Geometric Speed Limit of Accessible Many-Body State Preparation

We analyze state preparation within a restricted space of local control parameters between adiabatically connected states of control Hamiltonians. We formulate a conjecture that the time integral of energy fluctuations over the protocol duration is bounded from below by the geodesic length set by the quantum geometric tensor. The conjecture implies a geometric lower bound for the quantum speed limit (QSL). We prove the conjecture for arbitrary, sufficiently slow protocols using adiabatic perturbation theory and show that the bound is saturated by geodesic protocols, which keep the energy variance constant along the trajectory. Our conjecture implies that any optimal unit-fidelity protocol, even those that drive the system far from equilibrium, are fundamentally constrained by the quantum geometry of adiabatic evolution. When the control space includes all possible couplings, spanning the full Hilbert space, we recover the well-known Mandelstam-Tamm bound. However, using only accessible local controls to anneal in complex models such as glasses or to target individual excited states in quantum chaotic systems, the geometric bound for the quantum speed limit can be exponentially large in the system size due to a diverging geodesic length. We validate our conjecture both analytically by constructing counter-diabatic and fast-forward protocols for a three-level system, and numerically in nonintegrable spin chains and a nonlocal SYK model.


I. INTRODUCTION
The Quantum Speed Limit (QSL) is the minimum time, T QSL , required to prepare a quantum state with unit fidelity. Understanding the physics behind it is anticipated to lead to significant advances in the field of quantum computing [1], which is based to a large extent on the ability to reliably manipulate the population of quantum states. The QSL is also of prime importance for experimental quantum emulators, such as cold atoms [2][3][4], trapped ions [5][6][7], and superconducting qubits [8], which require preparing quantum states with high fidelity before they can be studied. The origin of its physical meaning is rooted deeply in the Heisenberg energy-time uncertainty principle [9], which implies that the time over which a quantum process occurs is intimately tied to the energy uncertainty ∆E it leads to. This was recognised by Mandelstam and Tamm [10][11][12][13], who used it to introduce the lower bound T QSL ≥ π/(2∆E).
In recent years quantum speed limits have been studied ever more extensively, and various improved bounds and alternative derivations have been proposed [14][15][16][17], including generalisations to mixed states [18]. In particular, it has been noticed that the bound can be sharpened by the absolute geodesic length L = arccos | ψ i |ψ * | [78], leading to for an initial state |ψ i and a target state |ψ * . Unfortu- * Electronic address: mgbukov@berkeley.edu nately, this bound is of limited practical use in quantum many-body systems, where ∆E ∼ √ L scales with the system size L, and hence in the thermodynamic limit the bound becomes trivially T QSL ≥ 0, misleadingly suggesting that it is possible to prepare any many-body state in no time.
It is not hard to see that this issue arises due to the lack of constraints on the allowed terms in the Hamiltonian used to prepare the target state. In other words, since the bounds are based on generic arguments, they must hold for any Hamiltonian. However, if one can fine tune the Hamiltonian arbitrarily, the quantum brachistochrone problem becomes almost trivial to solve [19]. In fact, the bound is tight, because the equality holds when the Hamiltonian is unconstrained: performing H * = i/ √ 2(|ψ * ψ i | − |ψ i ψ * |) effectively realizes the σ y Pauli matrix between the initial and the target states, saturating the bound. While this is admittedly not a problem in simple setups, such as a two-level system, where the control space is sufficiently small, it quickly becomes the bottleneck for many-body Hamiltonians, in which the realization of nonlocal terms like H * requires access to exponentially many couplings, and exponential sensitivity to fine-tune them. Indeed, realizable protocols only control local physical couplings and require much longer times, such that the bound (1) essentially becomes impractical.
Let us illustrate this point explicitly. Consider a system of L noninteracting qubits, prepared in some product initial state |ψ i = | ↓ · · · ↓ and subject to a Hamiltonian H = i H i . We want to transfer the population into the target product state |ψ * = | ↑ · · · ↑ . On the singlequbit level, it is optimal to do a π-pulse around the y-(or x-) axis, i.e. H i = ∆σ y i , such that T Clearly, the existence of L independent qubits does not make the process any faster. On the other hand, the energy fluctuations in the total system are ∆E = ∆ √ L, so the expression (1) suggests that it would be possible to rotate the spins faster. This fallacious argument shows how the standard QSL bounds are based on the premise that one can access the full Hilbert space to construct the optimal driving Hamiltonian. In the present example, this bound will be achievable only if one can realize the Hamiltonian H * ∝ i/ √ 2(| ↑ · · · ↑ ↓ · · · ↓ | − h.c.) which transfers the population from the initial into the target state by rotating it into a marcroscopic Schrödinger cat state at intermediate times. In experiments, where one only has local control over the system, one simply cannot implement this evolution. Moreover, in more complex interacting setups the structure of the target state itself is very complicated so H * will not only be non-local but exponentially complex. One intuitively expects that T (L) QSL should generically increase with L as it is usually much harder to prepare many-body states with a good fidelity, especially in complex systems.
In this work, we formulate a conjecture and give numerical and analytical evidence supporting the validity of a new, geometric lower bound on the QSL (cf. Eq. (2), (6) below). This bound implies that the QSL is controlled by the geodesic length between the initial and the target state in the eigenstate manifold set by the control parameter space. Based on this conjecture, we show that the adiabatic limit and the associated quantum geometry [30] constrain the time of possible unit-fidelity protocols both in single-particle and complex many-body systems. From our conjecture it also follows that the QSL for all protocols is bounded by the QSL for CD protocols, which generally cannot be implemented within the constrained control parameter space, but for which the geodesic bound can be rigorously proven using recent results from Ref. [59].
