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arXiv:2212.03805 · Physical review. B./Physical review. B

Perturbative analysis of quasiperiodic patterning of transmon quantum computers: Enhancement of many-body localization

Evangelos Varvelis, David Peter Divincenzo

Published Apr 10, 2024• 4 citations
arXivHTMLPDF

Abstract

To be useful for quantum computation, qubits in an array must remain localized -- that is, not ``dressed'' too much by nearby degrees of freedom. This localization has been previously achieved using disorder. But the authors observe here that a more effective route to localization is with a quasiperiodic patterning of the qubit array Hamiltonian. This paper develops two different perturbation approaches to calculate diagnostics for many-body localization in quasiperiodic superconducting transmon arrays. Perturbing in anharmonicity versus perturbing in hopping strength give complementary information.

DOI
10.1103/physrevb.109.144201

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To be useful for quantum computation, qubits in an array must remain localized -- that is, not ``dressed'' too much by nearby degrees of freedom. This localization has been previously achieved using disorder. But the authors observe here that a more effective route to localization is with a quasiperiodic patterning of the qubit array Hamiltonian. This paper develops two different perturbation approaches to calculate diagnostics for many-body localization in quasiperiodic superconducting transmon arrays. Perturbing in anharmonicity versus perturbing in hopping strength give complementary information.
From the perspective of many-body physics, the transmon qubit architectures currently developed for quantum computing are systems of coupled nonlinear quantum resonators. A certain amount of intentional frequency detuning ('disorder') is crucially required to protect individual qubit states against the destabilizing effects of nonlinear resonator coupling. Here we investigate the stability of this variant of a many-body localized phase for system parameters relevant to current quantum processors developed by the IBM, Delft, and Google consortia, considering the cases of natural or engineered disorder. Applying three independent diagnostics of localization theory - a Kullback-Leibler analysis of spectral statistics, statistics of many-body wave functions (inverse participation ratios), and a Walsh transform of the many-body spectrum - we find that some of these computing platforms are dangerously close to a phase of uncontrollable chaotic fluctuations.
The failure of quantum thermalization due to Many-Body Localization (MBL) has evolved from a theoretical concept in spin chains to an experimental reality in synthetic quantum platforms, most notably superconducting circuits based on transmon qubits. Despite its significance, the MBL transition has been studied primarily under purely random disorder and local couplings, leaving more complex and realistic configurations largely unexplored. Here, we investigate the quantum dynamics of transmon networks subject to the unavoidable competing effects of correlated disorder and network-mediated non-local interactions. First, we show how to engineer the physical parameters of the quantum hardware to systematically control the correlated disorder patterns emerging in the system. Then, we demonstrate that the MBL phase transition is robust against such correlations, which is essential for tuning localization properties in realistic transmon devices. This robustness is established through the analysis of the block entanglement entropy variance across disorder realizations, which precisely locates the MBL critical point. Independently, we introduce a local memory parameter, whose dynamics at long evolution times reveals memory retention in the localized phase and yields a critical point consistent with the entropy analysis. These results provide a framework for understanding localization in complex quantum network topologies, with potential implications for multi-qubit processor design.
We study the many-body localization (MBL) transition in an 1D exactly solvable system with long-range hopping and quasiperiodic on-site potential introduced in Phys. Rev. Lett. 131, 186303 (2023). Unlike other disorder or quasiperiodic model, an interaction-enhanced MBL happens in the moderate interaction regime, which is dubbed as the interaction-enhanced MBL. This counterintuitive phenomenon can be understood by noticing the fragility of the critical band lying at the bottom of the spectrum. The fragile band is localized by other localized states once the interaction is turned on. This mechanism can be verified by introducing a mean-field theory description which can derive highly excited states with high accuracy. The effectiveness of this mean-field theory is captured by the quasihole physics, validated by the particle entanglement spectra.
Coupling a one-dimensional quasiperiodic interacting system to a Markovian bath, we study the avalanche instability of the many-body localized phase numerically, finding that many-body localization (MBL) is more stable in pseudorandom quasiperiodic systems than the corresponding randomly disordered systems for a disorder strength W>8W>8W>8, potentially up to arbitrarily large system sizes. We support our conclusion by additionally developing real-space RG arguments, and we provide a detailed comparison between quasiperiodic and random MBL from the avalanche instability perspective, concluding that the two belong to different universality classes.
Local integrals of motion play a central role in the understanding of many-body localization in many-body quantum systems in one dimension subject to a random external potential, but the question of how these local integrals of motion change in a deterministic quasiperiodic potential is one that has received significantly less attention. Here we develop a powerful new implementation of the continuous unitary transform formalism and use this method to directly compute both the effective Hamiltonian and the local integrals of motion for many-body quantum systems subject to a quasiperiodic potential. We show that the effective interactions between local integrals of motion retain a strong fingerprint of the underlying quasiperiodic potential, exhibiting sharp features at distances associated with the incommensurate wavelength used to generate the potential. Furthermore, the local integrals of motion themselves may be expressed in terms of an operator expansion which allows us to estimate the critical strength of quasiperiodic potential required to lead to a localization/delocalization transition, by means of a finite size scaling analysis.
Superconducting circuits consisting of a few low-anharmonic transmons coupled to readout and bus resonators can perform basic quantum computations. Since the number of qubits in such circuits is limited to not more than a few tens, the qubits can be designed to operate within the dispersive regime where frequency detunings are much stronger than coupling strengths. However, scaling up the number of qubits will bring the circuit out of the regime, and this invalidates current theories. We develop a formalism that allows to consistently diagonalize the superconducting circuit Hamiltonian beyond the dispersive regime. This will allow to study qubit-qubit interaction unperturbatively, therefore, our formalism remains valid and accurate at small or even negligible frequency detuning; thus, our formalism serves as a theoretical ground for designing qubit characteristics for scaling up the number of qubits in superconducting circuits. We study the most important circuits with single- and two-qubit gates, i.e., a single transmon coupled to a resonator and two transmons sharing a bus resonator. Surprisingly, our formalism allows to determine the circuit characteristics, such as dressed frequencies and Kerr couplings, in closed-form formulas that not only reproduce perturbative results, but also extrapolate beyond the dispersive regime and can ultimately reproduce (and even modify) the Jaynes-Cumming results at resonant frequencies.