Evolution-Time Dependence in Near-Adiabatic Quantum Evolutions
aa r X i v : . [ qu a n t - ph ] M a y Evolution-Time Dependence in Near-Adiabatic Quantum Evolutions
Lucas T. Brady
Department of Physics, University of California, Santa Barbara, CA 93106-5110, USA
Wim van Dam
Department of Computer Science, Department of Physics,University of California, Santa Barbara, CA 93106-5110, USA (Dated: May 7, 2018)We expand upon the standard quantum adiabatic theorem, examining the time-dependence ofquantum evolution in the near-adiabatic limit. We examine a Hamiltonian that evolves along somefixed trajectory from ˆ H to ˆ H in a total evolution-time τ , and our goal is to determine how thefinal state of the system depends on τ . If the system is initialized in a non-degenerate groundstate, the adiabatic theorem says that in the limit of large τ , the system will stay in the groundstate. We examine the near-adiabatic limit where the system evolves slowly enough that most butnot all of the final state is in the ground state, and we find that the probability of leaving theground state oscillates in τ with a frequency determined by the integral of the spectral gap alongthe trajectory of the Hamiltonian, so long as the gap is big. If the gap becomes exceedingly small,the final probability is the sum of oscillatory behavior determined by the integrals of the gap beforeand after the small gap. We confirm these analytic predictions with numerical evidence from barriertunneling problems in the context of quantum adiabatic optimization. I. INTRODUCTION
The Quantum Adiabatic Theorem [1] is a powerful toolfor analyzing dynamical quantum systems. For slowlyevolving Hamiltonians it ensures the system will closelytrack its originally initialized energy state throughout theentire evolution. The key point of the adiabatic theoremis that it also gives a condition for how slowly the systemneeds to evolve.The adiabatic theorem is notable for many applica-tions in physics and chemistry. Of specific note for ourpurposes are Quantum Adiabatic Optimization [2] andQuantum Annealing which are a class of quantum algo-rithms for solving optimization problems. These algo-rithms initialize the system in an easily prepared eigen-state and rely on the adiabatic theorem to evolve it intoa desired state under the influence of a suitably designedHamiltonian. Quantum adiabatic computing is universalfor quantum computing in general [3], but it is currentlya matter of debate [4–16] how much of that power can becaptured by the model used in many applications, suchas the D-Wave machine [17].In this study, we will focus on the near-adiabaticregime of quantum evolution and quantum annealing,where the system evolves slowly enough to mostly stay inthe desired eigenstate but with noticeable leakage. Non-adiabatic evolution has garnered much interest becauseit can potentially lead to speed-ups that the adiabatictheorem does not account for. In quantum annealing,recent work has focused on a rapid diabatic speed-up incertain barrier tunneling models [13, 16, 18].While we couch our findings in the language of quan-tum computing, our results are more general. Many ofour assumptions and approximations are general and canapply to a large number of problems and settings. Quan-tum Adiabatic Optimization relies on ground state evolu- tion, and even this condition can be relaxed in our results.In the near-adiabatic limit, our findings show that inthe absence of a small spectral gap, the probability oftransitioning out of the initialized state oscillates as afunction of the total evolution time, τ . Furthermore,we confirm previous results [19–21] that show that thefrequency of this oscillation depends on the integral ofthe spectral gap over the evolution.Our new results add a layer of depth by consideringthe case where the spectral gap becomes small duringone portion of the evolution, as would occur in a Landau-Zener avoided crossing [22, 23]. This crossing is localized,so that it only effects the system during a short periodof time. We find that the avoided crossing effectivelysplits the evolution in two, resulting in a superpositionof oscillatory behavior in τ .This article is structured with a basic overview of theQuantum Adiabatic Theorem and its context in quantumcomputing in Section II. In Section III, we develop muchof the mathematical machinery that will be used in latersections.We study the large gap regime in Section IV, exam-ining how the oscillatory transition probability behaviorarises, and we back up these results with numerical evi-dence from quantum algorithm settings. In Section V, weadd in an avoided level crossing and explore analyticallyand numerically how this leads to a more complicatedsuperposition of oscillatory behavior in total evolutiontime, τ . Section VI shows an application of the large gaposcillations to the adiabatic version of Grover’s search.Finally we review our conclusions and discuss future av-enues of study in Section VII. II. ADIABATIC THEOREM
Suppose we have some quantum system obeying theSchr¨odinger equation i ddt | ψ i = ˆ H ( t ) | ψ i , (1)where time runs between t = 0 and τ , the evolution timeof the system or the run time in a quantum computingsetting. The Hamiltonian should follow the same tra-jectory even for different values of τ , so we can use the“normalized time” s ∈ [0 ,
