Generalized Shortcuts to Adiabaticity and Enhanced Robustness Against Decoherence
GGeneralized Shortcuts to Adiabaticity andEnhanced Robustness Against Decoherence
Alan C. Santos & Marcelo S. Sarandy
Instituto de F´ısica, Universidade Federal Fluminense, Av. Gal. Milton Tavares deSouza s/n, Gragoat´a, 24210-346 Niter´oi, Rio de Janeiro, BrazilE-mail: [email protected]
August 2017
Abstract.
Shortcuts to adiabaticity provide a general approach to mimic adiabaticquantum processes via arbitrarily fast evolutions in Hilbert space. For these counter-diabatic evolutions, higher speed comes at higher energy cost. Here, the counter-diabatic theory is employed as a minimal energy demanding scheme for speeding upadiabatic tasks. As a by-product, we show that this approach can be used to obtaininfinite classes of transitionless models, including time-independent Hamiltoniansunder certain conditions over the eigenstates of the original Hamiltonian. We applythese results to investigate shortcuts to adiabaticity in decohering environments byintroducing the requirement of a fixed energy resource. In this scenario, we show thatgeneralized transitionless evolutions can be more robust against decoherence than theiradiabatic counterparts. We illustrate this enhanced robustness both for the Landau-Zener model and for quantum gate Hamiltonians.
1. Introduction
The adiabatic theorem [1, 2, 3, 4] constitutes a successful strategy for eigenstate trackingin quantum information and quantum control (see, e.g., Ref. [5]). It states that a systeminitially prepared in an eigenstate of a time-dependent Hamiltonian H ( t ) will evolve tothe corresponding instantaneous eigenstate at a later time T , provided that H ( t ) variessmoothly and that T is much larger than a power of the relevant minimal inverse energygap. It is worth highlighting that the validity conditions of the adiabatic approximationhave been revisited (see, e.g. Refs [6, 7, 8, 9, 10, 11]), which has implied in enhancedformulations of the adiabatic theorem [12, 13] (see also Ref. [14] for a recent review).In a real open-system scenario, the performance of the adiabatic dynamics is upperbounded by the competition between the adiabatic time scale, which is favored by aslow evolution, and the typically short decoherence time scales. This interplay providesan optimal time scale for adiabatic processes in decohering environments [15, 16].The adiabatic dynamics can be reproduced by generalized transitionless evolutionsobtained via shortcuts to adiabaticity [17, 18, 19]. Such accelerated processes allow us toderive an exact adiabatic evolution at an arbitrary finite time. Shortcuts to adiabaticity a r X i v : . [ qu a n t - ph ] D ec eneralized Shortcuts to Adiabaticity τ ,a fundamental problem is then whether shortcuts to adiabaticity can provide a moreefficient performance in terms of fidelity than their adiabatic counterparts by adjustingits pace within the decoherence time scales. We address this question by considering ageneral counter-diabatic theory [38], which is here optimized for a minimum energyconsumption. As a by-product, we apply this general approach to obtain infiniteclasses of transitionless models, including time-independent Hamiltonians under certainconditions over the eigenstates of the original (adiabatic) Hamiltonian. Concerningrobustness against decoherence, we consider Markovian open systems and impose fixedenergy resources. This is a key point, since unlimited energy provides arbitrarily fastdynamics already for adiabatic evolution, through an arbitrarily large gap between theground and first excited states. It is then shown that a supremacy of the counter-diabaticdynamics can always be achieved by adjusting the evolution rate. This is illustrated inthe Landau-Zener model and in quantum gate Hamiltonians.
2. Generalized transitionless dynamics theory and energy cost
The starting point for the counter-diabatic theory is the evolution operator U ( t ), whichcan be defined as (see, e.g., Ref. [38]) U ( t ) = (cid:88) n e i (cid:82) t θ n ( ξ ) dξ | n t (cid:105)(cid:104) n | , (1)where θ n ( t ) is a set of arbitrary real phases [39, 40] and {| n t (cid:105) = | n ( t ) (cid:105)} is the setof eigenstates of the original (adiabatic) Hamiltonian H ( t ). Let us assume that thequantum system is initially prepared in an specific eigenstate | k (cid:105) of H ( t ), namely, | ψ (0) (cid:105) = | k (cid:105) . Then, the Hamiltonian H SA ( t ) = − iU ( t ) ˙ U † ( t ), which denotes the shortcut to the adiabatic Hamiltonian H ( t ), evolves the system to its instantaneouseigenlevel | ψ ( t ) (cid:105) = e i (cid:82) t θ k ( ξ ) dξ | k t (cid:105) . Explicitly, we write H SA ( t ) as ( (cid:126) = 1) H SA ( t ) = i (cid:88) n ( | ˙ n t (cid:105)(cid:104) n t | + iθ n ( t ) | n t (cid:105)(cid:104) n t | ) . (2)The functions θ n ( t ) have originally been identified with the adiabatic phase θ n ( t ) = − E n ( t ) + i (cid:104) n t | ˙ n t (cid:105) [41], which exactly mimics an adiabatic evolution.In this particular case, we can write H SA ( t ) as H ( t ) + H CD ( t ), where H ( t ) = (cid:80) n E n ( t ) | n t (cid:105)(cid:104) n t | is the Hamiltonian that drives the adiabatic dynamics and H CD ( t ) = i (cid:80) n ( | ˙ n t (cid:105) (cid:104) n t | + (cid:104) ˙ n t | n t (cid:105) | n t (cid:105) (cid:104) n t | ) is the counter-diabatic Hamiltonian. eneralized Shortcuts to Adiabaticity There is a number of situations for which we need not exactly mimic an adiabatic process,but only assure that the system is kept in an instantaneous eigenstate (independentlyof its associated quantum phase) [23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 37, 42, 43]. Thisgeneralized dynamics in terms of arbitrary phases θ n ( t ) will be denoted as a transitionless evolution. Now, we will show that θ n ( t ) can be nontrivially optimized in transitionlessevolutions both in terms of energy cost and robustness against decoherence effects. Inthis direction, we adopt as a measure of energy cost the average Hilbert-Schmidt normof the Hamiltonian throughout the evolution, which is given by [44, 23, 45, 46]Σ ( τ ) = 1 τ (cid:90) τ || H ( t ) || dt = 1 τ (cid:90) τ (cid:112) Tr [ H ( t )] dt , (3)where τ denotes the total evolution time. The energy cost Σ ( τ ) aims at identifyingchanges in the energy coupling constants and gap structure, which typically accountfor the effort of speeding up adiabatic processes. It is a well-defined measure for finite-dimensional Hamiltonians exhibiting non-degeneracies in their energy spectra. SuchHamiltonians describe the quantum systems within the scope of this work. Therefore,Eq. (3) is applicable, e.g., for generic systems composed of a finite number of quantumbits (qubits) under magnetic or electric fields (see Refs. [46, 44] for similar costmeasures). Note that Eq. (3) is non-invariant with respect to a change of the zeroenergy offset. However, by adopting a fixed reference frame, it can be used to quantifythe energy cost involved in attempts of accelerating the adiabatic path, either via anincreasing of the energy gap in the adiabatic approach or via a reduction of τ by adjustingthe relevant energy couplings in the counter-diabatic theory. In addition, as τ can be setby the quantum speed limit [47], Eq. (3) allows us to establish a trade-off between speedand energy cost for an arbitrary dynamics [23, 45]. For instance, in NMR experimentalsetups, the quantity || H ( s ) || represents how intense a magnetic field (cid:126)B ( s ) is expectedto be in order to control the speed of such a dynamics.Now, let us discuss how, for a fixed time τ , the energy cost in a transitionlessevolution can be minimized by a suitable choice of the arbitrary parameters θ n ( t ).Remarkably, this optimization can be analytically derived, which is established byTheorem 1 below. Its derivation is provided in Appendix A. Theorem 1.
