text: ∗ Blind quantum computation for QFT on multi-qubit states † Xiaoqian Zhang1, Jian Weng1, Weiqi Luo1, and Xiaoqing Tan2 ‡ and § 1 (Dated: May 30, 2020) After quantum computers come out, governments and rich companies will have the abilities to buy these useful quantum computers, meanwhile they are familiar with these technologies proﬁciently. If a client wants to perform quantum computing but she does not have quantum computers with relevant quantum technologies. She can seek help from the server and pay his salary, but she does not want to leak anything to the server. Blind quantum computing (BQC) give a good method for the client to realized her quantum computing. In this article, we propose a new BQC protocol of quantum fourier transform (QFT) performed on multi-qubit states with a trusted, a client and a server, where the trusted center can generate resource states, the client can delegate her quantum computing to a server who can perform universal quantum computing without knowing anything about the client’s inputs, algorithms and outputs. We ﬁrst give the BQC protocols of three-qubit QFT with the equivalently quantum circuits, Greenberg-Horne-Zeilinger(GHZ) entangled states and W entangled states as examples. Further, we extend them to multi-qubit QFT on multi-qubit with the equivalently quantum circuits. At last, we give the analyses and proofs of the blindness and correctness. PACS numbers: 03.67.Hk, 03.65.Ud. I. INTRODUCTION quantum computing, it is always inevitable to transmit qubits from a client to servers or from servers to a client, In recent years, various blind quantum computation it is a diﬃcult task to remove quantum channel noises. (BQC) protocols are proposed [1–21]. In the future, when These noises have already reduce transmission eﬃciency, large scale of quantum computers come out, their prices so many scholars started to solve this problem and pro- are very expensive such that few people can aﬀord a com- pose anti-noise BQC protocols [6, 23, 24] recently. puter. When they want to perform quantum comput- In above description, we only review some basic BQC ing in some cases, they need to delegate their quantum protocols, but it is an open question how to verify the cor- computing to some rich companies owned quantum com- rectness quantum inputs and quantum computing. It will puters. This is the reason that blind quantum quantum be solved in these veriﬁable BQC protocols [13–20, 25– computing becomes a hot topic in academia. BQC allows 28], where the client has the ability to realize the ver- a client who has few quantum abilities can delegate her iﬁability. Some important problems in BQC protocols quantum computation to servers who have full advanced have already been conquered by above references. Nat- quantum computers in which her quantum inputs, out- urally, we will consider the application of BQC to re- puts and algorithms keep unknown to the server. Broad- ﬂect their values. So many application of BQC proto- bent et al. [1] in 2009 ﬁrst proposed an universal BQC cols are proposed to complete various important tasks in protocol with brickwork states, where every single qubit [29–36]. In [35], Huang et al. implemented an experi- is randomly chosen from a ﬁnite set in which every qubit mental BQC protocol to factorize the integer 15 in which is randomly rotated the multiple of π. Further, Barz et two quantum servers can classically interact with each 4 al. [3] made an experiment to demonstrate blind quan- other and share three Bell states while the client is clas- tum computing by using brickwork state. Since it is per- sical. With the development of BQC, many outstanding fect BQC protocol based on measurement with the brick- achievements have been presented to the world, a people, work state, many proposed multi-server BQC protocols Fitzsimons, analyzed and summarized these protocols. In were simpliﬁed into single-server BQC protocol to con- [37], Fitzsimons reviewed some important BQC protocols struct the brickwork state and realize BQC in [4–6, 8]. and analyzed these protocols by security, state prepara- Quantum circuits play a key role in constructing BQC tion, semi-classical client, measurement, multiple servers, protocols, blind quantum computing based on circuits computing on encrypted data and homomorphic, physi- models were widely studied in [9–12, 22]. Since in blind cal implementations. This gives us a clear understanding of the development and knows that there exist problems in blind quantum computing. ∗ Since blind quantum computing can be used to realize † an algorithm securely, we consider the QFT and propose cryptjweng@gmail.com ‡ the corresponding BQC protocols. The quantum fourier § transform (QFT) is performed on multi-qubit states in- ##LTLine## ##LTLine##

text: Blind

text: quantum

text: computation

text: for

text: QFT

text: on

text: multi-qubit

text: states

text: Xiaoqian

text: Jian

text: Weiqi

text: and

text: Xiaoqing

text: and

text: (Dated:

text: May

label: date

text: After

text: quantum

text: computers

text: come

text: out,

text: governments

text: and

text: rich

text: companies

text: will

text: have

text: the

text: abilities

text: to

text: buy

text: these

text: useful

text: quantum

text: computers,

text: meanwhile

text: they

text: are

text: familiar

text: with

text: these

text: technologies

text: proﬁciently.

text: If

text: a

text: client

text: wants

text: to

text: perform

text: quantum

text: computing

text: but

text: she

text: does

text: not

text: have

text: quantum

text: computers

text: with

text: relevant

text: quantum

text: technologies.

text: She

text: can

text: seek

text: help

text: from

text: the

text: server

text: and

text: pay

text: his

text: salary,

text: but

text: she

text: does

text: not

text: want

text: to

text: leak

text: anything

text: to

text: the

text: server.

text: Blind

text: quantum

text: computing

text: (BQC)

text: give

text: a

text: good

text: method

text: for

text: the

text: client

text: to

text: realized

text: her

text: quantum

text: computing.

text: In

text: this

text: article,

text: we

text: propose

text: a

text: new

text: BQC

text: protocol

text: of

text: quantum

text: fourier

text: transform

text: (QFT)

text: performed

text: on

text: multi-qubit

text: states

text: with

text: a

text: trusted,

text: a

text: client

text: and

text: a

text: server,

text: where

text: the

text: trusted

text: center

text: can

text: generate

text: resource

text: states,

text: the

text: client

text: can

text: delegate

text: her

text: quantum

text: computing

text: to

text: a

text: server

text: who

text: can

text: perform

text: universal

text: quantum

text: computing

text: without

text: knowing

text: anything

text: about

text: the

text: client’s

text: inputs,

text: algorithms

text: and

text: outputs.

text: We

text: ﬁrst

text: give

text: the

text: BQC

text: protocols

text: of

text: three-qubit

text: QFT

text: with

text: the

text: equivalently

text: quantum

text: circuits,

text: Greenberg-Horne-Zeilinger(GHZ)

text: entangled

text: states

text: and

text: W

text: entangled

text: states

text: as

text: examples.

text: Further,

text: we

text: extend

text: them

text: to

text: multi-qubit

text: QFT

text: on

text: multi-qubit

text: with

text: the

text: equivalently

text: quantum

text: circuits.

text: At

text: last,

text: we

text: give

text: the

text: analyses

text: and

text: proofs

text: of

text: the

text: blindness

text: and

text: correctness.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: quantum

text: computing,

text: it

text: is

text: always

text: inevitable

text: to

text: transmit

text: qubits

text: from

text: a

text: client

text: to

text: servers

text: or

text: from

text: servers

text: to

text: a

text: client,

text: In

text: recent

text: years,

text: various

text: blind

text: quantum

text: computation

text: it

text: is

text: a

text: diﬃcult

text: task

text: to

text: remove

text: quantum

text: channel

text: noises.

