Quantum optimisation via maximally amplified states

T. Bennett and J. B. Wang∗

Department of Physics, The University of Western Australia, Perth, Australia(Dated: November 2, 2021)

This paper presents the Maximum Amplification Optimisation Algorithm (MAOA), a highly ef-ficient quantum algorithm designed for combinatorial optimisation in the restricted circuit depthcontext of near-term quantum computing. The algorithm is shown to produce maximum possibleamplification of optimal solutions, which grows quadratically in achievable circuit depth, and doesso without the computationally demanding variational procedure required by other near-term quan-tum algorithms, such as QAOA or QWOA. A simple but novel modification to the existing Groveradaptive search in the context of similar restricted circuit depths is also introduced, referred to asthe restricted Grover adaptive search (RGAS). The MAOA and RGAS are simulated on a practi-cal vehicle routing problem, a computationally demanding portfolio optimisation problem, and anarbitrarily large problem with normally distributed solution qualities. In all cases, the MAOA andRGAS are shown to provide substantial speedup over classical-sampling in finding optimal solutions,where the MAOA consistently outperforms the RGAS. More importantly, the MAOA is shown toprovide a speedup which is extraordinarily close to the upper bound of what is possible within theQWOA framework, which includes QAOA.

I. INTRODUCTION

Quantum computing provides a paradigm that ex-ploits quantum-mechanical principles, such as super-position and entanglement, to solve computationalproblems far more efficiently than current or futurecomputers based on classical principles [1–5]. A sig-nificant potential application of quantum computingis in finding high quality solutions to combinatorialoptimisation problems. This class of problems ap-pears often and across a broad range of contexts, forexample, in commercial settings such as vehicle rout-ing, in financial settings such as portfolio optimisa-tion, and even in medical research such as proteinfolding. Often these problems have solution spacesthat grow exponentially with increasing problem size,which makes finding the most optimal solutions forlarge problems classically intractable [6]. Quantumcomputers have the ability to operate on these ex-ponentially large solution spaces in quantum parallelby using superposition of computational basis states,one assigned to each solution, the total number ofwhich grows exponentially in the number of qubits.This ability is what allows optimisation algorithmssuch as the Grover adaptive search (GAS) [7–9] todeliver significant speed up in finding optimal solu-tions within unstructured solution spaces relative to aclassical search procedure.

However, due to environmental noise, decoherance,and an insufficient number of qubits for error protec-tion, current and near-term quantum computers arelimited with regards to practically achievable circuitdepths [10, 11]. The aforementioned GAS algorithm istherefore not likely to be implementable on near-termquantum devices, since it necessarily requires large cir-cuit depths of O(

√N), where N is the size of the solu-

tion space. Consequently, much of the recent quantumalgorithm research and development has been focusedon achieving quantum advantages while restricted tolow circuit depths. One such restricted circuit depth

∗ jingbo.wang@uwa.edu.au

algorithm, focused specifically on combinatorial opti-misation, is the Quantum Approximate OptimisationAlgorithm (QAOA) [12–16].

The initial motivation behind the alternating op-erator ansatz which underpins the QAOA is relatedto the quantum adiabatic theorem [17] and its use inthe quantum adiabatic algorithm (QAA) [18]. Thequantum adiabatic theorem states that a quantumsystem will remain in its ground state if its Hamilto-nian changes sufficiently slowly with time. The QAAinvolves preparing a system in the ground state ofa known Hamiltonian, then slowly evolving it to theground state of a Hamiltonian that encodes the costfunction of the optimisation problem. The QAOAseeks to approximate this process by instead apply-ing these two Hamiltonians on a quantum circuit inalternating fashion, where the application times arecontrolled and tuned via a classical optimisation pro-cess. Alternating application of the Hamiltonians is inessence an amplitude amplification process [19]. Theclassical optimiser seeks to improve the expectationvalue of solution quality as measured from the finalamplitude amplified state, hence increasing the prob-ability that a measurement of the state produces ahigh quality solution.

The Quantum Walk Optimisation Algorithm(QWOA) [20, 21] was developed as a generalisationof this process, where it was recognised that applica-tion of the two Hamiltonians was essentially equiva-lent to a continuous time quantum walk over a con-nected graph, and a quality-dependent phase shift ap-plied to each solution state on the graph. This theo-retical framework has proven extremely useful in sub-sequent research and indeed in the research presentedin this paper. For example, the QWOA was shownto provide a significant improvement in performanceover the QAOA for a portfolio optimisation problemwhen restricting the quantum walk mixing process tojust a subset of valid solutions [22]. In the context ofa capacitated vehicle routing problem (CVRP) witha solution space of approximately N = 400, 000, theQWOA was again shown to produce optimised ampli-tude amplified states, leading to significant speedup

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in finding optimal solutions over classical sampling ofthe non-amplified solution space [23].

It was also noted, however, that the variational pro-cedure amplifies sub-optimal and near-optimal solu-tions more than the most optimal solutions, present-ing a potential opportunity for further improvementof the algorithms overall performance. Moreover, thecomputational effort required for the variational pro-cedure in both the QAOA and QWOA algorithms is asignificant challenge to be addressed if either of thesemethods are to provide quantum advantage for com-binatorial optimisation in the restricted circuit depthcontext of near-term quantum computation. On theother hand, if there exists a method to bypass thisvariational procedure altogether, then achieving quan-tum advantage should follow accordingly. The pur-pose of this paper is to present such a method, re-ferred to from here on as the Maximum AmplificationOptimisation Algorithm (MAOA).

The remainder of this paper is organised as fol-lows. In Section II the QWOA framework is re-viewed, and it is shown how the underlying mechanicof the MAOA arises naturally from this framework.In Section III the connection between the MAOAand Grover’s search is outlined, a relationship thatis central in developing the MAOA. In Section IV, theMAOA is introduced and presented in detail, followedby a discussion on the GAS and RGAS in Section V, asthey will form a relevant baseline for the comparisonof algorithm performance. Numerical simulations, re-sults and analysis are then presented in Section VI andSection VII. We conclude with a general discussion onthe advantages unique to the MAOA in Section VIII.

II. JUSTIFYING THE MAXIMUMAMPLIFICATION OPTIMISATION

ALGORITHM

In this section, the QWOA framework is formallyintroduced. Within the framework, there are two pri-mary degrees of freedom, the graph structure used inthe mixing-walk procedure and the treatment of thequality distribution prior to the application of quality-dependent phase shifts. It is by analysing these twodegrees of freedom and their effects on both conver-gence rates into optimal solutions and the optimisa-tion landscape in general, that we arrive at the under-lying mechanics of the MAOA.

A. The QWOA framework

A detailed description on the theoretical frame-work of the Quantum Walk Optimisation Algorithm(QWOA) was given in a previous paper [23]; it is in-cluded here for clarity and completeness. Formally,we consider a mapping f : S −→ R, which returns ameasure of the quality associated with each possiblesolution in the solution space S, where S has cardinal-ity N .

The starting point of the QWOA is a quantum sys-tem with N basis states, one for each solution in S,

initialised in an equal superposition,

|s〉 =1√N

∑x∈S|x〉 . (1)

This initial state is then evolved through repeatedand alternating application of the quality-dependentphase-shift and quantum-walk-mixing unitaries. Thequality-dependent phase-shift unitary is defined as

UQ(γj) = exp(−iγjQ), (2)

where γj ∈ R and Q is a diagonal operator such thatQ |x〉 = f(x) |x〉. The quantum-walk-mixing unitaryis defined as

UW (tj) = exp(−itjA), (3)

where tj > 0, and A is the adjacency matrix of a circu-lant graph that connects the feasible solutions to theproblem. Note that the Laplacian matrix of the graphcould also be used, but it would produce equivalentbehaviour. The graphs are selected to have circulantconnectivity, because all circulant graphs are diago-nalised by the Fourier transform and hence can beefficiently implemented on a quantum computer [24–27].

The first unitary UQ applies a phase-shift at eachvertex proportional to the quality of the solution atthat vertex, with the proportionality constant givenby the parameter, γj . The second unitary UW canbe understood as performing a quantum walk overthe graph for time tj , mixing the amplitudes acrossvertices. Following the mixing of phase-shifted am-plitudes across the vertices of the graph, construc-tive and destructive interference will result in quality-dependent amplitude amplification, controlled by theparameters γj and tj . Application of UQ and UW isrepeated r times, resulting a final state of the systemgiven by

|γ, t〉 = UW (tr)UQ(γr)...UW (t1)UQ(γ1) |s〉 , (4)

where t = (t1, t2, ..., tr) and γ = (γ1, γ2, ..., γr).

