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What Is the Optimal Annealing Schedule in Quantum Annealing

Oscar Galindo and Vladik Krenovich

Department of Computer Science University of Texas at El Paso El Paso, Texas 79968, USA

• galindomo@miners.utep.edu, vladik@utep.edu

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1. Optimization Is Important

• In practical applications, we want to find:

– the best possible decision, i.e., – the decision for which the appropriate objective function attains its largest possible value.

• There exist many numerical algorithms for optimiza-

tion.

• How do we know that we have reached the maximum?
• We compare the quality of current alternative with the

qualities of several similar alternatives.

• Often, we have a situation of local maximum, when:

– an alternative is better than all nearby alternatives, but – is still worse than some other – distant – alterna- tive.

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2. How to Jump out of a Local Maximum?

• It is not clear in which direction we should go.
• All directions are, in this sense, equally good (or equally

bad).

• So a reasonable idea is to pick each direction at ran-

dom.

• The corresponding change should be significant – oth-

erwise, we will not get out of the local maximum.

• This is the main idea behind simulated annealing.
• We want this process to eventually stop.
• Thus, the probability of jumping decreases with time.
• How exactly this probability decreases with time is

known as the annealing schedule.

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3. Proper Selection of Annealing Schedule Is Very Important

• The success – or not – of simulated annealing depends
• n annealing schedule.
• If we decrease the probability too fast, we will not give

the algorithm enough time to jump out of a local max.

• On the other hand, if we decrease it too slowly, the

algorithm will work forever and never stops.

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4. Quantum Annealing

• Traditional computations are deterministic.
• So, to perform simulated annealing, we need to artifi-

cially introduce randomness.

• In contrast, quantum processes are, in general, proba-

bilistic.

• Thus, in quantum computing, to implement a random

deviation: – there is no need to artificially introduce random- ness, – it is sufficient to somewhat modify the system’s dy- namics.

• In general, the state of a quantum system is described

by a complex-valued function ψ(t).

• It is known as the wave function.

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5. Quantum Annealing (cont-d)

• The dynamics of a quantum system is described by

Schroedinger’s equations i · · ∂ψ ∂t = Hψ.

• Here i

def

= √−1 and H is a linear operator.

• In these terms, annealing-type modification means adding

additional terms – decreasing with time – to H: i · · ∂ψ ∂t = Hψ + γ(t) · H0ψ.

• Quantum annealing is the main idea behind D-Wave

Systems quantum computing devices.

• These are the only commercially available and practi-

cally useful computers that use quantum computing.

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6. Which Annealing Schedule Works Best

• Empirically, depending on the specific optimization prob-

lem, two schedules work the best: – the power law annealing schedule γ(t) = A · ta, for some A and a < 0; and – the exponential annealing schedule γ(t) = A · exp(a · t) for some A and a < 0.

• In this talk, we present a possible theoretical explana-

tion for these empirical facts.

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7. Physical Meaning of Annealing

• A physical system tends to end up in a state with the

smallest possible energy.

• For example, in a gravitational field, this means getting

to as low a position as possible.

• In principle, we can place a ball on top of the mountain

– which will constitute a local minimum of energy.

• However, if a strong wind blows and disturbs the ball,

it will start falling down.

• So, a natural way to use simulated physical phenomena

for optimization is to simulate a system for which: – for all the values of the parameters, – its energy is equal to the value of minimized objec- tive function.

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8. Physical Meaning (cont-d)

• In general, in Schroedinger equations, energy is repre-

sented by the operator H.

• So for quantum annealing, this operator must represent

the desired objective function.

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9. Need to Select a Family

• The original optimization problem is usually not for-

mulated in terms of energy.

• So, how we transform it into energy depends on our

choice of units.

• If we select a different unit, this means that in original

unit, instead of H, we will have C · H; – if for H, the best schedule was γ(t), – then for C · H, the best schedule is C · γ(t).

• Indeed, i·· ∂ψ

∂t = C·Hψ+C·γ(t)·H0ψ is equivalent to the original equation if we re-scale the time: t → t/C.

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10. Need to Select a Family (cont-d)

• The choice of the energy unit is rather arbitrary, so:

– with each such optimal schedule γ(t), – in different energy units, a schedule C · γ(t) is op- timal.

• Thus, we can only select a family {C · γ(t)}C>0 of an-

nealing functions.

• Here, a function γ(t) is fixed, and the parameter C can

take any positive value.

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11. What Does Optimal Mean?

• What is the best depends on the situation.
• We may want to get the results as fast as possible.
• We may want to get the maximum as close to the global

maximum as possible even if it takes more time.

• There are many other possible options.
• In all these cases, we have a criterion for comparing

families, i.e., saying when: – one family F1 is better than family F2 (we will de- note it by F2 < F1), – or F2 is better than F1 (F1 < F2), – or F1 and F2 are of the same quality with respect to the given criterion; we will denote this by F1 ∼ F2.

• Of course, if F1 is better than F2 and F2 is better than

F2, then F1 should be better than F3.