text: (BQC)

text: protocols

text: are

text: proposed

text: In

text: the

text: future,

text: when

text: These

text: noises

text: have

text: already

text: reduce

text: transmission

text: eﬃciency,

text: large

text: scale

text: of

text: quantum

text: computers

text: come

text: out,

text: their

text: prices

text: so

text: many

text: scholars

text: started

text: to

text: solve

text: this

text: problem

text: and

text: pro-

text: are

text: very

text: expensive

text: such

text: that

text: few

text: people

text: can

text: aﬀord

text: a

text: com-

text: pose

text: anti-noise

text: BQC

text: protocols

text: recently.

text: puter.

text: When

text: they

text: want

text: to

text: perform

text: quantum

text: comput-

text: In

text: above

text: description,

text: we

text: only

text: review

text: some

text: basic

text: BQC

text: ing

text: in

text: some

text: cases,

text: they

text: need

text: to

text: delegate

text: their

text: quantum

text: protocols,

text: but

text: it

text: is

text: an

text: open

text: question

text: how

text: to

text: verify

text: the

text: cor-

text: computing

text: to

text: some

text: rich

text: companies

text: owned

text: quantum

text: com-

text: rectness

text: quantum

text: inputs

text: and

text: quantum

text: computing.

text: It

text: will

text: puters.

text: This

text: is

text: the

text: reason

text: that

text: blind

text: quantum

text: quantum

text: be

text: solved

text: in

text: these

text: veriﬁable

text: BQC

text: protocols

text: computing

text: becomes

text: a

text: hot

text: topic

text: in

text: academia.

text: BQC

text: allows

text: where

text: the

text: client

text: has

text: the

text: ability

text: to

text: realize

text: the

text: ver-

text: a

text: client

text: who

text: has

text: few

text: quantum

text: abilities

text: can

text: delegate

text: her

text: iﬁability.

text: Some

text: important

text: problems

text: in

text: BQC

text: protocols

text: quantum

text: computation

text: to

text: servers

text: who

text: have

text: full

text: advanced

text: have

text: already

text: been

text: conquered

text: by

text: above

text: references.

text: Nat-

text: quantum

text: computers

text: in

text: which

text: her

text: quantum

text: inputs,

text: out-

text: urally,

text: we

text: will

text: consider

text: the

text: application

text: of

text: BQC

text: to

text: re-

text: puts

text: and

text: algorithms

text: keep

text: unknown

text: to

text: the

text: server.

text: Broad-

text: ﬂect

text: their

text: values.

text: So

text: many

text: application

text: of

text: BQC

text: proto-

text: bent

text: et

text: al.

text: in

text: ﬁrst

text: proposed

text: an

text: universal

text: BQC

text: cols

text: are

text: proposed

text: to

text: complete

text: various

text: important

text: tasks

text: in

text: protocol

text: with

text: brickwork

text: states,

text: where

text: every

text: single

text: qubit

text: In

text: Huang

text: et

text: al.

text: implemented

text: an

text: experi-

text: is

text: randomly

text: chosen

text: from

text: a

text: ﬁnite

text: set

text: in

text: which

text: every

text: qubit

text: mental

text: BQC

text: protocol

text: to

text: factorize

text: the

text: integer

text: in

text: which

text: is

text: randomly

text: rotated

text: the

text: multiple

text: of

text: π.

text: Further,

text: Barz

text: et

text: two

text: quantum

text: servers

text: can

text: classically

text: interact

text: with

text: each

text: al.

text: made

text: an

text: experiment

text: to

text: demonstrate

text: blind

text: quan-

text: other

text: and

text: share

text: three

text: Bell

text: states

text: while

text: the

text: client

text: is

text: clas-

text: tum

text: computing

text: by

text: using

text: brickwork

text: state.

text: Since

text: it

text: is

text: per-

text: sical.

text: With

text: the

text: development

text: of

text: BQC,

text: many

text: outstanding

text: fect

text: BQC

text: protocol

text: based

text: on

text: measurement

text: with

text: the

text: brick-

text: achievements

text: have

text: been

text: presented

text: to

text: the

text: world,

text: a

text: people,

text: work

text: state,

text: many

text: proposed

text: multi-server

text: BQC

text: protocols

text: Fitzsimons,

text: analyzed

text: and

text: summarized

text: these

text: protocols.

text: In

text: were

text: simpliﬁed

text: into

text: single-server

text: BQC

text: protocol

text: to

text: con-

text: Fitzsimons

text: reviewed

text: some

text: important

text: BQC

text: protocols

text: struct

text: the

text: brickwork

text: state

text: and

text: realize

text: BQC

text: in

text: and

text: analyzed

text: these

text: protocols

text: by

text: security,

text: state

text: prepara-

text: Quantum

text: circuits

text: play

text: a

text: key

text: role

text: in

text: constructing

text: BQC

text: tion,

text: semi-classical

text: client,

text: measurement,

text: multiple

text: servers,

text: protocols,

text: blind

text: quantum

text: computing

text: based

text: on

text: circuits

text: computing

text: on

text: encrypted

text: data

text: and

text: homomorphic,

text: physi-

text: models

text: were

text: widely

text: studied

text: in

text: Since

text: in

text: blind

text: cal

text: implementations.

text: This

text: gives

text: us

text: a

text: clear

text: understanding

text: of

text: the

text: development

text: and

text: knows

text: that

text: there

text: exist

text: problems

text: in

text: blind

text: quantum

text: computing.

text: Since

text: blind

text: quantum

text: computing

text: can

text: be

text: used

text: to

text: realize

text: an

text: algorithm

text: securely,

text: we

text: consider

text: the

text: QFT

text: and

text: propose

text: cryptjweng@gmail.com

text: the

text: corresponding

text: BQC

text: protocols.

text: The

text: quantum

text: fourier

text: transform

text: (QFT)