By tuning the parameters γ and t, it is possible toamplify the amplitudes corresponding with high qual-ity solutions, and therefore increase the probabilityof a measurement of the system collapsing it into ahigh quality solution. The process of tuning the pa-rameters is conducted iteratively through the use ofa classical optimisation algorithm which takes as itsobjective function the expectation value of the Q op-erator:

c(γ, t) = 〈γ, t|Q |γ, t〉 . (5)

The QWOA framework also assumes there existsan indexing algorithm which provides a one to onemapping from indices ∈ {0, 1, ..., N − 1} to solutionsin the solution space. This allows for the quantumwalk and graph connectivity to be restricted to justthe space of valid solutions.

3

B. Effect of graph structure and qualitydegeneracy on convergence rate into a single

most optimal state

Since the variational process within the QWOAframework operates on the expectation value for qual-ity, c(γ, t), it tends to amplify a large group of sub-optimal solutions more than a less numerous groupof most optimal solutions. The expectation value ofquality is more sensitive to amplification of the lessoptimal solutions simply due to their superior numberwithin the solution space. Moreover, it is also possi-ble for the least optimal solutions to be amplified asa secondary effect, since these are less numerous andhave less influence on the expectation value of quality.Both of these effects were seen clearly in our previousCVRP simulation results [23], included here in Fig. 1.It is for this reason that the focus in this section willbe instead on optimising convergence into specificallythe most optimal solutions, and to begin with, theconvergence rate into a single most optimal solution.

r

r

100

10

1

0.1

0.01

0.001

1000

10

1

0.1

0.01

0.001

0.0001

100

FIG. 1: Probability amplification of solutions as afunction of their cost after 10 and 35 QWOA

iterations [23].

In order to understand how the graph structure andthe quality distribution affect the amplification of themost optimal solution in a solution space, a meticu-lous numerical investigation has been carried out, dis-cussed in detail in Appendix A and Appendix B, withthe important results summarised as follows:

• Increased degeneracy in a quality distributionincreases the rate of amplification of the mostoptimal solution. As such, a binary mark-ing function applied to the quality distributionproduces the largest amplification of a singlemarked vertex for any graph.

• A binary marking function also produces max-imum amplification of the single marked vertexwith repeated applications of the same phase-shift and walk-time parameters, reducing theoptimisation landscape for the classically tuned

parameters to one that is 2-dimensional for anynumber of iterations.

• Specifically in the case of a binary marking func-tion, of all the investigated graphs, the com-plete graph produces the maximum amplifica-tion of the single marked vertex. Note that thecomplete graph is the graph in which each ver-tex/solution is connected to every other.

Besides the fact that a binary-marked completegraph produces the highest convergence rates into themost optimal solution with repeated application of thesame parameter pairs, it also has another significantadvantage, discussed in the next section.

C. The reduced complete graph

The complete graph presents a unique opportunityin that its behaviour can be greatly simplified whenthere is degeneracy in the distribution of qualitiesacross the vertices [28, 29]. The degenerate verticescan be combined through an edge contraction pro-cess to produce a graph with significantly fewer ver-tices which produces equivalent behaviour with re-spect to amplitude amplification within degenerategroups. This simplified graph will be referred to fromhere on as the reduced graph. The reason degener-ate vertices can be combined in this way is becauseeach of them are functionally equivalent within thegraph. They each receive identical phase shifts andhave identical connectivity with regards to neighbour-ing vertices of each quality. Fig. 2 illustrates the edgecontraction process and show how it produces a re-duced graph. Note the illustration is specifically foran example solution space containing solutions with 3distinct qualities. The solutions space can therefore bedivided into 3 subsets, SA, SB and SC , with respectivequalities, qA, qB and qC containing respectively, a, band c solutions each. In any case, the reduced graphis a weighted graph where each vertex represents onegroup of degenerate vertices from its parent graph.The weight of each regular edge corresponds with thesquare root of the number of edges connecting the re-spective degenerate groups in the parent graph. Theweight of each self loop is equal to one less than thenumber of vertices in the respective degenerate group.The self loops are necessary because they account forthe mixing of amplitude that occurs within vertices ofthe same group.

The reduced graph in Fig. 2 can be characterised bythe following quality operator Q3, adjacency matrixA3, and initial equal superposition state |s3〉, where

Q3 =

qA 0 00 qB 00 0 qC

, A3 =

a− 1√ab√ac√

ab b− 1√bc√

ac√bc c− 1

,

|s3〉 =

√

aN√bN√cN

.

4

a nodes

b nodes

c nodes

a− 1

b− 1

c− 1√

√ab√√

ac√

√bc

Edge

Con

traction

Subset A

Subset B

Subset C

qualities = qA

qualities = qB

qualities = qC

Complete graph, N = a+ b+ c

Weighted complete graph

Subset Aqualities = qA

Subset B

qualities = qB

Subset Cqualities = qC

FIG. 2: An illustration of the edge contraction processfor a quality distribution containing 3 distinct quali-ties, distributed over a complete graph.

This process displays quite clearly a rather signifi-cant advantage of the complete graph. Its behaviourwith regards to amplitude amplification within degen-erate groups of vertices is insensitive to solution place-ment within the solution space or across the verticesof the graph. This hints at the possibility for analyti-cally derived optimal parameters for amplification intoan optimal set of solutions, or at the very least, pa-rameters which are also insensitive to solution place-ment/ordering with respect to their resulting ampli-tude amplification.

D. Optimal parameters for a binary markingfunction on a complete graph

Up to this point, the focus has been on just a sin-gle most optimal solution and convergence into this.However, in reality, there is no way to know the num-ber of solutions marked by a binary marking functionon an unknown solution space, or similarly the num-ber of solutions in the most optimal partition for someother partitioning of the solution space. As such, thefocus will now be on investigating convergence intosome unknown fraction of marked solutions.

The two partition or binary marked problem on

a complete graph is characterised by the followingreduced graph adjacency matrix, quality operatorand initial state, where m represents the number ofmarked vertices (solutions), and N represents the to-tal number of vertices (solutions), namely

Q2 =

[1 00 0

], A2 =

[m− 1

√m(N −m)√

m(N −m) N −m− 1

],

|s2〉 =

[ √mN√

N−mN

].

Consider a single iteration amplified state,

|s′2〉 = UW (t)UQ(γ) |s2〉 .

Substituting m = ρN , where ρ is the ratio of markedsolutions (which is also the initial marked solutionprobability), it is possible to derive an expression forthe probability contained in the marked solutions ofthe amplified state. Dividing by ρ, then taking thelimit for small ρ, the final expression for amplifica-tion of the marked solutions after a single iterationbecomes

3 + 2(cosNt(cos γ − 1)− cos γ)− 2 sinNt sin γ (6)

which takes a maximum value of 9 for γ = π andt = π

N .

Since the above expression for amplification is inde-pendent of m, and any binary marked complete graphcan be characterised by Q2, A2 and |s2〉, the ampli-fication of marked vertices should be independent ofwhether we are looking at a single marked vertex, orany arbitrary number of marked vertices, so long asthe ratio of marked vertices is small. It was demon-strated that convergence into a single marked vertexwas maximised by repeated application of the sameparameters. This result is expected to remain true forgraphs with arbitrary marked-vertex ratios. In otherwords, repeated application of the derived parameters,γ = π and t = π

N , is expected to maximally amplifymarked vertices on the binary marked complete graph,for arbitrary r.

In order to show that the binary marking functionremains the most effective at producing convergenceinto a small group of marked vertices on a very largecomplete graph and across a range of iteration num-bers r, the performance of various partitions of thecomplete graph will be assessed. Namely, 2, 3, 5 and10 part partitions will be assessed. In addition, thederived parameters, γ = π and t = π

N , will be appliedto the binary partitioned complete graph to ensurethat they do indeed produce optimal amplification.In each case, the graph will have a total of N = 108

vertices, with 10 vertices in the marked partition (withquality 1). The remaining vertices will be partitionedinto equal sized groups to achieve the required totalnumber of partitions and each group assigned with asingle quality from those distributed uniformly overthe interval [0,1]. Note that amplified probabilitiesfor the marked vertices are computed using Eq. (4)

5

with the adjacency matrix, quality operator and ini-tial state taken as those for each respective reducedgraph. The process for optimising amplification intothe marked vertices for a given number of iterationsand a given partitioned graph consists of randomlygenerating 10,000 sets of 2r initial parameters.

The three sets of parameters that produce maxi-mum initial amplification are taken as the initial val-ues in a Nelder-Mead optimisation procedure. Thisprocess was repeated 24 times and the maximum prob-ability from the 72 optimised results was taken as thefinal most optimal probability. The results of thisanalysis are shown in Fig. 3, where it is clear thatthe binary (two-part) partition performs the best, andthat the rate of convergence decreases significantly aswe tend towards the typical QWOA process with in-creasing numbers of partitions (i.e. decreasing degen-eracy). The solid curve shows the maximal ampli-fication given by (2r + 1)2, which fits the observedmaximum amplification, a fact that will be addressedin Section III. Repeated application of the derivedoptimal parameters also clearly matches with the re-sults from the optimisation procedure, at least up un-til r = 15, after which point the optimisation pro-cedure is outperformed by the derived parameters,likely because the optimisation procedure is not rig-orous enough to find the global maxima in the higherdimensional optimisation landscapes.