1] to determine where in theHamiltonian’s evolution we are.The normalized time s relates to the actual timethrough t = sτ . Since the form of the Hamiltonian de-pends only on s and not τ , we can rewrite the Schr¨odingerequation as i dds | ψ i = τ ˆ H ( s ) | ψ i . (2)Now all the evolution time information has been pulledout into one parameter, τ , which we can vary to run theevolution more slowly or quickly. A. Quantum Adiabatic Theorem
The Quantum Adiabatic Theorem is an old result firstattributed to Born and Fock [1] but not treated fully rig-orously until more recently (e.g. [19, 24]). The theoremconcerns systems with a time dependent Hamiltonian,ˆ H ( s ), that is initialized in the i th eigenstate. If the i theigenstate has a non-zero spectral gap, ∆( s ), separatingit from other eigenstates for the entire time evolution,then a sufficiently slow evolution of the Hamiltonian willkeep the system in the i th eigenstate.The key idea in the adiabatic theorem is how slowlythe system must be evolved. Specifically, how large must τ be to ensure that a significant portion of the probabilityremains in i th eigenstate. An oft quoted folklore resultis that the adiabatic theorem holds if τ ≫ Z ds |h ϕ | d ˆ Hds | ϕ i| ∆( s ) , (3)where | ϕ i i is the i th energy state.This adiabatic condition is not the full rigorous con-dition for adiabatic evolution [19, 24, 25], but it is suf-ficient for most cases, especially in quantum computing.Additionally, the more rigorous versions of the adiabaticcondition still depend polynomially on the inverse of thespectral gap, ∆( s ) − , and matrix norms of the Hamilto-nian and its s derivatives.For our purposes, we will always focus on the groundstate. Therefore, the spectral gap ∆( s ) will just be theenergy difference between the first excited state and theground state. B. Quantum Adiabatic Optimization
One major application of the quantum adiabatic theo-rem is to quantum computing. Quantum Adiabatic Op-timization ( qao ) is a quantum algorithm introduced in[2] building upon previous quantum annealing models[26, 27]. In qao , a quantum system is initialized in theground state of a simple Hamiltonian, and the Hamilto-nian is then adiabatically evolved into one with a groundstate that solves a desired computational problem. Bymeasuring the final ground state, a solution to the com-putational problem can be obtained.The original framing of qao [2] works with an initialHamiltonian that is a sum of σ x terms on n qubitsˆ H = 12 n X i =1 σ ( i ) x (4)and a final Hamiltonian which is diagonal in the com-putational, z , basis and depends on some cost function f ( z ) ˆ H = X z ∈{ , } n f ( z ) | z ih z | . (5)The goal is to find a bit string that minimizes f ( z ); there-fore, we are looking for the ground state of ˆ H . The algo-rithm linearly interpolates between the two Hamiltoniansin total time τ : ˆ H ( s ) = (1 − s ) ˆ H + s ˆ H (6) qao relies on the adiabatic theorem to keep the systemin the ground state, but quantum annealing in generalcan run this algorithm faster than the adiabatic theoremrecommends. This paper can be interpreted in termsof quantum annealing as describing how a non-adiabaticevolution effects the final success probability of the algo-rithm. We examine this success probability as a functionof τ , and we find that the success probability dependsgreatly on the spectral gap.In particular, we look at situations where the spec-tral gap remains large, except possibly in isolated regionswhere avoided-level crossings are allowed. To this end,we study two different symmetric qubit problems. n -Qubit Barrier Tunneling Model The first model we examine is one that has been stud-ied in numerous articles [2, 11–16, 18, 25, 28] with a finalHamiltonian cost function f ( z ) = µ | z | + b ( | z | ) , (7)where | z | is the Hamming weight of the bit string z . Thefunction b ( | z | ) is some localized barrier function that haswidth and height that scale with n α and n β , respectively,for some constants α and β . Depending on the values of α and β , qao can adiabatically require run times, τ ,that grow with n in constant, polynomial, or exponentialways [12, 15, 28]. In this article, we take the barrier tobe localized around | z | = n and take the shape of thebarrier to be binomial; though, neither of these choicesare particularly relevant.The problem determined by Eq. 7 has a final groundstate at | z | = 0, but in order to reach the final state, theinstantaneous state must pass through or over the barriergiven by b ( | z | ). Adiabatically this is accomplished by atunneling event, and it appears in the spectrum as anavoided level crossing between the ground state energyand the n -fold degenerate first excited state. Thus, thisproblem, exhibits a large spectral gap for most time thatis well approximated by the b ( | z | ) = 0 case, except in thevicinity of the barrier where the problem takes on theform of a Landau-Zener avoided crossing. This problemis often studied in order to extract how much tunnelingeffects qao .Since the Hamiltonian, ˆ H ( s ) is symmetric betweenqubits, the symmetric subspace fully describes the eigen-spectrum of the system. Therefore, this 2 n dimensionalHamiltonian can be simulated using an n + 1 dimensionalsystem, described by a tridiagonal matrix. This reduc-tion of the size of the system allows for efficient calcula-tion of the spectrum and other properties of the systemnumerically, allowing much larger n to be studied.We utilize this simple barrier tunneling model in qao as an example of how our near-adiabatic evolution de-pends on the evolution or run time, τ . Our analyticapproximations and results are general and independentof this specific computational problem, but we use it asa numerical example to verify our analytic results.