II. GEOMETRIC BOUND CONJECTURE

Consider a system described by the Hamiltonian H(λ),
where λ is the control parameter which couples to a local operator. To simplify the discussion we assume that the control parameter has a single component [79]. At time t = 0 we prepare the ground state (GS) |ψ(t = 0) = |ψ 0 (λ i ) . We want to transfer the population with unit probability over a finite time span T from this initial state into a target state |ψ(t = T ) = |ψ 0 (λ * ) , which (up to an overall phase) is the GS of H(λ * ) [80]. In order to implement such a protocol we only allow Hamiltonians of the form H(t) ≡ H(λ(t)), which depend on time solely through the control function λ(t). Such constrained Hamiltonians, if they prepare the target state with unit fidelity, are called fast-forward (FF) Hamiltonians: H FF (t) ≡ H FF (λ((t)) [81].
Whenever preparing the target state with unit probability (or unit fidelity) is possible, the system is called controllable. By the adiabatic theorem, for any nondegenerate Hamiltonian H(λ) the problem becomes asymptotically controllable in the limit T → ∞. Notice that, in general, there may exist multiple protocols which yield unit fidelity. Any unit-fidelity protocol obtained using Optimal Control methods gives rise by definition to a FF Hamiltonian.
Conjecture.-Let us formulate the following conjecture: for any fast-forward Hamiltonian H FF (λ(t)) the energy fluctuations, averaged over the protocol duration, are larger than the geodesic length : where the parameter λ changes along a fixed unit-fidelity protocol in an arbitrary way, and are the energy variance δE 2 (t), which can be thought of as the time-time component of the geometric tensor δE 2 (t) = g tt , and the eigenstate Fubini-Study metric tensor, g λλ , respectively. Here |ψ 0 (λ) is the instantaneous ground state of H(λ), and A λ is the adiabatic gauge potential [30]. The ket |ψ(t) denotes the time-evolved initial state under the Hamiltonian H FF (t), which satisfies the boundary conditions |ψ(0) = |ψ 0 (λ i ) and |ψ(T ) = |ψ 0 (λ * ) . We emphasize the difference between the evolved and the instantaneous states: |ψ(t) = |ψ 0 (λ(t)) . The subscript c denotes the connected expectation value: To motivate the conjecture, notice that this bound is tight and can be saturated in the adiabatic limit. Indeed, from Adiabatic Perturbation Theory (APT) it follows that [30,60] Hence, for any monotonic λ(t) the bound (2) is satisfied in the adiabatic limit. Moreover, it is easy to see that at least for any real-valued Hamiltonian, satisfying instantaneous time-reversal symmetry, the next-order correction to Eq. (4) scales asλ 4 with a non-negative pre-factor, such that g tt −λ 2 g λλ ≥ 0. This fact follows immediately from the structure of APT where all the coefficients in the expansion of the wave function in the instantaneous basis in powers ofλ are imaginary in linear order, and real-valued in quadratic order [see Eq. (12) in Ref. [61]]: where |ψ 0 , |ψ (1) and |ψ (2) are real-valued functions. This observation, in turn, implies that there is no interference between theλ andλ 2 contributions to the energy variance. In particular there is noλ 3 contribution, and hence the quadratic and quartic terms above come from squares and are non-negative: Therefore, at least perturbatively, the bound is satisfied for any sufficiently slow protocol. We note that within APT, theλ 4 contribution is treated on the same footing as the squared acceleration termλ 2 because d t (λ) = λ∂ λ (λ) ∼λ 2 . Indeed the linear in acceleration correction to the wave function also comes imaginary [30,61]. Even though we formulated the bound for FF Hamiltonians, the conjecture is intimately related to CD driving protocols. In a recent work Funo et al. derived that, for any CD protocol with monotonic λ(t) the inequality Eq. (2) is always saturated [59]. This can be seen as follows: using the CD Hamiltonian, H CD = H(λ(t)) +λA λ , the system follows the instantaneous ground state of H(λ): |ψ(t) = |ψ 0 (λ(t) . Then evaluating the variance of H CD one can convince oneself that the only non-zero contribution comes from the gauge potential term: CD |ψ(t) c =λ 2 ψ 0 |A 2 λ |ψ 0 c =λ 2 g λλ and hence g tt =λ 2 g λλ . This leads to the interesting observation that the leading non-adiabatic contribution to the energy variance (and other energy cumulants), is identical to the extra contribution to the variance of energy coming from the gauge potential in the CD protocols. However, a major difference is that for CD protocols this result applies to arbitrarily fast protocols where APT does not hold. We point out that CD protocols usually require adding new control parameters, e.g. for any real-valued Hamiltonian H(λ(t)) the gauge potential is imaginary so that any CD protocol necessarily breaks instantaneous timereversal symmetry. Moreover gauge potentials for generic Hamiltonians are highly fine-tuned typically requiring hard-to-implement non-local operators. In certain simple cases it is possible to explicitly map CD protocols to fast-forward protocols by an extra unitary rotation [30] but in general this unitary is hard to find. This emphasizes the importance of establishing a QSL bound for the experimentally-relevant and more accessible FF Hamiltonians.
If correct, the conjecture has immediate far-reaching implications: (i) Minimum Time Bound : using the Cauchy-Schwarz inequality, we have where δE 2 is the time average energy variance over the protocol duration T . Combining this result with the conjecture, and setting T = T QSL to be the minimum time required to prepare the target state with unit fidelity, we obtain the following bound which holds for any optimal protocol. This bound is tight because it is saturated for slow geodesic protocols [62]. For these protocols the inequality (2) is saturated by the validity of APT. In addition, in geodesic protocols the energy variance is kept constant along the trajectoryλ 2 g λλ = const t , which sets the velocity profile. In this case the Cauchy-Schwarts inequality becomes an equality and hence the bound (6) is saturated.
