Quantum-Walk-Inspired Dynamic Adiabatic Local Search

We investigate the irreconcilability issue that arises when translating the search algorithm from the Continuous Time Quantum Walk (CTQW) framework to the Adiabatic Quantum Computing (AQC) framework. For the AQC formulation to evolve along the same path as the CTQW, it requires a constant energy gap in the Hamiltonian throughout the AQC schedule. To resolve the constant gap issue, we modify the CTQW-inspired AQC catalyst Hamiltonian from an XZ operator to a Z oracle operator. Through simulation, we demonstrate that the total running time for the proposed approach for AQC with the modified catalyst Hamiltonian remains optimal as CTQW. Inspired by this solution, we further investigate adaptive scheduling for the catalyst Hamiltonian and its coefficient function in the adiabatic path of Grover-inspired AQC to improve the adiabatic local search.


I. INTRODUCTION
Quantum technologies have advanced dramatically in the past decade, both in theory and experiment.From the view of theoretical computational complexity, Shor's factoring algorithm [1] and Grover's search algorithm [2] are well-known for their improvements over the best possible classical algorithms designed for the same purpose.From a perspective of universal computational models, Quantum Walks (QWs) have become a prominent model of quantum computation due to their direct relationship to the physics of the quantum system [3,4].It has been shown that the QW computational framework is universal for quantum computation [5,6], and many algorithms now are presented directly in the quantum walk formulation rather than through a circuit model or other abstracted method [3,7].Besides being search algorithms, CTQWs have been applied in fields such as quantum transport [8][9][10][11], state transfer [12,13], link prediction in complex networks [14] and the creation of Bell pairs in a random network [15].Some other well-known universal models include the quantum circuit model [16][17][18], topological quantum computation [19], adiabatic quantum computation (AQC) [20], resonant transition based quantum computation [21] and measurement based quantum computation [22][23][24][25].Each model might has its own bottleneck.Investigation on the relationship among the frameworks helps identify the violations when mapping frameworks and potential solutions.By studying the mapping, one can extend the techniques from one framework to another for some potential speedup [26].
In this work we investigate the irreconcilability issue that arises when translating the search algorithm from the Continuous-Time Quantum Walk (CTQW) framework to the Adiabatic Quantum Computing (AQC) framework as first pointed out by Wong and Meyer [27].
This irreconcilability issue can be described as follow.One first notes that the CTQW is the unique continuoustime quantum walk formulation of Grover's discrete search algorithm.While the CTQW search evolves the initial unbiased (equal amplitude) state to the unknown (marked) state on the order of time T ∼ O( √ N ) (where N is the size of search space), it does not follow the same evolution path (on the Bloch sphere) as that of Grover's algorithm.The uniqueness of the CTQW formulation stems from the fact that the unknown marked state only acquires a (time-dependent) phase from the oracle operation.Most importantly the marked states does not undergo evolution, and thus the CTQW effective employes a dichotomous "Yes/No" oracle, for which the discrete Grover's algorithm has been proven to be optimal.
The AQC formulation of the search algorithm with a non-uniform adiabatic evolution schedule [28] also finds the marked state in time T ∼ O( √ N ) while at the same time following the same path as Grover's algorithm.Thus, if one investigates what adiabatic Hamiltonian gives rise to the same evolution path as the CTQW formulation, one finds [27] the AQC formulations introduces an extra "catalyst" Hamiltonian which introduces structure beyond the standard "Yes/No" oracle employed in the CTQW or discrete (Grovers) search algorithm.A scaled version of the AQC Hamiltonian leads to a constant energy gap that implies that the marked state can be found in time T ∼ O (1).This discrepancy between the formulations of the two versions of a continuous time search algorithm was termed the "irreconcilability (difference) issue" between CTQW and AQC by Wong and Meyer [27].
In this work we address the CTQW/AQC search algorithm irreconcilability issue by modifying the constant energy gap Hamiltonian of the AQC formulation.Our contribution is twofold.We first adapt the result from the mapping of CTQW to AQC by selecting the regular oracle Z operator as the catalyst Hamiltonian and explore an alternative for the coefficient function for the catalyst Hamiltonian in order to attempt to avoid the Approved for Public Release, Distribution Unlimited: PA# AFRL-2022-1619.

