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ZZ and ZZZ coupling in a three-qubit system

Oskar Watsfeldt

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

This Master thesis investigates ZZ and ZZZ couplings in a system of three superconducting transmon qubits. Such couplings represent energy shifts arising from qubit-qubit interactions, and they are important for the performance of quantum computers. Adopting an analytical method based on the Schrieffer-Wolff transformation, we extend it with new insight and compute energy corrections associated with ZZZ and ZZ couplings.The thesis explores a known parameter region, where one ZZ coupling can be eliminated, to find if the ZZZ coupling can be made zero simultaneously. Lastly, an attempt is made to connect ZZZ and ZZ couplings with a simple linear relationship.

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This Master thesis investigates ZZ and ZZZ couplings in a system of three superconducting transmon qubits. Such couplings represent energy shifts arising from qubit-qubit interactions, and they are important for the performance of quantum computers. Adopting an analytical method based on the Schrieffer-Wolff transformation, we extend it with new insight and compute energy corrections associated with ZZZ and ZZ couplings.The thesis explores a known parameter region, where one ZZ coupling can be eliminated, to find if the ZZZ coupling can be made zero simultaneously. Lastly, an attempt is made to connect ZZZ and ZZ couplings with a simple linear relationship.
A major challenge for scaling up superconducting quantum computers is unwanted couplings between qubits, which lead to always-on ZZ couplings that impact gate fidelities by shifting energy levels conditional on qubit states. To tackle this challenge, we introduce analytical and numerical techniques, including a diagrammatic perturbation theory and a state-assignment algorithm. Together, these tools enable us to explain the emergence of ZZ coupling in three linked pictures, where each picture tells us more about the underlying mechanisms creating the ZZ coupling. These pictures generalize previous efforts, which focused on specific setups and a single mechanism. The deeper understanding that we provide of the mechanisms behind the ZZ coupling facilitate finding parameter regions of weak and strong ZZ coupling. We showcase our techniques for a system consisting of two fixed-frequency transmon qubits connected by a flux-tunable transmon coupler. There, we find three types of parameter regions with zero or near-zero ZZ coupling, all of which are accessible with current technology. We furthermore find regions of strong ZZ coupling nearby, which may be used to implement adiabatic controlled-phase gates and quantum simulations. Our framework is applicable to many types of qubits and opens up for the design of large-scale quantum computers with improved gate fidelities.
This study's modeling of superconducting quantum processors reveals that, contrary to popular belief, three-body ZZZ stray couplings can surpass parasitic ZZ interactions, posing a serious threat to gate fidelity and crosstalk. Even more striking, strongly decoupled qubits---long thought to be ideal---can be more problematic than loosely coupled ones, even when idle. These findings overturn traditional design principles and force us to rethink next-generation architectures for quantum processors.
We have analyzed and proposed coupling mechanisms between Three Josephson Junction Flux Qubits (3JJQ). For this, we have developed a numerical method to extract the effective Hamiltonian of a system of coupled qubits via the Schrieffer-Wolff transformation (SWT). We then give a comprehensive introduction to the 3JJQ, and study it analytically by approximating its potential with a Harmonic well. With a clear understanding of the 3JJQs, we use the SWT to gain intuition about their effective dipolar interaction with the electromagnetic field, and use that intuition to propose and study analytically and numerically the capacitive coupling of two 3JJQs via a non-tunable capacitor, and the inductive coupling of two 3JJQs via a tunable Josephson Junction (dc-SQUID), showing that we are able to reproduce non-stoquastic Hamiltonians in the strong-coupling regime.
In transmon qubits, the leading source of errors for quantum gates is the existence of higher energy levels, besides the |0> and |1> states which form the computational subspace. Several methods have been developed to eliminate these errors systematically by clever use of the available experimental controls. In this bachelor thesis we focus on performing quantum gates on multiple qubits simultaneously, when transition frequencies of different qubits are close to each other. The existing solution is designed to perform a quantum gate on one qubit, while eliminating all the effects on the other one. We develop a procedure to create new analytic pulse shapes which produce low-error gates for single qubits and we generalize this approach to multi-qubit systems to apply multiple quantum gates at the same time. For both single- and two-qubit gates these new pulses reduce errors by several orders of magnitude compared to simple driving pulses. In addition we combine these optimal analytical pulse shapes with multi-parameter pulses. The parameters of this additional pulse are optimized numerically, to produce even lower gate errors.
We have analyzed and proposed coupling mechanisms between Three Josephson Junction Flux Qubits (3JJQ). For this, we have developed a numerical method to extract the effective Hamiltonian of a system of coupled qubits via the Schrieffer-Wolff transformation (SWT). This method is a more efficient version of the one proposed by Ref. [1]. We then give a comprehensive introduction to the 3JJQ, and study it analytically by approximating its potential with a Harmonic well. With a clear understanding of the 3JJQs, we use the SWT to gain intuition about their effective dipolar interaction with the electromagnetic field, and use that intuition to propose and study analytically and numerically the capacitive coupling of two 3JJQs via a non-tunable capacitor, and the inductive coupling of two 3JJQs via a tunable Josephson Junction (dc-SQUID), showing that we are able to reproduce non-stoquastic Hamiltonians in the strong-coupling regime.
Overcoming the issue of qubit-frequency fluctuations is essential to realize stable and practical quantum computing with solid-state qubits. Static ZZ interaction, which causes a frequency shift of a qubit depending on the state of neighboring qubits, is one of the major obstacles to integrating fixed-frequency transmon qubits. Here we propose and experimentally demonstrate ZZ-interaction-free single-qubit-gate operations on a superconducting transmon qubit by utilizing a semi-analytically optimized pulse based on a perturbative analysis. The gate is designed to be robust against slow qubit-frequency fluctuations. The robustness of the optimized gate spans a few MHz, which is sufficient for suppressing the adverse effects of the ZZ interaction. Our result paves the way for an efficient approach to overcoming the issue of ZZ interaction without any additional hardware overhead.