Consider a closed quantum system under adiabatic evolution governedby a Hamiltonian H ( t ) . The energy cost to implement its generalized transitionlesscounterpart, driven by the Hamiltonian H SA ( t ) , can be minimized by setting θ n ( t ) = θ min n ( t ) = − i (cid:104) ˙ n t | n t (cid:105) . (4)In particular, for any evolution such that the quantum parallel-transport conditionis verified [41], the energy cost to implement a transitionless evolutions is alwaysoptimized by choosing θ minn ( t ) = 0. This approach is useful for providing both realisticand energetically optimal Hamiltonians in several physical scenarios. For example, by eneralized Shortcuts to Adiabaticity (cid:126)B ( t ) in a nuclear magnetic resonancesetup, the energy cost can be optimized by adjusting θ minn ( t ) such that the magnitude B ( t ) of the magnetic field is reduced, since || H ( t ) || ∝ B ( t ).As a by-product, the generalized counter-diabatic theory can be used as a toolto yield time-independent Hamiltonians for transitionless evolutions. In general, theHamiltonian H SA ( t ) has its form constrained both by the choice of the phases θ n ( t )and by eigenstates of the adiabatic Hamiltonian H ( t ). Thus, we can delineate underwhat conditions we can choose the set { θ n ( t ) } in order to obtain a time-independentHamiltonian for a transitionless evolution. To answer this question, we impose ˙ H SA ( t ) =0 considering arbitrary phases θ n ( t ). This leads to Theorem 2 below. Its derivation isprovided in Appendix B. Theorem 2.
Let H ( t ) be a discrete quantum Hamiltonian, with {| m t (cid:105)} denoting itsset of instantaneous eigenstates. If {| m t (cid:105)} satisfies (cid:104) k t | ˙ m t (cid:105) = c km , with c km complexconstants ∀ k, m , then a family of time-independent Hamiltonians H { θ } for generalizedtransitionless evolutions can be defined by setting θ m ( t ) = θ , with θ a single arbitraryreal constant ∀ m .2.2. Transitionless dynamics under decoherence Theorems 1 and 2 ensure both an energetically optimal counter-diabatic evolution andfamilies of possible time-independent transitionless Hamiltonians. A rather importantpoint for the generalized counter-diabatic theory is whether it is robust againstdecoherence. The robustness of the counter-diabatic dynamics and inverse engineeringschemes has recently been considered in the literature [48, 49, 50, 51]. Here, in orderto provide a comparison between adiabatic and generalized counter-diabatic dynamics,we will require identical energy resources for each implementation. More specifically, wewill consider the performance of transitionless evolutions in open systems described byconvolutionless master equations given by d s ρ ( s ) = − iτ [ H SA ( s ) , ρ ( s )] + τ L i [ ρ ( s )] , (5)where L i [ ρ ( s )] describes the decohering contribution to the quantum dynamics,which is parametrized by the normalized time s = t/τ , with τ the total time ofevolution and 0 ≤ s ≤
1. For Markovian evolution [52, 53], we have L i [ ρ ( s )] = (cid:80) i γ i ( s ) [2 L i ( s ) ρ ( s ) L † i ( s ) − { L † i ( s ) L i ( s ) , ρ ( s ) } ], with L i ( s ) denoting Lindbladoperators and γ i ( s ) (positive) decoherence rates. Here, we will consider, as anillustration, Lindblad operators for generalized amplitude damping (GAD) in theeigenbasis of the Hamiltonian, which reads L GAD ± ( s ) = U † ( s ) σ ± U ( s ) , (6)where U ( s ) is the unitary operator that diagonalizes the Hamiltonian and σ ± =( σ x ∓ iσ y ) /
2, with { σ x , σ y , σ z } denoting Pauli matrices. The GAD channel describesdissipation to an environment at finite temperature. Its decoherence rates γ + and γ − eneralized Shortcuts to Adiabaticity γ + = √ γ N th and γ − = (cid:112) γ ( N th + 1), where γ is the spontaneousemission rate and N th is the Planck distribution that gives the number of thermalphotons at a fixed frequency. For simplicity, we adjust the temperature such that N th = 1 / γ ≡ αω r , with α a dimensionless parameter and ω r some relevantfrequency associated with the quantum system. Then, we obtain γ + = (cid:112) αω r / γ − = √ γ + . In addition to GAD, we will also consider dephasing in the instantaneousHamiltonian eigenbasis, whose Lindblad operator reads L d ( s ) = U † ( s ) σ z U ( s ) , (7)with decoherence rate given by γ d ≡ αω r . Both GAD and dephasing are commondecohering processes in a number of physical realizations [53]. They will be used hereas probes to the counter-diabatic robustness in the open-system realm.In this paper we consider that any systematic error due to experimental deviationsof fields used to implement the Hamiltonian is negligible. In general, finding an optimaltransitionless scheme against arbitrary systematic errors is not a trivial task [56]. Inparticular, given a fixed class of error, we can obtain an optimal transitionless dynamicsfor such a class, but the associated dynamics may be not robust against other classes ofsystematic errors [56, 57, 58].