2. Cubic Potential Model
The second model we consider works with a final po-tential that is cubic but without an explicit barrier. Thismodel is the p = 3 case of the p -spin model that has beenused by numerous groups [29–34], and in the language ofHamming weight it is given by the final potential f ( z ) = n (cid:18) | z | n − (cid:19) . (8)This cost function still has the all-zero bit string asits ground state, so the annealing evolution should stilltake the ground state from being localized around | z | = n/ s = 0 to | z | = 0 at s = 1. This problem doesnot include a barrier in the final potential but can stillbe visualized as a barrier tunneling problem in a semi-classical large- n limit, using such methods as the Villaintransformation[28]. Notably, J¨org et al. [29] showed thatif the exponent p ≥
3, then the spectral gap becomesexponentially small in this problem. Therefore, findingthe ground state of this cost function through quantumadiabatic optimization is a difficult task. Many of the useful properties from the n -qubit bar-rier model also carry over to this system. The final costfunction remains symmetric between qubits, so symme-try simplifications can be employed to make this problemnumerically tractable to solve for large n . III. SETUP
We will start with the normalized time, s , version ofthe Schr¨odinger equation, Eq. 2. The next step involvesrewriting the equation in the eigenbasis of the Hamilto-nian. We take the instantaneous eigenbasis to be givenby | ϕ j ( s ) i with associated eigenenergies λ j ( s ). Then ageneral state of our system is written as | ψ ( s, τ ) i = X j C j ( s, τ ) | ϕ j ( s ) i . (9)In terms of the eigenbasis the Schr¨odinger equationgives i dC k ds + X j C j h ϕ k | dds | ϕ j i = τ λ k C k . (10)We want to know how the system evolves if we start thesystem in the ground state at s = 0. These same argu-ments work for higher excited states as well, but for ourpurposes, the ground state is sufficient and simpler. Theadiabatic theorem tells us that if τ is large enough, weremain in the ground state. We relax the adiabatic con-dition slightly and allow near-adiabatic evolution wherethe majority of the state remains in | ϕ ( s ) i , but somesmall amount leaks into the first excited states, with allother states being essentially unvisited.We require the ground state to be nondegenerate, butthe first excited eigenstate can be degenerate. We denotethis possible m -fold degeneracy with a superscript C ( a )1 .We want to restrict down Eq. 10 to just those probabil-ity amplitudes that are assumed to be relevant, namely,those close to the ground state. In doing this, we can re-member that h ϕ j | dds | ϕ j i = 0. We also shift the Hamilto-nian by an overall ( s -dependent) constant so that λ = 0,which means that λ = ∆ is just given by the spectralgap. Using all this information, we obtain the followingcoupled equations for the relevant amplitudes i dC ds + i m X a =1 C ( a )1 h ϕ | dds | ϕ ( a )1 i = 0 (11) i dC ( a )1 ds + iC h ϕ ( a )1 | dds | ϕ i + i X b = a C ( b )1 h ϕ ( a )1 | dds | ϕ ( b )1 i = τ ∆ C ( a )1 (12)For h ϕ ( a )1 | dds | ϕ ( b )1 i , we can freely choose our basis withinthe degenerate eigenspace, and it is possible and de-sireable to choose our degenerate eigenbasis such that h ϕ ( a )1 | dds | ϕ ( b )1 i = 0 for all a and b .Just from the definition of the eigenvalues of ˆ H ( s ) andthe orthonormality of its eigenvectors, we can look at ∂∂s (cid:16) ˆ H | ϕ i i (cid:17) = ∂∂s ( λ i | ϕ i i ) (13)˙ˆ H | ϕ i i + ˆ H ∂∂s | ϕ i i = ∂λ i ∂s | ϕ i i + λ i ∂∂s | ϕ i i . We can look at the inner product of this time derivativewith an eigenstate j = i and utilize the orthonormatilityof the eigenstates h ϕ j | ˙ˆ H | ϕ i i + h ϕ j | ˆ H ∂∂s | ϕ i i = ∂λ i ∂s h ϕ j | ϕ i i + λ i h ϕ j | ∂∂s | ϕ i ih ϕ j | ˙ˆ H | ϕ i i = ( λ i − λ j ) h ϕ j | ∂∂s | ϕ i i . (14)which means that h ϕ ( s ) | dds | ϕ ( a )1 ( s ) i = −h ϕ ( a )1 ( s ) | dds | ϕ ( s ) i (15)= h ϕ ( s ) | d ˆ Hds | ϕ ( a )1 ( s ) i ∆( s ) ≡ γ a ( s )∆( s ) . (16)Thus, our differential equations can be reduced to dC ds + m X a =1 C ( a )1 γ a ∆ = 0 (17) i dC ( a )1 ds − iC γ a ∆ = τ ∆ C ( a )1 . (18)At this point it is obvious that all the m -fold degener-ate first excited states will behave in the exact same way.Therefore, for notational convenience, we will drop the a index and make the m -fold degeneracy explicit in thesedifferential equations: dC ds + mC γ ∆ = 0 (19) i dC ds − iC γ ∆ = τ ∆ C . (20) IV. LARGE GAPA. Analytic Approximation
Now, we take the near-adiabatic limit by assuming thatthe vast majority of of the state remains in the groundstate. In this limit, we assume that C ≫ C so thatEq. 