Ultrafast critical ground state preparation via bang-bang protocols

The fast and faithful preparation of the ground state of quantum systems is a challenging task but crucial for several applications in the realm of quantum-based technologies. Decoherence poses a limit to the maximum time-window allowed to an experiment to faithfully achieve such desired states. This is of particular significance in critical systems, where the vanishing energy gap challenges an adiabatic ground state preparation. We show that a bang-bang protocol, consisting of a time evolution under two different values of an externally tunable parameter, allows for a high-fidelity ground state preparation in evolution times no longer than those required by the application of standard optimal control techniques, such as the chopped-random basis quantum optimization. In addition, owing to their reduced number of variables, such bang-bang protocols are very well suited to optimization purposes, reducing the increasing computational cost of other optimal control protocols. We benchmark the performance of such approach through two paradigmatic models, namely the Landau-Zener and the Lipkin-Meshkov-Glick model. Remarkably, the critical ground state of the latter model can be prepared with a high fidelity in a total evolution time that scales slower than the inverse of the vanishing energy gap.

e fast and faithful preparation of the ground state of quantum systems is a challenging task but crucial for several applications in the realm of quantum-based technologies. Decoherence poses a limit to the maximum time-window allowed to an experiment to faithfully achieve such desired states. is is of particular signi cance in critical systems, where the vanishing energy gap challenges an adiabatic ground state preparation. We show that a bang-bang protocol, consisting of a time evolution under two di erent values of an externally tunable parameter, allows for a high-delity ground state preparation in evolution times no longer than those required by the application of standard optimal control techniques, such as the chopped-random basis quantum optimization. In addition, owing to their reduced number of variables, such bang-bang protocols are very well suited to optimization purposes, reducing the increasing computational cost of other optimal control protocols. We benchmark the performance of such approach through two paradigmatic models, namely the Landau-Zener and the Lipkin-Meshkov-Glick model. Remarkably, the critical ground state of the la er model can be prepared with a high delity in a total evolution time that scales slower than the inverse of the vanishing energy gap.

I. INTRODUCTION
antum technologies have seen considerable progress in recent years [1], thanks to the unprecedented degree of isolation and manipulation capabilities achieved over individual quantum systems [2][3][4], paving the way to the development of novel technologies and furthering our fundamental understanding of quantum information processing [5]. Yet, continued development of these technologies requires fast and robust schemes to prepare and manipulate quantum states. In particular, reducing the preparation time of target quantum states would have a profound impact for several quantum technologies, embodying an area of active research [1,6]. e ability to prepare ground states of a given Hamiltonian is especially important for many reasons. On one hand, arbitrary states can be encoded as ground states of suitably arranged Hamiltonians, which is important for adiabatic quantum computation [7]. On the other hand, the ground state of quantum many-body systems is pivotal to the investigation of quantum phase transitions (QPTs) [8]. Indeed, close to the critical point of a second-order QPT, the ground states feature non-analytic behavior, and are very sensitive to variations of the underlying control parameter. is provides advantages for tasks such as quantum metrology [9][10][11]. Critical ground states of many-body systems also o en possess a large degree of entanglement, making them an invaluable resource for several quantum information tasks [12][13][14][15][16][17]. Nevertheless, the preparation of a critical ground state is experimentally challenging. is stems from the extremely long time required by adiabatic ground state preparation, due to the vanishing energy gap close to the critical point of a second-order QPT [8]. Devising fast and robust protocols for the generation of critical ground states is thus an important avenue of research. Such e orts would shed further insight into the study of QPTs, such as the experimental determination of their universality class and the fundamental time constraints posed by their vanishing energy gap. Here, we will focus on the preparation of the ground state of a second-order quantum critical model, aiming to shorten the time duration of the protocol.
Currently known fast state-preparation strategies include local adiabatic protocols [18][19][20], shortcuts to adiabaticity [21][22][23] and fast quasi-adiabatic ramps [24]. ese methods typically require the system to be analytically solvable or numerically treatable. In addition, further demanding control in the system, embodied for instance by additional timedependent parameters, is o en required.
