Counterdiabatic driving of the quantum Ising model
The system undergoes adiabatic evolution when its population in the instantaneous eigenbasis of its time-dependent Hamiltonian changes only negligibly. Realization of such dynamics requires slow-enough changes of the parameters of the Hamiltonian, a task that can be hard to achieve near quantum critical points. A powerful alternative is provided by the counterdiabatic modification of the Hamiltonian allowing for an arbitrarily quick implementation of the adiabatic dynamics. Such a counterdiabatic driving protocol has been recently proposed for the quantum Ising model [A. del Campo et al., Phys. Rev. Lett. 109, 115703 (2012)]. We derive an exact closed-form expression for all the coefficients of the counterdiabatic Ising Hamiltonian. We also discuss two approximations to the exact counterdiabatic Ising Hamiltonian quantifying their efficiency of the dynamical preparation of the desired ground state. In particular, these studies show how quantum criticality enhances finite-size effects in the counterdiabatic dynamics.
The ability to prepare a quantum system in its ground state is crucial in adiabatic quantum computing Far (), quantum simulation of strongly correlated systems with trapped atoms and ions Lewenstein et al. (2012); Qsi (), studies of macroscopic superpositions of quantum phases in these systems BDS (); Mor (); Mas (), etc.
One way to experimentally achieve this goal is to start from an easy to prepare ground state, and then drive the system towards the desired ground state that is hard to prepare in a different way. This can be done through the change of the parameter of the Hamiltonian. If such driving is slow enough, the excitation of the system is typically exponentially small in the driving rate Exp (). The adiabatic approximation tells us then that the wave-function during such evolution will be approximately given by
where is the (instantaneous) ground state of the system’s Hamiltonian , , and is the parameter driving the evolution.
The main limitation of such a procedure comes from the requirement of the slow change of the parameter of the Hamiltonian. The longer the evolution lasts, the more affected by decoherence will be the evolved state. This issue becomes especially troublesome in the quantum critical systems, where the vanishing energy gap near the critical point requires prohibitively slow time variations of the driving parameter Jac (a); Pol ().
One way to circumvent this problem is to temporarily modify the system’s Hamiltonian such that the adiabatic evolution (1) is strictly imposed regardless of the rate of driving. To achieve that, one adds a counterdiabatic term to the Hamiltonian Dem (); Ber ()
where are the eigenstates of the system’s Hamiltonian . It is then easy to see that the wave-function (1) satisfies time-dependent Schrödinger equation
where the modified Hamiltonian of the system is
In fact, not only the ground state but also all the other eigenstates of undergo a perfect adiabatic evolution with the Hamiltonian no matter how fast the parameter is changed. A comprehensive discussion of different techniques allowing for a quick preparation of the desired quantum state is presented in a recent review Ado (a); see also Refs. Ado (b, c). It is shown there that such techniques – also known as shortcuts to adiabaticity – are applicable to both single-particle and many-body systems of atoms, spins, etc.
We consider the Ising chain in the transverse field
where denote the usual Pauli matrices acting on the -th spin and is the number of spins in the chain. We assume periodic boundary conditions and choose even for which the ground state lies in the positive parity subspace BDf (a); ABQ (). We focus on magnetic fields . When the system is in the ferromagnetic phase, while for it is in the paramagnetic phase. The critical point is at the magnetic field . The quantum Ising model is exactly solvable. It is a prototypical model undergoing a quantum phase transition Sac (); Sachdev (2011). It serves as a test bed for different theoretical concepts in many-body quantum physics. Besides being of theoretical interest, such a model describes the magnetic properties of cobalt niobate CoNbO Col (). The number of spins in such a system is thermodynamically large and avoids precise experimental control.
The counterdiabatic term for the quantum Ising model has been found in Ref. Ado (d). It reads
where each term
describes interactions of adjoint spins. Thus, to implement the counterdiabatic protocol one needs to be able to: (i) prepare an Ising chain with a predefined number of spins; (ii) implement the multi-spin interaction terms. The former should be available in the near future in cold ion Mon (); Lan (), cold atom Gre (), and NMR LiP () simulators of spin systems. The latter is even more demanding, but may be possible with the techniques proposed in Ref. Casanova et al. (2012).