text: is

text: performed

text: on

text: multi-qubit

text: states

text: in-

text: ##LTLine##

text: ##LTLine##


text: 3 For three-qubit entangled states, GHZ entangled states and W entangled states are often used to complete com- | (cid:105) −−−→ √1 | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) 000 QFT ( 0 + 1 )( 0 + 1 )( 0 + 1 ), munication between remote parties. Therefore, we can 2 2 clearly get the universal BQC protocols of QFT on three- | (cid:105) −−−→ √1 | (cid:105) − | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) 001 QFT ( 0 1 )( 0 + eπ2i 1 )( 0 + eπ4i 1 ), qubit states by analyzing BQC protocols of QFT on GHZ 2 2 entangled states and W entangled states. In FIG. 3, the | (cid:105) −−−→ √1 | (cid:105) | (cid:105) | (cid:105) − | (cid:105) | (cid:105) | (cid:105) 010 QFT ( 0 + 1 )( 0 1 )( 0 + eπ2i 1 ), circuits of BQC protocols for three-qubit QFT performed 2 2 on GHZ states are presented. | (cid:105) −−−→ √1 | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) − | (cid:105) QFT 100 ( 0 + 1 )( 0 + 1 )( 0 1 ), 2 2 (1) | (cid:105) −−−→ √1 | (cid:105) − | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) 101 QFT ( 0 1 )( 0 + eπ2i 1 )( 0 + e5π4i 1 ), 2 2 | (cid:105) −−−→ √1 | (cid:105) | (cid:105) | (cid:105) − | (cid:105) | (cid:105) | (cid:105) 110 QFT ( 0 + 1 )( 0 1 )( 0 + e3π2i 1 ), 2 2 | (cid:105) −−−→ √1 | (cid:105) − | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) 011 QFT ( 0 1 )( 0 + eπ2i 1 )( 0 + e3π4i 1 ), 2 2 | (cid:105) −−−→ √1 | (cid:105) − | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) 111 QFT ( 0 1 )( 0 + e3π2i 1 )( 0 + e7π4i 1 ). 2 2 In this paper, we need to know the properties of X and T gates, the relationship between CS, CZ and CT, S, Z fig3-eps-converted-to.pdf and T. | (cid:105) → | ⊕ (cid:105) | (cid:105) → | (cid:105) X : j j 1 , T : j (ei4π )j j , (2) CZ = CS2 = CT4 , Z = S2 = T4. 12 12 12 What’s more important is that we use the combination of rotation operators to express a gate, so we must give the decomposition of rotation operators for gates H, T and X. π π H = ei2π Ry( 2 )Rz(π), T = ei8π Rz( 4 )Ry(0), (3) FtrIaGn.sf3o.rmThpeeerqfourimvaeldenotnqutahnretuemqucbirictsuiotsf fGorHqZuastnattuems. FIonurtiheer X = ei2π Ry(π)Rz(π). green box, three circuits are the composed circuits of rotation operations of gates H, X and T. where R (0) = R (2π) = R (π)8, R (π) = R (π)4 y y y 4 y y 4 | ± (cid:105) and R (π) = R (π)4. BQC protocol 1. For states GHZ , we give the BQC z z 4 000 III. BQC PROTOCOLS FOR MULTI-QUBIT protocol of QFT performed on qubits 123. Alice sends QUANTUM FOURIER TRANSFORM qubit 2 to Bob and Bob performs one round operation H (That is, Bob performs four rounds R (π) and two 2 z 4 In this section, we will give these BQC protocols in rounds Ry(π4 )). Bob returns the result to Alice and Alice detail. Since all BQC protocols will be performed cross- sends 3 to Bob. Bob performs H3 (That is, Bob performs ways, we ﬁrst show the procedures for all BQC protocols four rounds Rz(π4 ) and two rounds Ry(π4 )) and returns as follows. to Alice. Alice sends 12 to Bob and Bob performs 10 {| Step±1.(cid:105)A| trust±ed(cid:105)ce|nter p±re(cid:105)pa|res en±ou(cid:105)g}h initial states rounds CT12 (In eve−ry round CT operation, −Bob per- GHZ000 , GHZ001 , GHZ010 , GHZ100 and sends forms eight rounds C Ry(π4 ) and one round C Rz(π4 )), them to Alice. where C is controlled) and returns them to Alice. Al- Step 2. Alice disturbs the order of qubits and sends ice sends 13 to Bob and Bob performs 11 rounds CT13 qubits to Bob. (That is, in every round CT operation, Bob performs − − Step 3. Bob performs rotation operators and returns eight rounds C Ry(π4 ) and one round C Rz(π4 )). Bob the operated qubits to Alice. All gates are decomposed returns it to Alice and Alice sends 1 to Bob. Bob per- into two rotation operators Ry(θ) and Rz(ϑ) in Eq.(5). forms H1 (That is, Bob performs four rounds Rz(π4 ) and In these BQC prot−ocols, in fact, B−ob only performs gates two r|ounds R(cid:105)y(π4 )) and returns it to Alice. Alice obtains Ry(π4 ), Rz(π4 ), C Ry(π4 ) and C Rz(π4 ). QFT GHZ000 . | ± (cid:105) Step 4. They interact with each other multiple rounds BQC protocol 2. For GHZ , Alice sends qubit 1 to 001 by repeating Step 2 and Step 3 to completing the quan- Bob and Bob performs operation X (In fact, Bob per- 1 tum computing. forms four rounds R (π) and four rounds R (π)). Bob z 4 y 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: For

text: three-qubit

text: entangled

text: states,

text: GHZ

text: entangled

text: states

text: and

text: W

text: entangled

text: states

text: are

text: often

text: used

text: to

text: complete

text: com-

label: equation

text: √1

text: QFT

text: munication

text: between

text: remote

text: parties.

text: Therefore,

text: we

text: can

text: clearly

text: get

text: the

text: universal

text: BQC

text: protocols

text: of

text: QFT

text: on

text: three-

text: √1

text: QFT

text: qubit

text: states

text: by

text: analyzing

text: BQC

text: protocols

text: of

text: QFT

text: on

text: GHZ

text: entangled

text: states

text: and

text: W

text: entangled

text: states.

text: In

text: FIG.

text: the

text: √1

text: QFT

text: circuits

text: of

text: BQC

text: protocols

text: for

text: three-qubit

text: QFT

text: performed

text: on

text: GHZ

text: states

text: are

text: presented.

text: √1

text: QFT

text: √1

text: QFT

text: √1

text: QFT

text: √1

text: QFT

text: √1

text: QFT

text: In

text: this

text: paper,

text: we

text: need

text: to

text: know

text: the

text: properties

text: of

text: X

text: and

text: T

text: gates,

text: the

text: relationship

text: between

text: CS,

text: CZ

text: and

text: CT,

text: S,

text: Z

text: and

text: T.

text: X

text: j

text: j

text: T

text: j

text: )j

text: j

text: CZ

text: Z

text: What’s

text: more

text: important

text: is

text: that

text: we

text: use

text: the

text: combination

text: of

text: rotation

text: operators

text: to

text: express

text: a

text: gate,

text: so

text: we

text: must

text: give

text: the

text: decomposition

text: of

text: rotation

text: operators

text: for

text: gates

text: H,

text: T

text: and

text: X.

text: π

text: π

text: H

text: Ry(

text: )Rz(π),

text: T

text: Rz(

text: fGorHqZuastnattuems.

text: FIonurtiheer

text: X

text: Ry(π)Rz(π).

text: green

text: box,

text: three

text: circuits

text: are

text: the

text: composed

text: circuits

text: of

text: rotation

text: operations

text: of

text: gates

text: H,

text: X

text: and

text: T.

text: where

text: R

text: R

text: R

text: R

text: (π)

text: R

text: y

text: y

text: y

text: y

text: y

text: and

text: R

text: (π)

text: R

text: BQC

text: protocol

text: For

text: states

text: GHZ

text: we

text: give

text: the

text: BQC

text: z

text: z

text: III.