0 5 10 15 20 25 30QWOA iterations, r

0

5

10

15

20

25

30

35

Prob

abilit

y (1

05 )

(2r + 1)2 amplificationTwo part partitionThree part partitionFive part partitionTen part partitionOptimal binary parameters

FIG. 3: Optimised amplification of 10 markedvertices on a complete graph of 108 total vertices,with increasing iterations of the QWOA process.

It is important to note that, for a given circuitdepth, characterised by r QWOA iterations, maxi-mum amplification of a small group of high qualitysolutions can be achieved by using a binary mark-ing function over a complete graph and repeatedlyapplying the derived set of parameters, γ = π andt = π

N . This is the primary mechanism underlyingthe MAOA, and allows the variational component ofthe typical QWOA or QAOA approach to be avoidedentirely. To understand how this process can be imple-mented within a generalised optimisation procedure,it is first important to understand how it is related toGrover’s search.

III. CONNECTION WITH GROVER’SSEARCH

A. Grover’s rotation and diffusion operators

To understand how the MAOA works, it’s usefulto first understand how the alternating applicationof continuous-time quantum-walks over a completegraph and quality dependent phase shifts with a bi-nary marking function are related to Grover’s search[30]. Note that the following is outlined in furtherdetail by Marsh and Wang [29].

Consider a binary marking function which takes asits inputs a threshold quality, T , and a solution qual-ity, f(x), and returns a 1 if the solution quality issuperior to the threshold quality or a 0 otherwise. Ap-plication of this marking function would transform thediagonal Q operator to one with eigenvalues of 1 formarked states and 0 for non-marked states. Whencombined with the optimal phase shift parameter,γ = π, this transforms the quality-dependent phase-shift unitary, UQ(γ), into an operator which applies aπ phase shift to all marked states. Note that this isfunctionally equivalent to the Grover rotation opera-tor, which applies a π phase rotation to the markedstate(s) [30].

The adjacency matrix of the complete graph can beexpressed as, A = N |s〉 〈s| − I, so the quantum-walkunitary applied for time t = π

N can be expressed as

UW (π

N) = e−i

πNA = e−i

πN (N |s〉〈s|−I) = ei

πN (I−2 |s〉 〈s|).

(7)This is equivalent to the Grover diffusion operator [30]up to a global phase, and hence produces equivalentbehaviour with regards to mixing of states.

Given that the two QWOA unitaries applied in se-quence and with the derived parameters, γ = π andt = π

N , are equivalent to a single iteration of Grover’ssearch, and both processes operate on the initial equalsuperposition, then r such QWOA iterations producesthe same amplitude amplification of the marked solu-tions as a Grover’s search terminated after r rotations.The key difference between the two processes is thatGrover’s search aims for complete convergence and re-quires large circuit depths, where as, with restrictedcircuit depths, the MAOA will terminate the processearly and produce only partial convergence into themarked solutions.

B. The low-convergence regime of Grover’ssearch

The amplified probability of the marked states dur-ing a Grover’s search depends only on the number ofcompleted rotations (r) and the ratio of the markedsolutions to the total solution space (ρ = m

N ) [8, 31],as given by

P (r, ρ) = sin((2r + 1) arcsin(√ρ))

2. (8)

6

In the limit of small r and ρ, Eq. (8) reduces to

PLC(r, ρ) = ρ(2r + 1)2, (9)

which gives the probability of measuring marked so-lutions when the convergence into these states is low.This low-convergence estimate is accurate to within1% when the amplified probability is less than 1

40 . InFig. 4, we plot these probability expressions with in-creasing rotation counts, r, relative to the rotationcount required for complete convergence, i.e.

rc =π

4 arcsin√ρ− 1

2.

0.0 0.2 0.4 0.6 0.8 1.0Iterations, r

rc

0.0

0.2

0.4

0.6

0.8

1.0

Mar

ked

node

s pro

babi

lity

P(r, )PLC(r, )

FIG. 4: The amplified probability of the markedsolutions with increasing number of Grover rotations

r relative to rc.

The region within which the low-convergence ap-proximation is accurate shall be referred to fromhere on as the low-convergence regime. When apply-ing a certain rotation count, r, a sufficiently smallmarked-vertex ratio will guarantee that the ampli-fied state will be one that exists within this regime,and hence one that is maximally amplified. Theamplification applied to the marked solutions withinthe low-convergence regime can be read directly fromEq. (9), where ρ is the initial marked-solution prob-ability and hence (2r + 1)2 is the amplification rel-ative to the marked-solution probability in the non-amplified state. Once in this regime, adjusting thethreshold so as to further reduce the marked ratio willnot result in any further amplification of the markedstates, but will instead only reduce the total size ofthe marked set. Note that at this point it shouldbe apparent why the (2r + 1)2 amplification curvewas included in Fig. 3, as it shows perfect agreementwith the amplification produced by the application ofthe derived optimal parameters to the binary-markedcomplete graph.

IV. THE MAXIMUM AMPLIFICATIONOPTIMISATION ALGORITHM

In summary of what has so far been established,for a given circuit depth characterised by r QWOAiterations, a binary marking function on the completegraph produces the maximum possible amplificationof the marked states via repeated applications of theQWOA parameters, γ = π and t = π

N , so long asthe selected quality threshold for the marking func-tion produces a marked ratio small enough to guaran-tee that the amplified state is in the low-convergenceregime of a functionally equivalent Grover’s search ter-minated after r rotations. Note that with respect tofinding the most optimal solutions, repeated prepa-ration and measurement of this maximally amplifiedstate represents an upper bound to the speedup avail-able within the QWOA or QAOA frameworks. Muchof this section will therefore be focused on present-ing a method to reliably and efficiently find a qualitythreshold which will produce this maximally amplifiedstate.

Note that whether operating within the QWOAframework or that of the truncated Grover’s search,the MAOA remains functionally equivalent, so fromthis point on, the Grover framework will be adopted,i.e. r will be referred to as the number of rotations.

A. Threshold response curves

In order to understand how the MAOA locatesa maximally amplifying quality threshold, it is firstuseful to introduce the concept of a threshold re-sponse curve, which quantifies, for a fixed numberof rotations, r, how the probability of measuring amarked solution varies with the quality threshold, T .The threshold response curve is given by Eq. (8) asP (r, ρ(T )) where ρ(T ) is the marked-solution ratioproduced by the marking function. The threshold re-sponse of a system represents the rate of successfulmeasurements of marked solutions. A strong responsewould occur where the probability of measurement isclose to 1. An example threshold response curve forr = 128 is shown in Fig. 5. This curve is the re-sponse of a system which has a quality distributionmatching that of the standard normal distribution,i.e. mean quality of 0, standard deviation equal to1, and with a sufficiently large number of solutionssuch that the distribution of solution qualities is ap-proximately continuous. This response curve is for aminimisation problem, i.e. the marked set is all solu-tions with qualities less than T .

In general, the total combined number of peaks andtroughs at either side of the mean is equal to the ro-tation count, r, so the response curve in Fig. 5 con-tains 64 peaks and 64 troughs. It is useful to definethree different regions or regimes within the thresh-old response curve, each of which is labelled in Fig. 5.The chaotic regime refers to the range of thresholdswithin which there is a tightly spaced fluctuation be-tween high and low response. In terms of a tradi-tional Grover’s search, this is where the number of

7

6 5 4 3 2 1 0Quality threshold, T

0.0

0.2

0.4

0.6

0.8

1.0

Low-convergence

High-convergence

Chaotic P

(128

,(T

))Th

resh

old

resp

onse

cur

ve,

FIG. 5: Threshold response curve for r = 128amplified states over a solution space with qualitiesdistributed as per the standard normal distribution.The low-convergence, high-convergence and chaotic

regimes are all labelled accordingly.

rotations significantly exceeds the minimum numberrequired for complete convergence, r > 2rc. The high-convergence regime refers to the region around themost optimal quality at which we see peak response.Again, in terms of a traditional Grover’s search, thisis where the number of rotations first approaches thatrequired for complete convergence, 0.1rc < r < 2rc.Finally, the low-convergence regime refers to the re-gion of qualities in which solutions are too few in num-ber for r rotations to be sufficient to produce high-convergence. Note that this low-convergence regimeis identical to that defined earlier, where maximumamplification of marked solutions is achieved, and cor-responds with P (r, ρ(T )) < 1

40 and r < 0.1rc.So the goal of the first part of the MAOA is to

find a threshold, T , which is located within the low-convergence regime of the threshold response curveP (r, ρ(T )), where r corresponds with the restrictedcircuit depth at which the final process of repeatedstate preparation/measurement is to be carried out.Lowering the threshold and monitoring the responseof the system at the final value of r is not likely tobe a practical method, as it requires navigation ofthe chaotic regime in a controlled fashion, which con-tains r

2 − 1 peaks and requires a highly accurate esti-mate of the mean. On the other hand, there exists amuch more efficient method to navigate from an ini-tial threshold located loosely around the mean on ther = 1 response curve, through to a final thresholdwithin the low-convergence regime of the final r re-sponse curve, doubling the rotation count as required.