arXiv:2204.09830v2 [quant-ph] 20 Apr 2023
irreconcilability issue.Through the simulation, the modified model provides optimal results in terms of time required for search.We then apply this modification to adiabatic local search by adding an additional sluggish parameter δ which delineates the width of the adiabatic run time schedule over which the catalyst Hamiltonian effectively acts (i.e. the "slowdown" region in the vicinity of the system's smallest energy gap ∆).The sluggish parameter tracks the increase of running time t = t(s) with respect to schedule parameter 0 ≤ s ≤ 1 where δ = |d 2 t/ds 2 |.The catalyst is employed when δ ≥ δ 0 to facilitate the process, where we have found that the threshold value of δ 0 = 64 provides good results.
The outline of this work is as follows.The background information regarding CTQW and AQC is given in section II where the translation of CTQW to AQC is described in section III.The irreconcilability issue that occurs during the translation is explained in section III A and our proposed solution is provided in section III B. The mapping of Grover search to AQC as an adiabatic local search is summarized in section IV.We propose and describe the catalyst Hamiltonian mechanism in section IV A 2 and determine the sluggish interval where it is employed.We further explore three coefficient functions of the catalyst Hamiltonian in section IV A 3. The simulation results for proposed modifications are discussed in section V. Finally, our conclusions are given in section VI.

II. BACKGROUND A. Continuous-Time Quantum Walk
Given a graph G = (V, E), where V is the set of vertices and E is the set of edges, the CTQW on G is defined as follows.Let A be the adjacency matrix of G, the |V | × |V | matrix is defined component-wise as where i, j ∈ V .A CTQW starts with a uniform superposition state |ψ 0 in the space, spanned by nodes in V , evolves according to the Schrödinger equation with Hamiltonian A. After time t, the output state is thus The probability that the walker is in the state |τ at time t is given by | τ | e −iAt |ψ 0 | 2 .To find the marked node |ω starting from an initial state |ψ 0 via a CTQW, one has to maximize the success probability while minimizing the time t.For instance, initially at time t = 0, the success probability is The success probability is extremely small when the search space |V | = N is large and |ψ 0 is a uniform superposition state.
When applied to spatial search, the purpose of a CTQW is to find a marker basis state |ω [29,30].For this purpose, the CTQW starts with the initial state |ψ 0 = N i=1 1 √ N |i , and evolves according to the Hamiltonian [31] where γ is the coupling factor between connected nodes.The value of γ has to be determined based on the graph structure such that the quadratic speedup of CTQW can be preserved.Interested readers can refer to [29,31] for more details.

B. Adiabatic Quantum Computing
In the AQC model, H 0 is the initial Hamiltonian, H f is the final Hamiltonian.The evolution path for the timedependent Hamiltonian is where 0 ≤ s ≤ 1 is a schedule function of time t.For convenience, we denote s as s(t) and use them interchangeably.The variable s increases slowly enough such that the initial ground state evolves and remains as the instantaneous ground state of the system.More specifically, where λ k,t is the corresponding eigenvalue the eigenstate |λ k,t at time t and k labels for the k th excited eigen-state.The minimal eigenvalue gap is defined as where T a is the total evolution time of the AQC.Let |ψ(T a ) be the state of the system at time T a evolving under the Hamiltonian H(s(t)) from the ground state |λ 0,0 at time t = 0.The Adiabatic theorem [32,33] states that the final state |ψ(T a ) is -close to the real ground state |λ 0,Ta as provided that Approved for Public Release, Distribution Unlimited: PA# AFRL-2022-1619.
There are several variations of AQC to improve the performance.The variations are based on modifying the initial Hamiltonian and the final Hamiltonian [34,35] or adding a catalyst Hamiltonian H e [34], which is turned on/off at the beginning/end of the adiabatic evolution.In this work, we are interested in the catalyst approach.A conventional catalyst Hamiltonian assisted AQC path is expressed as where s(t) = sin 2 ( t √ N ) with or explicitily in the {|w , |r } basis as