The coefficients providing the strength of the -body interaction terms are given by111 The prefactor in Eq. (4) differs from the prefactor in the expression for in Ref. Ado (d) because we sum over positive momenta only, while the summation in Ref. Ado (d) goes over positive and negative momenta, i.e., in our notation.
where both here and everywhere below
It has been proposed in Ref. Ado (d) that in the thermodynamic limit one can approximate the exact coefficients with
This article is organized as follows. Section III is devoted to the derivation of the exact closed-form expression for all the coefficients and to the discussion of how they reflect the symmetries of the quantum Ising model. Sections IV and V discuss two approximations to the exact counterdiabatic Hamiltonian. The former studies the dynamics induced by the interaction-truncated counterdiabatic Hamiltonian, while the latter focuses on the dynamics resulting from the thermodynamic approximation (5). In both sections, we quantify the efficiency of the dynamical preparation of the desired ground states. Our findings are briefly summarized in Sec. VI.
Iii Exact counterdiabatic coefficients
We will prove in this section the exact closed-form expression for coefficients, the remarkably compact formula:
for , where the trivial case is added to simplify the subsequent calculations.
To prove it, we define one additional sum
and conjecture that
First, we stress that Eqs. (6) and (8) are correct for even only. Moreover, it is worth to notice that all we need for the counterdiabatic Hamiltonian is the expression for the coefficients for , so we will prove a bit more general result. Finally, we mention that although we assume in the proof, Eqs. (6) and (8) work for as well. Indeed, from Eqs. (4) and (7) one gets that and . The same mapping is exhibited by Eqs. (6) and (8).
The induction starts from , where we find that indeed Eqs. (6) and (8) hold: The former holds trivially, while the latter immediately follows from the following identity provided in Sec. 1.382 of Ref. Ryz ()
after substituting .
From these relations it is clear why we have to consider the additional series, , to inductively prove the conjectured expression for . Eqs. (10) can be written in such a compact form with the help of the following identities valid for integer
which agrees with the conjectured expressions (6) and (8) taken at . Such an inductive reasoning can be carried out for , for which Eqs. (11) – (13) hold. This concludes the inductive proof that Eqs. (6) and (8) are valid for . Note that the largest for which one can inductively prove Eq. (6) is indeed , because Eq. (6) gives clearly wrong result for . We will now point out some general features of expression (6).
The coefficients follow a simple law under the Kramers-Wannier duality transformation (this symmetry of the quantum Ising model is discussed in Ref. Kra ())
Indeed, using Eq. (6) one easily finds that
This equation provides the ferromagnetic/paramagnetic duality of the coefficients, i.e., a one-to-one mapping between their values in the ferromagnetic and paramagnetic phases. Thus, Eq. (14) explicitly shows how the counterdiabatic Hamiltonian reflects the symmetry of the system for which it is written down. Moreover, we mention that duality symmetry (14) implies that the maximum of the coefficients is not at the quantum critical point in any finite Ising chain (2). This follows from the observation that one can rewrite Eq. (14) to the form
A similar symmetry property have been found in the studies of the related quantity, fidelity susceptibility providing the overlap of two “nearby” ground states BDf (b). Just as , fidelity susceptibility is peaked near the critical point GuR (); GuE (). Besides the location of the critical point, it also encodes the universal critical exponent providing the scaling of the correlation length with the distance from the critical point ABQ (). We can also see here the influence of quantum criticality by noting that depends on and through the combinations
is the correlation length of the quantum Ising model Bar (); upper/lower sign corresponds to the paramagnetic/ferromagnetic phase. The first of these relations have been noticed in Ref. Ado (d) as it is properly captured by the thermodynamic approximation (5). Assuming that the divergence of the correlation length near the critical point is given by the standard expression
one may speculate that the magnitude of the -body counterdiabatic interactions in other critical systems will depend on the interaction range and the system size through the combinations and . It would be interesting to verify this simple conjecture.
Iv Truncated counterdiabatic dynamics
The main complications in the experimental implementation of the counterdiabatic dynamics should come from the need to engineer the multi-spin interactions. Suppose now that we can implement only the counterdiabatic terms coupling up to spins, i.e., we substitute
for in the Hamiltonian (3). If the full counterdiabatic protocol is implemented and an exact preparation of the desired ground state takes place. In the opposite limit, i.e. when , the whole counterdiabatic Hamiltonian disappears and we are left with the system susceptible to the non-adiabatic dynamics across the critical point Dor (); Jac (b). The purpose of this section is to quantify the probability of finding the system in the ground state by the end of time evolution, if the dynamics is driven by the truncated counterdiabatic Hamiltonian. The study of the density of defects created due to the truncation of the counterdiabatic Hamiltonian was performed in Ref. Ado (d). Thus, our work provides a complementary insight into the influence of the truncation on the dynamics of the Ising chain. It is also worth to mention that a different simplification of the counterdiabatic Ising Hamiltonian has been recently discussed in Ref. Sab ().