(ii) Note that the metric tensor can be expressed through the non-equal time correlation function [60,61]: where M λ (t) = −∂ λ H(t) is the conjugate force with respect to the parameter λ in the Heisenberg representation [30]. If we target ground states of systems with glassy dynamics or exact many-body excited states in generic systems satisfying the eigenstate thermalization hypothesis (ETH) [63], then the geodesic length scales exponentially with the system size L, while the energy variance is at most extensive. Therefore, the conjecture implies that at best the FF Hamiltonian with local control can reach the target state only at exponentially long times. Interestingly, according to this bound, isolated critical points can be crossed at non-extensive times, which can be seen as follows. The geodesic length scales as √ L for any phase transition with the correlation length exponent ν < 1 [60], and so does the energy variance (if we drive the system with some global coupling); therefore, the ratio in Eq. (6) is system-size independent. Intuitively, such finite-time protocols can be e.g. realized by driving the system fast everywhere except near the critical point [62,64].
(iii) Our results immediately generalize to systems with a multi-component parameter space λ. Then by in Eq. (6) one understands the geodesic length, which is defined as te minimum over all accessible paths connecting λ i and λ * .
(iv) We emphasize that the conjecture (2) applies to all optimal protocols with durations T ≥ T QSL , and does not apply to protocols of duration T < T QSL since the latter do not prepare the target state with strictly unit fidelity.
(v) Since both sides of Eq.
(2) represent well-defined quantities in the classical limit [30,31,59], the bound is expected to hold for classical systems as well. The same applies to the inequality (6). Note that with explicitly included into the equations, = dλ 2 g λλ and it is the product 2 g λλ which is well-defined in the classical limit [30].
Despite its plausibility, a direct proof of this conjecture has so far remained elusive due to the absence of a general procedure to obtain fast-forward Hamiltonians analytically. In the following, we demonstrate its validity beyond APT in a variety of systems of increasing complexity ranging from few-spin models to a non-integrable Ising chain: (i) analytically, using specific exactly solvable examples, showing a proof-of-concept strategy to deriver FF Hamiltonians by unitarily rotating CD protocols, and (ii) numerically, using Optimal Control algorithms.

III. ANALYTICAL VERIFICATION OF THE GEOMETRIC BOUND CONJECTURE
In this section we consider two exactly-solvable examples to analytically verify the validity of the conjecture. To this end, we first show how one can use CD driving to find a FF Hamiltonian. The first example will be a two-level system for which the conjecture reduces to the original Mandelstam-Tamm bound. We nevertheless want to show the proof as it highlights how going from a counter-diabatic to a fast-forward protocol increases the time length and hence the QSL. The second example is a three-level system, where the conjecture becomes much less trivial and gives a larger value of QSL than the Mandelstam-Tamm bound.

A. Two-Level System
Consider first the prototypical model of a two-level system (2LS) governed by the following Hamiltonian: where h z is a fixed magnetic field along the z-axis and λ(t) is an a priori unknown optimal protocol. We prepare the system in the GS |ψ i of H 2LS (λ i = −2h z ) and seek a function λ(t) which targets the GS |ψ * at λ * = +2h z in time T , following unitary evolution under H 2LS (t).
State preparation in this model has been discussed extensively in the context of various approaches, and analytical expressions for the optimal protocols have been derived [65]. As we mentioned we will use this example to highlight connections between counter-diabatic and fastforward protocols. Before we dive into this analysis, notice a quick but curious fact: the initial and target states are related by the rotation |ψ * = exp(−iπS z )|ψ i . Hence, the static Hamiltonian H FF (t) = −h z S z is a legitimate FF Hamiltonian for T = π (with λ(t) ≡ 0). Let us compute the left-hand side (LHS) and the right-hand side (RHS) in Eq. (2) separately. On the RHS, note that the geodesic length is = θ, where tan θ = λ i /h z . On the LHS, on the other hand, we have ψ(t)|(S z ) 2 |ψ(t) = 1/4 and ψ(t)|S z |ψ(t) = 1/2 cos θ, and hence t = 1/2 sin θ. Therefore, the inequality (2) yields π sin θ ≥ 2θ, which is indeed always true for θ < π/2, and hence the conjecture holds true for this special case.
The counter-diabatic (CD) protocol amounts to adding an extra (counter) term to the Hamiltonian which keeps it in the instantaneous ground state [21,24,26,28,30]: where is the (adiabatic) gauge potential with respect to the parameter λ (see e.g. Ref. [30] for details). However, the CD protocol kicks the Hamiltonian out of the original control space by adding a magnetic field along y-direction. In order to map the CD protocol to a valid FF protocol, we need to perform an additional unitary rotation, as was first discussed in Ref. [25]: where R(t) is a unitary change-of-frame matrix, which is equal to the identity in the beginning and in the end of the protocol: R(0) =1 = R(T ). In this case it is easy to see that the wave function |ψ(t) follows the ground state of a gauge equivalent Hamiltonian H (t) = R † H(λ(t))R. Therefore |ψ(t) coincides with the initial and target states in the beginning and in the end of the protocol.