A. The Irreconcilability Issue: Constant Gap Catalyst Hamiltonian and Small Norm
The main concerns that are raised from Eqn. ( 12) are twofold.The first issue is the factor 4 s(1−s) 4 2 N of H(s).The adiabatic theorem [36] states the system achieve a fidelity of 1 − to the target state, provided that Here dH dt 0,1 are the matrix elements of dH/dt between the two corresponding eigen-states.E 0 (t) and E 1 (t) are the ground energy and the first excited energy of the system at time t.Given the H(s) in Eqn.(12), one might conclude that a factor of O( 4 1/N ) significantly reduces the required time to achieve 1 − precision.This might be misleading as the g min of H(s) also carries the same factor.The second issue is that the catalyst H e provides power greater than a typical Yes/No oracle as it maps non-solution states to a solution state and a solution state to non-solution states.Provided initially the we start with a superposition state with amplitude of N −1 N for a non-solution, it takes time of O(1) for this catalyst to drive the initial (unbiased, equal amplitude) state to the solution state.In the following we will relax this constraint by using a normal oracle.For the rest of the paper, let us simply treat 1 as some small negligible constant.

B. Modified CTQW-Inspired Adiabatic Search
In Eqn.(12), the following parameters were computed during the mapping [27]: • the scaling factor 4 s(1−s) • the coefficient function of H e as s(1 − s).
In [37] the cost of the adiabatic algorithm was defined to be the dimensionless quantity (using = 1) where t f is the running time.To prevent the cost from being manipulated to be arbitrarily small by changing the time units, or distorting the scaling of the algorithm by multiplying the Hamiltonians by some size-dependent factor as shown in the irreconcilability concern [27], the norm of H(s) should be fixed to some constant, such as 1.
To address the irreconcilability issue, the scaling factor is dropped and the catalyst Hamiltonian H e is modified.
Since H e = 2 N −1 N iXZ in the {|ω , |r } basis provides more power than a standard Oracle, for our modification we remove the imaginary number i and the X operator.The operator Z alone behaves as a conventional "Yes / No" oracle in the {|ω , |r } basis.Let M = 2 N −1 N and choose the modified adiabatic path H m (s) as where f z (s) is our chosen s-dependent coefficient for catalyst Z.In addition to f z (s) = s(1 − s) that was used Approved for Public Release, Distribution Unlimited: PA# AFRL-2022-1619.
in [27], functions that reach its maximum when s = 1/2 are good candidates for f z (s), such as f z (s) = sin(sπ) 2 .The use of the factor 1/2 on the sine function is to offset the magnitude M to bound the norm of H e as described in Eqn.(16).

IV. GROVER SEARCH TO ADIABATIC LOCAL SEARCH MAPPING
In this section we consider the mapping of Grover's algorithm to an adiabatic search.Given the initial driving Hamiltonian H 0 and the final Hamiltonian H f as where in the {|ω , |r } basis.The adiabatic path [27,28] in the {|ω , |r } basis is given by Instead of employing a linear evolution of s(t), Eqn.(20) adapts the evolution ds/dt to the local adiabaticity condition [28] such that where g(t) is the energy gap of the system at time t.The running time t is then a function of schedule s such that The relation between the schedule s and the running time t is shown in Figure 1.It is clear that the system evolves quickly when the gap is large (s away from 1/2) and slowly when the gap is small (s 1/2) [28].In this example, the sluggish period s ∈ [0.4,0.6].For completeness, we provide the formal proof of the close form of the squared gap function g 2 (t) (second order in s) with respect to the schedule s in Appendix A.

A. Adaptive Scheduling
For a fixed schedule of a adiabatic path, the schedule s moves fast when the eigen-energy gap is large, and slowly when the gap is small.We desire to employ the catalyst Hamiltonians H e to amplify the eigen-energy gap during the "slow down" period such that the total time to pass through the sluggish period is reduced (s ∈ [0.4,0.6] in Fig. (1)).

FIG. 1.
Schedule s in terms of time t with N = 64 in adiabatic local search, as observed in [28].

Schedule Dependent Gap Function
In this section, we consider employing gap-dependent scheduling functions.Let H f be an arbitrary 2 by 2 Hermitian Hamiltonian.Let the time-dependent Hamiltonian H(s) be Operators σ x and σ z are chosen as catalyst Hamiltoni- where a, b, c, p, q, r are some given constants.The matrix form of the timedependent Hamiltonian is given by ) and the schedule-dependent gap can be analytically computed to yield (see see Appendix B for a derivation).By using Eqn.(22), the total running time T stp strt from s = s strt to s = s stp is thus where 0 ≤ s strt ≤ s stp ≤ 1.In brief, the time spent during a certain period of a schedule can be obtained by use of gap function.The gap function can be expressed via the entries of H 0 , H e , H f , schedule s and the coefficient functions of the catalyst Hamiltonians.

