You Can Use Qiskit to Control Cold Atom Systems

By Laurin Fischer, Rohit Prasad Bhatt, Fred Jendrzejewski, Daniel J. Egger

Figure 1. Qiskit Cold Atom introduces two new settings for quantum circuits: Fermionic modes and spins. The fermionic setting describes the occupation of fermionic atoms in tweezers or optical lattices. The spin setting describes the orientation of long spins formed by trapped bosons.

Cold Atoms: Quantum circuits and algorithms for quantum simulators?

Universal quantum computers may one day speed-up a variety of challenging problems. However, this universality comes at the price of an algorithmic overhead, such as the need for swap gates and trotterization. If we give up the constraint of a universal quantum computer, we can build specialized quantum computers for many problems of interest. Such devices are called quantum simulators and ultra-cold atoms have become a leading platform of such quantum simulators at a large scale. Already-demonstrated applications involve an enormous variety of condensed-matter problems [1] and, more recently, high-energy physics [4].

With their increasing degree of control, modern cold atomic quantum simulators are now suited for programmable quantum system evolution. Most of these quantum simulators are currently limited to an academic environment. This is in part due to the complexity in formulating the problems in a Hamiltonian language, plus the cumbersome manual compilation of programs on this type of hardware. Integrating cold atom systems with the gate-based software stack of Qiskit alleviates these issues. A gate-based treatment provides a convenient abstraction layer that facilitates the development of specialized quantum algorithms for these devices. This substantially lowers the barrier to entry for users outside of the quantum physics community.

The hardware platform

Cold atoms are gases that are well-isolated from the environment and are cooled to near-zero temperature. In this regard, they are quite close to trapped ion systems, which also heavily draw on the experimental techniques of atomic physics and optics. At such low temperatures, quantum effects are dominant in the behavior of the gas cloud. Thanks to their precise control and flexibility, cold atoms are now a well-established platform for the quantum simulation of complex quantum problems. Similar to quantum circuits with qubits, cold atomic quantum simulators can be described by a cycle with a state initialization, a subsequent programmable evolution, and a final measurement. The last step yields information on the quantum state of the atoms such as their location or spin.

The new Qiskit Cold Atom package integrates cold atoms into the Qiskit stack, thereby extending the concept of quantum circuits to heterogeneous quantum hardware which is not described by qubits. This allows us to show that quantum simulators are powerful platforms complementary to qubit architectures. Furthermore, integrating cold atomic setups as backends made available through a provider in a modern quantum computing software stack, such as Qiskit, facilitates running experiments and allows a broad community to access the devices. Qiskit Cold Atom will provide users classical simulators as well as the ability to control real-world experiments.

Cold atomic quantum circuits

In digital quantum computing, the fundamental unit of information is a two-level system called a qubit. In the circuit model, each qubit is assigned a wire. A measurement on a wire projects the qubit onto one of its internal states i.e. 0 or 1, see Fig. 1(a). By contrast, the internal states and the measurement in a cold atomic system depend on its implementation and requires us to revisit the concept of a wire. Qiskit Cold Atom currently supports two different architectures:

• A “fermionic” architecture where wires describe the occupations of fermionic modes, as realized by trapped fermionic atoms in arrays of optical tweezers, shown in Fig. 1(b).

• A ”spin” architecture where wires are long spins realized e.g. by the total spin state of trapped Bose-Einstein-Condensates, shown in Fig. 1(c).

In traditional quantum circuits, quantum gates define the unitary operations that are carried out on the subsystems formed by the wires which the gate acts on. These gates form an abstraction of the underlying quantum dynamics which are described by the Hamiltonian of the system. Cold atomic quantum systems are commonly described directly at the level of the Hamiltonians that they implement, which are different from the underlying qubit Hamiltonians. Hence, in Qiskit Cold Atom the gates of the circuits are defined through their generating Hamiltonians. The circuits that can be run on the cold atomic hardware thus use different gates which have no direct analogues in the qubit gates typically employed in Qiskit.

Figure 2: Example problem showcasing the capabilities of cold atomic systems to efficiently simulate physical systems. Here, the time evolution of a three site Fermi-Hubbard system is simulated with a fermionic circuit and a qubit circuit. The fermionic hardware directly implements the fermionic dynamics in a single global gate (a). By contrast, the qubit circuit simulates the time evolution with a Trotterization ansatz which results in a very deep circuit (b, c). Simulations of the local spin density after time evolution show that even the deep qubit circuit has trouble keeping up with the exact solution of the fermionic backend (d).

Fermionic circuits

In the fermionic architecture, the wires describe the occupations of individual fermionic modes which obey Fermi statistics. Due to the Pauli exclusion principle, a fermionic mode is either empty, i.e. in the ”vacuum state” |0>, or occupied |1> but no two particles can ever occupy the same mode. We therefore assign one wire to each fermionic mode. Experimentally, fermionic architectures can be realized by loading fermionic atoms into arrays of optical tweezers. A measurement then yields the occupation of each mode, projecting the internal state of each wire onto 0 or 1.