text: BQC

text: PROTOCOLS

text: FOR

text: MULTI-QUBIT

text: protocol

text: of

text: QFT

text: performed

text: on

text: qubits

text: Alice

text: sends

text: QUANTUM

text: FOURIER

text: TRANSFORM

text: qubit

text: to

text: Bob

text: and

text: Bob

text: performs

text: one

text: round

text: operation

text: H

text: (That

text: is,

text: Bob

text: performs

text: four

text: rounds

text: R

text: (π)

text: and

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text: 7 [17] M. Hajdusek, C. A. Pe´rez-Delgado, J. Fitzsimons, De[3v2ic]eZ-i.ndSeupne,nJd.enYtu,vPe.riWﬁaabnleg, bLl.inXdu,qSuyamntmumetriccoamllypuptraivtaiotne,information retrieval based on blind quantum computing, Phys. arXiv:1502.02563v2. Rev. A 91 (2015) 052303. [18] K. Fujii, M. Hayashi, Veriﬁable fault-tolerance in measureme[n3t3-]baHs.edL.qHuuanatnugm, Wco.mS.pButaaot,ioTn.,Lair,XFi.vG:1.6L10i,.0X5.2Q16.vF1u., S. Zhang, H. L. Zhang, X. Wang, Universal blind quantum computation [19] T. Morimae, Measurement-only veriﬁable blind quantum compuftoinrghywbirtihd qsyusatnetmum, QiunapnuttumverIinﬁfc.aPtiroonc,esPs.hy1s6. (R20e1v7. )A19994. (2016) 042301. [34] S. Aaronson, A. Cojocaruy, A. Gheorghiuz, E. Kasheﬁx, On the implausibility of classical client blind quantum computing, [20] A. Broadbent, How to verify a quantum computation, arXiv:1509A.0r9X1i8v0:1v730. 4.08482v1. [21] V. Dunjko, E. Kasheﬁ, Blind quantum computing with two a[l3m5]osHt.idLe.nHtiucaalnsgt,aQte.s.Z, haarXo,ivX:1.6F0.4.M01a5,8C6.. Liu, Z. E. Su, X. L. Wang, L. Li, N. L. Liu, B. C. Sanders, C. Y. Lu, J. W. Pan, [22] C. A. P´erez-Delgado, J. F. Fitzsimons, Iterated gate teleportationExapnedribmlienndtaqlubalnintudmqucaonmtpuumtactoiomnp,uPthinygs.foRreav.cLlaeststi.ca1l14client, Phys. Rev. Lett. 119 (2017) 050503. (2015) 220502. [36] K. Marshall, C. S. Jacobsen, C. Sch¨afermeier, T. Gehring, C. Weedbrook, U. L. Andersen, Continuous-variable quantum [23] Y.-B. Sheng, L. Zhou, Blind quantum computation with noise envcoirmonpmuteinntg, oanrXeinvc:1ry6p09te.0d8d90a2tav,1N. at. Comm. 7 (2016) 13795. [24] Y. Takeuchi, K. Fujii, R. Ikuta, T. Yamamoto, N. Imoto, B[3l7in]dJ.qFua. nFtiutzmsimcoomnsp,uPtartivioanteoqvuearnatucmollceocmtivpeu-tnaotiisoen:chaannninetlr,oduction to blind quantum computing and related protocols, npj Phys. Rev. A 93 (2016) 052307. Quantum Information 3 (2017) 1–11. [25] T. Kapourniotis, V. Dunjko, E. Kasheﬁ, On optimising q[3u8a]ntMum. Ac.oNmiemlsuenni,cIa.tiLo.nCihnuavnegri,ﬁQabulaentquumanCtuommpcuotmatpiountinagn,d Quantum Information, Cambridge University Press, 2000. arXiv:1506.06943. [39] Y. Gao, H. Zhou, D. Zou, X. Peng, J. Du, Preparation of greenberger-horne-zeilinger and w states on a one-dimensional [26] Y. Takeuchi, K. Fujii, T. Morimae, N. Imoto, Practically veriﬁiasbinleg bchlianidn bqyuagnltoubmal ccoonmtproult,aPtihoyns.wRitehv.aAcce8p7t(a2n0c1e3)ra0t3e2335. ampliﬁcation, arXiv:1607.01568. [40] J. Chen, H. Zhou, C. Duan, X. Peng, Preparing greenberger-horne-zeilinger and w states on a long-range ising spin model [27] E. Kasheﬁ, P. Wallden, Optimised resource construction for veriﬁabbyleglqoubaanltcuomntrcoolms,pPuhtaytsi.oRn,evJ.. PAh9y5s.(A20M17a)t0h3.2T3h4e0o.r. 50. [28] K. Honda, Publicly veriﬁable blind quantum computation, ar[4X1i]v:Y16.0S4..0N0a1m16,.R. Blu¨mel, Robustness of the quantum fourier transform with respect to static gate defects, Phys. Rev. A 89 [29] A. Mantri, C. A. Pe´rez-Delgado, J. F. Fitzsimons, Optimal blin(2d01q4u)an7t6u9m–77c2o.mputation, Phys. Rev. Lett. 111 (2013) 230502. [42] L. Ruiz-PerezEmail, J. C. Garcia-Escartin, Quantum arithmetic with the quantum fourier transform, Quant. Inf. Process. [30] V. Giovannetti, L. Maccone, T. Morimae, T. G. Rudolph, Eﬃcient16un(i2v0e1r7sa)l1b5l2in. d quantum computation, Phys. Rev. Lett. 111 (2013) 230501. [43] C. Moore, D. Rockmore, A. Russell, Generic quantum fourier transforms, Journal ACM Transactions on Algorithms [31] V. Dunjko, E. Kasheﬁ, A. Leverrier, Blind quantum computing (wTitAhLwGe)a2k(4c)oh(e2r0e0n6t) p7u0l7s–e7s,23P.hys. Rev. Lett. 108 (2012) 200502.