19 reduces to dC ds = 0. This reduction assumes thatthe gap ∆ does not become too small. If the gap becomessmall, then this problem can be approximated using dif-ferent techniques, such as the Landau-Zener transition in the next section, but we focus on the large gap case inthis section.Given this approximation that dC ds = 0, we assumethat it is a good approximation that C ( s ) = 1 for theentire evolution. Essentially we are saying that the ma-jority of the amplitude remains in the ground state withminimal changes to its value. With this assumption,Eq. 20 becomes i dC ds − i γ ∆ = τ ∆ C . (21)Notice that C is kept in this equation despite being dis-regarded in the C equation. It is kept both because weare now looking at the change in C itself and becausethis otherwise small term is multiplied by τ which is takento be large in the near-adiabatic limit. This differentialequation has an integral solution when C ( s = 0) = 0 C ( s, τ ) = Z s dx e − iτ R sx dz ∆( z ) ∆( x ) /γ ( x ) . (22)What we are actually going to care about is the finalamplitude after the total evolution time τ , so the quan-tity we work with is mainly C (1 , τ ) = Z ds e − iτ R s dz ∆( z ) ∆( s ) /γ ( s ) (23)Our goal is to approximate this integral in the limit oflarge τ so that we can find the probability amplitude forleaving the ground state and entering one of these excitedstates.Throughout this approximation, we need to assumethat ∆( s ) does not become exceedingly small. We canrewrite our integral as C (1 , τ ) = Z ds dds e − iτ R s dz ∆( z ) iτ ∆( s ) /γ ( s ) (24)Integration by parts yields C (1 , τ ) = " e − iτ R s dz ∆( z ) iτ ∆( s ) /γ ( s ) s =0 (25) − Z ds e − iτ R s dz ∆( z ) dds iτ ∆( s ) /γ ( s ) . By the properties of oscillatory integrals, the last integralhere is O ( τ − ), so we are left with C (1 , τ ) = 1 iτ ∆(1) /γ (1) − e − iτ R ds ∆( s ) iτ ∆(0) /γ (0) + O ( τ − ) . (26)For convenience, we will define ρ ( s ) = γ ( s ) / ∆( s ) , andthis value ρ ( s ) is related to the naive adiabatic conditionthat τ ≫ R ds | ρ ( s ) | . Therefore, this ρ ( s ) can be thoughtof as the gauge by which we can determine whether weare in the adiabatic limit. . . . . . P r ob a b ilit yo f F a il u r e τ µ = 0 . µ = 1 µ = 2 FIG. 1: The probability of transition to the first excited stateversus the evolution time τ for no barrier. The solid linesrepresent the theoretical predictions coming from Eq. 27, andthe circles represent data obtained from direct integration ofthe Schr¨odinger equation. Data is shown for various µ and n = 1; though this problem has decoupled qubits, so this isrepresentative of arbitrary n . Notice especially the oscillatorybehavior that depends on the integral of the spectral gap overthe entire evolution. Then the probability of transitioning into one of the m -fold degenerate first excited states is given by P ( τ ) m = ρ (1) + ρ (0) τ (27) − ρ (0) ρ (1) τ cos( ωτ ) + O ( τ − ) , where ω ≡ Z ds ∆( s ) . (28)Therefore, the final probability of failure (and success)are oscillating functions with a frequency dependent onthe integral of the spectral gap. This result is a pre-viously known result [19–21]. Additionally, Wiebe andBabcock [21] have proposed using this oscillating behav-ior to enhance quantum adiabatic computing.Notice that the rule of thumb for the adiabatic con-dition states that the evolution time, τ , needs to growwith ρ ( s ), and indeed our formula captures this since theprobability of failure depends on ρ ( s ) τ . The new and in-teresting behavior here is not the overall τ dependencebut the oscillating dependence. B. Numerical Confirmation
To numerically test the predictions of the previous sub-section, we return to the quantum computational prob-lem of Hamming weight barrier tunneling introduced insubsection II B. . . . . . . P r ob a b ilit yo f F a il u r e τ µ = 0 . µ = 1 µ = 2 FIG. 2: The probability of transition to the first excited stateversus the evolution time τ for a barrier that grows with α = β = 1 /
10. The solid lines represent the theoretical predictionscoming from Eq. 27, and the circles represent data obtainedfrom direct integration of the Schr¨odinger equation. Data isshown for various µ and n = 100. Even in this case withmore approximations than in Fig. 1, the oscillatory analyticprediction is still quite accurate. One important and instructive way to get a large gapout of the barrier tunneling problem is to set the barrierto zero, b ( | z | ) = 0. This decouples all the qubits fromeach other, making the problem effectively a two-levelsystem, independent of the number of qubits. As well,the spectral gap has a simple closed form expression forthis case ∆ NB ( s ) = p − s + (1 + µ ) s (29)This gap is also useful even in the cases with a barrierbecause it can well approximate the gap far away fromwhere the tunneling event occurs. Also note that in theno barrier case γ ( s ) = µ s ) . In later analyses of caseswith barriers, we use this expression for γ ( s ) again whenevaluating γ (0) and γ (1) which are far from the tunnelingevent.When there is no barrier, we can analytically integrateEq. 29 and obtain a form for the probability of failure,Eq. 27. In Fig. 1, we compare this analytic expressionfor the near-adiabatic probability of transitioning to anexcited state with the exact result obtained by numericalintegration of the Schr¨odinger equation. The analyticapproximations predicts the actual data extremely well.It should be noted that without a barrier, the qubits aredecoupled, so this is a two-level system, meaning manyof our approximations are exact.In Fig. 2, we look at a case, where the barrier is presentand the qubits are not decoupled. In this case, we takea barrier with scaling exponents α = β = 1 /
10. Basedon previous work [25], this barrier should be easy to tun-nel through, with only a constant gap as n increases. Inthis instance, we have taken n = 100 qubits and see thatgood agreement between the direct data and the analyticapproximation from Eq. 27 that was calculated using nu-merical integration of the spectral gap and approximat-ing ρ (1) and ρ (0) by the unperturbed value since theyare evaluated far from where the barrier is relevant. V. SMALL GAP
