Emulating Anyonic Fractional Statistical Behavior in a Superconducting Quantum Circuit

Demonstrating Path-Independent Anyonic Braiding on a Modular Superconducting Quantum Processor

Anyons, exotic quasiparticles in two-dimensional space exhibiting nontrivial exchange statistics, play a crucial role in universal topological quantum computing. One notable proposal to manifest the fractional statistics of anyons is the toric code model; however, scaling up its size through quantum simulation poses a serious challenge because of its highly entangled ground state. In this Letter, we demonstrate that a modular superconducting quantum processor enables hardware-pragmatic implementation of the toric code model. Through in-parallel control across separate modules, we generate a 10-qubit toric code ground state in four steps and realize six distinct braiding paths to benchmark the performance of anyonic statistics. The path independence of the anyonic braiding statistics is verified by correlation measurements in an efficient and scalable fashion. Our modular approach, serving as a hardware embodiment of the toric code model, offers a promising avenue toward scalable simulation of topological phases, paving the way for quantum simulation in a distributed fashion.

Observing and braiding topological Majorana modes on programmable quantum simulators

Electrons are indivisible elementary particles, yet paradoxically a collection of them can act as a fraction of a single electron, exhibiting exotic and useful properties. One such collective excitation, known as a topological Majorana mode, is naturally stable against perturbations, such as unwanted local noise, and can thereby robustly store quantum information. As such, Majorana modes serve as the basic primitive of topological quantum computing, providing resilience to errors. However, their demonstration on quantum hardware has remained elusive. Here, we demonstrate a verifiable identification and braiding of topological Majorana modes using a superconducting quantum processor as a quantum simulator. By simulating fermions on a one-dimensional lattice subject to a periodic drive, we confirm the existence of Majorana modes localized at the edges, and distinguish them from other trivial modes. To simulate a basic logical operation of topological quantum computing known as braiding, we propose a non-adiabatic technique, whose implementation reveals correct braiding statistics in our experiments. This work could further be used to study topological models of matter using circuit-based simulations, and shows that long-sought quantum phenomena can be realized by anyone in cloud-run quantum simulations, whereby accelerating fundamental discoveries in quantum science and technology.

Uniquely identifying topological order based on boundary-bulk duality and anyon condensation

Topological order is a new quantum phase that is beyond Landau's symmetry-breaking paradigm. Its defining features include robust degenerate ground states, long-range entanglement and anyons. It was known that R and F matrices, which characterize the fusion-braiding properties of anyons, can be used to uniquely identify topological order. In this article, we explore an essential question: how can the R and F matrices be experimentally measured? We show that the braidings, i.e. the R matrices, can be completely determined by the half braidings of boundary excitations due to the boundary-bulk duality and the anyon condensation. The F matrices can also be measured by comparing the quantum states involving the fusion of three anyons in two different orders. Thus we provide a model-independent experimental protocol to uniquely identify topological order. By using quantum simulations based on a toric code model with boundaries encoded in three- and four-qubit systems and state-of-the-art technology, we obtain the first experimental measurement of R and F matrices by means of an NMR quantum computer at room temperature.

Probing Operator Spreading via Floquet Engineering in a Superconducting Circuit

Operator spreading, often characterized by out-of-time-order correlators (OTOCs), is one of the central concepts in quantum many-body physics. However, measuring OTOCs is experimentally challenging due to the requirement of reversing the time evolution of systems. Here we apply Floquet engineering to investigate operator spreading in a superconducting 10-qubit chain. Floquet engineering provides an effective way to tune the coupling strength between nearby qubits, which is used to demonstrate quantum walks with tunable couplings, reversed time evolution, and the measurement of OTOCs. A clear light-cone-like operator propagation is observed in the system with multiple excitations, and has a nearly equal velocity as the single-particle quantum walk. For the butterfly operator that is nonlocal (local) under the Jordan-Wigner transformation, the OTOCs show distinct behaviors with (without) a signature of information scrambling in the near integrable system.

Realizing topologically ordered states on a quantum processor

[Figure: see text].

Bosonic Continuum Theory of One-Dimensional Lattice Anyons

Anyons with arbitrary exchange phases exist on 1D lattices in ultracold gases. Yet, known continuum theories in 1D do not match. We derive the continuum limit of 1D lattice anyons via interacting bosons. The theory maintains the exchange phase periodicity fully analogous to 2D anyons. This provides a mapping between experiments, lattice anyons, and continuum theories, including Kundu anyons with a natural regularization as a special case. We numerically estimate the Luttinger parameter as a function of the exchange angle to characterize long-range signatures of the theory and predict different velocities for left- and right-moving collective excitations.