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text: Comparing the above two equalities gi(cid:98)ves (cid:98) (cid:107) (cid:107) (cid:101) − (cid:104) − · −(cid:101) · − (cid:101)− (cid:105)(cid:101) εz 2T = (εp, δΦ)T εz εz, δΦ n β nδΨ τ (δΨ δΨ(cid:98)) T h h (cid:88)h h h h − (cid:104) 2 − ∂ h − (cid:105) + (δp, Π Φ)T + (y (u) y, Π Ψ)T + (τ τ )δz, Π Ψ P Ψ T V h h W h 2 2 W M ∂ h 7 = R . i i=1 { } Let C = max C, 1 , where C is the constant deﬁned in Lemma 2. For the terms R and R , 0 1 2 Lemma 8 gives R = −(εp, δΦ)T ≤ (cid:107)εp(cid:107)T (cid:107)δΦ(cid:107)T ≤ (E + κ(cid:107)εz(cid:107)T ) C ((cid:107)Φ(cid:107) + (cid:107)Ψ(cid:107) ) 1 ≤ h E h (cid:107) h(cid:107) h (cid:107) (cid:107) h h h 0 1 1 C C ( (cid:98)+ κ εz T ) εz T , (cid:98) 0 reg h h h h −(cid:104) − · − · − − (cid:105) R = εz (cid:98)εz, δΦ n β nδΨ τ (δΨ δΨ) T (cid:98) 2 h h 2 ∂ h ≤ (cid:107) − (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) εz εz T ( δΦ T + τ ∞ T δΨ T + τ ∞ T δΨ T ) ≤ hE h(cid:107)∂ (cid:107)h ∂ h(cid:107) (cid:107) 1 0, ,∂ h(cid:107) (cid:107) ∂ h 2 (cid:107)0, (cid:107),∂ h (cid:107) (cid:107)∂ h 3 ( + κ εz T ) (1 + τ ∞ T + τ ∞ T )C ( Φ + Ψ ) ≤ Eh h (cid:107) (cid:107) 1 0, (cid:107),∂ (cid:107)h 2 0, (cid:107),∂ (cid:107)h 0 (cid:107)1 (cid:107) 1 3C C ( + κ εz T ) (1 + τ ∞ T + τ ∞ T ) εz T . 0 reg h h 1 0, ,∂ h 2 0, ,∂ h h h For the terms R , R and R , we use the triangle inequality, the regularity estimate (31), and the ≤ 3 4 5(cid:101) (cid:101) assumption h 1 to give ≤ (cid:107) (cid:107) (cid:107) − (cid:107) (cid:107) (cid:107) R = (δp, Π Φ)T δp T ( Π Φ Φ T + Φ T ) 3 V V ≤ (cid:107) (cid:107) (cid:107)h (cid:107) (cid:107)h (cid:107) (cid:107) (cid:107) h h C δp T ( Φ + Ψ + Φ T ) ≤ 0 (cid:107)h (cid:107)(cid:101) 1(cid:107),Ω (cid:107) 1,Ω h (cid:101) 2C C δp T εz T , −0 reg h (cid:101)h h≤ (cid:107) − (cid:107) (cid:107) (cid:107) R = (y y (u), Π Ψ)T y y (u) T Π Ψ T 4 h W h h h W h ≤ (cid:107) − (cid:107) (cid:107) − (cid:107) (cid:107) (cid:107) y y (u) T ( Π Ψ Ψ T + Ψ T ) ≤ (cid:107) −h h(cid:107) (cid:107)W (cid:107) (cid:107)h (cid:107) (cid:107)h (cid:107) C y y (u) T ( Ψ + Φ + Ψ T ) ≤ 0 (cid:101)(cid:107)h −(cid:101) h (cid:107) 1(cid:107),Ω (cid:107) 1,Ω h 2C C y (cid:98) y (u) T εz T , 0 reg h h h h R = (cid:104)(τ −(cid:101)τ )δz, Π Ψ(cid:98)− P Ψ(cid:105) T (cid:98) 5 2 2 W M ∂ h ≤ (cid:107) − (cid:107) (cid:107) (cid:107) (cid:107) − (cid:107) τ τ ∞ T δz(cid:98) T δΨ δΨ T 2 2 0, ,∂ ∂ ∂ h h h ≤ (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) C β ∞ h1/2 δz T C ((cid:98) Ψ + Φ + Ψ ) β 1, ,Ω ∂ 0 1,Ω 1,Ω 1,Ω h ≤ (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) 2C C C β ∞ h1/2 δz T εz T . 0 reg β 1, ,Ω ∂ h h h Summing R1 to R5 gives (cid:98) (cid:107) (cid:107) ≤ C E (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) − (cid:107) (cid:107) (cid:107) εz T ( + κ εz T ) + C( δp T + y y (u) T + h1/2 δz T ), h h h h h h h ∂ h where C (cid:107) (cid:107) (cid:107) (cid:107) = 4C C (1 + τ ∞ T + τ ∞ T ). 0 reg 1 0, ,∂ 2 0, ,∂ h h Choose κ = 1C gives 2 (cid:107) (cid:107) (cid:46) | | | | | | | | εz T hk+1( q + y + p + z ). h h k+1 k+1 k+1 k+1 Finally, (29) and (33b) imply (33a). 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: √ h/ 2 1/8 1/16 1/32 1/64 1/128 (cid:107) − (cid:107) q q 1.3713e-02 3.5195e-03 8.8853e-04 2.2301e-04 5.5850e-05 h 0,Ω order - 2.00 2.00 2.00 2.00 (cid:107) − (cid:107) p p 3.5010e-02 8.9481e-03 2.2581e-03 5.6694e-04 1.4202e-04 h 0,Ω order - 2.00 2.00 2.00 2.00 (cid:107) − (cid:107) y y 1.1712e-02 2.9532e-03 7.4015e-04 1.8520e-04 4.6315e-05 h 0,Ω order - 2.00 2.00 2.00 2.00 (cid:107) − (cid:107) z z 2.3368e-02 5.9064e-03 1.4810e-03 3.7059e-04 9.2676e-05 h 0,Ω order - 2.00 2.00 2.00 2.00 Table 4: Example 2: Errors for the state y, adjoint state z, and the ﬂuxes q and p when k = 1. √ h/ 2 1/2 1/4 1/8 1/16 1/32 (cid:107) − (cid:107) q q 6.3167e-01 3.4472e-01 1.7715e-01 8.9373e-02 4.4778e-02 h 0,Ω order - 0.87 0.96 0.99 1.00 (cid:107) − (cid:107) p p 4.9907e-01 2.9505e-01 1.5339e-01 7.7393e-02 3.8724e-02 h 0,Ω order - 0.76 0.94 0.99 1.00 (cid:107) − (cid:107) y y 1.7959e-01 1.0026e-01 5.3061e-02 2.7275e-02 1.3646e-02 h 0,Ω order - 0.84 0.92 0.96 1.00 (cid:107) − (cid:107) z z 2.3121e-01 1.3646e-01 7.2318e-02 3.7004e-02 1.8587e-02 h 0,Ω order - 0.76 0.92 0.97 1.00 Table 5: Example 3: Errors for the state y, adjoint state z, and the ﬂuxes q and p when k = 0. √ h/ 2 1/2 1/4 1/8 1/16 1/32 (cid:107) − (cid:107) q q 9.2498e-02 2.7594e-02 7.4959e-03 1.9486e-03 4.8720e-04 h 0,Ω order - 1.75 1.90 1.94 2.00 (cid:107) − (cid:107) p p 1.8360e-01 5.3637e-02 1.3921e-02 3.5138e-03 8.7857e-04 h 0,Ω order - 1.80 1.95 1.99 2.00 (cid:107) − (cid:107) y y 4.4822e-02 1.1780e-02 2.9545e-03 7.3644e-04 1.8423e-04 h 0,Ω order - 1.93 2.00 2.00 2.00 (cid:107) − (cid:107) z z 9.1413e-02 2.7583e-02 7.3069e-03 1.8623e-03 4.6575e-04 h 0,Ω order - 1.73 1.92 1.97 2.00 Table 6: Example 3: Errors for the state y, adjoint state z, and the ﬂuxes q and p when k = 1. 21 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: [12] Bernardo Cockburn and Jiguang Shen. A hybridizable discontinuous Galerkin method for the p-Laplacian. SIAM J. Sci. Comput., 38(1):A545–A566, 2016. [13] Javier de Frutos, Bosco Garc´ı a Archilla, and Julia Novo. Local error estimates for the SUPG method applied to evolutionary convection-reaction-diﬀusion equations. J. Sci. Com- put., 66(2):528–554, 2016. [14] Hongfei Fu. A characteristic ﬁnite element method for optimal control problems governed by convection-diﬀusion equations. J. Comput. Appl. Math., 235(3):825–836, 2010. [15] Hongfei Fu and Hongxing Rui. A priori error estimates for optimal control problems governed by transient advection-diﬀusion equations. J. Sci. Comput., 38(3):290–315, 2009. [16] Hongfei Fu and Hongxing Rui. A characteristic-mixed ﬁnite element method for time- dependent convection-diﬀusion optimal control problem. Appl. Math. Comput., 218(7):3430– 3440, 2011. [17] Matthias Heinkenschloss and Dmitriy Leykekhman. Local error estimates for SUPG solutions of advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal., 47(6):4607–4638, 2010. [18] Weiwei Hu, Jiguang Shen, John R. Singler, Yangwen Zhang, and Xiabo Zheng. Hybridizable discontinuous Galerkin method for Dirichlet boundary control of elliptic PDEs. submitted. [19] Z. Kanar Seymen, H. Yu¨cel, and B. Karas¨ozen. Distributed optimal control of time-dependent diﬀusion-convection-reaction equations using space-time discretization. J. Comput. Appl. Math., 261:146–157, 2014. [20] Dmitriy Leykekhman and Matthias Heinkenschloss. Local error analysis of discontinuous Galerkin methods for advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal., 50(4):2012–2038, 2012. [21] A. Mart´ınez, C. Rodr´ıguez, and M. E. V´azquez-M´endez. Theoretical and numerical analysis of an optimal control problem related to wastewater treatment. SIAM J. Control Optim., 38(5):1534–1553, 2000. [22] N. C. Nguyen, J. Peraire, and B. Cockburn. An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diﬀusion equations. J. Comput. Phys., 228(9):3232– 3254, 2009. [23] N. C. Nguyen, J. Peraire, and B. Cockburn. An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diﬀusion equations. J. Comput. Phys., 228(23):8841– 8855, 2009. [24] N. C. Nguyen, J. Peraire, and B. Cockburn. A hybridizable discontinuous Galerkin method for Stokes ﬂow. Comput. Methods Appl. Mech. Engrg., 199(9-12):582–597, 2010. [25] M. Stanglmeier, N. C. Nguyen, J. Peraire, and B. Cockburn. An explicit hybridizable dis- continuous Galerkin method for the acoustic wave equation. Comput. Methods Appl. Mech. Engrg., 300:748–769, 2016. [26] Tongjun Sun. Discontinuous Galerkin ﬁnite element method with interior penalties for con- vection diﬀusion optimal control problem. Int. J. Numer. Anal. Model., 7(1):87–107, 2010. 24

