Introduction

The central question of quantum non-equilibrium dynamics revolves around exploring how quickly a quantum state can be modified1. It presents an elemental step towards understanding the rate of equilibration and thermalization of generic quantum systems2, the propagation speed of information3,4 as well as practical applications, such as quantum state transfer5,6, the minimal time of applying quantum gates7, and the analysis of optimal quantum control8,9. When putting the interpretation of the energy-time uncertainty relation10 on a firmer ground11, Mandelstam and Tamm (MT)12 realized that it sets the fundamental bound of the intrinsic time scale on how fast a unitary quantum dynamics can evolve.

Such bound constrains the unitary process of a (pure or mixed) initial state to its orthogonal counterpart, defining a minimal evolution time t⊥ obeying t⊥ ≥ tMT ≡ πℏ/(2ΔE), where \(\Delta E={(\langle {\hat{H}}^{2}\rangle -{\langle \hat{H}\rangle }^{2})}^{1/2}\) is the energy spread of the time-independent Hamiltonian \(\hat{H}\) governing the system and \(\langle \hat{H}\rangle \equiv E\) is the mean energy. Building on that, Margolus and Levitin (ML)13 further demonstrated that E with respect to the ground state energy \({E}_{\min }\) establishes a second orthogonalization bound, \({t}_{\perp }\ge {t}_{{{{\rm{ML}}}}}\equiv \pi \hslash /[2(E-{E}_{\min })]\). Together, they establish the quantum speed limit (QSL) concept, defining the lower bounds of the time required for the unitary dynamics of any quantum system to unfold. Substantial investigation across various domains, encompassing both isolated and open systems, unitary and non-unitary evolutions, and spanning the realms of fundamental14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36 as well as practical applications9,37,38,39,40,41,42,43,44,45,46,47,48 have since ensued. For isolated systems with a bounded energy spectrum (maximum spectral energy \({E}_{\max }\)), a time-reversal symmetric version of the ML bound (dubbed ML⋆) has also been characterized31, \({t}_{\perp }\ge {t}_{{{{{\rm{ML}}}}}^{\star }}\equiv \pi \hslash /[2({E}_{\max }-E)]\). Their combination, results in a tight49 (for pure states), unified31 minimal orthogonalization time \({t}_{\perp }\ge {t}_{{{{\rm{U}}}}}=\max \left[{t}_{{{{\rm{MT}}}}},{t}_{{{{\rm{ML}}}}},{t}_{{{{{\rm{ML}}}}}^{\star }}\right]\). QSL is also discussed in time-dependent systems; in the presence of external fields, e.g., in the context of counterdiabatic driving50, work, and its fluctuations may characterize the speed of evolution better than energy.

Experimental observation of these bounds has been put forward in single-particle systems, including a single atom on an optical trap29 and a single transmon qubit in a cavity resonator51. However, though a theoretical proposal of investigating QSL with ultracold gases52 has been contemplated, further experimental exploration of their generality and corresponding crossover in generic many-body systems remains uncharted territory. Likewise, also not often explored is the minimal speed at which the state overlap can change, a crucial aspect of quantum information processing. That is, the quantum dynamics of generic systems proceed, obeying maximal and minimal speeds. Fortunately, remarkable progress in simulating many-body Hamiltonians using superconducting quantum circuits53,54,55,56,57,58, accompanied by the flexibility of initial state preparations has poised us to meet this demand.

Here, we further leverage the tunability of the quantum circuits to examine the crossover among different bounds. Crucially, our investigation relies on noticing that a single Hamiltonian parameter can effectively govern which bound provides the tighter dynamics, even if fixing the initial state. Using this rationale, our study spans small (e.g., one qubit or one qutrit, Figs. 1a, 2a) and multi-qubit systems (such as one- and two-dimensional lattices shown in Figs. 3a, 4a), for which many-body effects are preponderant.

Fig. 1: Quantum speed limit in a qubit system.
figure 1

a Energy level representation of a driven qubit. Here, ω10 is the transition frequency between \(\left\vert 0\right\rangle\) and \(\left\vert 1\right\rangle\), ωd is the frequency of microwave drive, and Δ is the detuning between ω10 and ωd. b Theoretical QSL phase diagram in terms of the first and second moments of the energy, in the case of bounded energy spectra {En}, where the Bathia–Davis inequality59 applies. The different regions, MT, ML, and ML⋆ are further defined by whether the mean energy E compares to \(({E}_{\min }+{E}_{\max })/2\). The blue, purple, and orange markers correspond to Ω/2π = −2.5, 0, and 2.5 MHz, which are used in the experiment. c By choosing the initial state as \(\left\vert \psi (0)\right\rangle=\frac{1}{\sqrt{2}}(\left\vert 0\right\rangle+\left\vert 1\right\rangle )\) we theoretically contrast the Ω-dependency of ΔE, \(E-{E}_{\min }\) and \({E}_{\max }-E\); the smaller quantity at a given Ω defines the QSL region the dynamics is most tightly bounded at later times. d–f Dynamics of overlap for Ω/2π = −2.5, 0, and 2.5 MHz. The blue circles (lines) are the experimental (theoretical) results of \(| \left\langle \psi (0)| \psi (t)\right\rangle |\); dashed lines indicate the ML, ML⋆ and MT bounds. The yellow and green shaded lines mapped by the color bars are the generalized lower and upper ML bounds. The green and purple stars highlight the initial state and the local minima of the theoretical \(| \left\langle \psi (0)| \psi (t)\right\rangle |\), respectively. α is the parameter of the generalized ML (ML*) bound which is defined by a generic Lα-norm. Corresponding dynamics trajectories are plotted on Bloch spheres in the lower left corner. Error bars stem from 5 repetitions of measurements, and each repetition consists of 10,000 runs of the experimental circuits with single-shot measurements.

