Abstract
From the perspective of manybody 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 manybody 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 manybody wave functions (inverse participation ratios), and a Walsh transform of the manybody spectrum — we find that some of these computing platforms are dangerously close to a phase of uncontrollable chaotic fluctuations.
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Introduction
When subject to strong external disorder, wave functions of manybody 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 spin1/2 Heisenberg chain in a random zaxis magnetic field. Here, the disorder basis comprises the “physical” pqubits defined by the spin states, different due to spinexchange from the eigenbasis of “localized” lqubits^{1,2}. The latter are stationary but remain nontrivially correlated, including in the deeply localized phase.
Although it may seem paradoxical at first sight, intentional “disordering” and manybody 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 pqubit 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 intertransmon coupling required for computing functionality. This coupling makes the eigenlqubits of the system different from the pqubits. Considerable efforts are invested in the characterization and control of the induced correlations, known as ZZ couplings in the parlance of the QC community^{3}.
Connections between MBL and superconducting qubits have been considered earlier^{4}, 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 archenemy of quantum device control of any kind. Lowering the strength of disorder brings one closer to the MBLtochaos transition, heralded by the growth of lqubit 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 lqubits, 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 precisionengineered frequency alternation, e.g., … ABAB …. While this pattern efficiently blocks resonances between neighboring qubits, nextnearest 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 stateoftheart 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 Delft^{5}, Google^{6}, IBM^{7}, and others, assuming that device imperfections lead to random variations of individual qubit frequencies. Within this framework, we describe the diminishing localization of manylqubit wave functions, and the growth of lqubit correlations, upon lowering disorder. Considering small instances of multitransmon 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 manybody 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 localization of quantum information for given parameters characterizing the clean device.
The general conclusion of this work is that further progress towards larger QCs will be dependent on skirting the dangerous attributes of chaotic parts of the parameter space. We know from experience with general manybody systems that tendencies to longrange correlations and delocalization increase with increasing twodimensional system connectivity^{8}. On this basis, the monitors provided by the manybody localization theory may become an essential resource in the perfection of future transmonbased information devices.
Results
Overview
In what follows, we introduce our principal object of study: a transmon array modeled with realistic qubit parameters. Anticipating the importance of nonlinearities, we use effective “lowenergy Hamiltonians” solely to gain an intuition of the underlying physics but perform all subsequent computations avoiding any such approximations. We then introduce diagnostics inspired by MBL theory and apply them to detect signatures of chaos. We discuss enhanced tendencies to instability emerging in twodimensional geometries and address the question of whether the ideal of a stable and “perfectly clean” array can be reached by advanced qubit engineering.
Transmon array Hamiltonian
Our study begins with the wellestablished minimal model for interacting transmon qubits^{9,10}:
Here, n_{i} is the Cooperpair number operator of transmon i, conjugate to its superconducting phase ϕ_{i}. The transmon charging energy E_{C} is determined by the capacitance of the metal body of the transmon and is easily fixed at a desired constant, typically about E_{C} = 250 MHz (h = 1). The Josephson energy E_{J} is proportional to the critical current of the junction. Except in very recent work, it has been difficult to fix this constant reproducibly to better than a few percent. However, typical values lie in the vicinity of around 12.5 GHz, much larger than the charging energy. Finally, electrical coupling between the transmons, often via a capacitance, produces the charge coupling Tn_{i}n_{j}. The coefficient T has varied over a substantial range in 15 years of experiments^{11}; T values beyond 50 MHz are possible, but T < E_{C} is a fundamental constraint. Current experiments are often in the range T = 3–5 MHz, making T the smallest energy scale in the problem.
This model has been very concretely realized in experiments in many labs in recent years, but notably also in the many chips that have been made available for use in the IBM cloud service (https://www.ibm.com/quantumcomputing/systems/). These devices of the “Falcon” and “Hummingbird” generations have employed transmons laid out in the heavyhexagon lattice geometry of Fig. 1a. While these devices have fixed values of the coupling parameter T and of the charging energy E_{C}, their Josephson energy E_{J} varies from transmon to transmon. This effectively random variation is, in fact, crucially required to prevent the buildup of intertransmon resonances, and the compromising of quantum information; its role in the physics of presentday transmon device structures, with insights drawn from manybody localization theory, is the central theme of this paper.
