Introducing: A Software Package For Studying The Dissipative Dynamics of Quantum Hardware

By Haggai Landa¹, Grégoire Misguich², and Ryan Mandelbaum³

Simulating large quantum systems is difficult for classical computers — that’s why folks are building quantum computers in the first place. However, classical simulation of quantum dynamics is still crucial, not only for benchmarking, but for making predictions about the potential behaviors of quantum hardware. In other words, it’s important that we’re able to simulate quantum systems using both classical and quantum computers.

Simulating quantum systems on classical computers mostly relies on numerical methods, including exact solutions in the rare cases that they are available, brute-force methods, and various approximations. The amount of classical computational space required for brute-force methods scales exponentially with the number of qubits. In order to cope with this exponential increase, we can use approximations to compress the information that is necessary to represent quantum systems — somewhat similar to how we compress pictures and movies on our computer. One very powerful approach employs linear algebra tools, like tensors and matrices, to do this approximate compression.

In a recent paper (available as preprint), we introduced a solver capable of simulating continuous-time dynamics of noisy quantum systems. While we describe the state of a perfectly isolated quantum system using a multiqubit wave-function, we describe noisy quantum systems using a density matrix, and a commonly used equation for solving the evolution of the density matrix is called the Lindblad master equation. Our software, a solver of this equation, lets users simulate the dynamics of qubits and how they interact and undergo decoherence in the presence of noise. Using this solver, we were able to study the behavior of excitations propagating through systems with many qubits, similar to existing quantum devices. We did not simulate in detail a particular quantum processor; this is more of a first step in that direction, that has already revealed some interesting effects.

So how does the memory-compression of the quantum states work? Well-established algorithms known as matrix product states (MPS) replace the full multiqubit state with an array of matrices of a much lower dimension. Matrix product operators (MPO) are a generalization of MPS to represent quantum operators, including density matrices. We employ these representations in our code with the help of the open source package ITensor. The MPS construction is especially well-suited for describing a system of qubits arranged in a one-dimensional chain, and states in which qubits far away from each other are not strongly correlated or entangled. Nevertheless, our solver, called lindbladmpo, supports any connectivity of the qubits, and we have used it to simulate both chains of qubits and also configurations that are computationally much more expensive.

In short, lindbladmpo allows users to simulate the dynamics of two-level qubits in a uniform rotating frame (i.e., with time-independent coefficients in the master equation). General single-qubit Hamiltonian parameters are supported, together with two-qubit interactions of “flip-flop’’ or exchange (XY) type and of Ising (ZZ) coupling form with arbitrary connectivity, and three dissipative jump operators, those which describe energy exchange with a thermal bath and dephasing. The solver package consists of a high performance C++ core and a feature-rich Python interface.

Using this solver, we studied qubits with XY interactions where an edge qubit is driven continuously on-resonance. Such a resonant driving of single qubits is the fundamental workhorse of quantum information processing in many-qubit devices, being employed to realize single-qubit rotations and also to generate entanglement in some setups. Gate dynamics are often studied by focusing on systems with very few qubits, however, the focus of the current work is on the many-body, nonlocal correlations induced by the continuous drive. We study a plaquette configuration — that is, essentially, a ring of qubits with two additional edge qubits — where the frequencies of neighboring qubits alternate such that they are not resonantly interacting. This setup is motivated by currently deployed IBM Quantum devices accessible via the cloud using Qiskit. In those devices, the qubit connectivity is that of a “heavy-hexagonal’’ lattice, which consists of connected qubit plaquettes, and the fixed-frequency qubits are manufactured to have different frequencies for neighboring ones in order to reduce unwanted interactions (see Fig. 1).

Fig. 1. The main setup we studied in our current work is shown on the bottom and consists of a plaquette with single edge qubits, where neighboring qubits alternate in frequency as depicted schematically by their color (blue qubits have zero frequency, while red ones have a higher value). The bonds indicate qubits interacting with their nearest neighbors by a `flip-flop’ (XY) interaction term. Qubit 0 is being driven periodically. On the top is the connectivity map of one of IBM Quantum’s 27-qubit Falcon devices, where qubits are arranged in a configuration termed the heavy-hexagonal lattice. The color code indicates qubit frequencies, which can be considered as inhomogeneous or disordered.

Our main result is that even with off-resonant qubits that appear to be only weakly interacting, large two-qubit correlations can form between distant qubits. Moreover, we find cases in which some correlation functions increase with distance in the system, peaking on the driven qubit and the one farthest away from it (see Fig. 2). Qubits far from a driven qubit are typically assumed to remain uncorrelated with it, and hence mechanisms that result in nonlocal correlations are important to identify. Uncontrolled nonlocal correlations of qubits in large devices could be detrimental to their usage for computational tasks and are important to simulate and explore theoretically. Or, in short, if we do something to a qubit on one side of the device, we don’t want to see the behavior change in a qubit on the other side of the device.

Fig. 2. The two-qubit XY correlation function of all qubit pairs in one of the setups simulated in our work (as depicted in Fig. 1). This correlation function indicates how much each pair is statistically correlated — for example, the correlation function would be zero for any product state of the qubits. Typically, correlation functions decay as a function of distance between the two measured qubits, while here it is immediately clear that the driven qubit (qubit 0) becomes relatively strongly correlated with the qubits with which it is resonant (colored in blue in Fig. 1), such that the correlation function obtains a maximal magnitude on the edge qubit farthest from it.

The research in our paper also demonstrates the usage of the solver in studying a dynamical problem with large systems and analyzing a complex dependence on multiple parameters. We discuss aspects important when undertaking such a systematic analysis, part of which is based on comparing our results for small systems (here with up to ten qubits) with essentially exact simulations using the Python package qiskit-dynamics. Our solver comes with comprehensive documentation and some tutorials, and the source code used for generating the research discussed here is available as an example within the solver repository. This source code can be used as a starting point for a research project managing hundreds of simulations using a local dataframe in Python.

Our study can be considered as a first step towards the simulation of dynamics with realistic device connectivity and qubit parameters. In order to extend the solver’s capability to simulate quantum devices, we could in the future add support for d-level qubit dynamics (qubits that can access states beyond just 0 and 1, suitable for transmon qubits for example), with more general Hamiltonian and Lindbladian parameters, and time-dependence in the parameters allowing to directly integrate more complex driving protocols. And, as it’s an open source package, we encourage you to try out lindbladmpo, too, and contribute to it. It remains an open question to see what new many-body effects could be simulated and possibly observed in experiments with real quantum hardware. In particular, nonlocal correlations forming as a result of qubit driving such as those presented in our results could be of fundamental importance when considering quantum error correction codes on multiqubit devices.

¹ IBM Quantum, IBM Research — Israel, Haifa University Campus, Mount Carmel, Haifa 31905, Israel

² Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS, 91191 Gif-sur-Yvette, France

³ IBM Quantum and Qiskit