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Fig. 2: Quantum speed limit in a qutrit system.
figure 2

a Energy level representation of a driven qutrit. The qutrit’s nonlinearity is η/2π = −212 MHz, and the transition frequencies satisfy the relation ω21 = 2ω10 + η. Unlike the qubit case in Fig. 1, the qutrit is driven by a resonant microwave with ωd = ω10. b Theoretical QSL phase diagram with markers annotated based on the values of Ω. c The theoretical Ω-dependency of the quantities establishing the shortest orthogonalization time. d–f Dynamics of overlap for Ω/2π = −15, −5, and 8.5 MHz. The overlap is obtained by experimentally reconstructing the qutrit state with quantum state tomography (see Supplementary Note 2). Notations for the markers and lines are the same as the qubit case in Fig. 1. Error bars stem from five repetitions of measurements, and each repetition consists of 10,000 runs of the experimental circuits with single-shot measurements. Here, the initial state is \(\left\vert \psi (0)\right\rangle=\frac{1}{\sqrt{10}}[\left\vert 0\right\rangle+\frac{3}{\sqrt{2}}(\left\vert 1\right\rangle+\left\vert 2\right\rangle )]\). See Supplementary Note 2 for experimental data of longer time dynamics and the effects of imperfect pulse.

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Fig. 3: Quantum speed limit for the 1d qubit chain.
figure 3

a Schematic of the 1d qubit chain. Each circle represents a qubit with onsite potential  ± W. Neighbor qubits are coupled with coupling strengths J1/2π = −2 MHz, denoted by pink lines. b Theoretical QSL phase diagram. c Theoretically calculated W-dependency of the quantities establishes the shortest orthogonalization time bound. The initial state is chosen as the unbalanced superposition of two density wave product states \(\left\vert \psi (0)\right\rangle=\frac{1}{2}(\left\vert 101010\right\rangle+\sqrt{3}\left\vert 010101\right\rangle )\). d–f Dynamics of overlap for W/2π = 0, 1.8, and 8 MHz, respectively. Notation for markers and lines is similar to Figs. 1, 2. Error bars stem from five repetitions of measurements and each repetition consists of 3000 runs of the experimental circuits with single-shot measurements.

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Fig. 4: Quantum speed limit for 2d square qubit lattice.
figure 4

a 9-qubit (3 × 3) lattice with (next-) nearest-neighbor coupling J1/2π = −2 MHz (J2/2π = 0.597 MHz) in the presence of a checkerboard potential with amplitude W, which is tuned to establish the crossover from MT to an ML*-dominated bound. b Theoretical QSL phase diagram. c Theoretical W-dependency of the quantities establishing which is the shortest orthogonalization time. Here, the initial state is a single product state featuring five excitations initialized in the qubits with  + W energy. d–f Dynamics of overlap for W/2π = 0, 3.4, and 6.5 MHz; here notations for markers and lines are similar to Figs. 1, 2 and error bars stem from five repetitions of measurements and each repetition consists of 10,000 runs of the experimental circuits with single-shot measurements. g–i Density of Fock states in the rescaled energy spectrum, highlighting the mean energy E = Es (vertical dashed line) and the energy uncertainty ΔE (shaded orange region). j–l Dynamics in Fock space, in terms of the wave-packet probability at the Hamming distance d.

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Results

Quantum speed limits in single qubit or qutrit systems

For a general time-independent Hamiltonian with a bounded energy spectrum, \({E}_{\min }\le \{{E}_{n}\}\le {E}_{\max }\), the QSL is bounded by a unified limit \({t}_{{{{\rm{U}}}}}=\max [\frac{\pi \hslash }{2\Delta E},\frac{\pi \hslash }{2(E-{E}_{\min })},\frac{\pi \hslash }{2({E}_{\max }-E)}]\)31. As a result of the Bhatia–Davis inequality59, its energy uncertainty ΔE and mean energy E satisfy the relation \(\Delta E\le \sqrt{({E}_{\max }-E)(E-{E}_{\min })}\), which further establishes three dynamical regimes for the time evolution of any accessible initial state \(\left\vert \psi (0)\right\rangle\). As shown in Fig. 1b, if \(\Delta E \, < \, \min (E-{E}_{\min },{E}_{\max }-E)\) the dynamics is dominated by the MT bound at all times; otherwise, i.e., \(\Delta E \, > \, \min (E-{E}_{\min },{E}_{\max }-E)\), a second condition, \(E \, < \, ({E}_{\min }+{E}_{\max })/2\) [\(E \, > \, ({E}_{\min }+{E}_{\max })/2\)], emerges in which tML (\({t}_{{{{{\rm{ML}}}}}^{*}}\)) limits the shortest orthogonalization time scale (see Supplementary Note 1 for further discussion).