3. Transitionless dynamics in the Landau-Zener model
As a first application, let us consider the dynamics of a two-level quantum system, i.e.,a qubit, evolving under the Landau-Zener Hamiltonian H LZ0 ( s ) = − ω [ σ z + tan ϑ ( s ) σ x ] , (8)with tan ϑ ( s ) a dimensionless time-dependent parameter associated with Rabifrequency. This Hamiltonian describes transitions in two-level systems exhibiting anti-crossings in its energy spectrum [59]. In particular, it can be applied, e.g., to performadiabatic population transfer in a two-level system driven by a chirped field [17] andto investigate molecular collision processes [60]. The instantaneous ground | E − ( s ) (cid:105) andfirst excited | E + ( s ) (cid:105) states of H LZ0 ( s ) are | E − ( s ) (cid:105) = cos (cid:20) ϑ ( s )2 (cid:21) | (cid:105) + sin (cid:20) ϑ ( s )2 (cid:21) | (cid:105) , (9) | E + ( s ) (cid:105) = − sin (cid:20) ϑ ( s )2 (cid:21) | (cid:105) + cos (cid:20) ϑ ( s )2 (cid:21) | (cid:105) . (10)The system is initialized in the ground state | E − (0) (cid:105) = | (cid:105) of H LZ0 (0). Byconsidering a unitary dynamics and a sufficiently large total evolution time (adiabatictime), the qubit evolves to the instantaneous ground state | E − ( s ) (cid:105) of H LZ0 ( s ). In this section we will discuss the generalized transitionless dynamics theory forthe Landau-Zener model. For optimal energy cost, Eq. (4) establishes θ n ( t ) = eneralized Shortcuts to Adiabaticity (cid:104) d s E ± ( s ) | E ± ( s ) (cid:105) = 0 for the states in Eqs. (9) and (10). Therefore, the optimalHamiltonian H SA ( s ) is given by H SA ( s ) = H LZCD ( s ), with H LZCD ( s ) = i (cid:88) k = ± | d s E k ( s ) (cid:105)(cid:104) E k ( s ) | = d s ϑ ( s )2 τ σ y . (11)From Eq. (11), we can see that H LZ0 ( s ) satisfies the hypotheses of Theorem 2 if, andonly if, we choose the linear interpolation ϑ ( s ) = ϑ s . Thus, we adopt this choice forsimplicity and, consequently, we have H SA ( s ) = ( ϑ / τ ) σ y . We observe that a complete(avoided) level crossing picture for the Landau-Zener model is described by varying theparameter tan ϑ ( s ) from −∞ to + ∞ . Here, we are taking a narrower range for tan ϑ ( s ),which simplifies the description of the transitionless dynamics for the model. Now we will be interested in the performance of the transitionless evolution with optimalenergy resource so that we impose θ n ( s ) as in the Theorem 1. Thus, by consideringadiabatic evolution through the Hamiltonian H LZ0 ( s ), Theorem 1 establishes that theoptimal energy resource is performed by setting θ n ( s ) = 0. Considering the energy costas provided by Eq. (3), we getΣ Ad ( τ ) = √ | ω | (cid:90) | sec[ ϑ ( s )] | ds , (12)Σ SA ( τ ) = (cid:90) | d s ϑ ( s ) |√ τ ds = | ϑ (1) |√ τ . (13)where Σ Ad and Σ SA are the energy costs for the adiabatic and optimal shortcutHamiltonians, respectively. Remarkably, from Eqs. (12) and (13), it follows that theenergy cost for the counter-diabatic Landau-Zener model is independent of the pathfollowed by the system on the Bloch sphere, while its adiabatic counterpart dependson it. In particular, for obtaining Σ SA ( τ ), we have used tan ϑ (0) = 0 and, therefore, ϑ (0) = 0. Note that there is a range of values for τ for which the energy cost ofthe generalized transitionless dynamics is less than its adiabatic version. Indeed, byevaluating the relation between Σ Ad ( τ ) and Σ SA ( τ ) we get R ( τ ) = Σ Ad ( τ )Σ SA ( τ ) = | ω | τ (cid:82) | sec[ ϑ ( s )] | ds | ϑ (1) | . (14)By imposing identical energy cost, i.e. R ( τ ) = 1, we obtain | ω | τ = | ϑ (1) | (cid:82) | sec[ ϑ ( s )] | ds . (15)Therefore, identical energy cost can be obtained by adjusting ω according to the totalevolution time τ , as in Eq. (14). eneralized Shortcuts to Adiabaticity -4 -3 -2 -1 → Adiabatic counterparts ← Transitionlessmodels F ( τ ) τ [ ω -1 ] α = 0.00 α = 0.05 α = 0.10 α = 0.15 (a) -4 -3 -2 -1 → Transitionless ← Adiabatic counterpartsmodels F ( τ ) τ [ ω -1 ] α = 0.00 α = 0.05 α = 0.10 α = 0.15 (b) -2 -1 → Transitionless → Adiabatic → Energeticboundary line modelscounterparts F ( τ ω ) τω α = 0.00 α = 0.05 α = 0.10 α = 0.15 (c) -2 -1 → Transitionless models → Adiabatic counterparts → Energeticboundary line F ( τ ω ) τω α = 0.00 α = 0.05 α = 0.10 α = 0.15 (d) Figure 1. (Color online) Fidelity F ( τ ) under decoherence in the eigenstate basis forboth adiabatic (solid curves) and optimal transitionless dynamics (dashed curves) inthe Landau-Zener model. Left column: Fidelity F ( τ ) under GAD for (1a) identicaland (1c) different energy resources. Right column: Fidelity F ( τ ) under dephasing for(1b) identical and (1d) different energy resources. The vertical dashed line in (1c) and(1d) represents the boundary line between Σ Ad ( τ ) (cid:54) Σ SA ( τ ) and Σ Ad ( τ ) (cid:62) Σ SA ( τ ).We set ϑ = π/ We are now ready to compare the behavior under decoherence of both transitionlessand adiabatic models. The system is prepared in the ground state | E − (0) (cid:105) = | (cid:105) of theLandau-Zener Hamiltonian H LZ0 ( s ) at s = 0. Then, we let the system evolve aimingat the target state | E − (1) (cid:105) . We adopt the fidelity F ( τ ) = (cid:112) (cid:104) E − (1) | ρ (1) | E − (1) (cid:105) as a success measure of each protocol, with ρ (1) denoting the solution of Eq. (5) at s = 1. To settle the problem in a fair scenario, we shall submit both models to the samerequirements of energy cost and total evolution time τ . The robustness of adiabatic andoptimal transitionless evolutions under GAD and dephasing, for the same and differentenergetic resources, are shown in Fig. 1. To both situations the decoherence rate strengthis controlled by the dimensionless parameter α . For equal energy resource provided for eneralized Shortcuts to Adiabaticity ω r will be taken as follows. Weconsider a set { τ i | ≤ i ≤ n } of total evolution times. The total time τ i fixes the energyof the generalized transitionless evolution, with faster evolutions related to shorter times.For a given τ i , we adjust the corresponding frequency ω i of the Hamiltonian that drivesthe adiabatic evolution so that Σ Ad ( τ i ) = Σ SA ( τ i ), with Σ Ad ( τ i ) denoting the energy costof the adiabatic model. The relevant frequency ω r that sets the decoherence rates γ ± will then be defined by the average of ω i for the values of τ i considered. More specifically, ω r ≡ n (cid:80) ni =1 ω i , with