Transmon platform for quantum computing challenged by chaotic fluctuations

ARTICLE https://doi.org/10.1038/s41467-022-29940-y OPEN Transmon platform for quantum computing challenged by chaotic fluctuations 1234567890():,; Christoph Berke 1 ✉, Evangelos Varvelis David P. DiVincenzo 2,3,4 2,3, Simon Trebst 1, Alexander Altland 1 & 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. 1 Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany. 2 Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany. 3 Jülich-Aachen Research Alliance (JARA), Fundamentals of Future Information Technologies, 52425 Jülich, Germany. 4 Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, 52425 Jülich, Germany. ✉email: NATURE COMMUNICATIONS | (2022)13:2495 | https://doi.org/10.1038/s41467-022-29940-y | www.nature.com/naturecommunications 1 ARTICLE W NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-29940-y hen subject to strong external disorder, wave functions of many-body quantum systems may localize in states defined by (but not in trivial ways) the eigenstates of the disordering operators. A standard paradigm in this context is the spin-1/2 Heisenberg chain in a random z-axis magnetic field. Here, the disorder basis comprises the “physical” p-qubits defined by the spin states, different due to spin-exchange from the eigenbasis of “localized” l-qubits1,2. The latter are stationary but remain non-trivially correlated, including in the deeply localized phase. Although it may seem paradoxical at first sight, intentional “disordering” and many-body localization (MBL) in the above sense are a vitally important resource in the most advanced quantum computing (QC) platform available to date, the superconducting transmon qubit array processor. Physically, the transmon array is a system of coupled nonlinear quantum oscillators. At the low energies relevant to QC the system becomes equivalent to the negative U Bose Hubbard model. Site occupations 0 and 1 define the transmon p-qubit states, known as “bare qubits” in QC language. Randomization of the individual qubit energies maintains the integrity of these states in the presence of the finite inter-transmon coupling required for computing functionality. This coupling makes the eigen-l-qubits of the system different from the p-qubits. Considerable efforts are invested in the characterization and control of the induced correlations, known as ZZ couplings in the parlance of the QC community3. Connections between MBL and superconducting qubits have been considered earlier4, but mainly with a focus on applications of qubit arrays as quantum simulators of the bosonic MBL transition. Surprisingly, however, the obvious reverse question has not been asked systematically so far: What bearings may qubit isolation by disorder have on QC functionality? Reliance on strong disorder localization is a Faustian approach inasmuch as it invites the presence of quantum chaos, which is an arch-enemy of quantum device control of any kind. Lowering the strength of disorder brings one closer to the MBL-to-chaos transition, heralded by the growth of l-qubit correlations as early indicators for the proximity of the uncontrollable chaotic phase. Since the key requirement of QC, the execution of gate operations, requires ondemand rapid growth of entanglement between l-qubits, it is imperative that some definite amount of coupling is present. A crucial question that we confront, therefore, is under which circumstances the necessary levels of coupling keep us outside the chaotic zone. While this question does not have an easy overall answer, one general statement can be made with confidence: True to its Faustian nature, the invitation of disorder into the platform can be renegotiated, but not revoked. For instance, in its road map for future devices, IBM aims to replace random variations of qubit frequencies with a precision-engineered frequency alternation, e.g., … -A-B-A-B- …. While this pattern efficiently blocks resonances between neighboring qubits, next-nearest neighbors are now approximately degenerate. In a nonlinear system, such degeneracies are potent triggers for instabilities; the only way to control, or “localize” (qubit) states is with degeneracy lifting and translational symmetry breaking— in short, with the retention of some frequency disorder in the A and B sets. With this general situation in mind, the purpose of this paper is twofold. In its first part, we apply state-of-the-art diagnostic tools of MBL theory to investigate the role of disorder in transmon qubit arrays. We consider realistic models of qubit arrays employed in the remarkable experimental efforts by the groups of Delft5, Google6, IBM7, and others, assuming that device imperfections lead to random variations of individual qubit frequencies. Within this framework, we describe the diminishing localization 2 of many-l-qubit wave functions, and the growth of l-qubit correlations, upon lowering disorder. Considering small instances of multi-transmon systems, we find that the phase boundary between MBL and quantum chaos indeed may come dangerously close to the parameter ranges of current experiments. We also find that increasing the coordination number of the transmon lattice, as necessary for 2D connected transmon networks, increases many-body delocalization and the incipient chaos of the dynamics. In the second part of the paper, we apply this diagnostic machinery to address the question of whether precision engineering may be employed to ultimately realize ‘clean’ devices. Considering the abovementioned IBM alternating sequence as a case study, we find that it may indeed be operated at low values of randomness. However, for the reasons indicated above, residual frequency variations remain required to safeguard the stability of the device; further purification will not merely lead to little further improvement, but will actually be detrimental to its operation. Importantly, the diagnostic framework developed in the paper may be applied to predict levels of randomness which lead to optimal locali (...truncated)