To experimentally observe three dynamical regimes and the possible crossover between different bounds during the dynamics, we consider a qubit (qutrit) system under a microwave drive \({\hat{H}}_{d}=\Omega \cos ({\omega }_{d}t)\), where Ω is the drive amplitude and ωd is the drive frequency. In this scenario, we can conveniently select which bound limits the speed of the system’s dynamics by tuning a single Hamiltonian parameter, e.g., Ω, even if the initial state remains unchanged. Experimental calibration of Ω can be done by varying the drive amplitude and fitting the resulting probability oscillations; see Supplementary Note 2 for more details. In the rotating frame of drive frequency ωd, the Hamiltonian of a driven qubit is generically described by

$${\hat{H}}_{{{{\rm{qubit}}}}}/\hslash=\left(\begin{array}{cc}0&\Omega \\ \Omega &\Delta \end{array}\right)\,,$$
(1)

with Δ = ω10 − ωd being the detuning between the qubit transition frequency ω10 and the drive frequency ωd (Fig. 1a). By setting the initial state as \(\left\vert \psi (0)\right\rangle=(\left\vert 0\right\rangle+\left\vert 1\right\rangle )/\sqrt{2}\), we show in Fig. 1c the Ω-dependence of the three key quantities that classify which bound regime the dynamics belongs to: ΔE, \(E-{E}_{\min }\), and \({E}_{\max }-E\); the smallest of them defines the minimal orthogonalization time and establishes a constraint in the dynamics before tU is reached. In our experiments, we observe the dynamics of the overlap between the initial and time-evolved states, \({{{\mathcal{F}}}}\equiv | \langle \psi (0)| \psi (t)\rangle |\), which proceeds along the geodesic distance in the complex Hilbert space, \({l}_{{{{\rm{geo}}}}}=\arccos ({{{\mathcal{F}}}})\)29,60. It is bounded by31,61,62

$$\begin{array}{rc}&\arccos ({{{\mathcal{F}}}})\ge \hfill\\ &\max \left(\frac{\Delta Et}{\hslash },\sqrt{\frac{\pi (E-{E}_{\min })t}{2\hslash }},\sqrt{\frac{\pi ({E}_{\max }-E)t}{2\hslash }}\right)\,.\end{array}$$
(2)

By setting Δ/2π = 12.5 MHz, Ω/2π = −2.5, 0, and 2.5 MHz, we show in Fig. 1d–f how the experimentally measured dynamics under the Hamiltonian \({\hat{H}}_{{{{\rm{qubit}}}}}\) is tightly bounded by each of the constraints set above, at times t ≤ tU, stemming from either the MT, ML, or ML⋆ bounds—at short times, however, MT is always the prevalent bound, see Supplementary Note 1. Here, the overlap \({{{\mathcal{F}}}}\) is obtained by reconstructing the qubit state with quantum state tomography.

In particular, the perfect orthogonalization for Ω = 0 MHz can be understood via the expectation values of the corresponding spin-1/2 Pauli matrices \(\{{\hat{\sigma }}_{\alpha }\}\) (α = x, y, z). In this case, \({\hat{H}}_{{{{\rm{qubit}}}}}\) is diagonal and thus proportional to \({\hat{\sigma }}_{z}\); the initial state \(\left\vert \psi (0)\right\rangle=\left\vert+\right\rangle\), an eigenstate of \({\hat{\sigma }}_{x}\), evolves as described by the precession in the Bloch sphere around the z quantization axis (inset in Fig. 1e). At t = tU ≃ 40 ns, \(\left\vert \psi (t)\right\rangle=\left\vert -\right\rangle\), the second eigenstate of \({\hat{\sigma }}_{x}\), leading to \({{{\mathcal{F}}}}=0\) (Fig. 1e). For Ω ≠ 0 MHz, the corresponding representation of \({\hat{H}}_{{{{\rm{qubit}}}}}\) in terms of \(\{\hat{{\sigma }_{\alpha }}\}\) results in an effective spin Hamiltonian with generic precession axis, precluding orthogonalization of the state, as seen in Fig. 1d, f.

The next level of complexity one can perform a similar analysis is on a three-level Hamiltonian, a qutrit. A transmon-type qubit is a multi-level system with a weak anharmonicity, and thus, a higher-energy state \(\left\vert 2\right\rangle\) will be involved in the system dynamics under the large microwave drive, which yields a driven qutrit Hamiltonian,

$${\hat{H}}_{{{{\rm{qutrit}}}}}/\hslash=\left(\begin{array}{ccc}0&\Omega &0\\ \Omega &0&\sqrt{2}\Omega \\ 0&\sqrt{2}\Omega &\eta \end{array}\right),$$
(3)

where η = ω10 − ω21, is the nonlinearity—for our device, η/2π ≈ −212 MHz. Similarly, adjusting the drive amplitude allows the exploration of regimes in which the time evolution falls into the different QSL regions (Fig. 2b), by checking the smallest of \(\{\Delta E,E-{E}_{\min },{E}_{\max }-E\}\) (Fig. 2c). Here, the initial state is prepared as a superposition of the natural qutrit levels, \(\left\vert \psi (0)\right\rangle=\frac{1}{\sqrt{10}}[\left\vert 0\right\rangle+\frac{3}{\sqrt{2}}(\left\vert 1\right\rangle+\left\vert 2\right\rangle )]\). We note that, unlike the qubit case, the experimental calibration of qutrit gates needs to consider the AC Stark shift; see Supplementary Note 2 for more details.

