Simulating Topological Systems with Qiskit Pulse
By John P. T. Stenger, Nicholas T. Bronn, Daniel J. Egger, David Pekker
There’s a natural place where quantum computers have the potential to trounce classical computers: simulating quantum systems. While today’s quantum computers are noisy, small-scale devices, they are increasingly showing promise on these kinds of simulations. And simply by tackling these problems, we’re adding important new features to our quantum computing toolkit.
In our new paper, we successfully simulated a quantum topological condensed matter system on an IBM Quantum processor: the dynamics of braiding Majorana zero modes (MZM) on a tri-junction. While our research demonstrates the potential power of quantum computers for quantum simulation, it wouldn’t have been possible without Qiskit Pulse’s pulse-level control. Qiskit Pulse allowed us to overcome the device’s noise with specially-crafted controlled gates.
Observing the braiding of nonabelian anyons, such as the MZMs, in a physical system is an important outstanding problem in condensed matter physics. This process involves moving the anyons around each other in two-dimensions, forming a three-dimensional braid in spacetime. A key feature is that the evolution of the quantum state of the anyons depends only on the topology of the braid, i.e. the order in which the anyons are moved around each other, and not the details of how braiding is carried out. It is hoped that this insensitivity could be used in the future to construct topologically protected quantum computers. In our case, we focus on MZMs that we move around a Y-shaped structure called a tri-junction.
Completing the braid in this tri-junction system requires moving one MZM all the way around the other MZM. A pair of MZMs start on two of the three tips of the Y-shaped tri-junction. A MZM can be moved from the tip of one arm of the Y to another by tuning the couplings at the center of the Y. Using three steps, we can swap the two MZMs and using six steps we can complete one full braid in which one MZM goes all the way around the other.
Simulating braiding on a quantum computer requires a few computational tricks employed across quantum computing. First, using the Jordan-Wigner transformation we map this system of three fermions, the tri-junction Hamiltonian, onto a Hamiltonian of three qubits. From here, we apply the Suzuki-Trotter decomposition to break the time-evolution of the system down into one- and two-qubit gates acting in small discrete time steps. Altering the quantum gates using Qiskit is the simulation’s equivalent to changing the coupling between the tri-junction’s bars. We found that the optimal way to apply one of the braiding steps — one-sixth of the total braid — took three time-steps worth of qubit gates in Qiskit.
Herein arose a problem: using the hardware’s native gates, we could only perform these three time-steps worth of operations before qubit decoherence set in. Here, the CNOT gates entangling the qubits were responsible for most of the errors in the quantum computation. However, we didn’t need the full entanglement provided by the CNOT gate in each time-step to simulate the motion of the MZMs; we realized that Qiskit’s implementation of the fully entangling CNOT gate was imparting a longer microwave pulse than we needed. Because each time step of the Suzuki-Trotter decomposition requires a smaller amount of entanglement, we used Qiskit Pulse to design a scaled-down version of the CNOT gate. This was done by asking the IQX backend for the pulse schedule corresponding to the CNOT gate, and modifying it by shortening the entangling part to get just what is needed for the desired interaction. By modifying the pulses given by the backend, no further calibration is required, making it feasible to employ these custom pulses for simulation (the recent announcement of Qiskit Runtime will make custom calibrations less demanding in the future). By providing a fraction of the entanglement of the native CNOT gate, and therefore only a fraction of the pulse duration, more interactions could be simulated within the coherence time of the qubits. Thus these custom gates allowed us to simulate the motion of the MZMs the entire way around the tri-junction and successfully observe the signature of braiding.
This system doesn’t represent MZMs actually moving around a physical topological superconductor, but a new way to demonstrate braiding using a dynamically-driven Hamiltonian that maps the topological superconducting Hamiltonian onto superconducting qubits. Still, it’s an important result, demonstrating the utility of quantum computers as simulators of a topological system with the help of the dynamic time evolution of the Hamiltonian. And, crucially, our introduction of custom scaled entangling gates — which don’t require extra calibration! — allows users requiring smaller controlled rotations to perform more operations within the coherence time of the qubits.
We’re currently adding the methodology to create such scaled entangling gates to Qiskit and are excited to see what you can do with them. This all goes to show that simply being more efficient with our quantum resources, plus hardware-aware programming going down even to the pulse level, can let us do some pretty cool things with our noisy quantum hardware.
This research was performed as part of the IBM Quantum Experience for Researchers. Learn more here, and get started using Qiskit (and Qiskit Pulse) here.