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12. What Does Optimal Mean (cont-d)

• An optimal family Fopt is a family which is better (or
• f the same quality) that every other family F:

F < Fopt or F ∼ Fopt.

• What if we have two different optimal families?
• This happens in real-life optimization problems; e.g.:

– if we are minimizing time, – we may have two different annealing schedules that takes the exact same time on given examples.

• In this case, a natural idea is to use this non-uniqueness

to optimize something else.

• E.g., we can select a family with the more accurate

approximation to the actual maximum.

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13. What Does Optimal Mean (cont-d)

• In effect, we thus have a more complex optimality cri-

terion: F1 is better than F2 if: – either F1 is faster, – or if they take the same time but F1 is more accu- rate.

• If this new criterion still leaves several optimal families,

we can optimize something else.

• We can continue thus modifying the optimality crite-

rion: – until we get a final criterion, – for which there is exactly one optimal family.

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14. Need for Re-Scaling Time

• The numerical value of time also depends on the choice

if the measuring unit.

• If we replace the original unit of time by a λ times

smaller unit, then each value t is replaced by λ · t.

• For example, if we replace minutes by seconds, then 2

minutes becomes 60 · 2 = 120 seconds.

• It is reasonable to requires that the relative quality of

a family not change if we simply change the t-unit: – if {C · γ1(t)}C>0 < {C · γ2(t)}C>0, – then {C · γ1(λ · t)}C>0 < {C · γ2(λ · t)}C>0.

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15. Need for Re-Scaling Time (cont-d)

• The numerical value of time also depends on the choice
• f the starting point:

– if we replace the original starting point with the

• ne which is t0 units earlier,

– then all numerical values t are replaced with shifted values t + t0.

• The relative quality of two families should not depend
• n the choice of the starting point:

– if {C · γ1(t)}C>0 < {C · γ2(t)}C>0, – then {C · γ1(t + t0)}C>0 < {C · γ2(t + t0)}C>0.

• Now, we are ready to formulate our problem in precise

terms.

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16. Definitions

• By a family, we mean a class of functions {C ·γ(t)}C>0,

where γ(t) is a monotonically tends to 0 at infinity.

• By an optimality criterion, we mean a pair of relations

(<, ∼) on the class of all families for which: – for every pair (F1, F2), we have exactly one of the possible relations F1 < F2, F2 < F1, and F1 ∼ F2; – if F1 < F2 and F2 < F3, then F1 < F3; – if F1 < F2 and F2 ∼ F3, then F1 < F3; – if F1 ∼ F2 and F2 < F3, then F1 < F3; – if F1 ∼ F2 and F2 ∼ F3, then F1 ∼ F3.

• We say that a family Fopt is optimal with respect to

the optimality criterion (<, ∼) if for every family F: F < Fopt or F ∼ Fopt.

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17. Definitions (cont-d)

• We say that an optimality criterion is final if exactly
• ne family is optimal with respect to this criterion.
• For every λ > 0, by a re-scaling Tλ(F) of a family

F = {C · γ(t)}C>0, we mean a family {C · γ(λ · t)}C>0.

• For every t0, by a shift St0(F) of a family

F = {C ·γ(t)}C>0, we mean a family {C ·γ(t+t0)}C>0.

• We say that an optimality criterion is scale-invariant

if for all Fi and λ < 0: – F1 < F2 implies Tλ(F1) < Tλ(F2) and – F1 ∼ F2 implies Tλ(F1) ∼ Tλ(F2).

• We say that an optimality criterion is shift-invariant if

for all Fi and t0: – F1 < F2 implies St0(F1) < St0(F2) and – F1 ∼ F2 implies St0(F1) ∼ St0(F2).

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18. Results

• It seems reasonable to require that the optimality cri-

terion be both scale- and shift-invariant.

• However, this turned out to be impossible:
• Proposition. No final optimality criterion is scale-

and shift-invariant.

• When we require either scale-invariance or shift-invariance,

we get the empirically best annealing schedules.

• Proposition. For every scale-invariant optimality cri-

terion, the optimal annealing schedule is γ(t) = A · ta.

• Proposition. For every shift-invariant optimality cri-

terion, the optimal annealing schedule is γ(t) = A · exp(a · t).

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19. Proof of Scaling Proposition

• Let Fopt = {C · γopt(t)}C>0 be the family which is op-

timal w.r.t. the scale-invariant optimality criterion.

• Let us first prove that this family if itself scale-invariant,

i.e., that we have Tλ(Fopt) = Fopt for all λ > 0.

• Indeed, since the family Fopt is optimal, we have F <

Fopt or F ∼ Fopt for every family F.

• In particular, Tλ−1(F) < Fopt or Tλ−1(F) ∼ Fopt.
• One can easily see that Tλ(Tλ−1(F)) = F.
• Thus, due to scale-invariance, for every family F, we

have F < Tλ(Fopt) or F ∼ Tλ(Fopt).

• By definition of optimality, this means that Tλ(Fopt) is

also an optimal family.

