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2.3 – Perform computation using adiabatic evolution 11

Hamiltonian H

.ThesetwotimeindependentHamiltonianscanbecombinedbythe linear interpolation

(t)=

T − t

, (2.19)

where the parameter T is the entire evolution time. If the evolution time is large enough, there will be a large probability

that the system remains in its ground state throughout the evolution. To determine how large T should be under the

constraint of adiabatic eq u ality, we should ﬁrst separate and normalize time dependent Hamiltonian. The time dependent

Hamiltonians can be transferred into

(s)=(1− s)H

+ sH

, (2.20)

where s ≡ t/T .ThenletusreturntoSchrödingerequation,andwehave

H(s)=T

(t). (2.21)

Since the system is in the ground state at the beginning, the most possible transition would be to the ﬁrst excited state.

Using the inequality in (2.18) from the previous section and substituting the (2.21), we have

max

s∈[0,1]

⟨ψ

(s)|

∂(s)

T∂s

|ψ

(s)⟩

≪

min

s∈[0,1]

(s) − E

(s))

, (2.22)

where ψ

(s) and E

(s) are eigenvector and eigenvalue obtained from the time varying Hamiltonian H(s),

i.e., H(s)|ψ

(s)⟩ = E

(s)|ψ

(s)⟩,andE

(s) is the lowest eigenvalue or the ground state. Similarly, we can g et ψ

(s)

and E

(s),andE

(s) is the second lowest eigenvalue or the ﬁrst excited state.

And then we have the following inequa lity that r eveals the constrain on the evolution time T

max

s∈[0,1]

⟨ψ

(s)|

dH(s)

|ψ

(s)⟩

min

s∈[0,1]

(s) − E

(s)]

≪ T. (2.23)

We shou ld also be awa re that th e co nstrains on T in (2.23) are derived from the worst-case analysis. We also have an

alternative adiabatic cond ition (Schaller et al., 2006; WeiandYing,2007)thatderivedfromadifferentproofinprevious

section, it is obtained directly from the adiabatic theorem

n=0

|⟨φ

(t)|

∂H(t)

∂t

|φ

(t)⟩|

∆

(t)

≪ 1, for all t, (2.24)

万方数据

12 Background theory

where ∆

(t) ≡ λ

(t)λ

(t).Herewecanalsosubstitute(2.21)to(2.24)andsowehavetheconstrainsonevolutiontime,

n=0

|⟨φ

(s)|

∂H(s)

∂s

|φ

(s)⟩|

∆

(s)

≪ T, for all s, (2.25)

the sum here is also mainly dominated by the ﬁrst excited state, i.e.,thetermsundern =1.

Let us deﬁne the the energy gap ∆(s ) between the ground state and the ﬁrst excited state as

∆(s)=E

(s) − E

(s). (2.26)

The minimum gap in the interval s ∈ [0, 1] can b e written as

∆

min

= min

0<s<1

∆(s). (2.27)

The inequality (2.23) inspires us that to minimize the running time of adiabatic evolution (Liboff, 1980), one possible way

is maximizing the minimum gap ∆

min

.Wewilldiscusstheapproachesformaximizing∆

min

in more details in the next

chapters.

Moreover, to evaluate the p er formance of o u r approach, whichisactuallydesignedforcuttingdowntheevolution

time, the ideal metric could be the success probability deﬁned as

succ

(T ) ≡ |⟨φ

(T )|ψ(T )⟩|

, (2.28)

where |φ

(T )⟩ is the ground state of H

and |ψ(T )⟩ is the actual achieved state at the ﬁnal time. Note that minimum gap

is just a proxy for the success probability.

Although it has been demonstrated that the model of adiabaticquantumcomputationisauniversalproblem(Aharonov

et al., 2007; Mizel et al., 2007; Biamon te and Love, 20 0 8), theformulationforthefault-tolerantorﬁdelityofadiabatic

quantum computation remains to be an open and challenging problem (Lidar, 2008; Quiroz and Lidar, 2012; Young et al.,

2013; Sarovar and Young, 2013). Although there doesn’t existanystardardforfault-tolerantevaluating,extensiveefforts

have been paid for implementations of speciﬁc purpose adiabatic processors on quantum annealing, see refer en ce (Joh nson

et al., 2011; Boixo et al., 2014). Among these efforts, there exist some we want to highlight here, since several heuristics

have been developed in order to improve the reliability of adiabatic quantum computation and quantum annealing, such

as encoding or raising the degeneracy of th e problem Hamiltonian (Young et al., 2013; Pudenz et al., 2014; Zhuang, 2014;

Pudenz et al., 2015).

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