B. Navigating the threshold response curves

The process of navigating from the mean on ther = 1 response curve, to the low-convergence regimeon the final r response curve is illustrated in Fig. 6,

where the final value for r is taken at 8. This it-erative process of doubling the rotation count, andmoving from the high-convergence peak of one curveto the trough below it on the next curve, allows forthe threshold to remain in the well-behaved high-convergence regime. This allows for reliable naviga-tion from one peak to the next until the rotation countcorresponding with the desired final circuit depth isreached. For this method, the number of peaks thatmust be navigated to arrive at the final peak at rgrows with log2(r). For contrast, navigating the entirethreshold response curve at the final r would involvea number of peaks which would grow linearly withr, not to mention the fact that any initial thresholdwould not have a well-defined position on the thresh-old response curve due to the tight spacing of peaksaround the mean. The other benefit of navigatingthrough successive response curves is that state prepa-rations/measurements made during the early stages ofthe process, at low r, incur significantly lower compu-tational effort.

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.00.0

0.2

0.4

0.6

0.8

1.0r=1r=2r=4r=8

4 3 2Quality threshold, T

P(r,

(T))

Thre

shol

d re

spon

se c

urve

,

FIG. 6: An illustration of how the MAOA navigatesacross threshold response curves from a threshold

near the mean for the r = 1 curve to a final thresholdin the low-convergence regime of the r = 8 curve.

It is important to note the necessary connection be-tween doubling the rotation count and the peak totrough relationship between successive threshold re-sponse curves, which is not just a coincidental rela-tionship specific to the standard normal distribution.The peak in the high-convergence regime for a givenr occurs when the argument of the sine function inEq. (8) is equal to π

2 , conversely, the trough imme-diately preceding the high-convergence regime occurswhen this argument equals π. Since this argumentis proportional to r, the doubling of r at the neces-sary threshold, transforms from the peak in the high-convergence regime at r to the trough immediatelypreceding the high-convergence regime at 2r. This re-lationship becomes much tighter for large r, but stillholds sufficiently true at low r, as is made evident inFig. 6.

The peak finding method is presented in detail inAppendix C but essentially reduces to recording thenumber of successful marked solution measurementsmade in a row at steadily improving thresholds (skip-

8

ping to the next threshold anytime a measurementis unsuccessful at producing a marked solution). If20 successful measurements are made in a row, thethreshold is held, and the rotation count doubled(this happens most of the time and happens near thepeak). If 20 consecutive successful measurements arenot made, the process of checking successively improv-ing thresholds is continued until the threshold is pastthe peak, at which point a weighted mean of the suc-cess counts is performed to locate the peak. Assess-ing whether the peak has been passed, is done ini-tially by making use of the interquartile range of asmall sample of the non-amplified solution space, andeventually the peak to peak gaps of the previouslynavigated threshold response curves. These previouspeak to peak spacings are also used to derive an adap-tive threshold step-size which is updated throughout,which helps to account for the narrowing of peak topeak gaps with increasing r, or any other changes inthe distribution at larger separations from the mean.

When arriving at the final peak (within the high-convergence regime of the final r) an adaptive search[32, 33] with fixed circuit depth can be performed,knowing that any threshold where P (r, ρ(T )) ≤ 1

40is guaranteed to be located in the low-convergenceregime and hence will be one that generates maximumamplification in all marked solutions for the given r.This adaptive search procedure is shown in Fig. 6 asthe final descent on the r = 8 curve.

An important feature of this threshold finding pro-cess is that it operates independently of the exactshape of the underlying distribution in solution qual-ities. This is important, because if the distributionis known exactly, and can be accurately fit with arelatively efficient sampling process, then a thresholdwithin the low-convergence regime can be trivially de-duced. This may be appropriate in some cases, but thereality is, for larger rotation counts, the relevant qual-ity thresholds are located at large deviations from themean, and the behaviour of the quality distributionsin these regions becomes less certain without signifi-cant sampling effort. Even for problems with solutionspaces which are in general normally distributed, thedistributions can vary from the perfect normal distri-bution at large deviations from the mean, for example,in the vehicle routing problem within Section VI. Inany case, due to imperfections in a problem’s actualquality distribution compared to its idealised form,extrapolating from a classical sampling can result inselection of a threshold which is too close or too farfrom the mean. If too close to the mean, a thresholdcould be located within a trough in the chaotic regime,exhibiting a threshold response which would be oth-erwise identical to a maximally amplifying threshold,but one that produces significantly less amplification.If too far from the mean, there may not be any markedsolutions at all, or at least so few that they may notbe measured at all.

C. General strategy

To summarise, the general strategy for the MAOAconsists of two parts:

Part 1: For a given problem and restricted rota-tion count, r, find a quality threshold which producesmaximum amplification of the marked solutions(through the application of r Grover rotations).Note that this threshold will be one within the low-convergence regime and hence will amplify markedsolutions by the maximum factor of (2r + 1)2.

Part 2: Using the quality threshold acquired inpart 1, repeatedly prepare and measure the maxi-mally amplified state. The repeated measurement ofthis amplified state will produce random high qualitysolutions from the marked set with a probabilityof approximately 1

40 or less, depending on the finalmarked-solution ratio.

D. Computational effort

In order to assess the performance of the MAOA, itis important to be able to quantify the computationaleffort that it expends. Assuming that the computa-tion of solution qualities is a task that requires a sig-nificant fraction of the total computational effort, wewill use the number of calls to the quality function,f(x), as a measure of computational effort. Note thatin both the Grover and QWOA framework, each timethe marked solutions are phase shifted, the qualityfunction is effectively called twice, once to mark therelevant solutions, and once for the uncomputationprocess, to reset relevant ancillary qubits for subse-quent iterations. In addition, once a final solution ismeasured from the amplified state, its quality muststill be computed, adding one more call to qualityfunction. As such, the preparation and measurementof an amplified state incurs a computational effort of2r+1. In addition, a classical sampling of the solutionspace incurs a computational effort equal to the num-ber of samples, as it requires only a single call to thequality function for each randomly selected solution.Note that to fully quantify any speedup relative toclassical sampling, the computational effort involvedin the quantum-walk/mixing process should also beaccounted for. This will not be considered as partof this work, however, the number of walks/mixes isequal to the rotation count, so would pose only a linearoverhead on top of the computational effort associatedwith the objective function calls.

E. Note on the use of expectation value ofquality

Very recently, Golden et al. [34], propose a mod-ification of the QAOA which makes use of a GroverMixer combined with a binary marking function. Notethat the Grover mixer is effectively a continuous-timequantum walk over the complete graph [29], and so

9

the underlying mechanic of their algorithm is close tothat of the MAOA. The primary differences are thatthe performance of the amplified state is monitoredvia the expectation value of quality, the final pair ofparameters are left free for a tuning process, and oth-erwise the parameter corresponding with the Grovermixer is taken as π rather than π

N . As an aside, theperiodicity of the walk on the complete graph is such

that a walk time of (2i+1)πN , where i ∈ {0, 1, 2, . . . },

will produce maximum mixing, but a walk time whichis some multiple of 2π

N will complete an integer numberof cycles and produce no mixing and hence no ampli-fication. This can be observed in the amplificationexpression in Eq. (6). As such, the mixing parameterequal to π will function well only for a solution spacewith an odd number of solutions.

By monitoring the expectation value of quality,Golden et al. [34] choose a threshold for their markingfunction which maximises the expected quality pro-duced by the amplified state. The issue with this isthat a threshold which maximises expectation valueof quality will not produce maximum amplification ofthe highest quality solutions. In fact, measurementof a state in which the most optimal solutions aremaximally amplified will only occasionally produce amarked solution (approximately 1 in every 40 mea-surements or less), and hence will have an expectationvalue for quality which is close to the mean quality ofthe solution space. The expectation value is thereforenot a useful metric in determining whether a stateproduces maximum amplification of the most optimalsolutions. This is illustrated in Fig. 7, where a 128-rotation amplified state is prepared over a solutionspace with qualities distributed as per the standardnormal distribution. The expectation value of qual-ity is shown relative to a target quality correspondingto the minimum solution out of 100 million total so-lutions. The amplification of the marked set is alsoshown, relative to the maximum possible amplifica-tion.