In this paper, we show that even simple protocols can provide remarkable results, in some cases even outperforming algorithms as sophisticated as CRAB. We showcase this, in particular, for the task of ground state preparation close to a second-order QPT. We propose the use of a double-bang protocol, which consists of two constant evolutions under a Hamiltonian with xed parameters rather than a single one as considered in [50,51]. We focus on two paradigmatic models: the Landau-Zener (LZ) [52], and the Lipkin-Meshkov-Glick (LMG) one [53]. e la er describes an interacting quantum many-body system featuring a mean-eld second-order QPT [54][55][56][57][58]. Furthermore, we provide strong numerical evidence in support of the optimality of double-bang protocols. Remarkably, our approaches are computationally resourcee cient owing to the small number of parameters de ning the protocol. At the same time, they allow us to reach almostunit delities in quite short times, compared to pulse shapes obtained via state-of-the-art QOC methods such as CRAB. As further evidence of the good performances of bang-bang protocols, we show that the time required to achieve the criti-cal ground state of the LMG model with good delity scales slower than the inverse of the minimum energy gap, which is the type of scaling observed in previous analyses [40,59]. e remainder of this paper is organised as follows. In Sec. II we formulate the problem while in Sec. III we discuss the application of optimal control techniques to the ground state preparation problem, focusing on bang-bang protocols. In Sec. IV we showcase the performance and advantages of our method through its application to the LZ and LMG models. Finally, in Sec. V we summarise our main ndings and brie y discuss further avenues of investigation.

II. GROUND STATE PREPARATION AND FUNDAMENTAL QUANTUM LIMITS
Let us consider a Hamiltonian H which, without loss of generality, we can assume to depend on a single tunable and dimensionless parameter g according to the decomposition where H 0,1 are time-independent Hamiltonian operators. Given the initial and nal values g 0 and g 1 of the (externally controllable) parameter, our goal is to nd a time-dependent protocol g(t) such that |φ 0 (g 0 ) evolves into |φ 0 (g 1 ) in the shortest possible evolution time τ , where |φ 0 (g) denotes the ground state of H(g). In general, the associated dynamics cannot be solved exactly, making it necessary to resort to numerical optimization techniques. We can broadly identify two distinct dynamical regimes. Given a typical energy scale ω, for evolution times such that ωτ → ∞, any continuous ramp is su cient to achieve the target state, and the evolving state follows the instantaneous ground state of the system, as a consequence of the adiabatic theorem [60]. On the other hand, for very short evolution times ωτ 1, the evolution is far from adiabatic. In such regime, quantum speed limits [61][62][63][64][65][66][67][68][69] provide fundamental bounds on the minimum evolution time τ required to evolve between two states under a given time-independent dynamics. Such time is lower-bounded by a quantity proportional to the Bures angle between initial and nal states, and inversely proportional of either the variance or the average energy along the trajectory. It is worth stressing here that such quantum speed limits do not provide any information on the optimal dynamics implementing the target transition, but rather give an estimate of the evolution time for a given dynamics. e task of nding the optimal Hamiltonian achieving a given evolution is a more di cult problem, sometimes referred to as the quantum brachistochrone problem [70] or minimum control time [71]. e notion of control at the quantum speed limit has attracted considerable a ention [40,71,72]. In particular, it has been observed that the minimal evolution time to generate a ground state scales as τ * ∝ ∆ −1 min where ∆ min denotes the minimum energy gap of the Hamiltonian during the evolution [59]. is is particularly interesting for the LMG model, where ∆ min occurs at the QPT and vanishes as ∆ min ∝ N −z with N the size of the system and z = 1/3 the dynamical critical exponent [54]. However, we will provide examples in which this does not hold, and the minimal evolution time τ * scales slower than ∆ −1 min , namely, τ * ∆ min ∝ N −α with α > 0 a scaling exponent.

III. OPTIMAL CONTROL
To nd an optimal time-dependent protocol we de ne the cost function F X as the state delity between output and target state for a given protocol parameterisation g X is the output state corresponding to a dynamics with pulse shape g X and F X = | ψ X (τ )|φ 0 (g 1 ) | 2 is the state delity. Here, |φ 0 (g) is the ground state of H(g), so that |φ 0 (g 0 ) and |φ 0 (g 1 ) are initial and target states, respectively, while T is the Dyson time-ordering operator. Numerical optimization is used to maximise F X with respect to X. e di erent methods put forward to achieve this goal di er in how the function g X is parameterised, that is, on the choice of ansatz being considered. Common choices include CRAB [39,40], local adiabatic ramps [18,19] and bang-bang protocols [50,51].
