergodic Regime

By leveraging the hyperfine interaction between the rotational and nuclear spin degrees of freedom, we demonstrate extensive magnetic control over the electric dipole moments, electric dipolar interactions, and ac Stark shifts of ground-state alkali-dimer molecules such as KRb(X^{1}Σ^{+}). The control is enabled by narrow avoided crossings and the highly ergodic character of molecular eigenstates at low magnetic fields, offering a general and robust way of continuously tuning the intermolecular electric dipolar interaction for applications in quantum simulation, quantum sensing, and dipolar spinor physics.

Quantum Simulation of SU(3) Lattice Yang-Mills Theory at Leading Order in Large-N_{c} Expansion

Quantum simulations of the dynamics of QCD have been limited by the complexities of mapping the continuous gauge fields onto quantum computers. By parametrizing the gauge invariant Hilbert space in terms of plaquette degrees of freedom, we show how the Hilbert space and interactions can be expanded in inverse powers of N_{c}. At leading order in this expansion, the Hamiltonian simplifies dramatically, both in the required size of the Hilbert space as well as the type of interactions involved. Adding a truncation of the resulting Hilbert space in terms of local energy states we give explicit constructions that allow simple representations of SU(3) gauge fields on qubits and qutrits. This formulation allows a simulation of the real time dynamics of a SU(3) lattice gauge theory on a 5×5 and 8×8 lattice on ibm_torino with a CNOT depth of 113.

Microscopic Study on Superexchange Dynamics of Composite Spin-1 Bosons

We report on an experimental simulation of the spin-1 Heisenberg model with composite bosons in a one-dimensional chain based on the two-component Bose-Hubbard model. Exploiting our site- and spin-resolved quantum gas microscope, we observed faster superexchange dynamics of the spin-1 system compared to its spin-1/2 counterpart, which is attributed to the enhancement effect of multi-bosons. We further probed the nonequilibrium spin dynamics driven by the superexchange and single-ion anisotropy terms, unveiling the linear expansion of the spin-spin correlations, which is limited by the Lieb-Robinson bound. Based on the superexchange process, we prepared and verified the entangled qutrits pairs with these composite spin-1 bosons, potentially being applied in qutrit-based quantum information processing.

Realization of higher-order topological lattices on a quantum computer

Programmable quantum simulators may one day outperform classical computers at certain tasks. But at present, the range of viable applications with noisy intermediate-scale quantum (NISQ) devices remains limited by gate errors and the number of high-quality qubits. Here, we develop an approach that places digital NISQ hardware as a versatile platform for simulating multi-dimensional condensed matter systems. Our method encodes a high-dimensional lattice in terms of many-body interactions on a reduced-dimension model, thereby taking full advantage of the exponentially large Hilbert space of the host quantum system. With circuit optimization and error mitigation techniques, we measured on IBM superconducting quantum processors the topological state dynamics and protected mid-gap spectra of higher-order topological lattices, in up to four dimensions, with high accuracy. Our projected resource requirements scale favorably with system size and lattice dimensionality compared to classical computation, suggesting a possible route to useful quantum advantage in the longer term.

Molecular nanomagnets: a viable path toward quantum information processing?

Molecular nanomagnets (MNMs), molecules containing interacting spins, have been a playground for quantum mechanics. They are characterized by many accessible low-energy levels that can be exploited to store and process quantum information. This naturally opens the possibility of using them as qudits, thus enlarging the tools of quantum logic with respect to qubit-based architectures. These additional degrees of freedom recently prompted the proposal for encoding qubits with embedded quantum error correction (QEC) in single molecules. QEC is the holy grail of quantum computing and this qudit approach could circumvent the large overhead of physical qubits typical of standard multi-qubit codes. Another important strength of the molecular approach is the extremely high degree of control achieved in preparing complex supramolecular structures where individual qudits are linked preserving their individual properties and coherence. This is particularly relevant for building quantum simulators, controllable systems able to mimic the dynamics of other quantum objects. The use of MNMs for quantum information processing is a rapidly evolving field which still requires to be fully experimentally explored. The key issues to be settled are related to scaling up the number of qudits/qubits and their individual addressing. Several promising possibilities are being intensively explored, ranging from the use of single-molecule transistors or superconducting devices to optical readout techniques. Moreover, new tools from chemistry could be also at hand, like the chiral-induced spin selectivity. In this paper, we will review the present status of this interdisciplinary research field, discuss the open challenges and envisioned solution paths which could finally unleash the very large potential of molecular spins for quantum technologies.