Note that by tuning the applied parameters, it ispossible to produce expected qualities along the up-per envelope of the MAOA curve, as shown with thedashed curve in Fig. 7. This dashed curve matchesthe observed behaviour shown in Figure 1 of the pa-per by Golden et al. [34]. With fixed parameters, the128-rotation amplified state of the MAOA oscillates inaccordance with the chaotic regime of Grover’s search.As shown in Fig. 7, the threshold which produces thebest expectation value for quality corresponds with anamplification of the marked set which is on the orderof only 40% of the maximum possible amplification.The MAOA threshold, on the other hand, producesessentially the maximum possible amplification. Ig-noring the computational effort involved in the tuningof the phase shift and mixing parameters involved inthis binary threshold version of QAOA, as well as theprocess of optimising expected quality, this approachwould not be capable of producing the same speedupas the MAOA simply due to the fact that it producesless than half of the maximum possible amplificationof the marked solutions.

5 4 3 2 1 0Threshold (T)

0.0

0.2

0.4

0.6

0.8

1.0

Rela

tive

ampl

ifica

tion,

P(r,

(T))

(T)(2

r+1)

2

5 4 3 2 1 0

0.0

0.2

0.4

0.6

Rela

tive

expe

cted

qua

lity,

Q Qmin

MAOA stateTuned parameters

3M

AOA

Thre

shol

d

2 T

hres

hold

3Go

lden

et a

l.

FIG. 7: Threshold response for the expectation valueof quality over a quality distribution matching the

standard normal distribution.

V. THE GROVER ADAPTIVE SEARCH AS ABENCHMARK

As mentioned in the introduction, a highly effectivequantum optimisation algorithm already exists, calledthe Grover adaptive search (GAS) [7–9]. However, itis unlikely to be implementable on near-term quan-tum devices, due to a requirement for large rotationcounts of O(

√N). Nevertheless, the performance of

this algorithm makes for a useful baseline for compar-ison against the performance of the MAOA. The Durrand Høyer (randomised rotation count) variation ofthe algorithm will be employed for comparison pur-pose, which is summarised by Baritompa et al. [8].The GAS effectively uses the Grover’s search proce-dure to amplify the iteratively improving marked setand perform a hesitant adaptive search [33], whichis essentially a pure adaptive search [32] where theprobability of successfully finding an element in theimproving set is generally less than 1.

Since the GAS navigates the solution space in arandom manner, there is no way of knowing whether,for a particular rotation count, the chosen thresholdfalls within the chaotic regime or not, and hence therotation count must be randomised throughout. Asthe marked set improves, and hence the marked ratiodecreases, the rotation count required to suitably am-plify the marked set increases. So although the rota-tion count needs to be randomised, it is randomly se-lected from a uniform distribution which has a steadilygrowing upper-bound. The rate at which this rotation

10

count upper-bound grows is controlled by a parame-ter, λ. In their work, Baritompa et al. [8] show thatthe GAS performs optimally on a solution space withuniformly distributed qualities when λ = 1.34. As itwill become clear in Section VI, combinatorial optimi-sation problems often have solution spaces which pos-sess something closer to a normal distribution in quali-ties. We confirm via simulation that λ = 1.34 remainsoptimal for normally distributed solution spaces. Assuch, we will use this value in subsequent simulations.

A novel but quite natural modification of the GAS,making it more suitable for the restricted circuit depthcontext of near-term quantum computation, is sim-ply to place a limit on the maximum allowable rota-tion count, where the procedure is otherwise identical.This modified version of the GAS will now be referredto as the restricted Grover adaptive search (RGAS).As will be seen in Section VI, the RGAS remainseffective in providing speedup relative to a classicalsampling method in terms of finding most optimal so-lutions. The speedup is related to the limit placedon the rotation count, with larger rotation count re-strictions producing better performance, closer to thatof the original GAS, and unsurprisingly, smaller re-strictions performing worse. The RGAS will there-fore make for a useful comparison with the MAOA,as they can both be restricted to the same circuitdepth/rotation count, leveling the playing field, andensuring comparison between two algorithms whichare equally well suited to the context of near-termquantum computation. Note that we should expectthe MAOA to outperform the RGAS, because theRGAS requires a randomised rotation count, mean-ing that many of the amplified states will be preparedusing smaller rotation counts than the upper limit,and hence won’t make use of the maximum possi-ble amplification, whereas, once the final thresholdhas been located, the MAOA produces the maximallyamplified state every time. Note also that the typicalQWOA and QAOA procedures have not been includedfor comparison, since the MAOA achieves the upperlimit of amplification available within these frame-works, and does so without the need for an optimi-sation/variational procedure to arrive at the optimalphase and walk parameters. The MAOA does requirea variational procedure to arrive at the final qualitythreshold, but this is a single dimensional optimisationand is hence significantly less computationally inten-sive.

VI. NUMERICAL SIMULATION

Now that the Maximum Amplification Optimi-sation Algorithm (MAOA), Grover adaptive search(GAS) and its restricted circuit depth variant (RGAS)have all been established, their performances will becompared in the context of combinatorial optimisa-tion. Firstly, each of the three algorithms will beapplied, via numerical simulations, to a capacitatedvehicle routing problem. Next, they will be applied toa portfolio optimisation problem. Lastly, the MAOAand the RGAS will be compared in the limit of arbi-

trarily large normally distributed solution spaces. Inevery case, curves for success probability vs. com-putational effort will be computed from the results of10,000 simulations. Each simulation operates over theactual distribution of solution qualities for each par-ticular problem, where each of these distributions isprecomputed. The key assumptions underlying eachof the simulations are listed below, and are consistentwith the dynamics of a truncated Grover’s search:

1. A given threshold, T , produces a marked-solution ratio, ρ(T ), which is fully defined bythe precomputed quality distribution.

2. For a given threshold, T , and rotation count, r,the probability of measuring a marked solutionis given by P (r, ρ(T )) as per Eq. (8).

3. When a marked solution is successfully mea-sured, it has an equal probability of being anyone of the marked solutions, so the returnedsolution is randomly selected (uniformly) fromthe full set of marked solutions for the specifiedthreshold.

4. The computational effort required for eachpreparation and measurement of an amplifiedstate is quantified by 2r+1, as per Section IV D.

Note that in reality, the MAOA and RGAS may re-quire computational effort closer to 2r or (2r− 1) perstate preparation and measurement. This is partlybecause on the final rotation, the uncomputation pro-cess may not be required, but mainly because most ofthe time, measurement collapses the system into anunmarked state, of which the quality does not needto be computed. As such, it would suffice to mea-sure just the binary quality register, only needing tosubsequently measure the solution register and com-pute the quality when the amplified state collapsesinto the space of marked solutions. Regardless of this,2r+1, has been maintained as a measure for the com-putational effort for each measurement of an amplifiedstate, since the difference is not significant.

A. The capacitated vehicle routing problem

A detailed theoretical framework for the capaci-tated vehicle routing problem is presented by Bennettet al. [23] and adopted without modification here forthe purpose of generating a solution space and corre-sponding quality distribution for analysis. The prob-lem essentially involves seeking the lowest cost routesfor delivering supplies from a central depot to a num-ber of external locations. The cardinality of the solu-tion space is given by

NCV (l) =

l∑k=1

(l − 1

k − 1

)l!

k!, (10)

where l is the number of locations. For the followingsimulation, we set l = 10 giving N = 58, 941, 091.

11

250 275 300 325 350 375 400 425Cost

0.0

0.2

0.4

0.6

0.8

1.0Fr

eque

ncy

1e6

(a)

250 275 300 325 350 375 400 425Cost

100

101

102

103

104

105

106

Freq

uenc

y

(b)

FIG. 8: Distribution of solution qualities for therandomly generated 10-location vehicle routing

problem, shown with (a) a linear scale and (b) a logscale.

To generate a quality distribution, the package vec-tor was randomly generated from integers on the in-terval [5,30], a symmetric cost matrix was generatedwith depot to location costs randomly generated fromintegers [10,20] and inter-location costs from integers[1,15], and finally, the vehicle capacity was taken as20. Note that integer values were used in the costmatrix to increase degeneracy in the quality distribu-tion of the solution space, just to provide some va-riety compared to the portfolio optimisation problemwhich shows virtually no degeneracy. The distribu-tion of solution qualities is shown in Fig. 8a, where it isclear that the qualities are distributed in what approx-imates a normal distribution. The same distributionis shown with a log-scale in Fig. 8b, in which it be-comes more clear that the distribution is not perfectlynormally distributed, but rather, the distribution atlower costs is somewhat discontinuous and truncated.In fact, there are 12 solutions which share the lowestcost.