Here we focus on bang-bang and in particular double-bang protocols, benchmarking our results against those obtained via CRAB.
We also constrain the magnitude of the interaction g X (t), imposing |g X (t)| ≤ g max for all t. is ensures that the optimized protocols only require nite energy to be implemented, and ensures the existence of a maximum, i.e. non-zero, evolution time. We refer to App. A for the details about the employed optimization procedures.

A. Bang-bang protocols
Bang-bang protocols with bangs involve a piece-wise constant function of the form where χ I (t) = 1 for t ∈ I and χ I (t) = 0 otherwise. Here, t 1 = 0 and t +1 = τ are the xed initial and nal evolution times, respectively and X ≡ (g 1 , ..., g , t 2 , ..., t ) are the 2 − 1 optimization parameters (with the added constraints t i−1 ≤ t i ≤ t i+1 ). Note how bang-bang protocols involving + 1 bangs include as a subset bang-bang protocols with bangs.
In particular, the double-bang protocols we will use have the form where X ≡ (g A , g B , t B ). When clear from the context, we will omit the explicit functional dependence of g DB on its parameters, writing g X,(0,τ ) DB ≡ g DB . In double-bang protocols, the control parameter g(t) is thus instantaneously changed from g 0 to g A at the beginning of the protocol, then suddenly quenched to take value g B at some time t B , and nally changed into g 1 at the end of the evolution [73]. An example of a double-bang protocol is given in Fig. 1. It is worth stressing that our use of the term bang-bang di ers from the way it is used in the context of NMR, where it refers to a technique to avoid environmental interactions [74]. e piecewise-constant nature of the bang-bang protocols allows one to simplify the time-evolution operators, which can be wri en as is makes simulating the associated dynamics computationally easier, compared to simulating the evolution of a state through generic time-dependent dynamics, as required for instance by CRAB or Krotov protocols.

B. Chopped-random basis quantum optimization (CRAB)
In order to benchmark our results and highlight the advantages o ered by the bang-bang protocols, we compare them with the results obtained via CRAB. is method uses a time-dependent pulse shape wri en as a modulation of a linear ramp connecting initial and nal parameter values. is variation is wri en in terms of trigonometric functions with randomly chosen frequencies. More precisely, it uses the ansatz [39][40][41][42] Nc n=1 (x n cos(ω n t) + y n sin(ω n t)) , • g Lin (t) ≡ g 0 + (g 1 − g 0 )t/τ is the linear ramp connecting g 0 and g 1 in a total time τ .
• e integer N c is the total number of frequencies in the ansatz. Its value is set before the start of the evolution, together with the total evolution time τ .
• e frequencies ω n are uniformly sampled around the principal harmonics, ω n = 2πnω 0 (1 + ξ n ) with ξ n ∈ [−1/2, 1/2] independent uniform random numbers. e use of random frequencies implies that the functional basis being used is not constrained to be orthogonal, a feature that was found to sometimes enhance the performance of the search algorithm [39,40].
• e function b(t) is used to normalise the CRAB correction, ensuring g(0) = g 0 and g(τ ) = g 1 . A possible choice for this is b(t) = ct(t − τ ) for some constant c > 0.
e optimization is run on the 2N c parameters X ≡ (x 1 , ..., x Nc , y 1 , ..., y Nc ). Whereas g 0 , g 1 , t 0 , t 1 are set by the problem, the values of ω n (equivalently, ξ n ) are chosen empirically (o en randomly) before the evolution starts. e optimization algorithm is o en further run for di erent sets of frequencies ω n , keeping only the best result.
Notice that while the most general formulation CRAB in principle encompasses a large class of parameterisations [40] which include bang-bang protocols as a special case, in this work we refer to the most common CRAB methods based on truncated random Fourier basis.