Let us now take the extreme case of the fastest CD protocolλ → ∞, where the CD Hamiltonian reduces to the rate times the gauge potential (the calculation away from the infinite-speed limit is shown in App. A): where Θ(t) is the Heaviside step function. This transformation rotates S y to S z . Note that R(t) is constant except at t = 0, T giving rise to the pulse-like contributions from R † (t)∂ t R(t) to the FF Hamiltonian: (12) with δ(t) = ∂ t Θ(t) the Dirac delta function. Finally, to make the z-magnetic field time independent, we can rescale the time according to Then, using that δ(t) = δ(t )|dt /dt|, we find The total protocol time T ≡ T QSL , which sets the QSL in this case [65], can be found as Let us now check the conjecture for this QSL protocol. To evaluate the LHS of Eq. (2), notice first that both δfunction kicks can be interpreted as a free rotation under the Hamiltonian H = S x for the time π/2. Second, for a (piecewise) constant Hamiltonian the energy variance is (piecewise) conserved. Therefore, we need to add two contributions from the kicks and a contribution coming from the rotation around z-axis, leading to: where |ψ(0 + ) , is the wave function right after the first π/2 rotation around x-axis, which brings the spin to the xy-plane. Using that ψ(0 + )|(S z ) 2 |ψ(0 + ) c = 1/4 and On the RHS of the conjecture (2), we have the geodesic length = dλ √ g λλ , where leading to = θ such that t ≥ is indeed satisfied. We see that in this simple example the difference between t and can be attributed to an extra rotation required to bring (at the QSL, kick) the y-gauge potential term back to the allowed xz-plane.

B. Three-Level System I
With the exception of the two-level system example above and a few other free-particle systems [31], it is not known how to analytically compute the FF Hamiltonian or the quantum speed limit T QSL in more complicated systems. Below, we show that the ideas of mapping CD to FF driving protocols presented in Sec. III A, can be used to identify other controllable models and compute the corresponding value for T QSL . Along the way, we unveil the difficulty and hidden complexity behind constructing FF protocols in generic systems, and showcase a concrete example which features an intrinsic emerging dynamical gauge degree of freedom.
Consider the two-qubit system described by the Hamiltonian where, as before, h z and λ are the magnetic field components along the z and x-directions respectively, and J = 1 is the zz-interaction strength which sets the reference energy scale. Let the initial and target states be the ground states of H 3LS (λ) for λ i = −2h z = −λ * , respectively. Similar to Sec. III A, our goal is to find a protocol λ(t) which prepares the target state in time T , following evolution with the single-particle Hamiltonian H 2LS (t). Due to the qubit-exchange symmetry of both H 3LS and H 2LS (t), the problem represents effectively a three-level system (3LS) with SU (3) spanning the space of all possible observables. A priori, it is not clear whether such an optimal protocol exists, since the initial and target states are eigenstates of a fully interacting Hamiltonian, whereas during the evolution the system is non-interacting (decoupled). Note that, in general, this population transfer can only be achieved if and only if the entanglement entropy of each of the two qubits is the same in the initial and target states, as entanglement is preserved during evolution with the non-interacting Hamiltonian H 2LS . This condition is clearly satisfied in our setup, since the states are related by the transformation is a legitimate FF Hamiltonian for T = π, similar to the 2LS, c.f Sec. III A. It is straightforward to check that this FF Hamiltonian satisfies the geometric bound conjecture (2).
Unfortunately, this only works for the protocol duration T = π, which immediately puts an upper bound on the QSL. For T < π, one can formally rely on general theorems in Optimal Control for systems on compact Lie groups [66], to argue the existence of a finite QSL T QSL > 0 for this problem. However, since the proofs are non-constructive, one cannot use them to directly check the validity of the geometric bound conjecture. Nonetheless, as we demonstrate now, one can apply the same strategy as the 2LS example in Sec. III A. In particular, (i) we first compute the CD Hamiltonian, and then (ii) use the latter to derive a FF protocol. However, in practice finding the correct frame transformation in step (ii) is a particularly difficult problem, since there is no straightforward way to identify the correct time-dependent rotation to map the interacting CD Hamiltonian to the non-interacting H 2LS . The procedure requires the use of non-commuting rotations in the 8-dimensional operator manifold corresponding to the SU (3) group which, due to their intrinsic time-dependence, lead to unwanted extra Galilean terms that take the transformed Hamiltonian outside the parameter manifold of H 2LS . Moreover, an additional constraint is imposed by the boundary conditions imposing that the rotation reduces to the identity at t = 0, T , c.f. Sec. III A. In the following, we demonstrate how to circumvent all these issues and construct a FF Hamiltonian for the system in Eq. (16).
Due to the small dimensionality of the Hilbert space, it is possible to find the exact adiabatic gauge potential in the ground state manifold. Note that, since the Hamiltonian H 3LS (λ) is real-valued, one can choose the gauge potential to be purely imaginary [30,31]. There are only three linearly-independent imaginary matrices which can be shown to generate a SU (2) ⊂ SU (3). Hence, in the most general form we have where α = α(λ, b(λ)) and γ = γ(λ, b(λ)) are fixed functions, which depend on the model parameters, and can be computed using, e.g., a variational principle [31]. We leave details of such computation for the appendix App. B. Let us only comment that because we are looking into the gauge potential for the ground state manifold, i.e. he gauge potential which adiabatically evolves the ground state but allows mixing between the two excited states, the gauge potential is defined up to an emergent dynamical gauge degree of freedom b = b(λ), which we use to our advantage in finding the FF Hamiltonian.
Having computed the exact gauge potential, which governs the dynamics at the QSL, we now aim at finding a transformation R(t) that brings the CD Hamiltonian (17) to the original parameter manifold (8) with renormalised drive field and an overall time-dependent pre-factor. If we fix the dynamical gauge field b(λ) to satisfy the following nonlinear differential equation one can show [see App. B] that the non-abelian SU (3)rotation obeys the boundary conditions R(0) = 1 = R(T ). Using this time-dependent transformation leads to the following FF Hamiltonian at the QSL: We note in passing that the existence of H FF is equivalent to a constructive proof of controllability, i.e. a finite T QSL < ∞, since by definition all FF Hamiltonians prepare the target state with unit fidelity. Thus, the above result establishes the relation between CD, FF and Optimal Control for the problem of preparing interacting two-qubit states using a single-particle Hamiltonian. The above mapping works at the infinite-speed QSL, where H CD =λA λ . Unlike the 2LS discussed in Sec. III A and App. A, it is currently an open question how to construct the correct transformation away from the QSL for this problem. As in the 2LS example we can rescale the time as such that the z-magnetic field is constant. Then, following the same strategy as in the 2LS, we obtain the expression for the QSL.