The time-dependent modulation of the magnetic field is chosen such that at the beginning and the end of the time evolution – and , respectively – . Moreover, we assume that the magnetic field changes from to . These requirements impose the following boundary conditions
They are satisfied, for example, by a simple polynomial ansatz of the form
which we will use below. Such an ansatz has been used in previous studies as well Sab (). We start the evolution from , i.e. far away from the critical point in the paramagnetic phase, and evolve the system deeply into the ferromagnetic phase all the way to . This is the typical setup for the studies of the dynamics of the quantum Ising chain (see e.g. Refs. Jac (b); Dor (); Ado (d)).
Although the experimental implementation of the counterdiabatic Hamiltonian (3) is challenging due to the presence of the multi-spin interactions, the theoretical description of the dynamics of the system driven by the non-local counterdiabatic Hamiltonian is straightforward. This is seen by performing the Jordan-Wigner transformation:
where are fermionic operators, and then going to the momentum space through the substitution
After these transformations one obtains
The evolution starts from a ground state of the Ising Hamiltonian, i.e., from a product state
taken at . The state is annihilated by all operators.
Solving the Schrödinger equation with the initial wave-function , one finds that
If we would substitute the exact into Eq. (20), we would get
which guarantees the perfect preparation of the ground state by the end of time evolution Ado (d). Namely,
First, when , no truncation is involved, the perfect preparation of the ground state happens. Therefore, should be equal to one. Our numerical simulations provide for all the cases illustrated in Fig. 1. This shows that the numerical errors are negligible in our computations.
Second, we consider the case, when counterdiabatic dynamics is absent. We see that the probability of finding the system in the ground state quickly goes to zero as the system size increases. The need to counteract this tendency motivates the introduction of the counterdiabatic driving protocol that we discuss. It should be also said that this disappearance of comes from the fact that the gap in the excitation spectrum scales as near the critical point. As a result, bigger systems get more excited due to the dynamical crossing of the critical point. The excitations that are created in the course of such dynamics are known as kinks (see e.g. Ref. Dor (); Jac (b)).
Third, for large enough systems – larger than a few tens for the evolutions from Fig. 1 – we find that depends mostly on the ratio (see the lower panel). Using Fig. 1, we can correlate the range of the counterdiabatic interactions with the probability of ground state preparation. On the one hand, we notice that even substantially truncated counterdiabatic Hamiltonian greatly increases with respect to what one ends up with when no counterdiabatic driving is present. On the other hand, to prepare the desired ground state with probability larger than , virtually all the terms in the counterdiabatic Hamiltonian have to be present. This observation quantifies how demanding is the counterdiabatic dynamics if high-probability of ground state preparation is sought for.
V Thermodynamic approximation
We will discuss here the thermodynamic approximation (5) to the counterdiabatic Hamiltonian to illustrate the interplay between finite-size effects and quantum criticality in the counterdiabatic dynamics.
The numerics is performed in the same way as in the previous section, except we use the coefficients instead of . The system-size dependence of the probability of ground state preparation is illustrated in Fig. 2. This figure shows that the lack of finite-size effects in the counterdiabatic coefficients limits the maximum efficiency of the preparation of the ground state.
On the other hand, the efficiency of the driving with the thermodynamic Hamiltonian is quite substantial and equals about in the large system limit (Fig. 2). This can be compared to the truncation approximation studied in Sec. IV. To achieve the same probability of the ground state preparation with the latter approach, one needs to perfectly emulate all but one or two longest-range interaction terms (Fig. 1).
The fact that obtained through the thermodynamic approximation does not approach unity as the system size increases is a bit surprising. In fact, it has an interesting explanation that we will provide below.
We expect that such an approximation will work well when and that it will fail for of the order of . This expectation is based on the observation that when , the summand in Eq. (4) changes slowly in the former case and quickly in the latter one. Since there are simple analytical expressions for the thermodynamic (5) and exact (6) coefficients, the reader can easily verify these expectations. We illustrate these remarks in Fig. 3.