n = 200 in our numerical treatment.By considering the situation of identical energy cost [see Figs. (1a) and (1b)], thefidelity for unitary dynamics ( α = 0) in the adiabatic model is constant and smallerthan one. This is because of the requirement of fixed energy given by Eq. (15), whichimposes a fixed relationship between τ and ω . The relation between τ and ω keeps theadiabatic condition unchanged as we increase τ , since we will have to decrease ω at thesame pace. On the other hand, transitionless evolutions have fidelity close to 1, sincethey are not ruled by the adiabatic constraint. For non-unitary evolutions ( α > α within a range of values for τ . Note also that, for the GAD channelin Fig. (1a), fidelity decreases for intermediate times due to the population of excitedstates in a thermal environment and then is favored for long times due to the spontaneousemission effect in the energy eigenbasis. In particular, it approximates to the adiabaticfidelity for closed systems for τ → ∞ . Remarkably, the fidelity of the adiabatic curvesincreases under dephasing in the eigenstate basis, as shown in Fig. (1b). For this specificcase, this occurs due to the fact that the ground eigenprojection | E − ( s ) (cid:105)(cid:104) E − ( s ) | is aneigenstate of the Lindblad superoperator, which governs the adiabatic approximation inopen quantum systems [15, 16]. Since adiabaticity is governed by the eigenvalue scale ofthe Lindblad superoperator instead of the Hamiltonian eigenvalue scale, Eq. (15) doesnot prevent the increase of the adiabatic fidelity as it happens in the closed case. Indeed,decoherence enhances adiabaticity in this situation.Similar results are also shown in Figs. (1c) and (1d), where we allow for differentresource contents. For this case, the relevant frequency is simply adopted as ω r ≡ ω .Observe that the behavior of the fidelity curve on the right and left hand side of thevertical line shows that, even for more energy provided for the adiabatic model, thetransitionless dynamics can be more robust than the adiabatic dynamics for a fixed α .Therefore, generalized transitionless evolutions can be more robust in a real open-systemscenario even in situations for which the adiabatic implementation has more energyresource available. For all situations considered in Fig. 1, the crossing points delimit thesupremacy region of the optimal transitionless dynamics. This region depends of thecoupling strength between the qubit and its reservoir (as measured by the parameter α ). Therefore, in general, the advantage of the optimal transitionless evolution is anon-trivial problem, which depends on both the decoherence channel and the couplingstrength with the reservoir.We observe that the generalized counter-diabatic theory can be shown to be more eneralized Shortcuts to Adiabaticity N th = 1 / α in Fig. 1 already indicate that the advantageholds for distinct temperature regimes. More specifically, provided the expression forthe parameters γ + and γ − in terms both of the decoherence rate γ and the temperature(which is implicit in N th ), we can think of the different values for the parameter α either as a change in the decoherence rate γ (keeping N th fixed) or as a change in thetemperature parameter N th (keeping γ fixed). Therefore, different values of α can betaken as yielded by a change in the temperature of the bath.
4. Transitionless dynamics in the counter-diabatic gate model
Shortcuts to adiabaticity can be used to speed up adiabatic quantum gates. Morespecifically, they have been applied to perform universal quantum computation (QC)via either counter-diabatic controlled evolutions [23] or counter-diabatic quantumteleportation [24]. As hybrid models, these approaches provide a convenient digitalarchitecture for physical realizations while potentially keeping both the generalityand some inherently robustness of analog implementations. Experimentally, digitizedimplementations of quantum annealing processes have been recently provided [61], withcontrolled quantum gates adiabatically realized with high fidelity via superconductingqubits [62]. In this Section, by focusing on controlled evolutions, we will now showthat counter-diabatic QC can be more robust against decoherence than its adiabaticcounterpart as long as the gate runtime is suitably determined within a range of evolutiontimes.
Consider a bipartite system composed by a target subsystem T and an auxiliarysubsystem A , whose individual Hilbert spaces H T and H A have dimensions d T and d A ,respectively. The auxiliary subsystem A will be driven by a family of time-dependentHamiltonians { H k ( s ) } , with 0 ≤ k ≤ d T −
1. The target subsystem will be evolvedby a complete set { P k } of orthogonal projectors over T , which satisfy P k P m = δ km P k and (cid:80) k P k = . In a controlled adiabatic dynamics, the composite system T A will begoverned by a Hamiltonian in the form [63] H ( s ) = (cid:88) k P k ⊗ H k ( s ) , (16)with H k ( s ) = g ( s ) H ( f ) k + f ( s ) H ( b ) , where H ( b ) is the beginning Hamiltonian, H ( f ) k isthe contribution k to the final Hamiltonian, and the time-dependent functions f ( s ) and g ( s ) satisfy the boundary conditions f (0) = g (1) = 1 and g (0) = f (1) = 0.Suppose now we prepare T A in the initial state | Ψ init (cid:105) = | ψ (cid:105) ⊗ | ε b (cid:105) , where | ψ (cid:105) isan arbitrary state of T and | ε b (cid:105) is the (non-degenerate) ground state of H ( b ) . Then | Ψ init (cid:105) is the ground state of the initial Hamiltonian ⊗ H ( b ) . By applying the adiabatic eneralized Shortcuts to Adiabaticity H ( t ) will drive the system (upto a phase) to the final state | Ψ final (cid:105) = (cid:88) k P k | ψ (cid:105) ⊗ | ε k (cid:105) , (17)where | ε k (cid:105) is the ground state of H ( f ) k [63]. Note that an arbitrary projection P k overthe unknown state | ψ (cid:105) can be yielded by performing a convenient measurement over A . In particular, as will be shown in Subsection 4.2, by suitably designing the auxiliaryHamiltonians H k ( s ), such a dynamics can be used to adiabatically implement individualquantum gates.The counter-diabatic version of this controlled evolution has been built in Ref. [23],where it is shown that the transitionless Hamiltonian for the composite system T A reads H SA ( s ) = (cid:88) k P k ⊗ H SA ,k ( s ) , (18)where H SA ,k ( s ) is the piecewise Hamiltonian implementing the shortcut to adiabaticityfor the controlled dynamics. Universal sets of quantum gates can be implemented through a bipartite system