We can use the analytical results obtained above to verify the validity of the geometric bound conjecture (2). Once again, we shall compute the LHS and RHS separately. On the RHS, we need to compute the geodesic length . This requires some care for the current problem. Since we quench the interaction strength J at t = 0, T , so that for 0 < t < T the time evolution remains free (J = 0), we effectively have a two-parameter manifold (λ, J). Thus, as we specified in Sec. II, point (iii), the geodesic length on the RHS of (2) is the minimum length of all accessible paths connecting (λ i , J i ) and (λ * , J * ). An upper bound for this absolute minimum is given by the geodesic along J = 1, which can easily be obtained from the gauge potential (17). We emphasize that is independent of the choice for the dynamical gauge field b(λ), as expected. Hence, to verify the conjecture, it suffices to show that t ≥ , since ≥ .
Let us now focus on the LHS and the number t . Notice that the calculation is formally equivalent to the one we carried out for the 2LS in Sec. III A, due to the structure of the 3LS FF Hamiltonian (20). Thus, we just need to apply Eq. (14) using the FF Hamiltonian (20) and the QSL expression (22). Therefore, decomposing the LHS t according to Eq. (15), we arrive at where = T QSL ψ(0 + )| (S z ) 2 |ψ(0 + ) c is the geodesic length of the one-parameter manifold. This already proves the conjecture (2). In Sec. IV we formalize and generalize this procedure. Since the exact analytical expression for t is rather cumbersome and involves cubic roots, we refrain from showing it here. We can, however, instead check numerically how tight the conjecture bound is. One can evaluate this integral (22) numerically, and e.g. for J = 1 and λ i = −2h z = −λ * , we find T QSL h z ≈ 1.838, which agrees with the number we obtained using Optimal Control algorithms. Interestingly, this number is smaller than the corresponding one for the 2LS. This means that one can prepare the interacting states faster using a free Hamiltonian. Similarly, one can compute the exact geodesic length . Figure 1 shows the validity of the conjecture at T = T QSL as a function of the interaction strength J/h z .

IV. GENERALIZATION OF THE MAPPING OF FAST-FORWARD TO COUNTER-DIABATIC PROTOCOLS
The previous two examples were very instructive. In particular, we saw that at the QSL the protocols which can be obtained by rotating the gauge potential automatically satisfy the conjecture (2) because they consist of two pieces: the rotated gauge potential contribution (or more generally rotated CD Hamiltonian) and the extra kick contribution due to the rotation, cf. Eqs. (12), (20). The contribution of the first term to t gives precisely the geodesic length, or more accurately dλ √ g λλ along the chosen path λ(t), while the second, or kick term results in an extra positive contribution. Hence, for protocols of this form, the conjecture is automatically satisfied. Let us show that this scenario (and hence the validity of the bound) is generic for at least a broad class of FF protocols. To do this, we first prove that any FF protocol can be represented as a rotated CD protocol. Let us assume that there is a Hamiltonian H FF (t) ≡ H FF (λ(t)) such that the corresponding wave function |ψ(t) satisfies the boundary conditions |ψ(0) = |ψ i and |ψ(T ) = |ψ * . We further assume that λ(t i ) = λ i and λ(T ) = λ * . The latter assumption is not crucial because λ(t) is allowed to change discontinuously (but we assume that |ψ(t) is continuous and differentiable with respect to time).
The last term in Eq. (26) does not affect the ground state and reflects the gauge freedom in the choice of the gauge potential we discussed in Sec. III B above. We can simply absorb it into A λ via A λ → A λ + K/λ.
We thus see that any FF Hamiltonian can be written as the rotated CD Hamiltonian. Clearly, by applying an appropriate time-dependent phase transformation to R: R(t) → R(t)e iφ(t) , with φ(t) a time-dependent phase, we can get a similar mapping of FF to generic CD Hamiltonians where in Eq. (26) we will have to re-placeλA λ → H CD = H(λ) +λA λ .
Once we have established the equivalence of the FF and CD protocols we can examine which conditions we need in order to satisfy the conjecture (2). Since a sufficient condition for the conjecture is that R can be represented as a finite product of piecewise constant transformations: where each R j is time independent and acts only in the interval [T j−1 , T j ] (T N = T  (28) and the conjecture (2) follows immediately. At the moment we can not prove that in general there exist no smooth R such that the integral of expression (27) is smaller than the geodesic length. We do not know any general recipe for finding R. Moreover the matrix R is not unique because we only need to rotate a single state to the desired manifold. For this reason, we check the validity of our conjecture numerically on various examples as discussed in the next section.

V. NUMERICAL VERIFICATION OF THE GEOMETRIC BOUND CONJECTURE
In this section, we use algorithms from Optimal Control to numerically test the geometric bound conjecture in systems where analytical solutions are limited by the complexity arising from the enhanced dimensionality of their Hilbert spaces.

A. Three-Level System II
The FF Hamiltonian we found in Sec. III B is noninteracting. One might wonder how the physics of the 3LS discussed in Sec. III B changes if we look for an interacting FF Hamiltonian. In other words, as before we start from and target the GS of H 3LS , see Eq. (16), for λ i = −2h z = −λ * and J = 1, but this time we also evolve with H 3LS (t). Hence, the FF Hamiltonian for this problem must be in the same control parameter manifold as Eq. (16) for some optimal protocol λ(t).