Determining the Sluggish Interval for the Catalyst Hamiltonian
By using the condition f (s) = dt/ds = 1 g 2 (s) (see Appendix A), the region where the gap quickly significantly Approved for Public Release, Distribution Unlimited: PA# AFRL-2022-1619.decreases or increases is during the sluggish period of s.That is the portion of schedule the s where catalyst should be employed.The region where |df 2 (s)/ds 2 )| ≥ δ 0 is the sluggish period.The threshold value δ 0 = 64 was chosen because if we choose a threshold proportional to N , as N increases exponentially, the quantity d 2 t/ds 2 might never reach the N -dependent threshold within the adiabatic evolution schedule 0 ≤ s ≤ 1.By using this threshold, the starting point s slug strt and the stopping point s slug stp used to mark the sluggish period can be identified.
Using the example in [28], we can re-plot and get t as a function of s as t = f (s) and f (s) = dt/ds in Figure 2 -3 with N = 64.

Catalyst Coefficient Functions
As discussed in section III B, we are interested in the H e = Z case in Eqn.( 17) and its coefficient function f z (s).Three coefficient functions of the catalyst Hamiltonian Z are proposed as the following where 0 ≤ a, b ≤ 1 under the constraint that a 2 + b 2 = 1.In the grid search a increased from 0 to 1 by 0.1 in each iteration.From the 10 pairs of (a, b), we find the values of a, b that give the shortest sluggish time interval.

V. EXPERIMENT & RESULT
For our simulations we used (Wolfram) Mathematica (version 12.3 run on a Linux Ubuntu 20.04 LTS laptop).The code is available upon request.The running time is based on Eqn.(28).The size N (number of nodes) ranges from 2 5 , 2 6 , • • • to 2 25 .We observe the corresponding running time and sluggish time for each of the proposed models.The result of the original adiabatic local search serves as the baseline for comparison, which used N = 64 [28].In this work, we generalize the setting for any arbitrary size N .
Given an arbitrary complete graph of size N with coupling factor 1/N , one can compute the entries in the reduced Hamiltonian for H 0 and H f in the {|ω , |r } basis.The values of variables a, b, c, p, q and r as discussed in section IV A 1 can be obtained from Eqn.( 14) for the CTQW case, and from Eqn. (19) for the adiabatic local search.It is worth noticing that the ground state energy is −1 in the CTQW case, but is 0 in the adiabatic local search case.Based on the adiabatic path Eqn.(25), and the gap function in Eqn. ( 27) with given schedule s, coefficient function f z (s) for σ z , we perform the simulation with the running time computed from Eqn. (28).

A. Modified CTQW-Inspired Adiabatic Search Simulation
This experiment aimed to demonstrate that the modified adiabatic paths addressing the irreconcilable issues remain optimal.The three proposed modifications we explored are as follows: • H org (s) takes Eqn.(12) and drops the scaling factor as explained in section III B. The adiabatic path is • H m1 (s) replaces the computed catalyst Hamiltonian H e with an ordinary Z oracle operator and keeps the magnitude M .This was used to address the constant gap H e irreconcilability issue.
We have  catalyst Hamiltonian H e = M XZ in H org (s) indeed is a constant gap Hamiltonian.This also shows the irreconcilability issue as suggested in [27].From the simulations we can conclude that both H m1 (s), H m2 (s) perform optimally with respect to running time, namely T ∼ O( √ N ), similar to that of the original adiabatic local search but with a minor constant factor which can be ignored in the Big O notation.As the simulation suggests, both modified CTQW-inspired approaches outperform the original adiabatic local search.When the N ≤ 2 21 , the H m2 (s) outperforms H m1 (s).When problem size N is larger then 2 21 , H m1 (s) is a better choice over H m2 (s).