Albeit remarkably restricting, the bounds in Eq. (2) are not always the tightest in actual dynamics. Generalizations of the ML (ML*) bound31, based on a generic Lα-norm, \({E}_{\alpha }\equiv {\langle {(\hat{H}-{E}_{\min })}^{\alpha }\rangle }^{1/\alpha }\,({E}_{\alpha }^{*}\equiv {\langle {({E}_{\max }-\hat{H})}^{\alpha }\rangle }^{1/\alpha })\) with \(\alpha \in {{\mathbb{R}}}^{+}\), have been demonstrated15,63, resulting in generalized orthogonalization times \({t}_{{{{\rm{ML}}}},\alpha }\equiv \pi \hslash /({2}^{1/\alpha }{E}_{\alpha })\,({t}_{{{{{\rm{ML}}}}}^{*},\alpha }\equiv \pi \hslash /({2}^{1/\alpha }{E}_{\alpha }^{*})\left.\right])\). A simple derivation in the Methods shows that, in general, the shortest orthogonalization time satisfies,

$${t}_{\perp }\,\ge\, \left\{{\gamma }_{\alpha }{({{{\mathcal{F}}}},{\theta }^{{\prime} })}^{\frac{1}{\alpha }}\frac{\pi \hslash }{{2}^{1/\alpha }{E}_{\alpha }},{\gamma }_{\alpha }{({{{\mathcal{F}}}},{\theta }^{{\prime}{\prime}})}^{\frac{1}{\alpha }}\frac{\pi \hslash }{{2}^{1/\alpha }{E}_{\alpha }^{ * }}\right\},$$
(4)

where the prefactor \({\gamma }_{\alpha }({{{\mathcal{F}}}},{\theta }^{{\prime} })\equiv \max \{0,1-{{{\mathcal{F}}}}{d}_{\alpha }^{\prime}\}[{\gamma }_{\alpha }({{{\mathcal{F}}}},{\theta }^{{\prime}{\prime}})\equiv \max \{0,1-{{{\mathcal{F}}}}{d}_{\alpha }^{\prime \prime}\}]\). Here we define \({d}_{\alpha }^{{\prime} }\equiv \cos {\theta }^{{\prime} }-\frac{2\alpha }{\pi }\sin {\theta }^{{\prime} }\) (\({d}_{\alpha }^{{\prime}{\prime}}\equiv \cos {\theta }^{{\prime}{\prime}}-\frac{2\alpha }{\pi }\sin {\theta }^{{\prime}{\prime}}\)); \({\theta }^{{\prime} }=\theta+{E}_{\min }t\,\) and \({\theta }^{{\prime}{\prime}}=-\theta -{E}_{\max }t\) are determined by the time-dependent phase that accumulates in the overlap: \(\left\langle \psi (0)| \psi (t)\right\rangle={{{\mathcal{F}}}}{e}^{{{{\rm{i}}}}\theta }\). Similar to the more known bounds [Eq. (2)], the overlap itself is always bounded depending on the sign of \({d}_{\alpha }^{{\prime} }\) :

$$1\ge {{{\mathcal{F}}}}\ge {B}_{{{{\rm{low}}}}}^{{\prime},\alpha },\,{d}_{\alpha }^{{\prime} } \, > \, 0,$$
(5a)
$${B}_{{{{\rm{up}}}}}^{{\prime},\alpha }\ge {{{\mathcal{F}}}}\ge 0,\,{d}_{\alpha }^{{\prime} } \, < \, 0,$$
(5b)

where \({B}_{{{{\rm{low}}}}}^{{\prime},\alpha }\equiv \max [0,\frac{1-\frac{2{t}^{\alpha }}{{(\pi \hslash )}^{\alpha }}\langle {(\hat{H}-{E}_{\min })}^{\alpha }\rangle }{{d}_{\alpha }^{{{\prime} }}}]\) for \({d}_{\alpha }^{{\prime} } \, > \, 0\) [\({B}_{{{{\rm{up}}}}}^{{\prime},\alpha }\equiv \min [1,\frac{1-\frac{2{t}^{\alpha }}{{(\pi \hslash )}^{\alpha }}\langle {(\hat{H}-{E}_{\min })}^{\alpha }\rangle }{{d}_{\alpha }^{{{\prime} }}}]\) for \({d}_{\alpha }^{{\prime} } \, < \, 0\)]. Similarly, based on the sign of \({d}_{\alpha }^{{\prime}{\prime}}\) one can have another set of bounds:

$$1\ge {{{\mathcal{F}}}}\ge {B}_{{{{\rm{low}}}}}^{{\prime\prime},\alpha },\,{d}_{\alpha }^{{\prime}{\prime}} \, > \, 0,$$
(6a)
$${B}_{{{{\rm{up}}}}}^{{\prime\prime},\alpha }\ge {{{\mathcal{F}}}}\ge 0,\,{d}_{\alpha }^{{\prime}{\prime}} \, < \, 0,$$
(6b)

where \({B}_{{{{\rm{low}}}}}^{{\prime\prime},\alpha }\equiv \max [0,\frac{1-\frac{2{t}^{\alpha }}{{(\pi \hslash )}^{\alpha }}\langle {({E}_{\max }-\hat{H})}^{\alpha }\rangle }{{d}_{\alpha }^{{{\prime}{\prime}}}}]\) for \({d}_{\alpha }^{{\prime}{\prime}} \, > \, 0\) [\({B}_{{{{\rm{up}}}}}^{{\prime\prime},\alpha }\equiv \min [1,\frac{1-\frac{2{t}^{\alpha }}{{(\pi \hslash )}^{\alpha }}\langle {({E}_{\max }-\hat{H})}^{\alpha }\rangle }{{d}_{\alpha }^{{{\prime}{\prime}}}}]\) for \({d}_{\alpha }^{{\prime}{\prime}} \, < \, 0\)]. Combining Eqs. (5) and (6) gives the final bound of \({{{\mathcal{F}}}}\) for a specific α, summarized in Table 1. By calculating the bounds at different α, one obtains a set of lower and upper bounds for \({{{\mathcal{F}}}}\), which are depicted together with the MT, ML, and ML⋆ bounds in Figs. 1–4.