• However, out optimality criterion is final, which means

that there is only one optimal family.

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20. Proof (cont-d)

• Thus indeed Tλ(Fopt) = Fopt.
• By definition of scaling Tλ(F), the function γ(λ · t)

belongs to the family Tλ(Fopt).

• This family is equal to the original family

Fopt = {C · γopt(t)}C>0.

• So, the f-n γ(λ · t) belongs to this original family as

well: γ(λ · t) = C(λ) · γ(t) for some C depending on λ.

• It is known that all monotonic solutions of this func-

tional equation have the desired form A · ta.

• The proposition is proven.

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21. Comment

• The functional equation γ(λ · t) = C(λ) · γ(t) is easy

to solve if γ(t) is differentiable.

• Then, the function C(λ) is also differentiable – as a

ratio of two differentiable functions.

• So, we can differentiate both sides with respect to λ

and take λ = 1; we get t · dγ dt = a · γ, where a

def

= C′(1).

• By moving terms containing γ to one side and terms

containing t to another side, we get dγ γ = a · dt t .

• Integrating, we get ln(γ) = a · ln(t) + C0, where C0 is

the integration constant.

• By applying exp(z) to both sides, we get the desired

formula γ = A · ta, for A = exp(C0).

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22. Proof of Shift-Related Proposition

• Let Fopt = {C · γopt(t)}C>0 be the family which is op-

timal w.r.t. the shift-invariant optimality criterion.

• Let us first prove that this family if itself shift-invariant,

i.e., that we have St0(Fopt) = Fopt for all t0.

• Indeed, since the family Fopt is optimal, we have F <

Fopt or F ∼ Fopt for every family F.

• In particular, S−t0(F) < Fopt or S−t0(F) ∼ Fopt.
• One can easily see that St0(S−t0(F)) = F.
• Thus, due to shift-invariance, for every family F, we

have F < St0(Fopt) or F ∼ St0(Fopt).

• By definition of optimality, this means that St0(Fopt)

is also an optimal family.

• However, out optimality criterion is final, which means

that there is only one optimal family.

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23. Proof (cont-d)

• Thus indeed St0(Fopt) = Fopt.
• By definition of a shift St0(F), the function γ(t + t0)

belongs to the family St0(Fopt).

• This family is equal to the original family

Fopt = {C · γopt(t)}C>0.

• So, the function γ(t+t0) belongs to this original family

as well.

• Thus, γ(t + t0) = C(t0) · γ(t) for some C depending
• n t0.
• It is known that all monotonic solutions of this func-

tional equation have the desired form A · exp(a · t).

• The proposition is proven.

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24. Comment

• The functional equation γ(t + t0) = C(t0) · γ(t) is easy

to solve when γ(t) is differentiable.

• Then, the function C(t0) is also differentiable – as a

ratio of two differentiable functions.

• Thus, we can differentiate both sides w.r.t. t0 and take

t0 = 0; dγ dt = a · γ, where a

def

= C′(0).

• By moving terms containing γ to one side and terms

containing t to another side, we get dγ γ = a · t.

• Integrating, we get ln(γ) = a · t + C0, where C0 is the

integration constant.

• By applying exp(z) to both sides, we get the desired

formula γ = A · exp(a · t), for A = exp(C0).

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25. Conclusions

• One of the main problems in numerical optimization is

avoiding local extrema.

• One of the most widely used ways for such avoidance

is simulated annealing, when: – we jump out of a local extremum to some other location – with a probability depending on the values of the

• bjective function at different locations.
• In many optimization problems, quantum annealing

works better than the traditional (non-quantum) one.

• In particular, quantum annealing is used by D-Wave

computers.

• These are the only commercially available devices that

use quantum computing.

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26. Conclusions (cont-d)

• The efficiency of quantum annealing depends on the

annealing schedule – i.e., on: – how transition probabilities – change from one iteration to the next.

• Two classes of annealing schedules have been found out

to be empirically the best: – power law schedule and – exponential schedule.

• On several examples, these two classes of schedules

worked better than several other tested schedules.

• However, up to now, there was no theoretical explana-

tion for the empirical success of these schedules.

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27. Conclusions (cont-d)

• So, it was not clear:

– whether these schedules are indeed the best or – whether they are only better than the schedules against which they have been tested.

• It was not clear:

– whether the selected schedules are indeed better for all the optimization problems – or only for the problems on which they have been tested.

• In this paper, we provide a theoretical explanation for

the empirical success of selected classes of schedules.

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28. Conclusions (cont-d)

• This theoretical explanation gives us confidence:

– that these schedules will be efficient for other opti- mization problems as well, and – that the selected schedules are indeed better than all other possible schedules, – including schedules against which they have not yet been tested.

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29. Acknowledgments This work was supported in part by the US National Sci- ence Foundation grants:

• 1623190 (A Model of Change for Preparing a New Gen-

eration for Professional Practice in Computer Science);

• HRD-1242122 (Cyber-ShARE Center of Excellence).