For the small gaps, we examine the case where thegap remains large everywhere except in a region rightaround a critical s ∗ . At this critical s ∗ , the system hasan avoided level crossing, which traditionally is handledby a formalism such as the Landau-Zener problem. Inthis section, we focus on a system with a single avoidedlevel-crossing, but our methods can easily be generalizedto systems with multiple level crossings.Note that this section discusses the near-adiabaticlimit, so big-O notation is not appropriate here. In the τ → ∞ limit, the results of the large gap section are ac-curate. This section focuses on the behavior of the suc-cess probability in regions where the inverse gap is smallrelative to the evolution-time, τ . Thus, we examine anintermediate region, and all of our results for modifica-tions to C (1 , τ ) tend to zero faster than Eq. 26 in theasymptotic limit of τ . A. Frequency Splitting
An integral of the form Eq. 23 can often be treatedwith the stationary phase approximation. If the phasefunction, R s dz ∆( z ), is ever stationary as would occurwhen ∆( s ) = 0, then the stationary phase approximationsays that asymptotically, the integral is dominated by thevalue close to that stationary point.Unfortunately, since the gap never goes to zero, ∆( s ) =0, we never have a point of true stationary phase. How-ever, we have an avoided level crossing where the gap be-comes very small in the vicinity of s ∗ . Additionally, near s ∗ , the denominator of the integral is also small since itis proportional to ∆( s ) ( γ ( s ) also often depends inverselyon the gap). Therefore, the region around s ∗ should stillcontribute more to the integral than other regions, butsince this is not a true stationary phase point, the con-tribution near s ∗ is drowned out in the asymptotic limitof τ .Therefore, we will make an ansatz that the regionaround s ∗ also contributes significantly to the final prob-ability amplitude in the near-adiabatic limit. The con-tribution to the probability amplitude in the vicinity of s ∗ is roughly of the form λ ≡ e − iτ R s ∗ dz ∆( z ) Z s ∗ + εs ∗ − ε ds e − iτ R s ∗ s dz ∆( z ) ∆( s ) /γ ( s ) , (30)where we have pulled out the contribution to the phasedue to getting to the critical point. We can generalize this further (potentially allowing us to relax some of theassumptions that led to Eq. 23) to λ = e − iτ R s ∗ dz ∆( z ) Λ( τ ) . (31)Later in this section we present numeric evidence sup-porting this ansatz. Furthermore, our numerics indicatethat Λ( τ ) is real, allowing us to ignore any potential extraphases.Then, our conjectured probability amplitude in thenear-adiabatic limit is C (1 , τ ) ≈ ρ (1) iτ − ρ (0) e − iτ ( ω + + ω − ) iτ + Λ( τ ) e − iτω + , (32) ω + = Z s ∗ dz ∆( z ) , ω − = Z s ∗ dz ∆( z ) . This probability amplitude leads to a probability oftransition of P ( τ ) m ≈ Λ( τ ) + ρ (0) + ρ (1) τ (33)+ 2Λ( τ ) τ ( ρ (0) sin( ω − τ ) + ρ (1) sin( ω + τ )) − ρ (0) ρ (1) τ cos(( ω + + ω − ) τ )where m is the degeneracy of the first excited state.Most importantly the final probability amplitude nowhas sinusoidal motion dependent on two frequencies, ω ± .Thus, the final probability no longer has a simple sinu-soidal behavior but depends on the superposition of mul-tiple sinusoids.This splitting of the frequency, creating a superposi-tion of sinusoids when the gap is small, is a well realizedfeature in actual problems, as seen numerically in Sec-tion V C. In fact, this frequency splitting seems to per-sist even when most of the simplifying assumptions thatwent into Eqs. 27 & 33 fail. In the next section, we takea Landau-Zener approach to the small gap, but in othersmall gap models we examined, this frequency splittingpersisted. B. Analytic Approximation
In this section, we approximate our avoided level cross-ing as a Landau-Zener transition. In the language of theprevious subsection, we use the ansatz that Λ( τ ) is re-lated to the Landau-Zener transition probability. TheLandau-Zener problem works with Hamiltonians of theform ˆ H LZ ( s ) = v s − s ∗ )ˆ σ z + g σ x , (34)where g ≡ ∆( s ∗ ) is the minimum spectral gap, and v isslope of the spectral gap far from s ∗ . The Landau-Zenerformula says that the probability of transitioning fromthe ground state to the excited state going from t = −∞ to t = ∞ is P LZ = e − π g v τ , (35)where the τ is coming in because the Landau-Zener tran-sition is formulated in actual time, t , whereas Eq. 34 isformulated in s .We take the probability amplitude of transitionthrough our avoided level crossing to be proportional to √ P LZ . We also include a real parameter, A , to accountfor non-idealnesses in the Landau-Zener transition suchas the finite nature of our transition and the fact thatwe do not start the avoided level-crossing in exactly theground state. Therefore, we takeΛ( τ ) = Ae − π g v τ . (36)Then, our ansatz for the final failure probability whenthere is an avoided-level crossing is P ( τ ) m ≈ A e − π g v τ + ρ (0) + ρ (1) τ (37)+ 2 Ae − π g v τ τ ( ρ (0) sin( ω − τ ) + ρ (1) sin( ω + τ )) − ρ (0) ρ (1) τ cos(( ω + + ω − ) τ ) . In an actual setting, g , v , and ω ± can be determinedfrom the shape of the spectral gap of the problem inquestion. We leave A as a fitted parameter that accountsfor non-idealness in our system. In the next section, wedetermine all these parameters in specific computationalsettings. C. Numerical Confirmation
