"优化量子计算中的能级差值：云计算与绝热量子计算研究"。

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"云计算-绝热量子计算与量子退火中的能级差值最优化问题.pdf" 这篇论文探讨了量子退火和绝热量子计算在优化和计算问题中的应用，这两种方法目前备受关注，因为它们在量子领域中获得了优势，允许我们解决传统方法无法解决的问题。量子退火和绝热量子计算具有一定的随机性，从相关文献中可以了解到，在系统演化过程中增加系统的基态和第一激发态之间的最小能级差可以单调地提高成功概率。在这篇论文中，作者研究了如何优化能级差值，以提高量子计算的成功率。 AbstractQuantum annealing and adiabatic quantum computing have drawn much attention nowadays since the classical approaches to optimization and computation problems become possible to obtain advantage in quantum area. Quantum annealing and adiabatic quantum computing are probabilistic in nature, and from the related literature, increasing the minimum gap between the ground state and the first excited state of the system during evolution can monotonically raise the success probability. In this thesis, 总的来说，本论文探讨了如何通过优化能级差值来提高量子计算的成功率，重点讨论了量子退火和绝热量子计算在这一过程中的应用。这些方法在当前的研究中备受关注，因为它们为解决传统计算无法解决的问题提供了新的可能性。通过增加系统基态和第一激发态之间的最小能级差，在量子计算中提高成功率的探索，是当前研究的热点之一。本论文的研究结果将有助于深入理解量子计算过程中能级差值的优化问题，为未来量子计算的发展提供重要的参考。"

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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