This project explores the potential of multi-qubit gates for quantum computing on a superconducting qubit platform. The main goal is to develop superconducting architectures and control methods to efficiently generate multi-qubit states going beyond the current paradigm of decomposing all state manipulations into single and two-qubit gates. We design and realize parametric couplers [McKay et al., Phys. Rev. Applied 6 (2016)] connecting multiple superconducting qubits and investigate multi-qubit operations that allow us to entangle multiple qubits at the same time. We explore multi-qubit entangling interactions and evaluate the maximally possible number of qubits coupled to a single coupler. The final goal is to address the question if there is an advantage in using multi-qubit gates over traditional two-qubit gates in practical experiments. To this end we assess the efficiency of multi-qubit gates in an algorithmic context targeting, e.g., a quantum chemistry problem, such as determining the ground state and energy spectrum of a small molecule using the variational quantum eigensolver (VQE). The devices and methods developed in this project will enhance the scalability of superconducting qubit platforms and the efficiency of quantum algorithms.

### Funding

DFG/SNF Project ‘Multi-qubit gates for the efficient exploration of Hilbert space with superconducting qubit systems’

EU Cofund action ‘Qustec‘

### Team Members

We are currently searching for qualified Postdocs.

### Collaborators

Michael Hartmann (University Erlangen/Nürnberg)

### Recent results

##### Gate-Efficient Simulation of Molecular Eigenstates on a Quantum Computer – M. Ganzhorn et al. Phys. Rev. Applied **11**, 044092 (2019)

A key requirement to perform simulations of large quantum systems on near-term quantum hardware is the design of quantum algorithms with a short circuit depth that finish within the available coherence time. A way to stay within the limits of coherence is to reduce the number of gates by implementing a gate set that matches the requirements of the specific algorithm of interest directly in the hardware. Here, we show that exchange-type gates are a promising choice for simulating molecular eigenstates on near-term quantum devices since these gates preserve the number of excitations in the system. We report on the experimental implementation of a variational algorithm on a superconducting qubit platform to compute the eigenstate energies of molecular hydrogen. We utilize a parametrically driven tunable coupler to realize exchange-type gates that are configurable in amplitude and phase on two fixed-frequency superconducting qubits. With gate fidelities around 95%, we are able to compute the eigenstates to within an accuracy of 50 mHa (milliHartree) on average, a limit set by the coherence time of the tunable coupler.

##### Entanglement Generation in Superconducting Qubits Using Holonomic Operations – D. Egger et al. Phys. Rev. Applied **11**, 014017 (2019).

We investigate a nonadiabatic holonomic operation that enables us to entangle two fixed-frequency superconducting transmon qubits attached to a common bus resonator. Two coherent microwave tones are applied simultaneously to the two qubits and drive transitions between the first excited resonator state and the second excited state of each qubit. The cyclic evolution within this effective three-level Λ-type system gives rise to a holonomic operation entangling the two qubits. Two-qubit states with 95% fidelity, limited mainly by charge noise of the current device, are created within 213 ns. This scheme is a step toward implementation of a swap-type gate directly in an all-microwave controlled hardware platform. By extending the available set of two-qubit operations in the fixed-frequency qubit architecture, the proposed scheme may find applications in near-term quantum applications using variational algorithms to efficiently create problem-specific trial states. We illustrate this point by computing the ground state of molecular hydrogen using the holonomic operation.

##### Quantum optimization using variational algorithms on near-term quantum devices – N. Moll et al. Quantum Science and Technology 3, 3 (2019).

Universal fault-tolerant quantum computers will require error-free execution of long sequences of quantum gate operations, which is expected to involve millions of physical qubits. Before the full power of such machines will be available, near-term quantum devices will provide several hundred qubits and limited error correction. Still, there is a realistic prospect to run useful algorithms within the limited circuit depth of such devices. Particularly promising are optimization algorithms that follow a hybrid approach: the aim is to steer a highly entangled state on a quantum system to a target state that minimizes a cost function via variation of some gate parameters. This variational approach can be used both for classical optimization problems as well as for problems in quantum chemistry. The challenge is to converge to the target state given the limited coherence time and connectivity of the qubits. In this context, the *quantum volume* as a metric to compare the power of near-term quantum devices is discussed. With focus on chemistry applications, a general description of variational algorithms is provided and the mapping from fermions to qubits is explained. Coupled-cluster and heuristic trial wave-functions are considered for efficiently finding molecular ground states. Furthermore, simple error-mitigation schemes are introduced that could improve the accuracy of determining ground-state energies. Advancing these techniques may lead to near-term demonstrations of useful quantum computation with systems containing several hundred qubits.