T A composed by a target subsystem T and a single-qubit auxiliary system A . In ourprotocol, the target system works as our quantum processor, with any computationperformed on it. In others words, both the input and output state, as well as anyintermediate stage of the computation, should be encoded in the target system. Forexample, the target system for a two-qubit gate is composed by the target qubit andthe control qubit . On the other hand, the auxiliary qubit works as an ancilla qubit . Anyresult of measurements over such a qubit is not relevant for the computation result, butit is important for determining whether or not the computation has been successfullyrealized. Differently from the target system, any information encoded in the auxiliary system may be deleted after the measure. Therefore, quantum gates will be applied tothe target subsystem, as a result of a measurement performed on the auxiliary qubit.Let us begin by considering T as a single qubit and a single-qubit gate as anarbitrary rotation of angle φ around a direction ˆ n over the Bloch sphere. Under thisconsideration, the Hamiltonian that adiabatically implements such a single-qubit gatefor an arbitrary input state | ψ (cid:105) = a | (cid:105) + b | (cid:105) , with a, b ∈ C , is given by [63] H sg ( s ) = P + ⊗ H ( s ) + P − ⊗ H φ ( s ) , (19)where { P ± } is a complete set of orthogonal projectors over the Hilbert space of the targetqubit. The projectors can be parametrized as P ± = ( ± ˆ n · (cid:126)σ ) /
2, with ˆ n associated withthe direction of the target qubit on the Bloch sphere. In Eq. (19), each Hamiltonian H ξ ( s ) ( ξ = { , φ } ) acts on A , and is given by [63] H ξ ( s ) = − ω { σ z cos( ϕ s ) + sin( ϕ s )[ σ x cos ξ + σ y sin ξ ] } , (20) eneralized Shortcuts to Adiabaticity Figure 2. (Color online) Geometric representation of an arbitrary single qubitgate implemented through an adiabatic controlled evolution. Information about thequantum gate to be implemented is encoded in the angles ε and δ that set the vector | n + (cid:105) and in the angle φ that sets the Hamiltonian in Eq. (20). with ϕ denoting an arbitrary parameter that sets the success probability of obtainingthe desired state at the end of the evolution. This parameter plays a role in the energyperformance of counter-diabatic QC, with probabilistic counter-diabatic QC ( ϕ (cid:54) = π )being energetically more favorable than its deterministic ( ϕ = π ) counterpart [25]. Theprojectors { P ± } may be written in terms of two basis vectors {| n ± (cid:105)} in the Bloch sphereas { P ± } = | n ± (cid:105)(cid:104) n ± | , where | n + (cid:105) = cos( ε/ | (cid:105) + e iδ sin( ε/ | (cid:105) (21) | n − (cid:105) = − sin( ε/ | (cid:105) + e iδ cos( ε/ | (cid:105) . (22)Thus, a quantum gate is encoded as a rotation of φ around the vector | n + (cid:105) , as shownin the Fig. 2. Now, by expressing the state | ψ (cid:105) in the basis {| n ± (cid:105)} , we write | ψ (cid:105) = α | n + (cid:105) + β | n − (cid:105) , with | ˆ n ± (cid:105) being a state in the direction ˆ n and α, β ∈ C . We thenprepare the system in the initial state | Ψ(0) (cid:105) = | ψ (cid:105)| (cid:105) . Then, assuming an adiabaticdynamics, the evolved state | Ψ( s ) (cid:105) is given by the superposition | Ψ( s ) (cid:105) = α | n + (cid:105)| E − , ( s ) (cid:105) + β | n − (cid:105)| E − ,φ ( s ) (cid:105) = cos (cid:16) ϕ s (cid:17) | ψ (cid:105)| (cid:105) + sin (cid:16) ϕ s (cid:17) | ψ rot (cid:105)| (cid:105) , (23)with | ψ rot (cid:105) = α | n + (cid:105) + e iφ β | n − (cid:105) being the rotated desired state and the ground | E − ,ξ ( s ) (cid:105) and first excited | E + ,ξ ( s ) (cid:105) states of H ξ ( s ) given by | E − ,ξ ( s ) (cid:105) = cos( ϕ s/ | (cid:105) + e iξ sin( ϕ s/ | (cid:105) , (24) | E + ,ξ ( s ) (cid:105) = − sin( ϕ s/ | (cid:105) + e iξ cos( ϕ s/ | (cid:105) . (25)We observe that, due to the dynamics of the auxiliary qubit through two adiabaticpaths, there are quantum phases ϑ ( s ) and ϑ φ ( s ) accompanying the evolutions associatedwith | E − , ( s ) (cid:105) and | E − ,φ ( s ) (cid:105) , respectively. Then, relative phases should in principlebe considered in Eq. (23). However, as shown in the Ref. [63], such phases satisfy ϑ ( s ) = ϑ φ ( s ). Thus, they factorize as a global phase of the state | Ψ( s ) (cid:105) . At theend of the evolution, a measurement on the auxiliary qubit yields the rotated statewith probability sin ( ϕ /
2) and the input state with probability cos ( ϕ / eneralized Shortcuts to Adiabaticity Target system (Encoded input state)
Auxiliary qubit Entangled system Target system (Encoded output state)
Auxiliary qubit ?? ?
Measurement
QuantumDynamics SuccessfulMeasurement
Figure 3. (Color online) Protocol for a probabilistic implementation of a controlledevolution in a two-qubit state. Before the quantum evolution (either adiabatic ornonadiabatic) the input state is encoded in the target system. After the evolution, ameasurement (in computational basis) is performed on the auxiliary qubit. A successfulmeasurement corresponds to | (cid:105) . If the result is | (cid:105) , the system returns to its initialstate and a repetition of the process is required (until | (cid:105) is obtained as a result of themeasurement). computation process is therefore probabilistic , which succeeds if the auxiliary qubit endsup in the state | (cid:105) . Otherwise, the target system automatically returns to the inputstate and we simply restart the protocol. In the adiabatic scenario, the parameter ϕ can then be adjusted in order to obtain the optimal fidelity 1 by taking the limit ϕ → π ,implying in a deterministic computation.This model can be easily adapted to implement controlled single-qubit gates. Tothis end, the target system has to be increased from one qubit to two qubits, as shownin the scheme provided in Fig. 3. Here, we adopt that the single-qubit gate acts onthe target register if the state of the control register is | (cid:105) . With this convention, theHamiltonian that implements a controlled single-qubit gate is given by H cg ( s ) = ( − P , − ) ⊗ H ( s ) + P , − ⊗ H φ ( s ) , (26)where now the set the orthogonal projectors is given by P k, ± = | k (cid:105)(cid:104) k | ⊗ | n ± (cid:105)(cid:104) n ± | , where | k (cid:105) denotes the computational basis. The input state of the target system is now writtenas | ψ (cid:105) = a | (cid:105) + b | (cid:105) + c | (cid:105) + d | (cid:105) , with a, b, c, d ∈ C and | nm (cid:105) = | n (cid:105)| m (cid:105) denoting thecontrol and target register, respectively. By rewriting | ψ (cid:105) in terms of the basis | n ± (cid:105) ,we have | ψ (cid:105) = α | n + (cid:105) + β | n − (cid:105) + γ | n + (cid:105) + δ | n − (cid:105) , with α, β, γ, δ ∈ C . Therefore, byassuming adiabatic evolution, the system evolves from the state | Ψ (0) (cid:105) = | ψ (cid:105)| (cid:105) , tothe instantaneous state | Ψ( s ) (cid:105) = α | n + (cid:105)| E − , ( s ) (cid:105) + β | n − (cid:105)| E − , ( s ) (cid:105) + γ | n + (cid:105)| E − , ( s ) (cid:105) + δ | n − (cid:105)| E − ,φ ( s ) (cid:105) = cos (cid:16) ϕ s (cid:17) | ψ (cid:105)| (cid:105) + sin (cid:16) ϕ s (cid:17) | ψ (cid:105)| (cid:105) (27)with | ψ (cid:105) = α | n + (cid:105) + β | n − (cid:105) + γ | n + (cid:105) + e iφ δ | n − (cid:105) being the rotated desired state.We then see that the final state | Ψ(1) (cid:105) allows for a probabilistic interpretation forthe evolution and, consequently, the computation protocol can again be taken asprobabilistic ( ϕ (cid:54) = π ) or deterministic ( ϕ = π ). eneralized Shortcuts to Adiabaticity Let us now provide energetically optimal shortcuts to the adiabatic controlled dynamicspreviously introduced. The transitionless evolution for the quantum gate Hamiltonian H sg ( s ) is based on the Hamiltonian H ξ ( s ) as in Eq. (20), such that for single-qubit andcontrolled single-qubit gates we have [23] H SAsg = P + ⊗ H SA , + P − ⊗ H SA ,φ , (28) H SAcg = ( − P , − ) ⊗ H SA , + P , − ⊗ H SA ,φ , (29)respectively. Remarkably, each Hamiltonian H ξ ( s ) satisfies the conditions required byTheorems 1 and 2, so that we can obtain an optimal time-independent Hamiltonian.Thus, the generalized Hamiltonians associated with H ξ ( s ) for transitionless dynamicscan be directly derived from Eq. (2). Notice that, from Eq. (24) and (25), it is possibleshow that Theorem 2 holds, which implies in H SA ,ξ = 1 τ (cid:88) k = ± | d s E k,ξ ( s ) (cid:105)(cid:104) E k ( s ) | = ϕ τ [ σ y cos ξ − σ x sin ξ ] (30)where we have used that (cid:104) E k,ξ ( s ) | d s E k,ξ ( s ) (cid:105) = 0, with k ∈ { + , −} . The Hamiltonian inEq. (30) improves the gate Hamiltonian derived in Ref. [23]. More specifically, Eq. (30)is energetically optimal and given by a time-independent operator. -4 -3 -3 -3 ← Adiabatic → Transitionlessmodelscounterparts F ( τ ) τ [ ω -1 ] α = 0.00 α = 0.05 α = 0.10 α = 0.15 (a) -4 -3 -3 -3 ← Adiabatic → Transitionlessmodelscounterparts F ( τ ) τ [ ω -1 ] α = 0.00 α = 0.05 α = 0.10 α = 0.15 -4 -3 -3 -3 F ( τ ) τ [ ω -1 ] -4 -3 -3 -3 F ( τ ) τ [ ω -1 ] (b) Figure 4. (Color online) Fidelity F ( τ ) for the implementation of (4a) a CNOT gateto the state | + (cid:105)| (cid:105) and (4b) single qubit gates, provided by ( top ) a Hadamard gate tothe state | (cid:105) , ( bottom left ) a phase gate to the state | + (cid:105) , and ( bottom right ) a π -gateto the state | + (cid:105) . The gates are implemented via deterministic ( ϕ = π ) adiabatic QC(solid curves) and probabilistic ( ϕ ≈ . π ) counter-diabatic QC (dashed curves),for unitary and non-unitary evolutions under dephasing for identical energy resources. eneralized Shortcuts to Adiabaticity For a transitionless evolution, it is possible to show that a probabilistic process, with ϕ (cid:54) = π , is energetically better than the deterministic approach ϕ = π [25]. For thisreason, we will consider here the probabilistic model for the generalized counter-diabaticquantum gates. In this scenario, given a fixed amount of energy resource available, ouraim is to compare the best adiabatic protocol to implement quantum gates with its bestgeneralized transitionless counterpart. From Eq. (28), we can write the energy cost ofa single evolution to implement single-qubit gatesΣ SA,sg ( τ, ϕ ) = ϕ ωτ Σ sg , (31)where Σ sg = 2 ω corresponds to the adiabatic energy cost Σ Ad,sg ( τ ) and ϕ is the freeangle parameter. The energy cost of probabilistic optimal transitionless evolutions canbe obtained by defining the quantity (cid:104) N (cid:105) ≡ ( ϕ /
2) , (32)which is the average number of evolutions for a successful computation. Thus, theaverage energy cost to implement a probabilistic evolution is [25]¯Σ
SA,sg ( τ, ϕ ) = (cid:104) N (cid:105) Σ SA,sg ( τ, ϕ )= ϕ ωτ csc ( ϕ /
2) Σ sg . (33)Hence, an optimal scheme requires that the choice of ϕ is such that ¯Σ SA,sg ( τ, ϕ ) isminimized. In particular, this minimization is obtained for ϕ ≈ . π . It is importantmention that the energy cost in Eq. (33) is obtained by two processes: i) energyminimization through the quantum phase θ n ( t ) that accompanies the transitionlessevolution and ii) application of the probabilistic model of quantum gates.By considering the energy rate R ( τ, ϕ ) for adiabatic and generalized transitionlessprotocols [similarly as in Eq. (14)], we have R ( τ, ϕ ) = Σ Ad,sg ( τ )¯Σ SA,sg ( τ, ϕ ) = ωτϕ sin ( ϕ /
2) . (34)By imposing identical energy resource, i.e. R ( τ, ϕ ) = 1, we obtain ωτ = ϕ csc ( ϕ /
2) . (35)Remarkably, the energy cost for the implementation of a controlled single-qubit gate bythe transitionless Hamiltonian in Eq. (29) is simply Σ
SA,cg = √ SA,sg [23, 25]. Thefactor √ R ( τ, ϕ ) and therefore in the same constraint over ωτ provided by Eq. (35). eneralized Shortcuts to Adiabaticity From Eq. (34), the optimal transitionless quantum gate model will be more efficient fromthe energy point of view than its adiabatic counterpart for ωτ (cid:62) ϕ csc ( ϕ / F ( τ ) = (cid:112) (cid:104) ψ rot | ρ (1) | ψ rot (cid:105) ,with ρ (1) denoting the density operator for the target subsystem, obtained from Eq. (5). -3 -2 -1 → Adiabatic → Transitionlessmodels ← Energeticboundarycounterparts F ( τ ) τ [ ω -1 ] α = 0.00 α = 0.05 α = 0.10 α = 0.15 (a) -3 -2 -1 → Adiabatic → Transitionlessmodels ← Energeticboundarycounterparts F ( τ ) τ [ ω -1 ] α = 0.00 α = 0.05 α = 0.10 α = 0.15 -3 -2 -1 F ( τ ) τ [ ω -1 ] -3 -2 -1 F ( τ ) τ [ ω -1 ] (b) Figure 5. (Color online) Fidelity F ( τ ) for the implementation of (5a) a CNOT gateto the state | + (cid:105)| (cid:105) and (5b) single qubit gates, provided by ( top ) a Hadamard gate tothe state | (cid:105) , ( bottom left ) a phase gate to the state | + (cid:105) , and ( bottom right ) a π -gateto the state | + (cid:105) .The gates are implemented via deterministic ( ϕ = π ) adiabatic QC(solid curves) and probabilistic ( ϕ ≈ . π ) counter-diabatic QC (dashed curves),for unitary and non-unitary evolutions under dephasing for different energy resources. The Hadamard gate is a rotation of π/ y in the Bloch sphere,so that we set φ Had = π/ ε Had = δ Had = π/ π gates, we take them as rotations around the direction z of an angle π and π/