Before addressing the physics of the full model Eq. (1), let us consider its lowenergy limit. Applying a sequence of approximations (series expansion of the Josephson characteristic, rotating wave approximation) one arrives at the effective Hamiltonian
To leading order, this model describes the transmon as a harmonic oscillator, where the above choices of energy scales place the frequencies \({\bar{\nu }}_{i}\approx 6\) GHz on average in the middle of a microwave frequency band, convenient for precision control. The attraction term, a remnant of the \(\cos\)nonlinearity, is considerably smaller than the average harmonic term, which is desired for transmon operation. Finally, the characteristic strength of the nearest neighbor hopping coefficients, \( {t}_{ij} \approx \frac{T}{4\sqrt{2}}\sqrt{\frac{{E}_{J}}{{E}_{C}}}\) (often called J in the literature), continues to be the smallest energy scale in the problem.
Unless novel engineering techniques are used (see below), the abovementioned variations of the Josephson energy, δE_{J}, are in the few percent range; thus, at a minimum, there is variation in oscillator frequencies ν_{i} of around \(\delta {\nu }_{i}\approx (\bar{\nu }/2{E}_{J})\delta {E}_{J}\approx 120\) MHz, when the typical E_{J} ≈ 12.5 GHz. This scale is much larger than the particle hopping strength, which for the same parameter set is about ∣t_{ij}∣ ≈ 6 MHz.
From the perspective of manybody physics, these variations make Eq. (2) a reference model for bosonic MBL. For the above characteristic ratio δν/∣t∣ ~20 we can hope that the system is in the MBL phase, and we confirm this below. From the perspective of transmon engineering, the frequency spread blocks the buildup of local nearest neighbor or nextnearest neighbor interqubit resonances. Below, we will discuss how these two perspectives go together (and where they may depart from each other). However, before turning to the observable consequences of frequency spread, we note that there exist two broad design philosophies for its realization in transmon array structures, schemes a and b throughout.
Generally speaking, scheme a aims to suppress the frequency spread to the lowest possible values required for the stability of the structure, or dictated by limits in precision engineering. For example, Fig. 1b shows that the spread of Josephson energies in IBM devices is consistent with a Gaussian distribution (with no stringent correlations from site to site). These observations hold true for all current devices whose parameters are documented publicly by IBM. Figure 1c shows that the variance δE_{J} has in fact remained very constant over 9 realizations of “Falcon” chips (27 qubits) and the two latest “Hummingbird” chips (65 qubits). This ‘natural disorder’ regime was in use in many generations of quantum computer processors^{7} that IBM has provided on its cloud service since 2016. However, a significant reduction of disorder has been reported in a very recent line of research at IBM employing high precision laserannealing^{12} as a pattern engineering approach^{13} to be discussed towards the end of the manuscript.
The complementary scheme b embraces frequency disorder as a potent means of protecting qubit information, and in fact, works to effectively enhance it. Examples in this category include the recent reports from TU Delft on their extensible module for surfacecode implementation^{5}. Google devices such as its 53qubit processor^{6} contain engineered frequency patterns whose aperiodic variation effectively realizes a form of the synthetic disorder, where in addition the qubit coupling t_{ij} is lowered during idle periods. In this way, the ratio δν/t—the relevant scale for localization properties—is drastically enhanced.
Below, we will consider both schemes a and b, and investigate the incipient quantum localization and quantum chaos present in the two settings. In our model calculations, we represent the disorder by drawing independent samples from a Gaussian distribution with standard deviation δE_{J}, added to the mean Josephson energy E_{J}. As a representative a case we take δE_{J} ≈ 500 MHz (for a Josephson energy of E_{J} = 12.5 GHz, giving δν ≈ 120 MHz, as above), while for a b case we will take δE_{J} and δν some ten times larger (precise numbers are given below, see also Supplementary Note 1 for further discussion of experimental parameters). Note that from this point onward, we continue with the full model Eq. (1).