Table 1 How the overlap \({{{\mathcal{F}}}}\) is bounded by the generalized lower and upper bounds for α, and the sign of \({d}_{\alpha }^{{\prime} }\) and \({d}_{\alpha }^{{\prime}{\prime}}\)
Full size table

In Figs. 1d–f, 2d–f, we show how the generalized ML bounds in Table 1 limit the speed of overlap dynamics. First, the generalized bounds typically exhibit tighter lower bounds in overlap, particularly at large times. Second, the upper bound in the dynamics coming from Table 1 makes the instants where the smallest overlap is attainable unique: the dynamics have no room but to follow a tight speed bounded from above and below. Moreover, these tight regions can periodically occur in time, a direct consequence of the periodic nature of the precessing dynamics in the (generalized) Bloch sphere (see Supplementary Note 2).

Quantum speed limits in many-body systems

Nothing so far is unique to simple quantum mechanical systems. The only constraint is that the Hamiltonian governing the dynamics has a bounded spectrum so that ML, ML* bounds, and their generalizations, are well-defined. As a result, the constraints we derive should also apply to inherently many-body Hamiltonians, which can similarly be emulated using a network of superconducting qubits but have not been experimentally explored on what concerns the speed of evolution of its many-body states in the Hilbert space.

To that end, leveraging the inter-qubit coupling tunability, their onsite energies, and the flexibility in preparing multi-qubit entangled states (see Supplementary Note 2 for additional device information), we emulate the quantum XY-model in a one-dimensional (1d) chain of six qubits:

$${\hat{H}}_{{{{\rm{1d}}}}}/\hslash={J}_{1}\sum\limits_{\langle ij\rangle }({\hat{\sigma }}_{i}^{+}{\hat{\sigma }}_{j}^{-}+{\hat{\sigma }}_{i}^{-}{\hat{\sigma }}_{j}^{+})+\sum\limits_{i}{W}_{i}{\hat{\sigma }}_{i}^{+}{\hat{\sigma }}_{i}^{-}\,,$$
(7)

where J1/2π = −2 MHz (see Methods) gives the hopping energy scale for excitations of nearest-neighbor qubits; Wi is the on-site potential of the ith qubit. In this case, we take Wi = (−1)iW, forming a staggered potential pattern (Fig. 3a) and the initial state is prepared as an entangled state \(\left\vert \psi (0)\right\rangle=\frac{1}{2}(\left\vert 101010\right\rangle+\sqrt{3}\left\vert 010101\right\rangle )\) (see Supplementary Note 2 for the state preparation circuit). The single Hamiltonian parameter controlling the restriction on the dynamics given by different bounds is the amplitude W. In Fig. 3d–f, we show that the dominant bound can be changed by adjusting W/2π from 0 MHz to 8 MHz. For a 6-qubit system, quantum state tomography is time-consuming, so we measure the overlap of two-component entangled state with parity oscillations (see Supplementary Note 2).

We make a closer inspection by selecting three representative values, W/2π = 0, 1.8, and 8 MHz, regimes into which the most restrictive bound changes from MT to the ML one (see Fig. 3c for the corresponding values of ΔE, \(E-{E}_{\min }\) and \({E}_{\max }-E\) vs. W). For W/2π = 0 or 1.8 MHz, Fig. 3d, e, the lower generalized bounds are less restrictive than the original MT bound. However, as we delve deeper into the ML regime (Fig. 3f), the situation changes. Beyond a crossing time tc ≈ 7 ns, ML exhibits the tightest lower bound, being superseded by its generalized version at longer times. Notably, at the times of lowest overlap, the upper generalized bound constrains \({{{\mathcal{F}}}}\) such that its values are tightly limited from above and below, as for the qubit and qutrit Hamiltonians.

Additionally, this type of integrable Hamiltonian has been extensively investigated64 since one can recast it in terms of free fermions via the Jordan-Wigner transformation65; as a result, revivals of overlap are expected. While the generalized orthogonality times are limited and typically shorter than the ones provided by the standard bounds, they are significantly constraining when deep into the ML regime—the upper generalized bounds also exhibit revivals closely following the oscillations in the overlap (see Supplementary Note 3 for more details).

Departing from integrability, we next emulate a similar Hamiltonian, using a two-dimensional (2d) structure featuring nine qubits. The Hamiltonian, schematically represented in Fig. 4a, reads

$${\hat{H}}_{2{{{\rm{d}}}}}/\hslash=\, {J}_{1}\sum\limits_{\langle ij\rangle }({\hat{\sigma }}_{i}^{+}{\hat{\sigma }}_{j}^{-}+{\hat{\sigma }}_{i}^{-}{\hat{\sigma }}_{j}^{+})\\ +{J}_{2}\sum\limits_{\langle \langle ij\rangle \rangle }({\hat{\sigma }}_{i}^{+}{\hat{\sigma }}_{j}^{-}+{\hat{\sigma }}_{i}^{-}{\hat{\sigma }}_{j}^{+})+\sum\limits_{i}{W}_{i}{\hat{\sigma }}_{i}^{+}{\hat{\sigma }}_{i}^{-},$$
(8)

where 〈 ⋅ 〉 and 〈〈 ⋅ 〉〉 denote the nearest and next-nearest-neighbor hopping qubit excitations with amplitudes J1 and J2, respectively; \({W}_{i}={(-1)}^{{i}_{x}+{i}_{y}}W\) is the checkerboard on-site potential of the qubit i. This is a typical nonintegrable Hamiltonian, where thermalization takes place2 and the time evolution proceeds by exploring the Hilbert space diffusively.