Experimental Quantum Channel Discrimination Using Metastable States of a Trapped Ion

We present experimental demonstrations of accurate and unambiguous single-shot discrimination between three quantum channels using a single trapped ^{40}Ca^{+} ion. The three channels cannot be distinguished unambiguously using repeated single channel queries, the natural classical analogue. We develop techniques for using the six-dimensional D_{5/2} state space for quantum information processing, and we implement protocols to discriminate quantum channel analogues of phase shift keying and amplitude shift keying data encodings used in classical radio communication. The demonstrations achieve discrimination accuracy exceeding 99% in each case, limited entirely by known experimental imperfections.

Quantum and Classical Spin-Network Algorithms for q-Deformed Kogut-Susskind Gauge Theories

Treating the infinite-dimensional Hilbert space of non-Abelian gauge theories is an outstanding challenge for classical and quantum simulations. Here, we employ q-deformed Kogut-Susskind lattice gauge theories, obtained by deforming the defining symmetry algebra to a quantum group. In contrast to other formulations, this approach simultaneously provides a controlled regularization of the infinite-dimensional local Hilbert space while preserving essential symmetry-related properties. This enables the development of both quantum as well as quantum-inspired classical spin-network algorithms for q-deformed gauge theories. To be explicit, we focus on SU(2)_{k} gauge theories with k∈N that are controlled by the deformation parameter q=e^{2πi/(k+2)}, a root of unity, and converge to the standard SU(2) Kogut-Susskind model as k→∞. In particular, we demonstrate that this formulation is well suited for efficient tensor network representations by variational ground-state simulations in 2D, providing first evidence that the continuum limit can be reached with k=O(10). Finally, we develop a scalable quantum algorithm for Trotterized real-time evolution by analytically diagonalizing the SU(2)_{k} plaquette interactions. Our work gives a new perspective for the application of tensor network methods to high-energy physics and paves the way for quantum simulations of non-Abelian gauge theories far from equilibrium where no other methods are currently available.

Towards Quantum Computing Phase Diagrams of Gauge Theories with Thermal Pure Quantum States

The phase diagram of strong interactions in nature at finite temperature and chemical potential remains largely theoretically unexplored due to inadequacy of Monte-Carlo-based computational techniques in overcoming a sign problem. Quantum computing offers a sign-problem-free approach, but evaluating thermal expectation values is generally resource intensive on quantum computers. To facilitate thermodynamic studies of gauge theories, we propose a generalization of the thermal-pure-quantum-state formulation of statistical mechanics applied to constrained gauge-theory dynamics, and numerically demonstrate that the phase diagram of a simple low-dimensional gauge theory is robustly determined using this approach, including mapping a chiral phase transition in the model at finite temperature and chemical potential. Quantum algorithms, resource requirements, and algorithmic and hardware error analysis are further discussed to motivate future implementations. Thermal pure quantum states, therefore, may present a suitable candidate for efficient thermal simulations of gauge theories in the era of quantum computing.

A hybrid-qudit representation of digital RGB images

Quantum image processing is an emerging topic in the field of quantum information and technology. In this paper, we propose a new quantum image representation of RGB images with deterministic image retrieval, which is an improvement over all the similar existing representations in terms of using minimum resource. We use two entangled quantum registers constituting of total 7 qutrits to encode the color channels and their intensities. Additionally, we generalize the existing encoding methods by using both qubits and qutrits to encode the pixel positions of a rectangular image. This hybrid-qudit approach aligns well with the current progress of NISQ devices in incorporating higher dimensional quantum systems than qubits. We then describe the image encoding method using higher-order qubit-qutrit gates, and demonstrate the decomposition of these gates in terms of simpler elementary gates. We use the Google Cirq's quantum simulator to verify the image preparation in both the ideal noise-free scenario and in presence of realistic noise modelling. We show that the complexity of the image encoding process is linear in the number of pixels. Lastly, we discuss the image compression and some basic RGB image processing protocols using our representation.