Specifically, for schemea parameter ranges, the energy eigenvalues of Eq. (1) cluster into energy bundles corresponding to the total number of bosonic excitations, as seen in Fig. 2a. Looking inside the 5excitation band, we see (in Fig. 2b) a dense tangle of energy levels. However, only some of these levels are used to perform quantum computations in quantum processors; the identification of these levels, as shown in Fig. 2c and discussed in detail below, can only be done unambiguously if we are far away from the chaotic phase.
Having QC applications in mind, we are primarily interested in signatures of quantum chaos in the “computational subspace” of the bosonic Hilbert space, i.e., the space comprising local occupation numbers \({a}_{i}^{{{{\dagger}}} }{a}_{i}=0,1\), defining the pqubit states for QC. In that Hilbert space sector, the problem reduces to a disordered spin1/2 chain, another paradigm of MBL. Recent results from the MBL community indicate that the separation into a chaotic ergodic and an integrable localized phase is not as straightforward as previously thought, and that wave functions show remnants of extendedness and fractality even in the ‘localized’ phase^{14}.
Diagnostics
In the following, we analyze the Hamiltonian Eq. (1) with a combination of different numerical methods tailored to the description of localized phases:

Spectral statistics: According to standard wisdom, manybody spectra have Wigner–Dyson statistics in the phase of strongly correlated chaotic states, and Poisson statistics in that of uncorrelated localized states^{15}. Real systems show more varied behavior, quantified below in terms of a Kullback–Leibler divergence (see Methods for details). This produces a charting of parameter space indicating the chaos/MBL boundary and the rapidity with which the boundary is approached.

Wave function statistics: Focusing on the localization regime, we analyze how strongly the eigenstates differ from the localized states of the strictly decoupled system.

Walsh transforms: We quantify the correlations between lqubits (known in the QC community as ZZ couplings, and in the MBL community as τHamiltonian tensor coefficients) by application of a Walsh transform filter. To the best of our knowledge, this particularly sensitive tool has not been applied so far to the diagnostics of MBL.
We consider a system of N coupled transmons in a onedimensional chain geometry—a minimalistic setting that allows us to probe essential aspects of localization physics and quantum chaos using the above diagnostics and whose computational feasibility allows us to map out the broader vicinity of experimentally relevant parameter regimes. Typical system sizes vary between N = 5 and 10 sites, as detailed below.
Spectral statistics
We probe the spectral signatures of this coupled transmon system in an energy bundle of excited states (see Fig. 2b), which are generated by a total of N/2 = 5 excitations. For the N = 10 transmon chain at hand, this manifold contains a total of 2002 different states. States within this bundle that have local excitation numbers 0,1 can be viewed as typical representatives in the computational subspace. Zooming in on this midenergy spectrum, we plot its spectral statistics in the main panel of Fig. 3: The KL divergence vanishes when calculated with respect to the Poisson distribution for small transmon couplings, indicating perfect agreement with what is expected for an MBL phase. This is also corroborated by the striking visual match of the distributions in the corresponding inset of Fig. 3. But the KL divergence is maximal when compared to Wigner–Dyson statistics (red curve in Fig. 3). This picture is inverted for large transmon couplings T ≈ 70 MHz, where we find an extremely good match to Wigner–Dyson statistics—unambiguous evidence for the emergence of strongly correlated chaotic states. Probably even more important is the fact that these KL divergences allow us to quantify proximity to the diametrically opposite regimes for all intermediate coupling parameters. This includes a region of ‘hybrid statistics’ around the crossing point of the two curves, indicating an equal distance from both limiting cases, which we will discuss in more detail below.
By way of this KL divergence one can then map out an entire phase diagram, e.g., in the plane spanned by varying values of the transmon coupling and Josephson energy, while fixing the charging energy as shown in Fig. 4 (for schemea parameters). This allows us to clearly distinguish the existence of two regimes, the expected MBL phase (colored in blue) for small transmon coupling and a quantumchaotic regime (colored in red), where the level statistics follow Wigner–Dyson behavior (with delocalized, but strongly correlated states) for sufficiently strong transmon couplings. It is this latter regime that one surely wants to avoid in any experimental QC setting. But before we discuss the experimental relevance of our results, we want to characterize more deeply the quantum states away from this chaotic regime using additional diagnostics.