In this section, we present numeric data confirmingthe usefulness of the analytics in the rest of the section.Our approximations rely on two key assumptions, namelythat | C ( s ) | is close to one, meaning we primarily stayin the grounds state and that τ is large. We will testthese approximations as well as the Landau-Zener anstazmodification by comparing Eq. 37 to direct data.Each numeric simulation is based on direct Schr¨odingerevolution of the wavefunction from s = 0 to s = 1. Thisevolution then gives us data of P ( τ ) (actually we calcu-late 1 − P ( τ ), the probability of staying in the groundstate) versus τ . This data is then fit using a function ofthe form of Eq. 37 to obtain a fitted value of A . Acrossnumerous trials with different barrier shapes and sizes,we determine that A can be taken as real.For each of our simulations, we numerically calculatethe gap as a function of s and use it to extract g , v , and ω ± . For v , we base its value on the slope of ∆( s ) closeto the avoided level crossing where the gap is increasing . . . . . . . . . . S p ec t r a l G a p , ∆ ( s ) s FIG. 3: The spectral gap, ∆( s ), versus s , for a binomialbarrier with α = 0 . β = 0 . µ = 1, and n = 84. Theminimum spectral gap, g is obtained from the minimum here,and the Landau-Zener slope, v , is approximated by the almostlinear sections near the minimum gap. As well the frequencies, ω ± , come from numerical integration of this curve. . . . . . . T r a n s iti on P r ob a b ilit y τ Direct EvolutionAnalytic Approximation
FIG. 4: The probability of transitioning out of the groundstate as a function of the total evolution or run time, τ , for abinomial barrier with α = 0 . β = 0 . µ = 1, and n = 84.The red dots represent data obtained through direct evolutionof the Schr¨odinger equation, and the blue curve is the resultof applying Eq. 37 to the problem. To obtain the blue curve,the parameter A was fitted to A = 0 . or decreasing effectively linearly. The frequencies ω ± areobtained through numerical integration of the spectralgap before and after the critical s ∗ .We have done several simulations for the Hammingweight barrier problem with a binomial barrier. For arepresentative plot, see Fig. 4 which shows the close cor-respondence between the analytic expression in Eq. 37 . . . . . . . . T r a n s iti on P r ob a b ilit y τn = 64 n = 84 n = 104 n = 124 n = 148 n = 176 n = 200 FIG. 5: The probability of transitioning out of the groundstate versus the evolution time, τ , compared to the theoreticalpredictions from Eq. 37. The dots are the direct Schr¨odingerevolution data, and the lines are the thoeretical predictions.This data all comes from the Hamming weight barrier problemwith a binomial barrier with ehight and width both growingwith n . and µ = 1. The frequency behavior seems to becaptured by the theoretic predictions well, but the overallamplitude is less well-predicted especially for higher n wherethe probability of trnasitioning is higher. and the direct data, for a binomial barrier with heightand width scaling like n . and n . respectively and with µ = 1 and n = 84. Notably, the frequency splittingbehavior is evident here in the superposition of two si-nusoids, and the overall scaling matches well with theLandau-Zener probability of transition exponential.The fitted parameter A in Fig. 4 is A = 0 . A is 1; thefact that our value is less than 1, is likely an indicatorof the finite scale of our Landau-Zener region. A trueLandau-Zener transition occurs from t = −∞ to t = ∞ ,so we have a finite range, which might influence the valueof A . Also the A parameter is probably absorbing dis-crepancies caused by our other assumptions in derivingEq. 37, including those independent of the Landau-Zener-like transition.We performed similar trials for other barrier sizes andvalues of n and µ , and for each trial, we found a verygood correspondence between our anstaz and the directSchr¨odinger data. One of the largest assumptions we usein deriving Eq. 37 is that the majority of the probabilityremains in the ground state so that we can approximate C ( s ) ≈ | C ( s ) | ≈ . τ is large. In Fig. 5 we display data compared topredictions for a variety of n values at much lower τ .The agreement is not as clear as with larger τ , and espe-cially at larger n (thus larger transition probability), the . . . . S p ec t r a l G a p , ∆ ( s ) s FIG. 6: The spectral gap, ∆( s ), versus s , for the cubic poten-tial and n = 30. The frequencies, ω ± , come from numericalintegration of this curve. Notice that the avoided level cross-ing is assymmetric for n = 30, leaving us unable to properlyuse the Landau-Zener transition probability to approximatethis crossing. agreement is noticeably degraded. However, there is stillcorrespondence, and especially the frequencies, if not theamplitudes line up well.In Fig. 5, all this data is taken for a barrier with heightand width scaling with n . . The spectral gap ∆( s ) isvery similar for all these n values. As n increases, thegap around the avoided level-crossing changes, lowering g and raising v , but the majority of the spectral gap re-mains the same, meaning that ω ± are virtually the samefor different n . The persistence of the same ω ± leadsto similar frequency behavior across n , even though theenveloping probability scaling with τ changes with g , inaccordance with the standard adiabatic theorem.For the cubic potential, the