These systems naturally implement Fermi-Hubbard dynamics, where atoms can tunnel to neighboring sites and interact locally. Terms in the Hamiltonian such as the hopping strength or on-site interaction are tuned by the experimental control optics which allows us to define a set of parametrized gates to expose to users.

Such a fermionic backend can naturally simulate certain fermionic systems such as the famous Fermi- Hubbard model which is a paradigmatic model that contains a multitude of complex matter phases and potentially high-temperature superconductivity [5]. In comparison, noisy digital quantum computers struggle to simulate this model due to the mapping from fermions to qubits, and the large number of required Trotter steps, see Fig. 2.

Spin circuits

The Qiskit Cold Atom package also supports high-dimensional spin architectures. Here, the unit of information, i.e. the individual wires in the quantum circuit, are given as quantum angular momenta or spins of length S, where S has a positive integer or half-integer value. This generalizes the qubit setting (i.e. a qudit) where the qubit case corresponds to the smallest possible value S = 1/2. Each individual spin has 2S+1 internal states labeled from |0> to |2S>. In analogy to qubits, we use the convention that each wire is initialized in the |0> state at the start of a circuit. Measuring a wire projects the corresponding spin onto one of its eigenstates and returns the measured outcome from 0 to 2S.

Long spins can be experimentally realized in systems of trapped ions [6], Rydberg atoms [7] as well as cold atoms [3]. In cold atoms the long spins are realized by trapping a cloud of bosonic atoms with two internal states (i.e. effective spin-1/2-systems) in a Bose-Einstein-Condensate (BEC). The number of trapped atoms defines the length of the “collective spins” and may reach very high dimensions such as S ~10ˆ5. Collective spins can be prepared in the initial state |0>, i.e. all atoms pointing down, in an array of optical tweezers, and the spin state of each tweezer can be measured individually.

Single-wire spin gates include rotations of the individual spins and non-linear “squeezing” dynamics. Wires, i.e. spins, are coupled with an entangling gate. This gate set is known in theory to form a universal set on the state space of the spins.

The Qiskit Cold Atom Provider

Users can execute fermionic and spin circuits on the backends supported in Qiskit Cold Atom in similar fashion as is conventionally done with qubit-based backends. These backends are accessed through a provider included in Qiskit Cold Atom. This ColdAtomProvider currently includes simulator backends for a fermionic tweezer hardware and a collective spin hardware.

We now consider example circuits and how to run them on classical simulators of special purpose quantum computers which can, for example, simulate the “Synthetic Quantum Systems (SYNQS)” hardware located at Heidelberg University in Germany. Below, we construct circuits which perform x-rotations on a single collective spin, forming generalized Rabi oscillations between the extremal states 0 and 2S. The same circuits can either be run on the local simulator or be sent to a remote backend. Their construction and execution on the backends follows the standard Qiskit syntax. The results of the final measurements are plotted in Fig. 3.

Figure 3: Results of the measurement of Rabi oscillations formed by x-rotations of a single collective spin simulated both on a local and a remote backend.

Conclusion

Qiskit Cold Atom demonstrates that the established Qiskit API of quantum circuits, gates, providers and backends can handle alternative quantum hardware. This non-qubit-based hardware can be as heterogeneous as general fermionic or spinful cold atom experiments. The simulator backends included in Qiskit Cold Atom serve as tools to explore and learn how cold atomic circuits help certain applications and algorithms by leveraging the different nature of the gates and Hilbert spaces of these systems. Further components of the Qiskit stack, such as circuit transpilation and pulse-level control may be integrated in the future.

Cold atom experiments currently have a large barrier to entry due to the experimental intricacies of the hardware, and the necessity to formulate the physics in the hardware-native “language” on the basis of Hamiltonians. This limits the field of cold atoms to specialized academic circles. Qiskit’s simple circuit workflow lowers this entry barrier by enabling external users to experiment with cold-atom-based hardware without having to deal with the complex underlying dynamics. This makes an increasing part of the quantum technologies landscape accessible to a larger community.

How to get started

We encourage you to familiarize yourself with Qiskit Cold Atom by visiting the GitHub repository, specifically the set of tutorials aimed to introduce Qiskit Cold Atom to new users. The documentation can be found as GitHub pages and you can install the repository via pip install qiskit-cold-atom.

References

[1] Gross, Christian and Bloch, Immanuel Quantum simulations with ultracold atoms in optical lattices, Science, 357, 995–1001, 2017

[2] Murmann, Simon et al. Two Fermions in a double well: Exploring a fundamental building block of the Hubbard model Physical Review Letters 114, 080402, 2015

[3] Kasper, Valentin et al., Universal quantum computation and quantum error correction with ultracold atomic mixtures Quantum Sci. Technol. 7, 015008, 2022

[4] Aidelsburger, Monika, et al. Cold atoms meet lattice gauge theory Philosophical Transactions of the Royal Society A 380.2216 (2022): 20210064.

[5] Chiu, Christie et al. String patterns in the doped Hubbard model Science 365, 251 (2019)
[6] Martin Ringbauer et al. A universal qudit quantum processor with trapped ions, arXiv:2109.06903
[7] Jordi R. Weggemans et al. Solving correlation clustering with QAOA and a Rydberg qudit system: a

full-stack approach, arXiv:2106.11672