To compute the evolution of a state through a CRAB protocol we need to numerically simulate the dynamics through the time-dependent Hamiltonian. is is in general not as ecient as computing the evolution through piecewise-constant protocols. Notice that the dynamics must be simulated a large number of times while looking for the optimal protocol, which builds up to a signi cant di erence in computational times, as illustrated in the example addressed in Sec. IV A.

IV. APPLICATIONS
We here discuss the e ectiveness of bang-bang protocols to generate the ground state at the critical point of LZ and LMG models, comparing them in particular with the results achieved using CRAB protocols. In [75] we make available the data as well as all the corresponding parameter values employed to generate the results.
with H 0 ≡ ωσ x , H 1 ≡ ωσ z , and σ k the k-Pauli matrix (k = x, y, z). Without loss of generality, we set the initial state to be the ground state of H LZ (−5), and use as the ground state of H LZ (0), that is the ground state at the avoided crossing, as a target. e initial Hamiltonian H LZ (−5) is an approximation of the asymptotic one H LZ (−∞) ∼ −σ z . is approximation is sensible in this context, as the state delity between the ground states of H LZ (−5) and H LZ (−∞) is ∼ 0.99. We optimize over double-bang protocols for di erent evolution times. Our goal is to nd simple protocols achieving the transition between initial and target ground states in the shortest possible time. We thus scan di erent values of the evolution times τ , optimizing the protocol for each chosen value. As shown in [76], depending on the imposed constraints and the time τ the control landscape shows a rich structure. We test both the bang-bang and CRAB protocols with di erent numerical optimization algorithms, and nd that the doublebang protocols achieve be er results in shorter τ times, while requiring signi cantly less computational time. Studying the optimal protocols at several di erent times allows us to pinpoint the minimum value of τ required to reach the target state with our protocol, with a given delity. We show in Fig. 1 examples of such optimized bang-bang and CRAB protocols (based on N c = 4 and 10 frequencies). e rst point to appreciate is the di erence in the number of parameters FIG. 2. Representation of the dynamics corresponding to doublebang (red) and CRAB (orange) protocols, optimized to transport the ground state of a LZ model from H = ωσx + g0ωσz with g0 = −5 to H = ωσx with g1 = 0. Note that σx |± = ± |± , while |1 and |0 (|R and |L ) are the eigenstates of σz (σy) with eigenvalue +1 and -1, respectively. e total evolution time is ωτ = 1, and the CRAB protocol shown has Nc = 4 frequencies. For such evolution time, both protocols reach the target state up to numerical precision. that need to be optimized: while double-bang requires the management of only 3 parameters, CRAB with N c = 10 frequencies needs 20 coe cients (in addition to the frequencies in the optimization). is can be quite demanding for numerical optimization toolboxes, with di erences in optimization times going from the order of hours for CRAB to seconds or minutes for double-bang. A second point of notice is that, intuitively, the search space grows with N c , thus allowing CRAB to e ectively encompass double-bang protocols. However, this would also make the associated optimization task demanding enough to be practically unfeasible.
In Fig. 2 we give a representation of the state evolution in the Bloch sphere under the protocols addressed here, while in Fig. 3 we report the delities obtained optimizing doublebang and CRAB protocols to achieve the ground state at the avoided crossing. We nd that double-bang, despite its simplicity, realises the target transition with good delity faster than CRAB, achieving delities F > 1 − 10 −10 in time ωτ * ≈ 0.8. whereas, CRAB requires ωτ * ≈ 0.9 to reach similar delities. We nd that increasing the number of frequencies N c in CRAB does not bring about signi cant improvements, while making the optimization considerably more computationally demanding.
To test further the minimum control time, we also performed the optimization with di erent protocols. In particular, we tested a variation of CRAB in which initial and nal values of the protocol are also included in the optimization, as well as triple-bang protocols. We nd that both such approaches achieve F > 1 − 10 −10 at a shorter time ωτ * ≈ 0.76. is suggests that the sub-optimality of CRAB for this particular case might be partly due to the xed initial and nal parameter values and the inherent analyticity of the ansatz.