We show results for five excitations, described as an initial product state where the excitations are initialized in the qubits with onsite energy  + W. As before, W is the tuning parameter that, if increased, leads to a regime whose dynamics, constrained initially by the MT bound, become limited by the ML* one at a value W/2π = 3.4 MHz (Fig. 4b). It is easy to see that this is the case: The initial product state, which is not an eigenstate of Hamiltonian (8), has associated mean energy \(E=\left\langle \psi (0)\right\vert {\hat{H}}_{2{{{\rm{d}}}}}\left\vert \psi (0)\right\rangle=5W\). For sufficiently large W, this state exhibits E close to the maximal spectral energy of \({\hat{H}}_{2{{{\rm{d}}}}}\), such that \({t}_{{{{{\rm{ML}}}}}^{\star }}\equiv \pi \hslash /[2({E}_{\max }-E)]\) is large. We note that the overlap between the initial product state and any other state is obtained by directly measuring the probability of this product state in our experiment.

The overlap dynamics for representative values of W (Fig. 4d–f) shows that bounds are typically less stringent, with the exception being when deep in the ML* regime, wherein generalized ML* bounds (Table 1) lower-limit it, exhibiting tighter bounds than the traditional ML* case [Eq. (2)]. That the bounds are typically less constraining follows from the extensiveness in system size of the energy and its uncertainty ΔE, i.e., for large system sizes preserving the density of excitations, one expects the orthogonalization times to become vanishingly small (see Supplementary Note 3). Yet, simultaneously, the rate of change of overlap increases in this limit such that the bounds at \({{{\mathcal{F}}}} \, > \, 0\) are always relevant for the corresponding dynamics.

For finite system sizes, the crossover of the bounds can be characterized by inspecting how the contribution of the different Fock states \(\left\vert {{{\bf{s}}}}\right\rangle\) to \(\left\vert \psi (t)\right\rangle\) evolves with time and with the amplitude of the checkerboard potential W. The latter renders that the Fock state energies turn stratified, counting the number of excitations in the  ± W qubits, \({E}_{{{{\bf{s}}}}}={\sum}_{i}{W}_{i}\langle {\hat{\sigma }}_{i}^{+}{\hat{\sigma }}_{i}^{-}\rangle\). At W/2π = 0 MHz, all product states are degenerate with Es = 0 MHz (Fig. 4g); the energy uncertainty encompasses all of them, allowing the dynamics to explore all possible states. At W/2π = 3.4 MHz, wherein \({t}_{{{{\rm{MT}}}}}\simeq {t}_{{{{{\rm{ML}}}}}^{*}}\), ΔE is sufficient to encompass only the initial state and the Fock states that exhibit one single excitation in a  −W qubit and the remaining ones in the  +W qubits with Es = 3W, forming a 20-fold degenerate subspace. This conclusion, that ΔE ≃ 2W at the crossover point, carries over to larger system sizes, also in the one-dimensional case (see Supplementary Note 3). Finally, further increasing the checkerboard energy offset to W/2π = 6.5 MHz makes Es to split further such that the energy uncertainty only encloses the initial state with Es = 5W.

Verification of Fock space exploration can be quantified by the L1-norm distance (Hamming distance) from the initial state \(\left\vert \psi (0)\right\rangle \equiv \left\vert {{{{\bf{s}}}}}_{0}\right\rangle\) to any other Fock state \(\left\vert {{{\bf{s}}}}\right\rangle\): \(D({{{\bf{s}}}},{{{{\bf{s}}}}}_{0})={\sum}_{i}| {s}^{i}-{s}_{0}^{i}|\), where \({s}^{i}\equiv \left\langle {{{\bf{s}}}}\right\vert {\hat{\sigma }}_{i}^{+}{\hat{\sigma }}_{i}^{-}\left\vert {{{\bf{s}}}}\right\rangle=0\) or 1, denotes the presence or absence of an excitation in qubit i. The many-body Fock space dynamics are identified by the probability of finding an excitation at a distance D(s, s0) ≡ d at an instant of time t66,

$${{\Pi }}(d,t)=\sum\limits_{{{{\bf{s}}}}\in \{D({{{\bf{s}}}},{{{{\bf{s}}}}}_{0})=d\}}\left| \left\langle {{{\bf{s}}}}\left| {e}^{-\frac{{{{\rm{i}}}}{\hat{H}}_{2{{{\rm{d}}}}}t}{\hslash }}\right| {{{{\bf{s}}}}}_{0}\right\rangle \right| ^{2}.$$
(9)

This is shown in Fig. 4j–l, which describes the propagation of the wave-packet over the whole Fock space within the MT-limited regime (W/2π = 0 MHz), whereas, in the ML* governed one, this exploration is significantly limited to the states with d ≲ 4, i.e., approximately in the manifold Es = 1W, 3W, and 5W.