Wave function statistics
One particularly potent measure of the degree to which a given wave function is localized or delocalized, is its inverse participation ratio (IPR), i.e., the second moment
where the angular brackets denote averaging over disorder realization, and ∑_{{n}} is a symbolic notation for the summation over a chosen basis (in the present case, the Fock basis). An IPR of 1 indicates a completely localized state (as in our example for vanishing coupling T = 0), while an IPR less than 1 indicates the tendency of a wave function towards delocalization^{16}.
Here we consider the IPR measured as an average over all states in one of the energy bundles illustrated in Fig. 2a, e.g., the manifold of typical states with N/2 = 5 bit flips considered in the spectral statistics above. Figure 4b shows the IPR in the same parameter space as in (a). What is most striking here is that the IPR rapidly decays—the contour lines in the panel indicate exponentially decaying levels of 1/2, 1/4, 1/8, …1/128—showing that the wave functions quickly delocalize. Note in particular, that the IPR has dropped to a value of less than 10% in the region of “hybrid statistics” identified in the level spectroscopy above.
Walshtransform analysis
The MBL phase is the right place to be for quantum computing since computational qubits (the lqubits above) retain their identity there. But, as indicated by the drop in IPR, even the localized phase may be problematic. We, therefore, apply another diagnostic that is specifically adapted to identifying problems with running a quantum computation in the MBL phase. It begins with the expectation, announced in^{1,2}, that the Hamiltonian of the multiqubit system, in the lqubit basis, can be expressed as
Eq. (4), the “τHamiltonian” of MBL theory^{1,2}, embodies the observation that a diagonalized Hamiltonian can be written in a basis of diagonal operators \({\tau }_{i}^{z}\), which are the same as the PauliZ operators (Z_{i}) in the quantuminformation terminology of Eq. (5). Here the sum is over Nbitstrings b = b_{1}b_{2}…b_{N}, where each b_{i} is 0 or 1.
A system described by the τHamiltonian can be an excellent data carrier for a quantum computer, particularly if the highweight terms are small. If only the onebody terms in Eq. (4) are nonzero, the system is an ideal quantum memory: In the interaction frame, defined by the nonentangling unitary transformation \(U(t)=\exp (it\ {\sum }_{i}{h}_{i}{\tau }_{i}^{z})\), all quantum states, including entangled ones, remain stationary. Unfortunately, the expectation of MBL theory is that the twobody and higher interaction terms are nonzero and grow as the chaotic phase is approached.
We have performed a numerical extraction of the parameters of Eqs. (4–5) for a fivetransmon chain. We find that problematic departures from full localization do indeed occur already at rather small values of the qubitqubit coupling parameter T. This reinforces the message, in a basisindependent way, of our IPR study. But this extraction must begin with a very nontrivial step, namely the identification of the qubit eigenenergies of the transmon Hamiltonian. Since this Hamiltonian Eq. (1) is bosonic, it has a much larger Hilbert space than the spin1/2 view embodied in Eqs. (4)–(5). The qubit states, those with bosonic occupations limited to 0 and 1, are not separated in energy from the others but are fully intermingled with states of higher occupancy, as illustrated in Fig. 2c. It would thus appear that this truncation is rather unnatural—but it is in fact crucial to the whole quantum computing program with transmons. It is essential to pick out, from all the eigenlevels E_{α} of the full Hamiltonian Eq. (1) as shown in Fig. 2, just the subset of levels E_{b} that can be associated with a bitstring label b (cf. Eq. (5)).
Having performed such a state identification (as discussed in the Methods) and tagged the subset of eigenlevels E_{b}(T) that can be identified as qubit states, the coefficients of the τHamiltonian are easily obtained by a Walsh–Hadamard transform^{17}:
Being a kind of Fourier transform, the Walsh transform functions to extract correlations, in this case in the correlations of the computational eigenstates (in energy). Figure 5 shows these coefficients vs. T. For small T, many of the expectations from MBL theory^{1,2} are fulfilled: There is a clear hierarchy according to the locality of the coefficients. Thus, nearestneighbor ZZ interactions are the largest, followed by secondneighbor ZZ and contiguous ZZZ couplings, and so forth. Jumps occur in these coefficients, initially very small, which arise from the switching of labeling at anticrossings.