The simulation results of the MAOA, GAS andRGAS applied to this 10-location CVRP are shownin Fig. 9 along with the classical sampling method,where the success probability refers to the probabilityof one of the 12 highest quality (lowest cost) solu-tions being measured. The GAS method is includedto show how an algorithm unrestricted in circuit depth

would perform. The rotation count that would be re-quired to produce complete convergence into a singlemost optimal solution over a solution space of thissize is given by rc = 6, 029. Since a user knows thesize of the solution space, but not the degeneracy ofthe most optimal solution, this is the user-specifiedmaximum rotation count required for the GAS in thisinstance. In contrast, the MAOA and the RGAS aretested with restricted circuit depths corresponding torotation counts of r ∈ {8, 16, 32, 64, 128}. In all cases,the quantum algorithms provide speedup over a clas-sical sampling method, but the amount of speedup in-creases with increasing rotation counts. The MAOAalso outperforms the RGAS, as predicted in Section V.To clarify, due to the higher upfront computationalexpense of the threshold finding process, the RGASbegins to perform better than the MAOA as r ap-proaches rc, but this is not likely to be relevant in thecontext of large solution spaces and restricted circuitdepth near-term quantum computing.

0 100000 200000 300000 400000 500000Effort

0.0

0.2

0.4

0.6

0.8

1.0Su

cces

s pro

babi

lity

r = 8 r = 16

, r = 32 r = 64

r = 128

Grover adaptive search (unrestricted), rmax = 6029

Restricted Grover adaptive search Classical sampling

Maximum Amplification Optimisation Algorithm

FIG. 9: Simulation results for a large vehicle routingproblem.

B. Portfolio optimisation

It was demonstrated by Slate et al. [22] that a port-folio optimisation problem based on the Markowitzmodel [35] was a suitable candidate for application ofthe QWOA framework, as such, it makes an equallysuitable candidate for the MAOA and GAS methods.This work will give a somewhat different treatmentto the problem, however, by treating the two com-ponents of the objective function, risk and expectedreturn, seperately. Given n different stocks/assets,a particular choice of portfolio can be expressed byz = (z1, z2, ..., zn), where for each asset i we havezi ∈ {−1, 0, 1}, where each value corresponds with ashort position, no position, or long position, respec-tively. In addition, the portfolio positions are con-

12

0.50 0.25 0.00 0.25 0.50 0.75 1.00 1.25Return

1.0

1.5

2.0

2.5

3.0

Risk

0.0

0.5

1.01e6

0.5 1.01e6

104

103

102

101

10050.0

FIG. 10: Distribution of expected returns and risks associated with each portfolio choice for a large portfoliooptimisation problem.

strained by the net position, I, such that:

n∑i=1

zi = I. (11)

Under the Markowitz model, using data for thedaily expected percentage return of asset i, Ri, andthe co-variance values between assets i and j, σij , fora group of n assets, the expected return and associatedrisk for each portfolio choice can then be characterisedby:

Return =

n∑i=1

Rizi (12)

Risk =

n∑i,j=1

σijzizj . (13)

The number of unique and valid portfolio choicesavailable, or the cardinality of the solution space, isgiven by:

NP (n, I) =

bn−I2 c∑s=0

(n

I + s

)(n− I − s

s

), (14)

where s represents the number of possible shorts, thefirst term represents the placement of longs within z,and the second term represents the subsequent place-ment of the shorts. For the purpose of generatingan example solution space and corresponding qualitydistribution, data for the daily adjusted close pricesfrom 01/01/2019 to 31/12/2020 was analysed for 20

different stocks from the ASX.20 index: AMP, ANZ,AMC, BHP, BXB, CBA, CSL, IAG, MQG, GMG,NAB, RIO, SCG, S32, TLS, WES, BKL, CMW, HUB,ALU.

The net position, I, was taken to be 7, such thatthe total number of solutions or portfolio choices wasN(20, 7) = 61, 757, 600. The distribution of risks andreturns for the resulting solution space is shown inFig. 10, where risks are scaled down by a factor of 100.The distribution resembles a 2D Gaussian, skewedtowards high risk portfolios. In order to navigatethe 2-dimensional optimisation landscape, the mark-ing function will use two thresholds, one for risk andone for return. It is presumed that a balance betweenoptimising for low risk while still maximising returnsis desirable. As such, the risk threshold will be set andfixed at that corresponding to the lowest 10% of all so-lutions, transforming the optimisation problem into amaximisation of expected return within the subspaceof low-risk solutions. As is discussed in more detail inSection VII, the MAOA presents a unique ability to beable to navigate multidimensional optimisation land-scapes, but because the GAS/RGAS do not have thesame ability, the problem must be transformed into a1 dimensional problem to allow for effective compari-son between the different methods.

The simulation results for this problem are shownin Fig. 11, where the probability of success is takenas the probability of finding the single highest returnportfolio from those within the lowest 10% for risk.The user-specified maximum rotation count requiredfor the GAS in this instance is given by rc = 6, 172,which is derived directly from the known size of thesolution space. In contrast, the MAOA and the RGASare tested with restricted circuit depths correspond-

13

ing to rotation counts of r ∈ {32, 64, 128, 256, 512}.The reason these rotation counts are higher than forthe CVRP problem, is because here, there is only asingle most optimal solution, compared to 12 in thecase of the CVRP problem. The smaller ratio of mostoptimal solutions requires higher rotation counts forcomparable performance. In any case, the results areconsistent with those for the CVRP simulations. Inall cases, the quantum algorithms provide speedupover a classical sampling method, with the amountof speedup increasing with increasing rotation counts.The MAOA also once again consistently outperformsthe RGAS.

Grover adaptive search (unrestricted), rmax = 6172

r = 32

Grover adaptive search (unrestricted), rmax = 6172

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Effort 1e6

0.0

0.2

0.4

0.6

0.8

1.0

Succ

ess p

roba

bilit

y

Restricted Grover adaptive search Classical sampling

r = 64 r = 128

, r = 256 r = 512

Maximum Amplification Optimisation Algorithm

FIG. 11: Simulation results for a large portfoliooptimisation problem.

C. Simulating an arbitrarily large problem

The above problems are of a scale where the qual-ity distributions can be readily computed on a desk-top computer. In reality, the MAOA is designed tobe applied to problems with significantly larger so-lution spaces, which are intractable through classicalmethods. It is therefore valuable to understand howthe MAOA performs in the limit of very large prob-lems. As can be seen in the CVRP and portfolio opti-misation problems, combinatorial optimisation prob-lems often possess solution spaces which have qualitiesthat are normally distributed. It is therefore possibleto simulate an arbitrarily large problem by using thestandard normal distribution. It is not unreasonableto think that a large enough problem would have aquality distribution which is normally distributed andwhich approximates a continuous distribution, even atlarge deviations from the mean. For such a problem,it is therefore possible to take the marked-solution ra-tio, ρ(T ), as the cumulative distribution function ofthe standard normal distribution at T . This allows

for the RGAS and MAOA to be simulated in optimis-ing an arbitrarily large normally distributed problem.

For the purpose of assessing the behaviour of theRGAS and MAOA algorithms in the limit of largeproblems, they will be analysed for a constant re-stricted rotation count, r = 64. In each case, theywill be seeking a solution within a certain most op-timal target group, forming a fraction of the solu-tion space, referred to as the target ratio, µ. TheRGAS does this by sequentially partitioning the solu-tion space into smaller and smaller improving subsets,until a solution within this target group is measured.On the other hand, for a maximum rotation countof r = 64, the MAOA does this by repeatedly mea-suring from a state prepared with a threshold whichproduces maximum amplification. Since this is knownto occur when the probability of successfully measur-ing a marked solution, P (r, T ) ≤ 1

40 , and also when

the probability is amplified by a factor of (2r + 1)2,the final marked-solution ratio will be approximatedby:

ρ(r) =1

40(2r + 1)2. (15)

So in this case, the MAOA will be somewhat regu-larly measuring marked solutions within roughly thetop ρ = 1.5× 10−6, regardless of the target ratio, µ. Itis through this method that the MAOA seeks to finda solution from the target group. The performanceof each algorithm has been simulated over a rangeof target ratios, µ ∈ {10−6, 10−7, 10−8, 10−9, 10−10},and the results are shown in Fig. 12.

2 3 4 5 6 7 8 9log(Effort)

0.0

0.2

0.4

0.6

0.8

1.0

Succ

ess p

roba

bilit

y

Probability of finding a target solution (r = 64, SND)

o = 1e-6Restricted Grover adaptive search, r = 64, Target ratio =

Target ratio = 1e-6 Target ratio = 1e-7 Target ratio = 1e-8

Target ratio = 1e-9 Target ratio = 1e-10

Maximum Amplification Optimisation Algorithm, r=64

Ma Am Am Op : predicted performance, r = 64

FIG. 12: The simulated performance of both theMAOA and RGAS in optimising arbitrarily large, nor-mally distributed solution spaces. Theoretically pre-dicted curves for the MAOA are also included, whichare derived in Section VII.