Next, we substitute the coefficients into Eq. (20) to see how the inaccuracies of the thermodynamic approximation affect time evolution. We get
where the upper/lower sign should be used in the ferromagnetic/paramagnetic phase. The first term in this expression reproduces the exact form of the factor (21) needed for a perfect adiabatic preparation of any eigenstate of the Ising Hamiltonian (2). The second one is the correction resulting from the above-described deficiencies of the thermodynamic approximation. Rewriting as , we see that this correction plays a role only in the quantum critical regime, where the correlation length is comparable or greater than the system size [note that this is possible as stands for the correlation length of the infinite system (17)].
These remarks give us the following picture of the dynamics driven by the thermodynamically-approximated counterdiabatic Hamiltonian. Suppose the evolution starts far away from the critical point in the paramagnetic phase. The population of the instanteneous eigenstates of the time-dependent Ising Hamiltonian (2) does not change until we reach the neighborhood of the critical point, where . For such magnetic fields the second term in Eq. (22) excites the system. Soon after leaving the neighborhood of the critical point, the adiabatic dynamics resumes and the excited system is driven without the change of its population in the instanteneous eigenbasis. This is illustrated in Fig. 4.
We have found the exact-closed form expression for all the coefficients of the counterdiabatic Ising Hamiltonian (6). Our results are very compact and provide the ultimate simplification of this Hamiltonian. We have also discussed how the Kramers-Wannier duality and quantum critical properties of the Ising chain are imprinted onto the counterdiabatic Hamiltonian (14,16). It should be also stressed that our Eqs. (6) and (8) provide neat summation formulas that can be used in other contexts as well.
The main complications in the experimental implementation of the counterdiabatic Hamiltonian come from its reliance on the many-body interactions that are hard-to-induce in real physical systems. We have truncated the exact counterdiabatic Hamiltonian removing its longest-range interaction terms. Using such a Hamiltonian, we have quantified the efficiency of the preparation of the desired ground state. We have shown that the truncated Hamiltonian can significantly improve the probability of finding the system in the ground state by the end of time evolution. However, if the goal is to prepare the ground state with a substantial probability, say greater than , then this approximation becomes inefficient as virtually all the multi-spin interaction terms have to be kept to achieve such a goal.
We have also considered a simple thermodynamic approximation to the counterdiabatic Hamiltonian coefficients and explained its limited efficiency in the dynamical preparation of the desired ground state. It should be noted that there is no incentive to actually use that approximation given the compactness of our exact result (6). We think, however, that the study of this approximation provides some insights into the role of the finite-size effects in the counterdiabatic dynamics. It shows that the thermodynamic approximation leads to the undesired excitation of the system in the quantum critical regime, where finite-size effects cannot be overlooked if one aims at high-probability preparation of the target ground state.
This work has been supported by the Polish National Science Centre (NCN) grant DEC-2013/09/B/ST3/00239. I would like to thank Adolfo del Campo for several useful suggestions and stimulating discussions.
Appendix A Conjecture
We describe below how one can conjecture fairly complicated Eqs. (6) and (8). The main difficulty here is that Eqs. (4) and (7) depend on both the system size and the range of counterdiabatic interactions . In the following, we factor out the -dependence exactly, leaving the -dependence for an easy educated guess. We focus on showing the key steps in this calculation leaving the details to the reader.
First, we notice that the denominator in Eq. (4) can be rewritten as
where . The calculations can be more efficiently performed in the -parameterization.
Second, we derive the recurrence relation for
To solve this relation, we find from identity (9) that
which is valid for (the sum on the right-hand side yields zero for ).
Third, we expand numerator in Eq. (4). We obtain
for . This equation is fairly complicated and we do not know how to simplify it. It has, however, the -dependence in a closed-form, which greatly facilitates the following considerations. Substituting into Eq. (28) one easily guesses that the general expression for is provided by Eq. (6). It should be stressed that it would be very difficult to conjecture Eq. (6) right from Eq. (4).
Appendix B Auxiliary sums
Integrating it over from to we get
where comes from the evaluation of and . Substituting and restricting the range of to we obtain Eq. (11).
Derivation of Eq. (12). Just as above, one finds that
Repeating the same steps as above, we get
which is equivalent to Eq. (12) for and integer .
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