The twobody (J_{ij}/ZZ) terms are known and carefully analyzed in transmon research^{3,18,19}. Their troublesome consequences, including dephasing of general qubit states, and failure to commute with quantum gate operations, add overhead which is ultimately found to be insupportable, enforcing a practical upper limit of J_{ij} ~ 50–100 kHz. This limit, marked (dashed line) in Fig. 5b, is exceeded already at T = 3 MHz. Even though the transition to chaos is still a long way off, quantum computing becomes very difficult above this limit.
Scheme b: spread frequency distribution
Other techniques for executing entangling gates leave considerably more freedom to increase the disorder, with δE_{J} values in the GHz range. This option can forestall the growth of problematic precursors of chaotic behavior. A good example of a quantum computer that uses this freedom is the surface7 device of TU Delft^{5}. During gate operation, the qubit frequencies are temporarily tuned into resonant conditions that “turn on” entanglement generation. This is done in a pattern that does not lead to any extensive delocalization. In the 53qubit quantum computer of Google^{6}, this tuning is also available, but in its operation, an additional strategy is used: extra hardware is introduced to also make T tunable. Being able to set the effective T to zero (although only in a perturbative sense) of course eliminates the problem of delocalization, and in this latest Google work, δE_{J} has been returned to a small value. Google made major changes in its “Hamiltonian strategy” in recent years^{6} that have led them to their recent success. In Supplementary Note 2, we discuss schemeb parameters (as found in recent Delft chips^{5}) quantitatively using our three diagnostics.
Transmon arrays in higher dimensions
All the conclusions of the last few sections have been reached by calculations for transmons coupled in a onedimensional chain geometry. However, actual quantum information architectures are two dimensional, and we have therefore also simulated the surface7 layout^{5,20} as well as a 3 × 3 transmon array as minimal examples in this category. The surface7 chip comprises a pair of square plaquettes, which is obtained from a chain of seven transmons by including two additional couplings, see top panel of Fig. 6. (Google’s 53qubit layout extends this principle to a large square array extension.)
The study of twodimensional geometries, or of onedimensional arrays shunted by longrange connectors, is motivated by the realization that the case of strictly onedimensional chains is exceptional: in one dimension, “rare fluctuations” with anomalously strong local disorder amplitudes may block the correlation between different parts of the system, enhancing the tendency to form a manybody localized state. In higher dimensions, such roadblocks become circumventable, which makes disorder far less efficient in inhibiting quantum transport. For an indepth discussion of the effects of dimensionality on MBL, we refer to ref. ^{8}.
Our simulations of the surface7 and 3 × 3 architectures, where we have chosen schemea parameters, are summarized in Fig. 6b, c. The spectral and wave function statistics data indicate that, not surprisingly, chaotic traces are rather more prominent than in the onedimensional simulations (and despite the fact that to get the surface7 geometry, we nominally added only two extra couplings in comparison to a purely onedimensional geometry, see Fig. 6a). The bottom line is that the comfort zone introduced by disorder schemes is considerably diminished when including higherdimensional couplings.
Qubit frequency engineering
Until very recently, process variations in E_{J} have led to an inevitable spread in qubit frequencies, as described by the effectively Gaussian distributions employed above (see Fig. 1b). However, the development of a high precision laserannealing technique^{12} (LASIQ, see Fig. 1c and Supplementary Note 3) has changed the situation and is opening the prospect to clone qubits with unprecedented precision. IBM proposes^{13} to use this freedom and realize arrays with ABAB, or ABAC (Fig. 7) frequency alternation, effectively blocking unwanted hybridization between neighboring qubits. However, even then a residual amount of random frequency variation remains essential for the functioning of the device. For example, a perfect ABAB sequence would block nearest neighbor hybridization at the expense of creating dangerous resonance between the nextnearest qubits; more formally, perfectly translationally invariant arrays would have extended Bloch eigenstates, different from the localized states required for computing.