Recently, methods from Shortcuts to Adiabaticity were applied to study related setups of three-level systems [67][68][69][70][71][72]. The physics of this optimisation problem below the QSL, i.e. for T < T QSL , was analysed extensively in Ref. [43], where it was shown that the state preparation problem close to optimality exhibits genuine quantum control phase transitions as a function the protocol duration T , including symmetry breaking, which introduce sharp changes in the functional form of the optimal protocols.
Despite the similarity of the current setup to the one in Sec. III B, for this initial value problem, we were unable to find the corresponding rotation of the CD Hamiltonian to its FF counterpart analytically, cf. Secs. III A and III B. Nevertheless, the existence of a finite QSL can be argued using Optimal Control theorems [66] and, variational FF protocols can been constructed which put an upper bound on the QSL [43]. This motivates the search for an approximate FF protocol λ(t) using Optimal Control algorithms.
Although within the scope of some numerical limitations, Optimal Control allows us to test the validity of the conjecture (2). Indeed, applying GRAPE [39,46,73], results in an (almost) optimal protocol which, in turn, defines a proper FF Hamiltonian. To find it, we fix a protocol duration T and discretise time in N T = 100 equal steps. We then use GRAPE, which is based on gradient ascend, to find the best possible value for the control field λ(t) at each time step in the range λ(t) ∈ [−256h z , 256h z ],which optimizers the fidelity of being in the target state at the end of the protocol t = T . In order to minimize the probability of getting stuck in a local fidelity maximum, we repeat the procedure a total of two hundred times and post-select the best outcome.
The optimal protocol enables us to test the geometric bound conjecture (2) numerically. To this end, we first identify the QSL within numerical precision, which allows us to safely focus on protocol durations T > T QSL [note that the conjecture holds only above the QSL, where we can achieve unit fidelity]. In this regime, we also make sure that the approximate FF protocol indeed prepares the target state with fidelity F h (T ) = | ψ(T )|ψ * | 2 of at least 99.99%. To evaluate the LHS t , we use the FF Hamiltonian with λ(t) obtained using GRAPE. The quantity t , related to the time-averaged energy fluctuations of H FF , is then computed numerically. On the RHS of (2), we determined independently by using (i) the geodesic length computed from the analytical gauge potential (17) and (ii) -a very slow ramp in the adiabatic limit (T = 100J), where the bound is saturated. We found excellent agreement between the two approaches. (2), across the quantum speed limit of preparing the interacting ground state of the Hamiltonian H3LS following evolution generated by H3LS(t). The parameters are λi/hz = −2 = −λ * /hz. The optimal control algorithm used is GRAPE [39,46]. Figure 2 shows the result which confirms the validity of the geometric bound conjecture for the interacting 3LS setup.

B. Nonintegrable Ising Chain: Ground State Physics
The previous examples we discussed all share in common a few-dimensional Hilbert space. A natural question to ask is whether the Conjecture (2) holds for many-body systems. In this section, we study a non-integrable Ising chain with emphasis on the ground state physics. Nonintegrability here implies both the absence of a closedform solution for the gauge potential, and the presence of locally thermalising quantum dynamics which obeys the Eigenstate Thermalization Hypothesis (ETH) [63]. Hence, this model represents a generic quantum manybody system, and our goal below is to test the geometric bound conjecture (2) on it.
Consider the non-integrable transverse-field Ising model (TFIM) in a longitudinal field, described by the Hamiltonian In the following, we set J = 1 as a reference energy scale. Once again, λ(t) denotes the control field. The initial and target states are the interacting GS for λ i = −2h z = −λ * , respectively, and the protocol duration is denoted by T . Quantum state preparation in this setup has been studied extensively using Reinforcement Learning in Ref. [42], and this state preparation problem has been shown to have glassy optimization complexity [74]. Due to the lack of a closed-form solution of the stationary Schrödinger equation, it is not possible to obtain a the ground state manifold of the system as a function of λ analytically. Therefore, we restrict the analysis of this initial value problem to the methods of Optimal Control. Because of the extensivity of the spectrum of manybody systems, it is unphysical to allow for unbounded drive fields λ(t), since local control does not grant access over extensively large energy scales. Therefore, we consider the experimentally relevant situation of a bounded drive λ(t) ∈ [−4h z , 4h z ]. As before, we discretise the protocol duration T in time steps δt, and study the problem using two different control algorithms (see Sec. V A for details): (i) GRAPE looks for continuous protocols, while (ii) Stochastic Descent (SD) has proven useful to look for the so-called bang-bang protocols, i.e. protocols which take values on boundary of the allowed domain: λ ∈ {±4}. Although discontinuous, the family of bangbang protocols are known to contain an optimal solution as a consequence of Pontryagin's maximum principle.
It is not known what the QSL for this problem is, nor whether it is finite in the thermodynamic limit. Therefore, we make sure to consider only optimal protocols with durations T , which allow for enough time to prepare the target state with many-body fidelity F h (T ) = | ψ(T )|ψ * | 2 of at least 99.99%. In this respect, it is important to mention that close to optimality finite-size effects have been shown to be negligible for this problem setup, starting from a system size of L > 6 sites, see Ref. [42], and hence we restrict to L = 10 for the results presented here.