spectral gap still goesthrough an avoided level crossing as shown in Fig. 6,but in this case, the avoided level crossing is much moreassymetric for n = 30. The slopes on either side of theavoided level crossing are different, making it difficult todetermine v . Therefore, in this case, we leave v as afitted parameter. This essentially eliminates the benefitof our Landau-Zener ansatz, but the frequency splittingof Eq. 33 is still valid. We also calculate ρ (0) and ρ (1)numerically using equivalent diagonalization methods tohow we calculate the spectral gap itself.Using Eq. 37 with fitted A and v , we predict how thetransition probability changes as a function of τ in Fig. 7.The correspondance between our prediction and the di-rect evolution data is quite good, but much of this is dueto two fitted parameters. The important part of this fig-ure is that the oscillatory behavior matches quite wellbetween the predicition and direct data.Also, Fig. 7 once again matches well even at the rela-tively high transition probability of around 0 .
24, indicat-ing that the frequency splitting behavior is fairly robustto our exact approximations. The correspondance in this . . . . . . . . . T r a n s iti on P r ob a b ilit y τ Direct EvolutionAnalytic Approximation
FIG. 7: The probability of transitioning out of the groundstate as a function of the total evolution or run time, τ , forthe cubic potential and n = 30. The red dots represent dataobtained through direct evolution of the Schr¨odinger equa-tion, and the blue curve is the result of applying Eq. 37 tothe problem. Due to the assymetric gap during the avoidedlevel crossing as seen in Fig. 6, the fit needed more than just A as a fitted parameter. We also included v as a fitted pa-rameter to obtain this agreement. Notice that the oscillatorypattern is still correct, confirming the robustness of the fre-quency splitting behavior from section V A. cubic potential problem as well as the barrier problem in-dicate the correctness of our frequency splitting ansatz. VI. ADIABATIC GROVER SEARCH
The Grover search algorithm [35, 36] is a digital quan-tum algorithm for searching an unstructured set of N elements for one of M target elements, where M ≪ N .Classically, O ( N/M ) queries must be made to find oneof the target states, but Grover’s algorithm requires only O ( p N/M ) oracular calls to find a target with high prob-ability [36]. Notably, the likelihood of success of thisalgorithm is periodic in the number of queries with pe-riod ∝ p N/M . Therefore, timing needs to be exact toachieve success; though, there are methods of alleviat-ing this periodicity in favor of a larger constant scalingfactor.An adiabatic version of Grover’s search algorithm hasbeen developed [37, 38] that mimics the O ( p N/M ) scal-ing. However, previous studies of this algorithm have re-lied on studying the adiabatic theorem’s asymptotic scal-ing, and to our knowledge, no groups have looked intowhether the periodic nature of digital quantum searchcarries over to adiabatic Grover in some way. In this sec-tion, we demonstrate that the Grover oscillations exist inadiabatic search and are a direct result of the large gaposcillations described in section IV.
A. Adiabatic Grover Background
The adiabatic grover algorithm is setup on a Hilbertspace with dimension N (not to be confused with n qubitsdiscussed in previous sections). The initial Hamiltonianis just a simple connection Hamiltonian between all basisstates: ˆ H = − N N X i,j =1 | i ih j | . (38)The ground state of this Hamiltonian is just the uniformsuperposition of all states. For ease of analysis, typicallythe first M basis states are chosen to be the target states,and the final Hamiltonian gives them a preferential en-ergy term such thatˆ H = ˆ I − M X m =1 | m ih m | . (39)Note that the ground state of the final Hamiltonianis M -fold degenerate; whereas, the ground state of theinitial Hamiltonian is non-degenerate. Seemingly, thismeans that the spectral gap would go to zero at somepoint during the evolution; however, this is not an issuedue to the nature of the degeneracy. The true groundstate throughout the evolution is symmetric between tar-get states so that in the end it is a uniform superposi-tion of all targets. The degeneracy in the final groundstate is due to states that are non-symmetric in targetstates transitioning between the first excited state andthe ground state. Since the Grover Hamiltonian and ini-tial ground state are symmetric between target states,and since we are concerned with coherent evolution, thesenon-symmetric states are inaccessible, meaning that wecan largely ignore them.A linear interpolation between these two Hamiltoniansdoes not lead to the square root speedup of Grover’s al-gorithm [37]. Instead, a more generalized interpolationneeds to be consideredˆ H ( s ) = (1 − g ( s )) ˆ H + g ( s ) ˆ H . (40)The spectral gap for this Hamiltonian is given by∆( s ) = r − − g ( s )) g ( s ) ( N − M ) N . (41)The optimal annealing schedule [37], utilizes the adi-abatic condition, Eq. 3, slowing down when the spectralgap is small and speeding up when it is large. By opti-mizing for the amount of time spent in the small region,the ideal annealing schedule uses g ( s ) = 12 − tan (cid:16) (1 − s ) tan − (cid:16)q N − MM (cid:17)(cid:17)q N − MM (42)This optimal annealing schedule leads to an adiabaticruntime that is O (cid:16)p N/M (cid:17) .0 . . . . . . .