Emulating Quantum Teleportation of a Majorana Zero Mode Qubit

Topological quantum computation based on anyons is a promising approach to achieve fault-tolerant quantum computing. The Majorana zero modes in the Kitaev chain are an example of non-Abelian anyons where braiding operations can be used to perform quantum gates. Here we perform a quantum simulation of topological quantum computing, by teleporting a qubit encoded in the Majorana zero modes of a Kitaev chain. The quantum simulation is performed by mapping the Kitaev chain to its equivalent spin version and realizing the ground states in a superconducting quantum processor. The teleportation transfers the quantum state encoded in the spin-mapped version of the Majorana zero mode states between two Kitaev chains. The teleportation circuit is realized using only braiding operations and can be achieved despite being restricted to Clifford gates for the Ising anyons. The Majorana encoding is a quantum error detecting code for phase-flip errors, which is used to improve the average fidelity of the teleportation for six distinct states from 70.76±0.35% to 84.60±0.11%, well beyond the classical bound in either case.

Manipulating Complex Hybrid Entanglement and Testing Multipartite Bell Inequalities in a Superconducting Circuit

Quantum correlations in observables of multiple systems not only are of fundamental interest, but also play a key role in quantum information processing. As a signature of these correlations, the violation of Bell inequalities has not been demonstrated with multipartite hybrid entanglement involving both continuous and discrete variables. Here we create a five-partite entangled state with three superconducting transmon qubits and two photonic qubits, each encoded in the mesoscopic field of a microwave cavity. We reveal the quantum correlations among these distinct elements by joint Wigner tomography of the two cavity fields conditional on the detection of the qubits and by test of a five-partite Bell inequality. The measured Bell signal is 8.381±0.038, surpassing the bound of 8 for a four-partite entanglement imposed by quantum correlations by 10 standard deviations, demonstrating the genuine five-partite entanglement in a hybrid quantum system.

Digital Simulation of Topological Matter on Programmable Quantum Processors

Simulating the topological phases of matter in synthetic quantum simulators is a topic of considerable interest. Given the universality of digital quantum simulators, the prospect of digitally simulating exotic topological phases is greatly enhanced. However, it is still an open question how to realize the digital quantum simulation of topological phases of matter. Here, using common single- and two-qubit elementary quantum gates, we propose and demonstrate an approach to design topologically protected quantum circuits on the current generation of noisy quantum processors where spin-orbital coupling and related topological matter can be digitally simulated. In particular, a low-depth topological quantum circuit is performed on both the IBM and Rigetti quantum processors. In the experiments, we not only observe but also distinguish the 0 and π energy topological edge states by measuring the qubit excitation distribution at the output of the circuits.

Deterministic Entanglement Swapping in a Superconducting Circuit

Entanglement swapping, the process to entangle two particles without coupling them in any way, is one of the most striking manifestations of the quantum-mechanical nonlocal characteristic. Besides fundamental interest, this process has applications in complex entanglement manipulation and quantum communication. Here we report a high-fidelity, unconditional entanglement swapping experiment in a superconducting circuit. The measured concurrence characterizing the qubit-qubit entanglement produced by swapping is above 0.75, confirming most of the entanglement of one qubit with its partner is deterministically transferred to another qubit that has never interacted with it. We further realize delayed-choice entanglement swapping, showing whether two qubits previously behaved as in an entangled state or as in a separable state is determined by a later choice of the type of measurement on their partners. This is the first demonstration of entanglement-separability duality in a deterministic way.

Propagation and Localization of Collective Excitations on a 24-Qubit Superconducting Processor

Superconducting circuits have emerged as a powerful platform of quantum simulation, especially for emulating the dynamics of quantum many-body systems, because of their tunable interaction, long coherence time, and high-precision control. Here in experiments, we construct a Bose-Hubbard ladder with a ladder array of 20 qubits on a 24-qubit superconducting processor. We investigate theoretically and demonstrate experimentally the dynamics of single- and double-excitation states with distinct behaviors, indicating the uniqueness of the Bose-Hubbard ladder. We observe the linear propagation of photons in the single-excitation case, satisfying the Lieb-Robinson bounds. The double-excitation state, initially placed at the edge, localizes; while placed in the bulk, it splits into two single-excitation modes spreading linearly toward two boundaries, respectively. Remarkably, these phenomena, studied both theoretically and numerically as unique properties of the Bose-Hubbard ladder, are represented coherently by pairs of controllable qubits in experiments. Our results show that collective excitations, as a single mode, are not free. This work paves the way to simulation of exotic logic particles by subtly encoding physical qubits and exploration of rich physics by superconducting circuits.