The question thus presents itself as to how to optimally navigate a landscape defined by the extremes of Bloch extended, chaotic, and manybody localized wave functions for absent, intermediate, and strong disorder, respectively. In Supplementary Note 4, we apply the diagnostic framework introduced earlier in the paper to address this question in quantitative detail. To summarize the results, we observe that for diminishing disorder the transmonarray Fock space disintegrates into a complex system of mutually decoupled subspaces, reflecting the complexity of the AB or ABAC “unit cells”. The strength of the residual disorder determines whether wave functions are chaotically extended or localized within these structures. Employing the inverse participation ratio as a quality indicator, we find that recent IBM engineering succeeded in hitting the optimum of near localized states with IPRs close to unity. However, it is equally evident that further reduction of the disorder would delocalize these states over a large number of qubit states and be detrimental to computing; disorder remains an essential resource, including in devices of the highest precision.
Discussion
The subject of this study has been an application of the stateoftheart methodology of manybody localization theory to realistic models of presentday transmon computing platforms. In the mindset of the localization theorist, all transmon quantum information hardware has in common that it operates in a regime where the tendency of quantum states to spread by interqubit coupling is blocked by the detuning of qubit frequencies—the manybody localized phase. Within this phase, there is considerable freedom for the realization of localizationprotected architectures; different strategies include a: weak coupling at weak detuning (IBM), or b: strong, intentionally introduced detuning interspersed by sporadic and incomplete coupling (Delft/Google). In this background, we have explored the integrity of different localized phases, in view of the omnipresent phase boundary to a chaotic sea of uncontrollable state fluctuations in the limit of too weak detuning and/or too strong coupling.
The single most important insight of this study is just how extended the twilight zone of partially compromised quantum states already is before reaching the boundary to hard quantum chaos. One may object that the existence of a crossover zone is owed to the smallness of the transmon arrays of 5–10 units studied in this work. However, it has to be kept in mind that the dimension of the random Hilbert spaces in which the manybody quantum states live is exponential in these numbers and large by any (numerical) standard of localization theory. This indicates that ‘finitesize effects’ in these systems are notorious and must be kept in mind for computing architectures of technologically relevant scales.
A second unexpected finding is that early indicators of chaotic fluctuations show in different ways in different observables. Among these, the least responsive observable is manybody spectral statistics, the most frequently applied diagnostics of the MBL/chaos transition. However, the computational states themselves respond far more sensitively to departures from the limit of extreme localization. We have observed this tendency in the standard observable for wave function statistics, the inverse participation ratios, where Fig. 4 shows tendencies to strong wave function spreading already in parameter regimes where spectral statistics suggests complete safety. Surprisingly, however, the Walsh transform diagnostic—which is uncommon in MBL theory, but highly relevant as an applied quality indicator for the integrity of physical qubit states—responds even more sensitively to parameter changes away from the deep localization limit. Expressed in the language of quantum information technology, it indicates strong ZZ coupling and the onset of ZZZ coupling already in regimes where the participation ratios are asymptomatic.
We note that the most recent experimental work has been strongly focused on the necessity to break the linkage between larger T coupling and the appearance of ZZ coefficients. On the hardware side, ingenious new coupler schemes^{21} show nearestneighbor ZZ reduced to well below the danger level of 100 kHz of earlier work. It is further shown that control techniques, involving advanced refocusing strategies, also strongly diminish the effect of nearestneighbor ZZ for a given T^{22}. But our work provides a warning that these innovations may ultimately not be enough: all other couplings in the τHamiltonian hierarchy, including next neighbor ZZ and ZZZ, remain neither diagnosed nor ameliorated in the current experiments.