4, respectively, so that we take ε pha = ε π = 0, φ pha = π , and φ π = π/ σ x (flipgate), which can be viewed as a rotation of π around the x direction. Thus we set ε CNOT = π/ δ CNOT = 0, and φ CNOT = π . Therefore, from Eq. (30), we have the eneralized Shortcuts to Adiabaticity -4 -3 -3 -3 ← Adiabatic → Transitionlessmodelscounterparts F ( τ ) τ [ ω -1 ] α = 0.00 α = 0.05 α = 0.10 α = 0.15 (a) -4 -3 -3 -3 ← Adiabatic → Transitionlessmodelscounterparts F ( τ ) τ [ ω -1 ] α = 0.00 α = 0.05 α = 0.10 α = 0.15 -4 -3 -3 -3 F ( τ ) τ [ ω -1 ] -4 -3 -3 -3 F ( τ ) τ [ ω -1 ] (b) Figure 6. (Color online) Fidelity F ( τ ) for the implementation of (6a) a CNOT gateto the state | + (cid:105)| (cid:105) and (6b) single qubit gates, provided by ( top ) a Hadamard gate tothe state | (cid:105) , ( bottom left ) a phase gate to the state | + (cid:105) , and ( bottom right ) a π -gateto the state | + (cid:105) . The gates are implemented via deterministic ( ϕ = π ) adiabatic QC(solid curves) and probabilistic ( ϕ ≈ . π ) counter-diabatic QC (dashed curves),for unitary and non-unitary evolutions under GAD for identical energy resources. following counter-diabatic Hamiltonians H CD,0 = (cid:126) ϕ τ σ y , H CD , π = − (cid:126) ϕ τ σ x , (36) H CD,0 = (cid:126) ϕ τ σ y , H CD ,π = − (cid:126) ϕ τ σ y , (37) H CD,0 = (cid:126) ϕ τ σ y , H CD , π = (cid:126) ϕ √ τ ( σ y − σ x ) , (38) H CD,0 = (cid:126) ϕ τ σ y , H CD ,π = − (cid:126) ϕ τ σ y , (39)for Hadamard, phase, π , and CNOT gates, respectively. In order to study the robustnessof single qubit gates we have considered the input stats | ψ Had (cid:105) = | (cid:105) for Hadamardoperation and | ψ pha (cid:105) = | + (cid:105) = (1 / √ | (cid:105) + | (cid:105) ) for phase and π -gate. On the otherhand, for the CNOT gate, we consider the initial state | ψ (0) (cid:105) = | + (cid:105)| (cid:105) and apply thegate Hamiltonian to create a Bell state | ψ (cid:105) = (1 / √ | (cid:105) + | (cid:105) ). Fidelity is thenobtained from the explicit solution of the Lindblad equation for ρ (1). For instance, forCNOT, we have F ( τ ) = (cid:112) (cid:104) ψ | ρ (1) | ψ (cid:105) .The robustness of the universal set of quantum gate under dephasing is illustratedin Figs. 4 and 5 for identical and different resources imposed, respectively. In theseplots, we compare the optimal adiabatic (deterministic) implementation with its optimaltransitionless version (probabilistic computation and optimal quantum phases). Notethat the generalized transitionless approach shows a higher fidelity for fast dynamics, butthere are regimes for which the adiabatic approach shows a better fidelity for a fixed α .Energy optimization in the optimal transitionless model is achieved for ϕ ≈ . π [25].Figs. 6 and 7 show similar results result for non-unitary evolution under GAD, withequivalent and different resources provided to the adiabatic and optimal transitionless eneralized Shortcuts to Adiabaticity -3 -2 -1 → Adiabatic → Transitionlessmodels ← Energeticboundarycounterparts F ( τ ) τ [ ω -1 ] α = 0.00 α = 0.05 α = 0.10 α = 0.15 (a) -3 -2 -1 → Adiabatic → Transitionlessmodels ← Energeticboundarycounterparts F ( τ ) τ [ ω -1 ] α = 0.00 α = 0.05 α = 0.10 α = 0.15 -3 -2 -1 F ( τ ) τ [ ω -1 ] -3 -2 -1 F ( τ ) τ [ ω -1 ] (b) Figure 7. (Color online) Fidelity F ( τ ) for the implementation of (7a) a CNOT gateto the state | + (cid:105)| (cid:105) and (7b) single qubit gates, provided by ( top ) a Hadamard gate tothe state | (cid:105) , ( bottom left ) a phase gate to the state | + (cid:105) , and ( bottom right ) a π -gateto the state | + (cid:105) . The gates are implemented via deterministic ( ϕ = π ) adiabatic QC(solid curves) and probabilistic ( ϕ ≈ . π ) counter-diabatic QC (dashed curves),for unitary and non-unitary evolutions under GAD for different energy resources. model, respectively. In any case of decoherence and energetic resource, there alwaysexist dynamical regimes for which the optimal transitionless evolutions are more robustand therefore a preferred approach in a decohering physical environment.
5. Conclusion
In summary, we have developed a generalized minimal energy demanding counter-diabatic theory, which is able to yield efficient shortcuts to adiabaticity via fasttransitionless evolutions. Moreover, we have investigated the robustness of adiabaticand counter-diabatic dynamics under decoherence by introducing the requirement offixed energy resources, so that a comparison is settled down in a fair scenario. Then, wehave shown both for the Landau-Zener model and for quantum gate Hamiltonians thatthere always exist dynamical regimes for which generalized transitionless evolutionsare more robust and therefore a preferred approach in a decohering setup. This hasbeen shown both for the dephasing and GAD channels acting on the eigenstate bases.It is also possible to show the advantage in other bases, such as the computationalbasis. The general picture is that the gain will typically occur during some finitetime range, disappearing in the limit of long evolution times. These results areencouraging for the generalized transitionless approach in the open-system realm as longas local Hamiltonians are possible to be designed. In the specific case of quantum gateHamiltonians, this approach can be applied, e.g. to derive robust local building blocksfor analog implementations of quantum circuits (see, e.g., Refs. [62, 61]). Experimentalrealizations, extensions for dealing with systematic errors, and generalized shortcuts viareservoir engineering are further directions left for future research. eneralized Shortcuts to Adiabaticity Acknowledgments
We acknowledge Gonzalo Muga and Tameem Albash for useful discussions. A.C.S.is supported by CNPq-Brazil. M.S.S. acknowledges support from CNPq-Brazil (No.303070/2016-1), FAPERJ (No. 203036/2016), and the Brazilian National Institute forScience and Technology of Quantum Information (INCT-IQ).