B. Lipkin-Meshkov-Glick model e LMG model [53], originally introduced in the context of nuclear physics, describes a fully long-range interaction of N spin-1/2 subjected to a transverse magnetic eld. anks to its experimental realisation with cold atoms [77] and trapped ions [78], the model has gained renewed a ention [79][80][81][82][83], and has served as a test bed to study several aspects of quantum critical systems [84][85][86][87][88][89][90]. e model is described by the Hamiltonian with S k = 1 2 i σ i k the k = x, y, z collective angular momenta operators. e model exhibits a second order meaneld QPT at a critical value g c = 1 [54][55][56][57][58] and belongs to the same universality class of the quantum Rabi [91,92] and the Dicke [93] models. We focus on the task of driving the ground state of H LMG (g 0 = 0) to the ground state at the critical point, H LMG (g c = 1). See also Ref. [94] for a similar task using a variational quantum-classical simulation [95]. As shown in Fig. 4, in line with the results achieved for the LZ model, the double-bang protocols achieve the target transition with Optimization results generating the ground state at the avoided crossing of an LZ model [75]. We given the optimized delity F when using both double-bang and CRAB protocols for di erent total evolution times ωτ . In each optimization, we constrain the available energy imposing |g(t)| ≤ 10 ∀t. Each point gives the delity obtained optimizing a double-bang (blue circles) or CRAB (orange crosses and green triangles) protocol to evolve the ground state of HLZ(−5) to the ground state of HLZ(0). e shaded region marks results for which the numerical precision starts being an important factor, and additional care must be taken to maintain the required level of accuracy while simulating the state. All the points shown in the gure correspond to F > 1 − 10 −14 . Optimizations with up to Nc = 10 CRAB frequencies do not provide a signi cant improvement in the delity. high delity faster than CRAB, and with scaling behavior be er than the expected speed-limit scaling τ * ∼ ∆ −1 min . More precisely we nd, with double-bang protocols, delities F 0.999 for very short evolution times ωτ ∼ 1. While F 0.99 for ωτ = 0.75 with a double-bang protocol, CRAB with N c = 10 frequencies only achieves F 0.9 for the same τ . Appendix B reports further details and results of the performance corresponding to the use of double-bang protocols.
Increasing the system size leads to a closing of the energy gap at the critical point according to ∆ min ∼ N −z with z = 1/3 [54]. Hence, larger systems exhibit a smaller gap, which translates into longer evolution times to faithfully prepare the ground state at g c . In Fig. 5 we plot the results upon optimizing double-bang protocols for di erent system sizes. Without loss of generality, we choose a constraint |g(t)| ≤ g max = 1.7. As argued in Ref. [40], an optimized protocol will only be able to nd F ≈ 1 for protocols of duration τ ≥ τ * ∝ ∆ −1 min . Since the energy gap scales as N −z , it follows that τ * ∼ N z . Hence, τ * ∆ min = O(1) should remain constant when the protocol operates at the quantum speed limit. We nd that the double-bang approach allows to prepare critical ground states with delities F 0.999 in a time τ * ∆ min ∝ N −α with α > 0. We obtain an estimate of τ * in two di erent ways: rst, as the time at which the delity surpasses F = 0.998 and, second, as the time at which the kink displayed in Fig. 5 in the delity is reached. Both criteria for τ * lead to the same scaling, as shown in Fig. 6 where we nd τ * ∆ min ∝ N −α with α = 0.21 (1).
We also study the dependence of the minimal evolution time on the energy constraints imposed on the protocol. As shown in Fig. 7, increasing the allowed energy decreases the minimum control time. Another interesting observed phenomenon is the existence of a threshold, at around F ∼ 0.999, above which it is harder to push the delity. We nd that the maximum delity, for both bang-bang and CRAB protocols, increases rapidly at rst, but then hits a threshold, at which the increase is very slow with τ . Moreover, this threshold seems to be una ected by the allowed energy, suggesting that it cannot be avoided by simply pumping more energy into the system, being instead related to the constraints inherent to the model under consideration. is same behavior can be seen also in Figs. 4 and 5.