text: Bernardo

text: Cockburn

text: and

text: Jiguang

text: Shen.

text: A

text: hybridizable

text: discontinuous

text: Galerkin

text: method

text: for

text: the

text: p-Laplacian.

text: SIAM

text: J.

text: Sci.

text: Comput.,

text: Javier

text: de

text: Frutos,

text: Bosco

text: Garc´ı

text: a

text: Archilla,

text: and

text: Julia

text: Novo.

text: Local

text: error

text: estimates

text: for

text: the

text: SUPG

text: method

text: applied

text: to

text: evolutionary

text: convection-reaction-diﬀusion

text: equations.

text: J.

text: Sci.

text: Com-

text: put.,

text: Hongfei

text: Fu.

text: A

text: characteristic

text: ﬁnite

text: element

text: method

text: for

text: optimal

text: control

text: problems

text: governed

text: by

text: convection-diﬀusion

text: equations.

text: J.

text: Comput.

text: Appl.

text: Math.,

text: Hongfei

text: Fu

text: and

text: Hongxing

text: Rui.

text: A

text: priori

text: error

text: estimates

text: for

text: optimal

text: control

text: problems

text: governed

text: by

text: transient

text: advection-diﬀusion

text: equations.

text: J.

text: Sci.

text: Comput.,

text: Hongfei

text: Fu

text: and

text: Hongxing

text: Rui.

text: A

text: characteristic-mixed

text: ﬁnite

text: element

text: method

text: for

text: time-

text: dependent

text: convection-diﬀusion

text: optimal

text: control

text: problem.

text: Appl.

text: Math.

text: Comput.,

text: Matthias

text: Heinkenschloss

text: and

text: Dmitriy

text: Leykekhman.

text: Local

text: error

text: estimates

text: for

text: SUPG

text: solutions

text: of

text: advection-dominated

text: elliptic

text: linear-quadratic

text: optimal

text: control

text: problems.

text: SIAM

text: J.

text: Numer.

text: Anal.,

text: Weiwei

text: Hu,

text: Jiguang

text: Shen,

text: John

text: R.

text: Singler,

text: Yangwen

text: Zhang,

text: and

text: Xiabo

text: Zheng.

text: Hybridizable

text: discontinuous

text: Galerkin

text: method

text: for

text: Dirichlet

text: boundary

text: control

text: of

text: elliptic

text: PDEs.

text: submitted.

text: Z.

text: Kanar

text: Seymen,

text: H.

text: Yu¨cel,

text: and

text: B.

text: Karas¨ozen.

text: Distributed

text: optimal

text: control

text: of

text: time-dependent

text: diﬀusion-convection-reaction

text: equations

text: using

text: space-time

text: discretization.

text: J.

text: Comput.

text: Appl.

text: Math.,

text: Dmitriy

text: Leykekhman

text: and

text: Matthias

text: Heinkenschloss.

text: Local

text: error

text: analysis

text: of

text: discontinuous

text: Galerkin

text: methods

text: for

text: advection-dominated

text: elliptic

text: linear-quadratic

text: optimal

text: control

text: problems.

text: SIAM

text: J.

text: Numer.

text: Anal.,

text: A.

text: Mart´ınez,

text: C.

text: Rodr´ıguez,

text: and

text: M.

text: E.

text: V´azquez-M´endez.

text: Theoretical

text: and

text: numerical

text: analysis

text: of

text: an

text: optimal

text: control

text: problem

text: related

text: to

text: wastewater

text: treatment.

text: SIAM

text: J.

text: Control

text: Optim.,

text: N.

text: C.

text: Nguyen,

text: J.

text: Peraire,

text: and

text: B.

text: Cockburn.

text: An

text: implicit

text: high-order

text: hybridizable

text: discontinuous

text: Galerkin

text: method

text: for

text: linear

text: convection-diﬀusion

text: equations.