14 60 80 100 120 140 160 180 200 T r a n s iti on P r ob a b ilit y τN = 100 , M = 1 N = 100 , M = 5 N = 100 , M = 10 FIG. 8: The probability of transitioning out of the groundstate as a function of the total evolution or run time, τ , forthe adiabatic Grover search problem. Notice that there isgood agreement between the analytics and the data when theprobability of transitioning is low but that the agreement isworse for shorter evolution times. The analytics here are exactand included no fitting parameters. B. Large Gap Oscillations
Our contribution is to show that the large gap oscil-lations described in section IV lead to the same peri-odic behavior in the adiabatic algorithm as in the stan-dard digital Grover algorithm. Note that the gap in theGrover problem does become small at s = 1 /
2, where∆(1 /
2) = p M/N , but in this section we consider onlythe large gap oscillations. Also, Wiebe and Babcock[21], have examined the large gap oscillations of adia-batic Grover in the case of M = 1. There, they showthat timing the adiabatic algorithm based on the largegap oscillations improves the performance of the algo-rithm; however, they never explicitly state the oscillationfrequency or connect it back to digital Grover. Further-more, our results make the simple extension to M > τ ∈ O (cid:16)p N/M (cid:17) , andour main goal is to determine what the fine structure ofthe oscillations in this limit are.Using the spectral gap and the annealing schedule,Eqs. 41 & 42, we can calculate the frequency of the largegap oscillations as described by Eq. 28: ω = r MN tanh − q N − MN tan − q N − MM (43)The period of oscillations T ≡ /ω in the limit of large N/M is given by T → π NM r NM . (44)Therefore, up to logarithmic factors, the period of os-cillations for adiabatic Grover’s search is e O ( q NM ) whichmatches the digital equivalent.Additionally, we can look at the amplitudes of the os-cillations in the probability of transitioning, Eq. 27. Inorder to get ρ ( s ), we need ∆( s ) which was listed in theprevious subsection and γ ( s ). Assuming the annealingschedule in Eq. 41 γ ( s ) = 4 M tan − r N − MM ! (45) × (cid:16)q N − MM − tan (cid:16) (1 − s ) tan − (cid:16)q N − MM (cid:17)(cid:17)(cid:17) N − M + N cos (cid:16) − s ) tan − (cid:16)q N − MM (cid:17)(cid:17) . The truly important part of this equation is that this γ ( s ) is symmetric about s = 1 / γ (0) = γ (1). Since the spectral gap is ∆(0) = ∆(1) = 1 atthe end points as well, this leaves us with ρ ≡ ρ (0) = ρ (1). Therefore, the probability of transitioning out ofthe ground state reduces to P ( τ ) = 4 ρ τ sin (cid:16) ωτ (cid:17) + O ( τ − ) . (46)Notably, this probability goes to O ( τ − ) periodicallyaccording to ω . Therefore, adiabatic Grover’s search canbe timed to get perfect success probability in the adia-batic limit. In Fig. 8, we show the agreement between theanalytic predictions of Eq. 46 and direct evolution data.There is good agreement between the analytics and thedata, especially for lower transition probabilities. VII. CONCLUSION
We have worked in the near-adiabatic limit, expandingupon the evolution time dependence. Specifically we haveexplored how the probability of transitioning out of theground state depends on τ , the total evolution time.In the adiabatic limit, the probability of transition de-creases with τ − , but we explored the structure on topof this basic decay, looking at the sinusoidal behaviorsuperposed on top. In the absence of a small gap, the si-nusoidal behavior has a frequency that depends only theintegral of the spectral gap along the evolution. Whenan avoided level crossings occur, the sinusoidal behaviorbecomes the superposition of frequencies that depend onthe integrals of the spectral gap, broken up at the avoidedcrossings.We back up our analytics with numerics that veryclosely match up with our predictions. Specifically our1numerics are in the context of quantum computing andannealing, where this work can predict how long to runthe algorithm to lead to oscillatory enhancements to thesuccess probability.Our work on frequency splitting in Sec. V A relieslargely on an ansatz inspired by the stationary phase ap-proximation and the Landau-Zener transition. Furtherwork could be done to remove the need for an ansatzhere, working more from first principles. The fact thatthe numerics match the analytics even when the approx- imations are less well-founded, indicates that strongeranalytic work could be possible. Acknowledgements
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