Experimental observation of classical analogy of topological entanglement entropy

Long-range entanglement is an important aspect of the topological orders, so efficient methods to characterize the long-range entanglement are often needed. In this regard, topological entanglement entropy (TEE) is often used for such a purpose but the experimental observation of TEE in a topological order remains a challenge. Here, we propose a scheme to observe TEE in the topological order by constructing specific minimum entropy states (MESs). We then experimentally construct the classical microwave analogs of the MESs and simulate the nontrivial topological order with the TEE in Kitaev toric code, which is in agreement with theoretical predictions. We also experimentally simulate the transition from Z₂ topologically ordered state to topologically trivial state.

Piezo-driven sample rotation system with ultra-low electron temperature

Piezo-driven rotator is convenient for tilted magnetic field experiments due to its precise angle control. However, the rotator itself and the sample mounted on it are difficult to be cooled down because of extra heat leaks and presumably bad thermal contacts from the piezo. Here, we report a piezo-driven sample rotation system designed for ultra-low temperature environment. The sample, as well as the rotating sample holder, can be cooled to as low as 25 mK by customized thermal links and thermal contacts. More importantly, the electron temperature in the electrical transport measurements can also be cooled down to 25 mK with the help of home-made filters. To demonstrate the application of our rotation system at ultra-low electron temperature, a measurement revealing tilt-induced localization and delocalization in the second Landau level of two-dimensional electron gas is provided.

Demonstration of Topological Robustness of Anyonic Braiding Statistics with a Superconducting Quantum Circuit

Anyons are quasiparticles occurring in two dimensions, whose topological properties are believed to be robust against local perturbations and may hold promise for fault tolerant quantum computing. Here we present an experiment of demonstrating the path independent nature of anyonic braiding statistics with a superconducting quantum circuit, which represents a 7-qubit version of the toric code model. We dynamically create the ground state of the model, achieving a state fidelity of 0.688±0.015 as verified by quantum state tomography. Anyonic excitations and braiding operations are subsequently implemented with single-qubit rotations. The braiding robustness is witnessed by looping an anyonic excitation around another one along two distinct, but topologically equivalent paths: Both reveal the nontrivial π-phase shift, the hallmark of Abelian 1/2 anyons, with a phase accuracy of ∼99% in the Ramsey-type interference measurement.

10-Qubit Entanglement and Parallel Logic Operations with a Superconducting Circuit

Here we report on the production and tomography of genuinely entangled Greenberger-Horne-Zeilinger states with up to ten qubits connecting to a bus resonator in a superconducting circuit, where the resonator-mediated qubit-qubit interactions are used to controllably entangle multiple qubits and to operate on different pairs of qubits in parallel. The resulting 10-qubit density matrix is probed by quantum state tomography, with a fidelity of 0.668±0.025. Our results demonstrate the largest entanglement created so far in solid-state architectures and pave the way to large-scale quantum computation.

Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit

Geometric phase, associated with holonomy transformation in quantum state space, is an important quantum-mechanical effect. Besides fundamental interest, this effect has practical applications, among which geometric quantum computation is a paradigm, where quantum logic operations are realized through geometric phase manipulation that has some intrinsic noise-resilient advantages and may enable simplified implementation of multi-qubit gates compared to the dynamical approach. Here we report observation of a continuous-variable geometric phase and demonstrate a quantum gate protocol based on this phase in a superconducting circuit, where five qubits are controllably coupled to a resonator. Our geometric approach allows for one-step implementation of n-qubit controlled-phase gates, which represents a remarkable advantage compared to gate decomposition methods, where the number of required steps dramatically increases with n. Following this approach, we realize these gates with n up to 4, verifying the high efficiency of this geometric manipulation for quantum computation.

Geometric phase is of fundamental interest and has practical application in quantum computation. Here the authors observe continuous-variable geometric phase in a superconducting circuit and demonstrate a multi-qubit controlled phase gate protocol based on this geometric effect.

Experimental Identification of Non-Abelian Topological Orders on a Quantum Simulator

Topological orders can be used as media for topological quantum computing-a promising quantum computation model due to its invulnerability against local errors. Conversely, a quantum simulator, often regarded as a quantum computing device for special purposes, also offers a way of characterizing topological orders. Here, we show how to identify distinct topological orders via measuring their modular S and T matrices. In particular, we employ a nuclear magnetic resonance quantum simulator to study the properties of three topologically ordered matter phases described by the string-net model with two string types, including the Z_{2} toric code, doubled semion, and doubled Fibonacci. The third one, non-Abelian Fibonacci order is notably expected to be the simplest candidate for universal topological quantum computing. Our experiment serves as the basic module, built on which one can simulate braiding of non-Abelian anyons and ultimately, topological quantum computation via the braiding, and thus provides a new approach of investigating topological orders using quantum computers.