text: J.

text: Comput.

text: Phys.,

text: N.

text: C.

text: Nguyen,

text: J.

text: Peraire,

text: and

text: B.

text: Cockburn.

text: An

text: implicit

text: high-order

text: hybridizable

text: discontinuous

text: Galerkin

text: method

text: for

text: nonlinear

text: convection-diﬀusion

text: equations.

text: J.

text: Comput.

text: Phys.,

text: N.

text: C.

text: Nguyen,

text: J.

text: Peraire,

text: and

text: B.

text: Cockburn.

text: A

text: hybridizable

text: discontinuous

text: Galerkin

text: method

text: for

text: Stokes

text: ﬂow.

text: Comput.

text: Methods

text: Appl.

text: Mech.

text: Engrg.,

text: M.

text: Stanglmeier,

text: N.

text: C.

text: Nguyen,

text: J.

text: Peraire,

text: and

text: B.

text: Cockburn.

text: An

text: explicit

text: hybridizable

text: dis-

text: continuous

text: Galerkin

text: method

text: for

text: the

text: acoustic

text: wave

text: equation.

text: Comput.

text: Methods

text: Appl.

text: Mech.

text: Engrg.,

text: Tongjun

text: Sun.

text: Discontinuous

text: Galerkin

text: ﬁnite

text: element

text: method

text: with

text: interior

text: penalties

text: for

text: con-

text: vection

text: diﬀusion

text: optimal

text: control

text: problem.

text: Int.

text: J.

text: Numer.

text: Anal.

text: Model.,


text: Once γ is computed, α and β can be quickly and easily computed using (14). 3 Error Analysis Next, we provide a convergence analysis of the above HDG method for the optimal control ∈ ∞ problem. Throughout this section, we assume β [W 1, (Ω)]d, Ω is a bounded convex polyhedral ≤ domain, h 1, and the solution of the optimality system (4) is smooth enough. 3.1 Main result For our theoretical results, we require the stabilization functions τ and τ are chosen to satisfy T 1 2 (A1) τ is piecewise constant on ∂ . 1 h · (A2) τ = τ + β n. 1 2 ∈ T − · | (A3) For any K , min (τ 1β n) > 0. h 1 2 ∂K We note that (A2) and (A3) imply 1 · | ∈ T min (τ + β n) > 0 for any K . (17) 2 ∂K h 2 Theorem 1. We have (cid:107) − (cid:107) (cid:46) | | | | | | | | q q T hk+1( q + y + p + z ), h h k+1 k+1 k+1 k+1 (cid:107) − (cid:107) (cid:46) | | | | | | | | p p T hk+1( q + y + p + z ), h h k+1 k+1 k+1 k+1 (cid:107) − (cid:107) (cid:46) | | | | | | | | y y T hk+1( q + y + p + z ), h h k+1 k+1 k+1 k+1 (cid:107) − (cid:107) (cid:46) | | | | | | | | z z T hk+1( q + y + p + z ), h h k+1 k+1 k+1 k+1 (cid:107) − (cid:107) (cid:46) | | | | | | | | u u T hk+1( q + y + p + z ). h h k+1 k+1 k+1 k+1 3.2 Preliminary material Next, we introduce the projection operators Π and Π deﬁned in [7] that we use frequently V W ∈ T in our proof. The value of the projection on each element K is determined by requiring that h the components satisfy the equations (Π q + βΠ y, r) = (q + βy, r) , (18a) V W K K (Π y, w) = (y, w) , (18b) W K K (cid:104) · · (cid:105) (cid:104) · · (cid:105) Π q n + β nP y + τ Π y, µ = q n + β ny + τ y, µ , (18c) V M 1 W e 1 e ∈ P × P × P for all (r, w, µ) − (K) − (K) (e) and for all faces e of the simplex K. Here, P k 1 k 1 k M denotes the L2-orthogonal projection from L2(ε ) into M satisfying h h (cid:104) − (cid:105) ∀ ∈ ∀ ∈ P y y, µ = 0, e ε , µ M . (19) M e h h 7 ##LTLine## ##LTLine##

text: Once

text: γ

text: is

text: computed,

text: α

text: and

text: β

text: can

text: be

text: quickly

text: and

text: easily

text: computed

text: using

label: section

text: Error

text: Analysis

text: Next,

text: we

text: provide

text: a

text: convergence

text: analysis

text: of

text: the

text: above

text: HDG

text: method

text: for

text: the

text: optimal

text: control

text: problem.

text: Throughout

text: this

text: section,

text: we

text: assume

text: β

text: [W

text: (Ω)]d,

text: Ω

text: is

text: a

text: bounded

text: convex

text: polyhedral

text: domain,

text: h

text: and

text: the

text: solution

text: of

text: the

text: optimality

text: system

text: is

text: smooth

text: enough.

text: Main

text: result

text: For

text: our

text: theoretical

text: results,

text: we

text: require

text: the

text: stabilization

text: functions

text: τ

text: and

text: τ

text: are

text: chosen

text: to

text: satisfy

text: T

text: τ

text: is

text: piecewise

text: constant

text: on

text: h

text: τ

text: τ

text: β

text: n.

text: T

text: For

text: any

text: K

text: min

text: (τ

text: 1β

text: n)

text: h

text: ∂K

text: We

text: note

text: that

text: and

text: imply

text: T

text: min

text: (τ

text: β

text: n)

text: for

text: any

text: K

text: ∂K

text: h

text: Theorem

text: We

text: have

text: q

text: q

text: T

text: q

text: y

text: p

text: z

text: h

text: h

text: p

text: p

text: T

text: q

text: y

text: p

text: z

text: h

text: h

text: y

text: y

text: T

text: q

text: y

text: p

text: z

text: h

text: h

text: z

text: z

text: T

text: q

text: y

text: p

text: z

text: h

text: h

text: u

text: u

text: T

text: q

text: y

text: p

text: z

text: h

text: h

text: Preliminary

text: material

text: Next,

text: we

text: introduce

text: the

text: projection

text: operators

text: Π

text: and

text: Π

text: deﬁned

text: in

text: that

text: we

text: use

text: frequently

text: V

text: W

text: T

text: in

text: our

text: proof.

text: The

text: value

text: of

text: the

text: projection

text: on

text: each

text: element

text: K

text: is

text: determined

text: by

text: requiring

text: that

text: h

text: the

text: components

text: satisfy

text: the

text: equations

text: (Π

text: q

text: βΠ

text: y,

text: r)

text: (q

text: βy,

text: r)

text: V

text: W

text: K

text: K

text: (Π

text: y,

text: w)

text: (y,

text: w)

text: W

text: K

text: K

text: Π

text: q

text: n

text: β

text: nP

text: y

text: τ

text: Π

text: y,

text: µ

text: q

text: n

text: β

text: ny

text: τ

text: y,

text: µ

text: V

text: M

text: W

text: e

text: e

text: P

text: P

text: P

text: for

text: all

text: (r,

text: w,

text: µ)

text: (K)

text: (K)

text: (e)

text: and

text: for

text: all

text: faces

text: e

text: of

text: the

text: simplex

text: K.