To check the geometric bound conjecture (2), we compute numerically the LHS and RHS. Once the optimal FF protocol λ(t) has been determined, the numerical computation of t on the LHS is straightforward. On the RHS, we can no longer calculate the geodesic length exactly, since we do not have the exact expression for the adiabatic gauge potential. Nevertheless, as we argued in Sec. II and verified numerically in Sec. V A, we can obtain the geodesic length from evolution in the adiabatic limit. Figure 3 shows the ratio t / between the timeaveraged energy fluctuations of the FF Hamiltonian corresponding to the optimal protocol and the geodesic length, as a function of the protocol duration T . It is an interesting observation that, even though both the bangbang protocols (dashed line) and the continuous GRAPE protocols (solid line) satisfy the conjecture, the average energy variance t is kept smaller by the GRAPE protocols. We recall that, according to Pontryagin's maximum principle, one can always find a bang-bang protocol to achieve (at least) the same fidelity as with any continuous protocol. We attribute the fact that the two families of protocols differ in terms of the average energy variance they create during the evolution, to their robustness properties: while bang-bang protocols might be optimal they have recently been shown to be unstable to small perturbations [42]. Mathematically bang-bang protocol result in a larger energy variance and hence larger t because they the Hamiltonian changes very rapidly between the bangs, while the state does not have time to follow. If we associate t with the fluctuating energy cost following Ref. [59] then clearly bang-bang protocols are more costly than smooth protocols.

C. Nonintegrable Ising Chain: Excited States Physics
It is well known that some properties of low-energy states differ significantly from those of their excited states counterparts. Most notably, in many systems, the ground state physics is protected by a finite gap in the energy spectrum, which renders the adiabatic limit well-defined. In contrast, the energy level spacing for excited states is usually exponentially suppressed in the system size, and for spin-1/2 chains scales as 2 −L . Consequently, the time scales for the adiabatic limit are exponentially longer for excited states. On the other hand, fast-forward protocols are allowed to excite the system during the evolution before they prepare the target state. One can imagine harnessing this additional freedom to improve on the time scales for adiabatic state preparation. This raises the question whether the conjecture (2) is not violated for excited states.
To test this, we add a small y-field to the non- integrable Ising chain and consider the Hamiltonian: with h y /J = −0.1, h x /J = 1, and a driving protocol λ(t)/J ∈ [−2, 2]. We pick for an initial state an infinite-temperature state, characterized by energy which is closest to zero at λ i = −2J, see Fig. 4 (purple line). The target state is the adiabatically connected state at λ * = 2J. This choice for the initial state is motivated by ETH, according to which the states in the middle of the spectrum are the first one which become chaotic and lead to thermalization of the system under generic dynamics like dynamics governed by the Hamiltonian (30).
To ensure a finite geodesic length and a well defined adiabatic limit, we introduced a small magnetic field in the y-direction which breaks the emergent integrability crossings along the adiabatic trajectory.
To test the conjecture for excited states, we consider two spin chains of length L = 6 and L = 8, respectively. Imposing periodic boundary conditions, the only two symmetries in the Hamiltonian (30) are translation invariance and parity (reflection about the middle of the chain). Without loss of generality, we work in the zero-momentum sector of positive parity, containing the GS, which allows us to consider only those states that are coupled during the time evolution. The corresponding symmetry-reduced Hilbert sub-spaces have sizes dimH = 13 and dimH = 30, respectively. Figure 4 shows parts of the instantaneous energy spectrum of the model, including the adiabatic trajectory from the initial into the target state. The magenta line in the middle marks the adiabatically connected state. One can clearly observe a number of avoided crossings, which are responsible for large protocol durations required to find te system in the adiabatic limit. For instance, to prepare the target state with 99.999% probability adiabatically in the Hamiltonian (30) requires ramp durations on the order of T = 4 × 10 4 for L = 6, and T = 10 5 for L = 8.
To compute the LHS of the geometric bound conjecture (2), we used GRAPE to find an (almost) optimal protocol sequence of 100 time steps at a number of fixed protocol durations of order JT ∼ O (10). The nonadiabatic character of these protocols allows for a protocol duration much shorter than the adiabatic ones, yet we made sure that all GRAPE protocols prepare the target state with at least 99% fidelity. Figure 5 demonstrates that the Conjecture (2) holds even for the excited states of generic many-body models. Interestingly unlike in the two-spin case the ratio t / increases with the protocol time T , c.f. Fig. 2. As we argued we anticipate that in the limit T → ∞ the bound is saturated for any state, ground or excited, because of applicability of APT, so we expect that at longer T the ratio t / with reach a maximum and will go down. But in order to see this we need to reach much longer protocol times, which are out of reach of optimal control methods. Nevertheless we clearly see that the inequality t > holds at all protocol times considered. This result comes with an important consequence. In generic systems satisfying ETH, the geodesic length for excited states exponentially diverges with the system size L so the conjecture implies that any FF protocol is exponentially long.
Since FF protocols excite the system in the basis of the instantaneous Hamiltonian [before they de-excite it to reach the target state with unit probability], one may naïvely think that by using such out-of-equilibrium protocols it is possible to circumvent the restrictions in the adiabatic limit imposed by the size of the energy gaps in the vicinity of the adiabatically-connected state. However, the validity of the geometric bound conjecture shows that this is not the case. Hence, equilibrium properties impose geometric constraints on the out-ofequilibrium dynamics.