14

VII. ANALYSIS OF THE MAXIMUMAMPLIFICATION OPTIMISATION

ALGORITHM IN THE LARGE PROBLEMLIMIT

To understand how well the MAOA performs rel-ative to a classical sampling of the solution space, itis useful first to quantify the probability of success-fully finding a target solution for the classical case.Since the behaviour of interest is that in the limit ofvery large solution spaces, it is suitable to ignore theremoval of sampled solutions from the solution space(i.e. sampling with repeats). Note this assumption isonly reasonable when the target ratio is much largerthan that for a single target solution, µ � 1

N . Giventhis assumption, the equation for the probability ofsuccess in the classical case is given by:

PC(e, µ) = 1− (1− µ)e. (16)

Note that e refers to the computational effort, butin this case, can also be understood as the numberof classical samples taken, since they are both equiva-lent. The equation is best understood as being derivedfrom the complement of a successful measurement, i.e.the probability of success after e samples is equal toone subtract the probability of no successes after esamples. The probability of failing to sample a targetsolution e times in a row is clearly (1− µ)e, since µ isthe probability of a successful sample.

The equation for probability of success for theMAOA can be derived in a similar fashion, Sincethe MAOA, once an appropriate threshold has beenfound, essentially reduces to repeated preparation andsubsequent measurement of the maximally amplifiedstate. The equation is therefore given by:

PQ(e, µ, r) = 1− (1− µ(2r + 1)2)e

2r+1 . (17)

The (2r + 1)2 term relates directly to the maxi-mum amplification of marked states due to r rota-tions. The power, e

2r+1 , gives the number of measure-ments, since each state preparation and measurementrequires computational effort (2r+1). As can be seenin Fig. 12, this analytically derived expression for thesuccess probability is consistent with simulation re-sults in the limit of small target ratios (i.e. thosesignificantly smaller than the final value for ρ).

Rearranging each of these equations for the effortin the classical case, eC , and the quantum case, eQ,taking their ratio and taking the limit in small targetratios, an expression for the speedup of the MAOAover classical sampling can be derived:

limµ→0

eCeQ

= limµ→0

log(1− µ(2r + 1)2

)(2r + 1) log(1− µ)

= 2r + 1. (18)

Note that this result can also be understood intu-itively, since each state preparation provides amplifi-cation of (2r + 1)2 at the expense of (2r + 1) compu-tational effort, what remains is a speedup of (2r+ 1).This result essentially implies that the MAOA is capa-ble of producing speedup (over classical sampling) infinding near-optimal solutions to large combinatorial

optimisation problems, and that this speedup growslinearly in the achievable circuit depth. This outcomeappears to represents the upper limit of speedup avail-able by an amplitude amplification algorithm in thecontext of restricted circuit depths and large prob-lems. Though the speedup is not as significant asthat for an unrestricted GAS, it represents the bestachieved by any current algorithm which is equallyconstrained in terms of circuit depth, and as such,the MAOA seems to be the best existing algorithmfor combinatorial optimisation in the context of near-term quantum computation.

The MAOA presents additional useful features be-yond providing maximum speedup for a given re-stricted circuit depth:

1. The analytically derived expression in Eq. (17)can be used to inform a user how likely theyare to have found a solution within a particulartarget ratio, µ, after a specified amount of com-putational expense. Note that due to its randomnature, the RGAS has no such feature.

2. The MAOA also provides additional flexibil-ity for multi-dimensional optimisation problems.For example, in the portfolio problem, thepeak finding process, implemented over the riskthreshold, can be used to isolate a known frac-tion of the lowest risk options. Fixing the riskthreshold and transitioning to optimisation overthe return threshold then allows the user to op-timise within this space of lowest risk options.Note that this multi-stage optimisation proce-dure can be generalised to other problems too.

3. Repeated sampling of the MAOA amplified stateproduces a large set of near optimal solutions. Incontrast, the RGAS only produces one solutionat each of the measured improving qualities. Atthe tail end of the RGAS procedure, these nearoptimal/improving solutions would be measuredextremely rarely. On the other hand, for theMAOA, the near optimal solutions are measuredregularly throughout the duration of sampling.Note that the ratio of these regularly sampledmarked solutions can be approximated as perEq. (15). This is beneficial in acquiring a signif-icantly larger group of near-optimal solutions,which may be of interest in some cases. For ex-ample, it may allow one to then select betweensolutions for features not accounted for withinthe optimisation procedure or alternatively tosimply have access to back-up high quality solu-tions.

VIII. CONCLUSION

This paper serves as a comprehensive introductionto the Maximum Amplification Optimisation Algo-rithm (MAOA), a near-term quantum algorithm de-signed for finding high quality solutions to large com-binatorial optimisation problems, while constrained torestricted circuit depths. Other existing near-term

15

algorithms, QAOA and QWOA, focus on producingamplified states in which the expected value of qual-ity has been optimised. When measuring from suchan amplified state, we expect to find a solution of highquality. As we have demonstrated, however, the high-est quality solutions in such a state are not amplifiedto the maximum extent possible.

The MAOA shifts the paradigm by seeking an am-plified state in which the highest quality solutions aremaximally amplified, rather than focusing on max-imising the expectation value of the quality function.The power of the MAOA comes from repeated sam-pling of the maximally amplified state. Since thehighest quality solutions are amplified to the maxi-mum extent possible, subject to a given circuit depth,the frequency with which they will be measured isalso maximised, hence delivering maximum possiblespeedup. Moreover, the MAOA is shown to providea speedup that is extraordinarily close to the upperbound of what is possible within the QAOA/QWOAframework.

We also demonstrate that these maximally ampli-fied states can be produced via a known set of pa-rameters, which removes entirely the computationallyexpensive variational process typically associated withthe QAOA and QWOA algorithms. A number of otherfeatures which are advantageous to the MAOA arealso discussed, including an expression which accu-

rately predicts the probability of success for findingsolutions of a specified minimum quality. It is arguedtherefore, that the MAOA represents the best exist-ing quantum algorithm for combinatorial optimisationproblems in the context of near-term quantum compu-tation. Future work is likely to involve analysis of theMAOA’s sensitivity to noise, as well as a focus on ac-tual physical implementation on near-term quantumhardware.

IX. ACKNOWLEDGEMENTS

We would like to thank Sam Marsh and EdricMatwiejew for their important contribution to thiswork. To Sam, thank you for providing a number ofimportant insights, including inspiration for the graphreduction process, as well as pointing out the connec-tions between QWOA and Grover’s search. To Edric,thank you for all your work developing and assisting inthe use of the QWOA simulation software with whichthe proficiency of a binary marking function was firstnoticed, leading the way to the MAOA’s development.This work was also supported with resources providedby the Pawsey Supercomputing Centre, with fundingfrom the Australian Government and the Governmentof Western Australia.

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Appendix A: Investigation of convergence rateinto a single most optimal state

In order to understand how a graph’s structure mayeffect its capacity to produce convergence into a singlevertex, it is important to first define some parameters.The first is the average degree of a graph, which is theaverage number of edges connected to each vertex onthe graph, referred to from here on by the variable,D. The second parameter considered, relates to thespectral quality of the graph’s adjacency matrix, ormore specifically, the number of distinct eigenvaluespossessed by the graph’s adjacency matrix, from hereon referred to as spectral count, E. In referring todifferent graphs within the following figures, the pa-rameters will be combined into a single label: DxEy,where x is the value for D, and y is the value for E.

In order to assess how the convergence rate varieswith these parameters, a number of 24-vertex graphs(N = 24) have been analysed. The reason for choosing24 vertices in particular is because of its large num-ber of divisors, allowing for a large number of uniquecirculant graph structures. Each graph has been char-acterised by an adjacency matrix consistent with somespecific values for D and E. Three iterations of theQWOA process were applied to each graph to gen-erate the final state of each graph, |γ, t〉, given byEq. (4) in combination with 6 variational parameters,t = (t1, t2, t3) and γ = (γ1, γ2, γ3). Each graph wasassigned 48 different randomly generated quality dis-tributions with each quality randomly sampled froma uniform distribution over [0, 1). The 6 parameterswere optimised in each case to produce maximum am-plification in the most optimal vertex. The optimisa-tion procedure consists of producing 10,000 randomlygenerated sets of initial parameters. Each of the 10parameter sets that produce maximum amplificationis used as the initial set of values in a Nelder-Meadoptimisation procedure. The most optimal of theseis taken to be the final amplified probability for thatparticular graph and quality distribution. This op-timisation procedure was repeated for the 48 differ-ent quality distributions assigned to each graph, fromwhich the mean and standard deviation of the ampli-fied probability was computed.

Perhaps the first thing to clarify, is that graphswith consistent values for D and E produce consis-tent convergence rates, to ensure there is not someother important factor requiring consideration. Assuch, 9 distinct graphs, each with values D = 12 andE = 12, were generated randomly from all such circu-lant graphs. The final amplified probabilities for eachof these 9 graphs is shown in Fig. 13a, from which itcan be concluded that there is no significant differ-ence in the convergence rates between them. As such,it seems likely that circulant graphs with the sameaverage degree and spectral count produce consistentbehaviour in terms of convergence rate into a singlemost optimal vertex.