Our ndings suggest the optimality of double-bang protocols for this task. Even allowing for more complex protocols, we never nd be er delities than those achieved using the simple double-bang. More precisely, we analyzed bang-bang protocols involving three and four bangs [cf. Eq. (4)  no improvement with respect to the performance of doublebang protocols. Indeed, it appears that the optimal protocols use only two distinct values of the parameter (as opposed to the three values allowed for by triple-bang protocols). is strongly suggests the optimality of a double bang for this task. As in the LZ case, this hints at a possible explanation of the sub-optimality of CRAB, which is constrained to use xed initial and nal values of g(t). Optimal paths that involve a sudden quench at the beginning and/or end of the protocol are hardly a ainable with a continuous CRAB with a nite number of frequencies. As further evidence in this direction, we considered a variation of CRAB in which the endpoints g 0 , g 1 are also optimized. Consistently with our conjecture, this improves the results, pushing the minimal time for N = 50 spins to ωτ * ≈ 1. As it can be seen in Fig. 4, ωτ * obtained with optimized endpoints lies between the ωτ * achieved with double-bang and that of CRAB with xed initial and nal points.

V. CONCLUSIONS
We have shown that simple double-bang protocols can be employed for a fast and faithful ground state preparation. In particular, we have explicitly addressed the paradigmatic LZ and LMG models, the la er to illustrate the possibility to reliably prepare a critical ground state. In these models, optimized double-bang protocols can perform be er than well-established optimal control techniques, such as CRAB. Owing to their nature, these double-bang protocols are very well suited for optimization purposes, o ering a large computational advantage with respect to other optimal control methods.
In the LMG model, double-bang protocols allow the preparation of the ground state at the critical point in a time that scales slower than the inverse of the energy gap at the QPT. Other quantum critical models can be investigated following similar routes. Although distinct optimal control techniques may reach similar results than those reported here under a double-bang scheme, the o en large number of variables to be maximised makes these protocols very di cult to be optimized, thus hindering this key observation.
Our results motivate further theoretical studies in the realm of quantum speed limits in many-body systems. It is worth stressing that our double-bang protocol can be readily implemented in di erent experimental setups, allowing for the fast preparation of interesting quantum states, such as highly entangled states of a large number of ions [51]. Appendix A: Implementation details e optimizations reported here have been carried out in P , using the algorithms provided by the S scienti c library [96]. We used the Nelder-Mead [97] and Powell [98] optimization methods, which were found to give the best performances. Nelder-Mead, in its adaptive variant [99], is found to give be er results when using the CRAB protocol, while Powell gives be er results to optimize bang-bang protocols. In each plot we report the delity corresponding to a saturated double-bang protocol, in which the interaction strength is gmax for times ranging from 0 to ωτ1, and −gmax between ωτ1 and ωτ . All plots use the same color scale, with dark blue corresponding to values close to zero and bright red to delities F > 0.99. e dashed vertical green lines are only used to mark the values ωτ = 0.5, 1.0 and 1.5. For each total evolution time ωτ and value of gmax, we report the corresponding delity. e optimal value of the delity is achieved for all times at values of gmax between 0.5 and 0.9. Recall that the QPT takes place at g = 1.

Appendix B: Saturated-boundary double-bang protocols
In the phase in which the optimal delity increases quickly, before the saturation point, the optimal double-bang protocols are found to be of the following form: g(t) = g max for t ∈ [0, τ 1 ], for τ 1 some threshold time, and g(t) = −g max for t ∈ [τ 1 , τ ], with τ the total evolution time. We analyse this further in Fig. 8, where for di erent energy constraints g max , we show the delities for the di erent possible saturated double-bang protocols, by varying ωτ and τ 1 /τ to explore the di erent possible shapes. We nd that the saturation threshold observed in Figs. 4, 5 and 7 corresponds to a marked change in behavior of the delity. Although not explicitly shown, we analyse the scaling of the optimal time ωτ * as a function of the energy constraint g max , which is found to follow ωτ * = 1.819 · g −0.559 max , where the values are determined via a numerical t. For completeness, we also give in Fig. 9 the delities obtained using a constant protocol based on the use of a g max value. As expected, in this simple model it is not possible to exploit the available energy to speed up the transition, and delities F > 0.99 are only possible for small energies, and the times always larger than those obtainable using double-bang.