B. Adaptive Adiabatic Local Search Simulation
With Various Coefficient Functions In the previous section V A, the proposed modifications are optimal, in the sense that T ∼ O( √ N ) up to a minor constant factor.For further improvement, the adaptive scheduling scheme is applied.The adiabatic path to be explored is therefore where f (s) ∈ [f sine z , f ss z , f grid z ] as seen in Eqn.(29).The catalyst Hamiltonian Z operator is only employed during the sluggish period and hence f (s) = 0 when s / ∈ [s slug strt , s slug stp ].The H 0 and H f are based on Eqn.(19).As the catalyst is only employed within the sluggish period, to compare the performance of each proposed modification, one only needs to compute the running time within this period.In Figure 5, f ss z provides the minimal reduced sluggish time while f sine z and f grid z provide significant improvements.The difference in the runtimes becomes significant for N ≥ 2 15 .In Figure 6

VI. CONCLUSION
In this work, we investigated different Hamiltonians for resolving the irreconcilability issue [27] when mapping the CTQW search algorithm to AQC.We modified the time-dependent Hamiltonian by (1) removing the original scaling CTQW factor 4 s(1−s) 4 2 N , and (2) replacing i X Z → Z in the original catalyst H e Hamiltonian obtained from mapping CTQW to AQC.These modification were made in order to resolve the irreconcilability issue.We further optimized the schedule s of the CTQW-inspired adiabatic path by an adaptive scheduling procedure.
The modified CTQW-inspired adiabatic search simulation experiment demonstrates that indeed the H e with-Approved for Public Release, Distribution Unlimited: PA# AFRL-2022-1619.out any modification leads to a constant time in the total running time, regardless of the search space size N .This result echoes the irreconcilability issue stated in [27].On the other hand, the modified CTQW-inspired adiabatic path with catalyst Hamiltonian coefficient sin(sπ) 2 behaves similarly to the behavior of the optimal adiabatic local search.Furthermore, the modifications are optimal and outperform the original adiabatic local search.
Lastly, in the adaptive adiabatic local search simulation with various coefficient functions experiment, we further investigated how to reduce the time wasted in the sluggish period of an adiabatic local search path.As our numerical experiments show, the function f sin z (s) and f grid z (s) provide significant improvement and both outperform the original adiabatic local search.Even though the grid search f grid z (s) approach could have further reduced the length of the sluggish ("slow down") interval, the benefit was offset by the additional cost incurred from implementation over that of the other two methods.

FIG. 2 .
FIG. 2. Time t as a function of schedule s for adiabatic local search with N = 64.

2 M
s) uses sin(sπ) 2 as the coefficient function for the catalyst Hamiltonian Z.The adiabatic path is H m2 (s) = (1 − s)H 0 + sin(sπ) Z + sH f For the above three models, simulations were run on Hamiltonian of size N ∈ [2 5 , 2 6 , • • • , 2 25 ].In the following figures, the abscissa is log 2 N while ordinate is the required total running time T .The time is computed based on Eqn.(28).As the dimension of the Hamiltonian increases, the difference in running times for the three models considered are magnified.The simulation results are shown in Figure 4.It is clear to see that H org is a constant time scheme as it does not scale as the size N increases.This indicates the original Approved for Public Release, Distribution Unlimited: PA# AFRL-2022-1619.
, both f sine z and f grid z have a reduced sluggish time over 75% in comparison to the original adiabatic local search when N reaches 2 25 .f sine z gradually outperforms the original adiabatic local search after N = 2 10 and remains almost as good as f grid z till N = 2 23 .When N = 2 25 , the sluggish time of f sine z is only twice that of

FIG. 5 .
FIG. 5. Case when N ∈ [2 5 , 2 25 ] and time spent in during the sluggish period for adiabatic paths with (f ss z , f sine z , f grid z ) coefficient functions where the original adiabatic local search serves as the baseline.
σ z .They should be good candidates for the catalyst perturbation in the AQC path.Similarly, if the Hamiltonian has an imaginary part in the off diagonal entries, |λ+ − λ − | = (α − β) 2 + 4(γ 2 + d 2 ).(B4)The Hamiltonian H (with no imaginery entries) can be expressed in terms of Pauli matrices as ) [27] set to 1/N and |ψ 0 is the uniform superposition of all states in the search space.State |r is the uniform superposition of non-solution states, state |ω is the solution state.Treating the state evolving in CTQW system as the time-dependent ground state of H(s), one constructs H(s) in the {|ω , |r } basis as[27]