Third and finally, our study of precisionengineered qubit arrays has revealed a general structure which we believe is ubiquitous in weakly disordered transmon arrays: a restructuring of the “total Hilbert space” into subspaces which are weakly crosscorrelated, but strongly (‘chaotically’) correlated within themselves. These hierarchies include spaces of fixed excitation number, or still further refined subspaces of these, distinguished by specific qubit permutation symmetries. Where this splintering occurs, a twofold task presents itself: first, identify the pattern of relevant spaces, and second apply the diagnostic tools discussed in earlier sections within these small spaces. Particular care must be exercised in cases where the relevant spaces overlap in energy. Absent considerable interspace correlations one may then be tricked into the conclusion of Poissonian level statistics (localization!) where in fact wave functions are chaotically extended over the basis of a computationally relevant space. In Supplementary Note 4, we detail all this intricate phenomenology, using the case study of the LASIQ engineered ABAB pattern as an example illustrating these principles and the development of reliable predictions for the integrity of qubit wave functions.
What is the applied significance of these observations? One bottom line is that further reduction of the frequency variance may be dangerous. All transmonbased quantum technology operates in a tension field defined by the desire to optimally protect (detune) and efficiently operate (couple). There are different approaches to resolving this conflict of interests, scheme a “weak coupling/weak detuning” and scheme b “transientincomplete coupling/strong detuning” defining two master strategies. Our study indicates that the a approach is more vulnerable to chaotic fluctuations. We go so far as to speculate that it may not sustain the generalization to larger and twodimensionally interconnected array geometries required by more complex applications. However, regardless of what hardware is realized, the findings of this work indicate that the shadows of the chaotic phase are much longer than one might have hoped and that careful scrutiny of chaotic influences should be an integral part of future transmon device engineering.
Methods
Spectral statistics via Kullback–Leibler divergence
To quantitatively analyze the spectral statistics, we look at the distribution of the ratios of adjacent level spacings r_{n} = ΔE_{n}/ΔE_{n+1}^{23} (in order to avoid level unfolding) by comparing to what is expected for these ratios in Poisson or Wigner–Dyson statistics. This is often done via qualitative observations, such as focusing on the limit of r → 0, where the distribution exhibits a maximum for Poisson statistics, but vanishes for Wigner–Dyson statistics (see, e.g., the insets of Fig. 3). However, a recent study^{24} (of a Fock space localization transition) has shown that such inspections may trick one into false conclusions and that the Kullback–Leibler (KL) divergence^{25} provides a far more reliable quantitative alternative. The KL divergence
defines an entropic measure quantifying the logarithmic difference between two distributions P and Q. In our case, the p_{k} are extracted from the numerical spectrum for a given set of parameters, while the q_{k} follows one of the two principal spectral statistics considered here. Note that in Fig. 3 we plot \({R}_{n}=\min ({r}_{n},1/{r}_{n})\) in order to restrict to the range [0, 1].
Data collapse and phase transition
A particularly consistent picture emerges if one performs a simple rescaling of the numerical data in both panels of Fig. 4. As shown in Fig. 8, the individual traces of both the KL divergence and the IPR for varying values of the Josephson energy E_{J} (shown in the insets) all collapse onto one another when rescaling the coupling parameter \(T\to T{E}_{J}^{\mu }\) with the exponent μ being the single free parameter. Such a data collapse is typically considered strong evidence for the existence of a phase transition, i.e., we can manifestly separate the MBL phase for small transmon couplings from a truly chaotic phase for sufficiently large couplings.
This also allows us to mark a Rubiconian line on our phase diagram (indicated by the green line in the top panel of Fig. 4) that should not be crossed in any quantum computation scheme, as all exquisitely prepared quantum information would be lost quickly upon entering the realm of quantum chaos lying beyond. The data collapse at a value μ ≃ 0.5 follows from a simple argument: thinking of the wave functions as states living on a high dimensional lattice defined by the occupation number configurations n = (n_{1}, n_{2}, …, n_{N}) (n_{i} = 0, 1 for the computational subspace), individual occupation number states n are connected to a large number of neighbors via the “hopping matrix elements”, t_{ij}. Wave functions hybridize over a pair of configurations n, m, provided ∣t_{ij}∣ ≳ ∣Δϵ_{nm}∣, where Δϵ_{nm} is the energy difference between the two configurations in the limit t → 0. Inspection of Eq. (2) shows that Δϵ_{nm} depends on the E_{J}s as \({E}_{{J}_{i}}^{1/2}{E}_{{J}_{j}}^{1/2} \sim {E}_{J}^{1/2}({E}_{{J}_{i}}{E}_{{J}_{j}})\). In the analysis of Fig. 8, the random deviations are scaled such that \(({E}_{{J}_{i}}{E}_{{J}_{j}}) \sim {E}_{J}^{1/2}\), such that Δϵ_{nm} is effectively independent of E_{J}. However, \({t}_{ij} \sim T{E}_{J}^{1/2}\), indicating that \(T{E}_{J}^{1/2}\) is the relevant scaling variable for the transition where the two parameters T and E_{J} are concerned.