It may be natural to suspect a higher average de-gree might increase convergence rates of a graph, butin fact, it appears to make very little difference, asshown in Fig. 13b, for which the plot was produced by

fixing the spectral count at 13, and randomly selectinggraphs with increasing average degrees ranging fromD = 2 to D = 21. This may be because even with lowaverage degrees, continuous time quantum walks areable to mix across arbitrarily large distances betweenvertices.

(a)

(b)

(c)

FIG. 13: Amplified probabilities of a single mostoptimal vertex for graphs with varying spectral

count and average degree.

On the other hand, spectral counts do appear tohave a noticeable effect on a graph’s capability to pro-duce convergence into a single most optimal vertex.By fixing the average degree at 12, and selecting ran-dom graphs with increasing spectral counts rangingfrom E = 3 to E = 13, the plot in Fig. 13c was pro-duced. Note that an additional graph, D23E2, whichis the complete graph, K24, was included because this

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is the only graph with only 2 distinct eigenvalues (afact which is consistent for graphs with any number ofvertices). From these results, it appears that for ran-domly distributed quality distributions, graphs withsmaller spectral counts are less effective at producingconvergence into a single most optimal vertex.

Referring to Fig. 14, it turns out that this depen-dence of convergence rate on spectral count is actu-ally highly sensitive to degeneracy within the qualitydistribution, with the relationship reversing for caseswith higher degeneracy, where graphs with lower spec-tral counts produce higher convergence rates. Furtherto this, and perhaps most importantly of all, degener-acy in the quality distribution appears to be the singlemost critical feature in terms of influencing conver-gence rate for graphs in general, with higher degener-acy allowing for higher convergence rates into singlemost optimal vertices. Note that the analysis for vary-ing levels of degeneracy was completed in the sameway as for the no degeneracy case, except that the 24qualities were generated by repetition of values froma smaller set of randomly generated values, assigningthe most optimal quality to only one vertex, and ran-domising the arrangement of the final set of qualitiesacross the 24 vertices.

FIG. 14: Amplified probabilities of a single mostoptimal vertex for graphs of varying levels of

degeneracy in the quality distributions.

Taking all of this together, it appears that the com-bination of quality distributions with high degeneracyon graphs with low spectral count provide the highestamplification into a single marked vertex. The fastestconvergence rate therefore appears to occur with a bi-nary marking function over a complete graph, at leastfor the case of a single most optimal vertex.

Appendix B: Investigation of the optimisationlandscape

Under the QWOA framework it’s also important toconsider how a graph’s structure and quality distri-bution effect the ruggedness of its optimisation land-scape, and the variability in its local extrema. It maystill be possible that a graph which produces lowerconvergence than another, still might be superior if

an optimal set of parameters can be found with suffi-ciently less computational effort.

In order to assess the optimisation landscapes, eachgraph in a series of graphs with varying spectralcounts is assigned the same uniformly distributedqualities over the interval [0, 1], with a single markedvertex assigned a quality of 1. This is repeated forquality distributions with varying levels of degener-acy. In each case, 240 initial parameter sets are gen-erated and taken as initial values is a Nelder-Meadoptimisation procedure. From the 240 optimisationresults, we take the mean and standard deviation forthe maximised probability in the single marked vertex.As such, the mean value is a measure of the averagequality (defined by the amplified probability) of localmaxima, and the standard deviation is a measure ofhow much local maxima vary in quality across the op-timisation landscape. The results of this procedureare shown in Fig. 15.

FIG. 15: Amplified probabilities of a single markedvertex for varying levels of degeneracy in the qualitydistributions.

The general result is that high degeneracy qualitydistributions combined with graphs of low spectralcount show the best performance in terms of the aver-age amplified probability produced at each local max-ima. Even though they still have comparable variabil-ity in local maxima quality, this would make the opti-misation procedure easier in terms of finding a “goodenough” solution. A more specific result is that all 240initial parameter sets converge to the same peak am-plified probability for the binary E=2 and E=3 cases,as shown by the zero offset error bars. As we alreadydemonstrate, binary marking on the complete graphproduces the best amplification, but it also appearsto do so with an optimisation landscape in which thelocal maxima are all equally optimal, meaning thatan optimisation procedure for this case is likely to behighly efficient. It’s worth noting the wide rangingperformance of the complete graph (D23E2) and theother low spectral count graphs. They perform poorlyin the general case (small amount of degeneracy) butexceedingly well in the binary case, and in varying de-gree between these two extremes for the intermediatecases.

In fact, there is another thing that is special abouta binary marking function, which is not obvious from

19

the results shown in Fig. 15. When applied to graphsin general (at least in the case of a single marked ver-tex), they appear to be capable of producing opti-mal amplification with repeated applications of thesame phase and walk parameters. So in other words,a binary quality distribution is capable of effectivelyreducing arbitrarily large optimisation problems toonly a 2 dimensional optimisation landscape. Refer toFig. 16a and Fig. 16b for a visual display of how therestricted 2D optimisation landscape produces highamplification in the binary case but is unable to inthe case of a quality distribution without degener-acy (both for the 24-vertex complete graph and for3 QWOA iterations). In addition, Fig. 17 shows theoptimised probabilities for the binary case contrastedagainst the no-degeneracy case (with uniformly dis-tributed qualities) subject to r = 3 repeated parame-ter pairs for a range of 24-vertex graphs and the resultis consistent across all of them, not just the completegraph. Note that the optimisation procedure consistsof producing 1,000 randomly generated sets of initialparameter pairs. The 10 parameter pairs that producemaximum initial amplification are taken as the initialvalues in a Nelder-Mead optimisation procedure. Themost optimal of these is taken to be the final ampli-fied probability for that particular graph and qualitydistribution.

All this is to say that the a binary marking functionproduces the highest rates of convergence into a singlemarked vertex for any graph, but also does so with re-peated applications of the same parameter pairs thatcan be easily acquired via an optimisation procedure.The fact that repeated applications of the same pa-rameter pairs can achieve maximum amplification alsohints that it may be possible to find these parametersanalytically. Out of all the graphs and quality distri-butions, the binary-marked complete graph seems toproduce the highest rate of convergence.

0 1 2 3 4 5 6Phase

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FIG. 16: Maximum amplified probability achievedwith repeated parameters is significantly larger for (a)the binary quality distribution (Pmax = 0.98) com-pared with (b) the uniform distribution of qualities(Pmax = 0.24).

FIG. 17: Amplified probabilities of a single markedvertex after repeated application of the same phaseand walk parameters.

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Appendix C: Algorithm pseudocode

The following algorithm finds a suitable qualitythreshold which produces a maximally amplified statefor the final rotation count, rf , where rf is a powerof 2. Note that Ψr(T ) refers to the amplified stateprepared via r Grover rotations of a binary-markedsolution space with quality threshold, T . Also notethat the algorithm has been written for a minimisa-tion problem.

1: function FinalThreshold(rf )2: r ← 13: sample← 200 randomly sampled solutions4: mean← mean of qualities in sample5: Qu1 ← Quartile 1 of qualities in sample6: stepsize← (mean−Qu1)/107: T ← mean8: T1 ← FindPeak(r, T, stepsize)9: r ← 2

10: T2 ← FindPeak(r, T1, stepsize)11: r ← 412: while r < rf do13: stepsize← (Tr/4 − Tr/2)/1014: Tr ← FindPeak(r, Tr/2, stepsize)15: r ← 2 ∗ r16: end while17: stepsize← (Tr/4 − Tr/2)/1018: T ← ThresholdForAS(r, Tr/2, stepsize)19: return AdaptiveSearch(r, T )20: end function21:

22: function FindPeak(r, T, stepsize)23: sum← 024: weights← 025: for i = 1 to 20 do26: T ← T − stepsize27: count← 028: while count < 20 do29: x← Measure[Ψr(T )]30: if f(x) < T then31: count← count+ 132: else33: sum← sum+ T ∗ count434: weights← weights+ count4

35: break while loop

36: end if37: end while38: if count = 20 then39: return T40: end if41: end for42: return sum/weights43: end function44:

45: function ThresholdForAS(r, T, stepsize)46: best← T47: for i = 1 to 20 do48: T ← T − stepsize49: x← Measure[Ψr(T )]50: if f(x) < best then51: best← f(x)52: end if53: end for54: return best55: end function56:

57: function AdaptiveSearch(r, T )58: count← 059: while count < 40 do60: count← 061: hit← False62: while hit = False do63: x← Measure[Ψr(T )]64: count← count+ 165: if f(x) < T then66: hit = True67: T ← f(x)68: end if69: end while70: end while71: return T72: end function

The output, T , of FinalThreshold(rf ), is a thresh-old suitable for the second part of the MAOA, wherethe state, Ψrf (T ), produces solutions in the markedset which are maximally amplified. As such, the usercan repeatedly prepare and measure from this statein order to deliver maximum speedup with respect tofinding optimal solutions within the marked set.