Appendix A. Proof of Theorem 1Theorem 1.
Consider a closed quantum system under adiabatic evolution governedby a Hamiltonian H ( t ) . The energy cost to implement its generalized transitionlesscounterpart, driven by the Hamiltonian H SA ( t ) , can be minimized by setting θ n ( t ) = θ min n ( t ) = − i (cid:104) ˙ n t | n t (cid:105) .Proof. We adopt as a measure of energy cost the Hamiltonian Hilbert-Schmidt norm,which reads Σ SA ( τ ) = 1 τ (cid:90) τ (cid:113) Tr (cid:2) H SA2 ( t ) (cid:3) dt , (A.1)Then, we obtain H ( t ) = (cid:125) (cid:88) n (cid:104) | ˙ n t (cid:105)(cid:104) ˙ n t | + θ n ( t ) | n t (cid:105)(cid:104) n t | + iθ n ( t ) ( | n t (cid:105)(cid:104) ˙ n t | − | ˙ n t (cid:105)(cid:104) n t | ]) (cid:105) . (A.2)By taking the trace of H ( t ) in Eq. (A.2), we haveTr (cid:2) H ( t ) (cid:3) = (cid:88) m (cid:104) m t | H ( t ) | m t (cid:105) = (cid:125) (cid:88) n (cid:104) (cid:104) ˙ n t | ˙ n t (cid:105) + θ n ( t ) + 2 iθ n ( t ) (cid:104) ˙ n t | n t (cid:105) (cid:105) . (A.3)Then Σ SA ( τ ) = 1 τ (cid:90) τ (cid:114)(cid:88) n (cid:104) ˙ n t | ˙ n t (cid:105) + Γ n ( θ n ) dt , (A.4)where we have Γ n ( θ n ) = θ n ( t ) + 2 iθ n ( t ) (cid:104) ˙ n t | n t (cid:105) .We can now find out the functions θ n ( t ) that minimize the energy cost intransitionless evolutions. For this end, we minimize the quantity Σ SA ( τ ) for theHamiltonian H SA ( t ) with respect to parameters θ n ( t ), where we will adopt it beingindependents. By evaluating ∂ θ n Σ ( τ ), we obtain ∂ θ n Σ SA ( τ ) = 12 τ (cid:90) τ ∂ θ n { Tr[ H ( t )] } (cid:112) Tr [ H ( t )] dt . (A.5)We then impose ∂ θ n { Tr[ H ( t )] } = 0 for all time t ∈ [0 , τ ], which ensures ∂ θ n Σ SA ( τ ) =0. Thus, by using Eq. (A.3), we write ∂ θ n { Tr[ H ( t )] } = 2 θ n ( t ) + 2 i (cid:104) ˙ n t | n t (cid:105) = 0 . (A.6) eneralized Shortcuts to Adiabaticity θ n ( t ) = θ min n ( t ) = − i (cid:104) ˙ n t | n t (cid:105) . (A.7)From the second derivative analysis, it follows that the choice for θ n ( t ) as in Eq. (A.7)necessarily minimizes the energy cost, namely, ∂ θ n Σ SA ( τ ) | θ n = θ min n >
0, which concludesthe proof.
Appendix B. Proof of Theorem 2Theorem 2.
Let H ( t ) be a discrete quantum Hamiltonian, with {| m t (cid:105)} denoting itsset of instantaneous eigenstates. If {| m t (cid:105)} satisfies (cid:104) k t | ˙ m t (cid:105) = c km , with c km complexconstants ∀ k, m , then a family of time-independent Hamiltonians H { θ } for generalizedtransitionless evolutions can be defined by setting θ m ( t ) = θ , with θ a single arbitraryreal constant ∀ m .Proof. By taking the time derivative of the Hamiltonian H SA ( t ), we obtain˙ H SA ( t ) = i (cid:88) n ddt (cid:104) | ˙ n t (cid:105)(cid:104) n t | + iθ n ( t ) | n t (cid:105)(cid:104) n t | (cid:105) . (B.1)Then, the matrix elements of ˙ H SA ( t ) in the eigenbasis {| m t (cid:105)} of the Hamiltonian H ( t )read (cid:104) k t | ˙ H SA ( t ) | m t (cid:105) = i (cid:104) k t | ¨ m t (cid:105) + i (cid:88) n (cid:104) k t | ˙ n t (cid:105)(cid:104) ˙ n t | m t (cid:105)− (cid:104) ˙ θ k ( t ) δ km + θ m ( t ) (cid:104) k t | ˙ m t (cid:105) + θ k ( t ) (cid:104) ˙ k t | m t (cid:105) (cid:105) . Now, by using (cid:104) k t | ˙ n t (cid:105) = −(cid:104) ˙ k t | n t (cid:105) , we write (cid:104) k t | ˙ n t (cid:105)(cid:104) ˙ n t | m t (cid:105) = (cid:104) ˙ k t | n t (cid:105)(cid:104) n t | ˙ m t (cid:105) and thus (cid:104) k t | ˙ H SA ( t ) | m t (cid:105) = i ddt [ (cid:104) k t | ˙ m t (cid:105) ] − (cid:110) ˙ θ k ( t ) δ km + [ θ m ( t ) − θ k ( t )] (cid:104) k t | ˙ m t (cid:105) (cid:111) . (B.2)For k = m in Eq. (B.2), we impose the vanishing of the diagonal elements of ˙ H SA ( t ),namely, (cid:104) k t | ˙ H SA ( t ) | k t (cid:105) = 0. This yields˙ θ m ( t ) = i ddt [ (cid:104) m t | ˙ m t (cid:105) ] , (B.3)On the other hand, for k (cid:54) = m in Eq. (B.2), we now impose the vanishing of theoff-diagonal elements of ˙ H SA ( t ), namely, (cid:104) k t | ˙ H SA ( t ) | m t (cid:105) = 0 ( k (cid:54) = m ). This yields i ddt [ (cid:104) k t | ˙ m t (cid:105) ] = [ θ m ( t ) − θ k ( t )] (cid:104) k t | ˙ m t (cid:105) ( k (cid:54) = m ) . (B.4)By taking (cid:104) m t | ˙ m t (cid:105) ≡ c mm in Eq. (B.3), with c mm denoting by hypothesis complexconstants, we get θ m ( t ) = θ m (0) ≡ θ m , namely, θ m ( t ) is a constant function ∀ m .Moreover, by using (cid:104) k t | ˙ m t (cid:105) ≡ c km in Eq. (B.4), with c km denoting nonvanishing complex eneralized Shortcuts to Adiabaticity θ k = θ m , ∀ k, m . If c km = 0, then θ k and θ m are not necessarilyequal, but Eq. (B.4) will also be satisfied by this choice. Therefore, it follows that θ m ( t )can be simply taken as θ m ( t ) = θ m = θ ∀ m , (B.5)with θ a single real constant. This concludes the proof. References [1] Born M and Fock V 1928
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