State identification for Walsh transform
We find that there is a workable procedure for identifying computational states in the bosonic spectrum, which however starts to become problematic long before we reach the MBLchaotic phase boundary. We adopt the following assignment procedure: at T = 0 all states have exact bosonic quantum numbers, so the 2^{N} eigenstates with bitstring label b (the Walsh transforms of the bitstrings in Eq. (5)) are immediately identified there. We increase T; as long as no nearcrossings of energy levels occur, the labeling remains unchanged. We then find that the first nearcrossings that occur have the character of isolated anticrossings with very small gaps. In this situation, we can confidently associate the label b with the diabatic state (i.e., the one that goes straight through the anticrossing). We show this by the coloring of Fig. 2c. Empirically, this identification procedure works through many anticrossings, up to about halfway to the phase transition; gradually, gaps become larger, and the identification of qubit states becomes more ambiguous (Fig. 2d). Naturally, in the chaotic phase, eigenstate thermalization says that there is no hope of consistently identifying any eigenstates as informationcarrying multiqubit states.
For clarity of visualization, we do not show all Walsh coefficients c_{b} in Fig. 5 and in Supplementary Fig. 2, but average over those with bitstring labels of an equal maximal distance of two 1’s, as illustrated in Fig. 9 for a fivetransmon chain with the coefficients of maximal distance four.
Simulation parameters
All simulation data shown in the main manuscript were calculated for E_{C} = 0.25 GHz. For schemea disorder, we use δν ~ E_{C}/2, resulting in a Josephson energy spread \(\delta {E}_{J}=\sqrt{{E}_{C}{E}_{J}/8}\). The Walshtransform analysis was performed for E_{J} = 12.5 GHz.
Classical chaos in transmon arrays
We may incidentally remark that this work started from a project study^{26} on classical chaos in the transmon system. Classically, the transmon Hamiltonian Eq. (1) describes a system of coupled mathematical pendula of mass m = 1/8E_{C} and gravitational acceleration g = 8E_{C}E_{J} (in units where ℏ = 1 and ℓ = 1 (pendulum length)). Nonlinearly coupled pendula generally show a transition from integrable motion at low energies to hard chaos at high energies. (There are desktop gimmicks with just two coupled masses demonstrating the phenomenon.) The principal observation of the project was that already the classical two transmon Hamiltonian showed tendencies to chaos when excited to sufficiently high energies. The generalization to ten coupled oscillators made the situation worse, with Lyapunov exponents signaling uncontrollable dynamics for energies matching those of QC applications with 0 and 1 qubit states, and at time scales way below typical coherence times.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The code used to generate the data used in this study is available from the corresponding author upon reasonable request.
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Acknowledgements
We thank S. Börner for collaboration on an initial project^{26} studying incipient chaos in classical transmon systems. We acknowledge partial support from the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 390534769 (E.V., S.T., A.A., and D.P.D.) and within the CRC network TR 183 (project grant 277101999) as part of projects A04 and C05 (C.B., S.T., and A.A.). The numerical simulations were performed on the CHEOPS cluster at RRZK Cologne and the JUWELS cluster at the Forschungszentrum Jülich.
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S.T., A.A., and D.P.D. conceived the project and coordinated the research. C.B. and E.V. performed all the simulations. All authors discussed and analyzed the results and contributed to writing the manuscript.
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Berke, C., Varvelis, E., Trebst, S. et al. Transmon platform for quantum computing challenged by chaotic fluctuations. Nat Commun 13, 2495 (2022). https://doi.org/10.1038/s4146702229940y
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DOI: https://doi.org/10.1038/s4146702229940y
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