Discussion

Quantum speed limits have been long introduced12,13, and their importance relies on the bounds they provide to any operation performed on a quantum state. Applications of that range from how fast a quantum gate operates in a quantum computer7,47,67 to investigating the spectral form factor68, and also on how to maximize the charging speed of a quantum battery40,69,70,71. Besides the maximum rate of change typically investigated29,51, we showed here the existence of (generalized) minimal speeds a quantum state needs to obey under unitary conditions—a related idea has been investigated for the minimal discharge speed of a quantum battery72. Consequently, in various isolated quantum systems, state space exploration proceeds at a restricted speed from above and below. In complex state spaces, like those of large many-body quantum systems, bounds can be less stringent and restricted to short time scales—a direct outcome of the fact that orthogonalization is facilitated with a larger number of states to explore.

Our systematic experimental characterization in structured single or many-body problems establishes the stage for future analysis under dissipation, whether it is engineered or not73,74. In particular, it also allows testing the cases in which an open quantum system described by a Markovian bath has dynamics at short times defined by an effective non-Hermitian Hamiltonian75,76 and the associated quantum speed limit derived for these cases77,78.

Methods

Generalization of the ML bound

Ref. 15 shows a generalized ML bound for t⊥, which reads

$${t}_{\perp }\ge \frac{\pi \hslash }{{2}^{1/\alpha }{\langle {(\hat{H}-{E}_{\min })}^{\alpha }\rangle }^{1/\alpha }},\,\alpha \, > \, 0.$$
(10)

Based on this, we can derive another bound similar to Eq. (10) except that there is an additional prefactor valid for all t. Given the Hamiltonian \(\hat{H}\) and its eigenstates \(\left\vert {E}_{n}\right\rangle\), initial and time-evolved states are expanded as \(\left\vert \psi (0)\right\rangle={\sum}_{n}{c}_{n}\left\vert {E}_{n}\right\rangle\) and \(\left\vert \psi (t)\right\rangle={\sum}_{n}{c}_{n}{e}^{-{{{\rm{i}}}}{E}_{n}t}\left\vert {E}_{n}\right\rangle\) (ℏ = 1). Following a similar procedure as in the Appendix of Ref. 14, we define

$$\left\langle \psi (0)| \psi (t)\right\rangle={{{\mathcal{F}}}}{e}^{{{{\rm{i}}}}\theta },$$
(11)

where \({{{\mathcal{F}}}}=| \langle \psi (0)| \psi (t)\rangle |\) is the overlap defined in the main text. For simplicity, we can promote an energy shift, such that one defines a semi-positive definite Hamiltonian \({\hat{H}}^{{\prime} }=\hat{H}-{E}_{\min }\) (\({E}_{\min }\) is the ground state energy of \(\hat{H}\)) and the corresponding overlap

$$\left\langle \psi (0)| {\psi }^{{\prime} }(t)\right\rangle={{{\mathcal{F}}}}{e}^{{{{\rm{i}}}}{\theta }^{{\prime} }},$$
(12)

where \(\left\vert {\psi }^{{\prime} }(t)\right\rangle={\sum}_{n}{c}_{n}{e}^{-{{{\rm{i}}}}({E}_{n}-{E}_{\min })t}\left\vert {E}_{n}\right\rangle\) —we can see that \({\theta }^{{\prime} }=\theta+{E}_{\min }t\). Additionally, we can further have

$$ \sum\limits_{n}| {c}_{n}{| }^{2}\cos [({E}_{n}-{E}_{\min })t]=\, {{{\mathcal{F}}}}\cos {\theta }^{{\prime} },\\ \sum\limits_{n}| {c}_{n}{| }^{2}\sin [({E}_{n}-{E}_{\min })t]=\, -{{{\mathcal{F}}}}\sin {\theta }^{{\prime} }.$$
(13)

Now, borrowing the trigonometric inequality15

$${x}^{\alpha }-\frac{{\pi }^{\alpha }}{2}+\frac{{\pi }^{\alpha }}{2}\cos x+\alpha {\pi }^{\alpha -1}\sin x\ge 0,$$
(14)

which is valid for x ≥ 0, α > 0, we replace x with \(({E}_{n}-{E}_{\min })t\) to obtain

$$\begin{array}{rc}&\sum\limits_{n}| {c}_{n}{| }^{2}\left(\cos [({E}_{n}-{E}_{\min })t]+\frac{2\alpha }{\pi }\sin [({E}_{n}-{E}_{\min })t]\right)\\ &\ge 1-\frac{2{t}^{\alpha }}{{\pi }^{\alpha }}\sum\limits_{n}| {c}_{n}{| }^{2}{({E}_{n}-{E}_{\min })}^{\alpha }\\ &=1-\frac{2{t}^{\alpha }}{{\pi }^{\alpha }}\left\langle {(\hat{H}-{E}_{\min })}^{\alpha }\right\rangle .\end{array}$$
(15)

Combining Eqs. (13) and (15), we obtain

$${{{\mathcal{F}}}}(\cos {\theta }^{{\prime} }-\frac{2\alpha }{\pi }\sin {\theta }^{{\prime} })\ge 1-\frac{2{t}^{\alpha }}{{\pi }^{\alpha }}\left\langle {(\hat{H}-{E}_{\min })}^{\alpha }\right\rangle \,,$$
(16)

leading to

$${t}^{\alpha }\ge [1-{{{\mathcal{F}}}}(\cos {\theta }^{{\prime} }-\frac{2\alpha }{\pi }\sin {\theta }^{{\prime} })]\frac{{\pi }^{\alpha }}{2\left\langle {(\hat{H}-{E}_{\min })}^{\alpha }\right\rangle }.$$
(17)