D. Fully-Connected Ising Model
Potential candidates that violate the conjecture are Hamiltonians which have small ground state gaps along their adiabatic path but have a lot of symmetry such that the ground state phases are trivially found by inspection. In those cases one could wonder whether numerical methods from optimal control theory can find protocols that violate our conjecture. Here we check one example and show that it does not. Consider a quantum p-spin model without disorder: For any p > 2, this model has a mean-field like first order transition from paramagnet to ferromagnet with a gap [75] exponentially closing with the system size L This makes it hard to adiabatically cross the transition but at the same time the ground states in the two phases are trivial Z and X polarized product states. Note that the ground state is unique for any odd p. Moreover, the Hamiltonian conserves total angular momentum S 2 such that the effective Hilbert space dimension is only L + 1. Figure 6 shows the low energy spectrum of an L = 14 spin model for large p. Even though the gap closes exponentially, the geodesic length does not exponentially grow with system size. In contrast, in the thermodynamic limit, it undergoes a jump of π/2 at the critical point. The latter reveals the simple Landau-Zener nature of the problem, with essentially only two states participating in the transition. Once again we use GRAPE to numerically find closeto-unit fidelity protocols that cross the quantum phase transition, i.e. they start at λ = −2 and end at λ = 0. The small gap and the highly non-local nature of the Hamiltonian seem to make the optimal control problem significantly harder than any other models considered so far in this work. Typical protocols, obtained from a random initial seed for the GRAPE, have energy fluctuations which are about two orders of magnitude larger than the conjecture bound (2). In order to obtained good protocols with small energy variance we therefore bias the algorithm to the right corner of phase space by starting from the geodesic protocol with some small random noise part. This results in much better protocols which, in the adiabatic limit saturate the bound, see Fig. 7. For shorter times, when the inverse time becomes comparable to the minimum gap along the trajectory, we can still find almost-unit-fidelity protocols but their energy variance grows rapidly with decreasing time. Numerical optimal control results thus suggest that our conjecture is also satisfied for mean-field like first order quantum phase transitions.

VI. DISCUSSION/OUTLOOK
By reconciling ideas of Adiabatic Perturbation Theory, Counter-Diabatic driving, and Optimal Control, we conjectured that the time length t for any fast-forward protocol, equal to the time integral of the instantaneous energy fluctuations, is bounded from below by the geodesic length imposed by the geometry of the instantaneous eigenstate manifold. While proving this statement for generic quantum systems remains an open problem, we have provided substantial evidence for the validity of the corresponding conjecture (2), and proved it in certain limits amenable to analytic treatment. In the exactlysolvable two-and three-level systems, we demonstrated that one can find a FF Hamiltonian at the quantum speed limit analytically using ideas from CD driving. By identifying and exploiting a residual dynamical gauge degree of freedom, we showed that the three-level system at the infinite-speed limit can be mapped to a single noninteracting collective spin degree of freedom. We also  (31). The model has a first order transition with an exponentially small gap separating the two phases. In the thermodynamic limit, the geodesic length jumps by π/2 at the critical point.
showed that any FF Hamiltonian can be obtained from a counter-diabatic Hamiltonian by a unitary rotation. The mapping might allow one to prove the conjecture in general. For a non-integrable Ising chain, we used optimal control algorithms to numerically verify the universality of the proposed geometric bound not only for the ground state but also for excited states.
The conjecture (2) likely applies to mixed states as well. Consider a process where we start from some stationary ensemble in a mixed state, described by initial the occupations ρ n , and let us require that the system ends up in the adiabatically connected target state with identical occupation numbers ρ n [83]. Then one can formulate a similar geometric bound for mixed states as

follows:
T 0 dt n ρ n ψ n (t)|H 2 FF (λ(t))|ψ n (t) c , ≥ λ * λi dλ n ρ n ψ n (λ)|A 2 (λ)|ψ n (λ) c , (32) where {|ψ n (λ) } are the instantaneous eigenstates, satisfying H(λ)|ψ n (λ) = E n (λ)|ψ n (λ) . The RHS of (32) stands for the work fluctuations, averaged over the initial probability distribution ρ n . For a single eigenstate, work and energy fluctuations are the same, while for a mixed state this need not be true [84]. We also note that the general mapping between CD and FF protocols discussed in Sec. IV also applies to the mixed states. The only difference is that the unitary matrix R is defined by requiring that system of equations is satisfied R(t)|ψ (n) (t) = |ψ n (λ(t)) , where |ψ (n) (t) is the state propagated by the fastforward Hamiltonian satisfying the boundary conditions |ψ (n) (0) = |ψ n (λ i ) and |ψ (n) (T ) = |ψ n (λ * ) . Because the Hamiltonian time evolution does not break orthonormality of the states |ψ (n) (t) at any time, the unitary matrix R always exists. So all general considerations from Sec. IV extend to the mixed states. An interesting observation, which comes from Eq. (3), is that the energy fluctuations can be interpreted as the time component of the non-equilibrium quantum metric tensor g tt = δE 2 (t), since the latter describes the distance between wave functions at two consecutive moments of time t and t + δt: g tt = ψ(t)|H 2 FF (t)|ψ(t) c = ∂ t ψ|∂ t ψ c , | ψ(t + δt)|ψ(t) | 2 ≈ 1 − g tt δt 2 . Then the conjecture (2) applied to a short time interval states that, for any time evolution, the control is always time-like g tt −λ 2 g λλ ≥ 0. The geometric bound conjecture can then be seen as a constraint imposed by causality on the optimal quantum state preparation protocols.
The universal geometric bound conjectured and checked in this paper can be used to define complexity of a dynamical control problem through the geometric length, which is a property of the ground state manifold. In particular, one can say that the problem is computationally hard if the equilibrium distance between the initial and final stated determined through the quantum geometric tensor is exponentially large in the number of degrees of freedom. This definition of complexity makes no reference to particular protocols, which can be say realized on a quantum computer. There are very few other examples we are aware of where equilibrium properties constrain the possible behaviour of a system away from equilibrium. One of them is the famous Jarzynski equality which constraints the work distribution done on a system in an arbitrary non-equilibrium process by the equilibrium free energy difference [76]. Such results are remarkable in their nature, because they demonstrate the conservative character of physical laws, and usually point towards deeper connections between seemingly unrelated phenomena.