Solving independent set problems with photonic quantum circuits

An independent set (IS) is a set of vertices in a graph such that no edge connects any two vertices. In adiabatic quantum computation [E. Farhi, et al., Science 292, 472-475 (2001); A. Das, B. K. Chakrabarti, Rev. Mod. Phys. 80, 1061-1081 (2008)], a given graph G(V, E) can be naturally mapped onto a many-body Hamiltonian [Formula: see text], with edges [Formula: see text] being the two-body interactions between adjacent vertices [Formula: see text]. Thus, solving the IS problem is equivalent to finding all the computational basis ground states of [Formula: see text]. Very recently, non-Abelian adiabatic mixing (NAAM) has been proposed to address this task, exploiting an emergent non-Abelian gauge symmetry of [Formula: see text] [B. Wu, H. Yu, F. Wilczek, Phys. Rev. A 101, 012318 (2020)]. Here, we solve a representative IS problem [Formula: see text] by simulating the NAAM digitally using a linear optical quantum network, consisting of three C-Phase gates, four deterministic two-qubit gate arrays (DGA), and ten single rotation gates. The maximum IS has been successfully identified with sufficient Trotterization steps and a carefully chosen evolution path. Remarkably, we find IS with a total probability of 0.875(16), among which the nontrivial ones have a considerable weight of about 31.4%. Our experiment demonstrates the potential advantage of NAAM for solving IS-equivalent problems.

Emergent Glassy Behavior in a Kagome Rydberg Atom Array

We present large-scale quantum Monte Carlo simulation results on a realistic Hamiltonian of kagome-lattice Rydberg atom arrays. Although the system has no intrinsic disorder, intriguingly, our analyses of static and dynamic properties on large system sizes reveal emergent glassy behavior in a region of parameter space located between two valence bond solid phases. The extent of this glassy region is demarcated using the Edwards-Anderson order parameter, and its phase transitions to the two proximate valence bond solids-as well as the crossover towards a trivial paramagnetic phase-are identified. We demonstrate the intrinsically slow (imaginary) time dynamics deep inside the glassy phase and discuss experimental considerations for detecting such a quantum disordered phase with numerous nearly degenerate local minima. Our proposal paves a new route to the study of real-time glassy phenomena and highlights the potential for quantum simulation of a distinct phase of quantum matter beyond solids and liquids in current-generation Rydberg platforms.

Native qudit entanglement in a trapped ion quantum processor

Quantum information carriers, just like most physical systems, naturally occupy high-dimensional Hilbert spaces. Instead of restricting them to a two-level subspace, these high-dimensional (qudit) quantum systems are emerging as a powerful resource for the next generation of quantum processors. Yet harnessing the potential of these systems requires efficient ways of generating the desired interaction between them. Here, we experimentally demonstrate an implementation of a native two-qudit entangling gate up to dimension 5 in a trapped-ion system. This is achieved by generalizing a recently proposed light-shift gate mechanism to generate genuine qudit entanglement in a single application of the gate. The gate seamlessly adapts to the local dimension of the system with a calibration overhead that is independent of the dimension.

Generalized Toffoli Gate Decomposition Using Ququints: Towards Realizing Grover's Algorithm with Qudits

Qubits, which are the quantum counterparts of classical bits, are used as basic information units for quantum information processing, whereas underlying physical information carriers, e.g., (artificial) atoms or ions, admit encoding of more complex multilevel states-qudits. Recently, significant attention has been paid to the idea of using qudit encoding as a way for further scaling quantum processors. In this work, we present an efficient decomposition of the generalized Toffoli gate on five-level quantum systems-so-called ququints-that use ququints' space as the space of two qubits with a joint ancillary state. The basic two-qubit operation we use is a version of the controlled-phase gate. The proposed N-qubit Toffoli gate decomposition has O(N) asymptotic depth and does not use ancillary qubits. We then apply our results for Grover's algorithm, where we indicate on the sizable advantage of using the qudit-based approach with the proposed decomposition in comparison to the standard qubit case. We expect that our results are applicable for quantum processors based on various physical platforms, such as trapped ions, neutral atoms, protonic systems, superconducting circuits, and others.

Emergent Z_{2} Gauge Theories and Topological Excitations in Rydberg Atom Arrays

Strongly interacting arrays of Rydberg atoms provide versatile platforms for exploring exotic many-body phases and dynamics of correlated quantum systems. Motivated by recent experimental advances, we show that the combination of Rydberg interactions and appropriate lattice geometries naturally leads to emergent Z_{2} gauge theories endowed with matter fields. Based on this mapping, we describe how Rydberg platforms could realize two distinct classes of topological Z_{2} quantum spin liquids, which differ in their patterns of translational symmetry fractionalization. We also discuss the natures of the fractionalized excitations of these Z_{2} spin liquid states using both fermionic and bosonic parton theories and illustrate their rich interplay with proximate solid phases.

Quantum algorithms for quantum dynamics

Among the many computational challenges faced across different disciplines, quantum-mechanical systems pose some of the hardest ones and offer a natural playground for the growing field of quantum technologies. In this Perspective, we discuss quantum algorithmic solutions for quantum dynamics, reporting on the latest developments and offering a viewpoint on their potential and current limitations. We present some of the most promising areas of application and identify possible research directions for the coming years.