As a result, any time t satisfies,

$$t\ge \gamma {({{{\mathcal{F}}}},{\theta }^{{\prime} })}^{1/\alpha }\frac{\pi }{{2}^{1/\alpha }{\left\langle {(\hat{H}-{E}_{\min })}^{\alpha }\right\rangle }^{1/\alpha }}\,,$$
(18)

where \(\gamma ({{{\mathcal{F}}}},{\theta }^{{\prime} })\equiv \max [0,1-{{{\mathcal{F}}}}(\cos {\theta }^{{\prime} }-\frac{2\alpha }{\pi }\sin {\theta }^{{\prime} })]\). In particular, these equations lead to (generalized) overlap bounds even before orthogonalization occurs. When \(\cos {\theta }^{{\prime} }-\frac{2\alpha }{\pi }\sin {\theta }^{{\prime} } \, > \, 0\), we can have, based on Eq. (16), a lower bound for the overlap \({{{\mathcal{F}}}}\)

$${{{\mathcal{F}}}}\ge \frac{1-\frac{2{t}^{\alpha }}{{\pi }^{\alpha }}\left\langle {(\hat{H}-{E}_{\min })}^{\alpha }\right\rangle }{\cos {\theta }^{{\prime} }-\frac{2\alpha }{\pi }\sin {\theta }^{{\prime} }},$$
(19)

and when \(\cos {\theta }^{{\prime} }-\frac{2\alpha }{\pi }\sin {\theta }^{{\prime} } \, < \, 0\) we can have an upper bound

$${{{\mathcal{F}}}}\le \frac{1-\frac{2{t}^{\alpha }}{{\pi }^{\alpha }}\left\langle {(\hat{H}-{E}_{\min })}^{\alpha }\right\rangle }{\cos {\theta }^{{\prime} }-\frac{2\alpha }{\pi }\sin {\theta }^{{\prime} }}.$$
(20)

Similarly, we can also extract the dual generalized ML bound as below

$${{{\mathcal{F}}}}\ge \frac{1-\frac{2{t}^{\alpha }}{{\pi }^{\alpha }}\left\langle {({E}_{\max }-\hat{H})}^{\alpha }\right\rangle }{\cos {\theta }^{{\prime}{\prime}}-\frac{2\alpha }{\pi }\sin {\theta }^{{\prime}{\prime}}},$$
(21)

when \(\cos {\theta }^{{\prime}{\prime}}-\frac{2\alpha }{\pi }\sin {\theta }^{{\prime}{\prime}} \, > \, 0\) and

$${{{\mathcal{F}}}}\le \frac{1-\frac{2{t}^{\alpha }}{{\pi }^{\alpha }}\left\langle {({E}_{\max }-\hat{H})}^{\alpha }\right\rangle }{\cos {\theta }^{{\prime}{\prime}}-\frac{2\alpha }{\pi }\sin {\theta }^{{\prime}{\prime}}},$$
(22)

when \(\cos {\theta }^{{\prime}{\prime}}-\frac{2\alpha }{\pi }\sin {\theta }^{{\prime}{\prime}} \, < \, 0\). Here we define \(\left\vert {\psi }^{{\prime}{\prime}}(t)\right\rangle={e}^{-{{{\rm{i}}}}({E}_{\max }-\hat{H})t}\left\vert \psi (0)\right\rangle\) and \(\left\langle \psi (0)| {\psi }^{{\prime}{\prime}}(t)\right\rangle={{{\mathcal{F}}}}{e}^{-{{{\rm{i}}}}({E}_{\max }-\hat{H})t}={{{\mathcal{F}}}}{e}^{{{{\rm{i}}}}{\theta }^{{\prime}{\prime}}}\); thus \({\theta }^{{\prime}{\prime}}=-(\theta+{E}_{\max }t)\). Equations (19) and (22) give the generalized ML bound.

Control and measurement of coupling strengths

Observation of quantum speed limits requires precise control and measurement of the coupling strengths between qubits. Our chip is a rectangular lattice, on which we only consider the actively controllable nearest-neighbor coupling and uncontrollable cross-coupling.

We can individually control nearest-neighbor coupling strengths for different qubit pairs by utilizing tunable couplers79. The effective coupling strength between a qubit pair is

$$\tilde{g}\approx \frac{{g}_{1}{g}_{2}}{2}\left(\frac{1}{{\Delta }_{1}}+\frac{1}{{\Delta }_{2}}\right)+{g}_{12},$$
(23)

where gi is the coupling strength between coupler and the qubit i and g12 the capacitive coupling strength of the two qubits. Δi = ωi − ωc is the detuning of the qubit i from the coupler. By adjusting Δ1 and Δ2, the effective coupling strength \(\tilde{g}/2\pi\) can be dynamically tuned up to about -20 MHz.

As for measuring coupling strengths, we follow the procedure described in Ref. 6. For nearest-neighbor couplings, we keep each qubit pair at the interaction frequency ωI, making all other qubits detuned Δ/2π = ± 120 MHz (or  ±70 MHz for one qubit limited by its maximum frequency) away from ωI at the same time. For cross-couplings, we do the same for a set of detunings to obtain a set of effective coupling strengths, which can be used to fit the actual cross-coupling strengths.