text: Here,

text: P

text: k

text: k

text: k

text: M

text: denotes

text: the

text: projection

text: from

text: into

text: M

text: satisfying

text: h

text: h

text: P

text: y

text: y,

text: µ

text: e

text: ε

text: µ

text: M

text: M

text: e

text: h

text: h

text: ##LTLine##

text: ##LTLine##


text: ∈ × × Lemma 4. For any (v , w , µ ) V W M (o), we have h h h h h h B (v , w , µ ; v , w , µ ) 1 h h h h h h (cid:104) − 1 · − − (cid:105) = (v , v )T + (τ β n)(w µ ), w µ T \ h h h 1 2 h h h h ∂ h ε∂h (cid:104) − 1 · (cid:105) + (τ β n)w , w , 1 2 h h ε∂h B (v , w , µ ; v , w , µ ) 2 h h h h h h (cid:104) 1 · − − (cid:105) = (v , v )T + (τ + β n)(w µ ), w µ T \ h h h 2 2 h h h h ∂ h ε∂h (cid:104) 1 · (cid:105) + (τ + β n)w , w . 2 2 h h ε∂h Proof. We only prove the ﬁrst identity; the second can be obtained by the same argument. B (v , w , µ ; v , w , µ ) 1 h h h h h h − ∇ · (cid:104) · (cid:105) = (v , v )T (w , v )T + µ , v n T \ − h h h ∇h h (cid:104)h · h h ∂ (cid:105)h ε∂h (v + βw , w )T + v n + τ w , w T (cid:104) h · − h h h(cid:105) h 1 h h ∂ h + (β n τ )µ , w T \ 1 h h ∂ ε∂ − (cid:104) · · h h − (cid:105) v n + β nµ + τ (w µ ), µ T \ h h 1 h h h ∂ ε∂ − ∇ (cid:104) (cid:105)h h = (v , v )T (βw , w )T + τ w , w T (cid:104)h h· h− h (cid:105) h h − (cid:104)1 ·h h ∂ h − (cid:105) + (β n τ )µ , w T \ β nµ + τ (w µ ), µ T \ . 1 h h ∂ ε∂ h 1 h h h ∂ ε∂ h h h h Moreover, ∇ ∇ · (cid:104) · (cid:105) − ∇ (βw , w )T = ( (βw ), w )T = β nw , w T (βw , w )T , h h h h h h h h ∂ h h h h which implies ∇ 1(cid:104) · (cid:105) (βw , w )T = β nw , w T . h h h 2 h h ∂ h Then we obtain B (v , w , µ ; v , w , µ ) 1 h h h h h h (cid:104) − 1 · − − (cid:105) = (v , v )T + (τ β n)(w µ ), w µ T \ h h h 1 2 h h h h ∂ h ε∂h (cid:104) − 1 · (cid:105) − 1(cid:104) · (cid:105) + (τ β n)w , w β nµ , µ T \ . 1 2 h h ε∂h 2 h h ∂ h ε∂h Since µ is single-valued across the interfaces, we have h −1(cid:104) · (cid:105) β nµ , µ T \ = 0. 2 h h ∂ h ε∂h This completes the proof. Next, we give a property of the HDG operators B and B that is critical to our error analysis 1 2 of the method. 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lemma

text: For

text: any

text: (v

text: w

text: µ

text: V

text: W

text: M

text: (o),

text: we

text: have

text: h

text: h

text: h

text: h

text: h

text: h

text: B

text: (v

text: w

text: µ

text: v

text: w

text: µ

text: h

text: h

text: h

text: h

text: h

text: h

text: (v

text: v

text: )T

text: (τ

text: β

text: n)(w

text: µ

text: w

text: µ

text: T

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: ε∂h

text: (τ

text: β

text: n)w

text: w

text: h

text: h

text: ε∂h

text: B

text: (v

text: w

text: µ

text: v

text: w

text: µ

text: h

text: h

text: h

text: h

text: h

text: h

text: (v

text: v

text: )T

text: (τ

text: β

text: n)(w

text: µ

text: w

text: µ

text: T

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: ε∂h

text: (τ

text: β

text: n)w

text: w

text: h

text: h

text: ε∂h

text: Proof.

text: We

text: only

text: prove

text: the

text: ﬁrst

text: identity;

text: the

text: second

text: can

text: be

text: obtained

text: by

text: the

text: same

text: argument.

text: B

text: (v

text: w

text: µ

text: v

text: w

text: µ

text: h

text: h

text: h

text: h

text: h

text: h

text: (v

text: v

text: )T

text: (w

text: v

text: )T

text: µ

text: v

text: n

text: T

text: h

text: h

text: h

text: ∇h

text: h

text: h

text: h

text: ε∂h

text: (v

text: βw

text: w

text: )T

text: v

text: n

text: τ

text: w

text: w

text: T

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: (β

text: n

text: τ

text: )µ

text: w

text: T

text: h

text: h

text: ε∂

text: h

text: h

text: v

text: n

text: β

text: nµ

text: τ

text: (w

text: µ

text: µ

text: T

text: h

text: h

text: h

text: h

text: h

text: ε∂

text: h

text: (v

text: v

text: )T

text: (βw

text: w

text: )T

text: τ

text: w

text: w

text: T

text: h·

text: h−

text: h

text: h

text: h

text: ·h

text: h

text: h

text: (β

text: n

text: τ

text: )µ

text: w

text: T

text: β

text: nµ

text: τ

text: (w

text: µ

text: µ

text: T

text: h

text: h

text: ε∂

text: h

text: h

text: h

text: h

text: ε∂

text: h

text: h

text: h

text: h

text: Moreover,

text: (βw

text: w

text: )T

text: (βw

text: w

text: )T

text: β

text: nw

text: w

text: T

text: (βw

text: w

text: )T

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: which

text: implies

text: (βw

text: w

text: )T

text: β

text: nw

text: w

text: T

text: h

text: h

text: h

text: h

text: h

text: h

text: Then

text: we

text: obtain

text: B

text: (v

text: w

text: µ

text: v

text: w

text: µ

text: h

text: h

text: h

text: h

text: h

text: h

text: (v

text: v

text: )T

text: (τ

text: β

text: n)(w

text: µ

text: w

text: µ

text: T

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: ε∂h

text: (τ

text: β

text: n)w

text: w

text: β

text: nµ

text: µ

text: T

text: h

text: h

text: ε∂h

text: h

text: h

text: h

text: ε∂h

text: Since

text: µ

text: is

text: single-valued

text: across

text: the

text: interfaces,

text: we

text: have

text: h

text: β

text: nµ

text: µ

text: T

text: h

text: h

text: h

text: ε∂h

text: This

text: completes

text: the

text: proof.

text: Next,

text: we

text: give

text: a

text: property

text: of

text: the

text: HDG

text: operators

text: B

text: and

text: B

text: that

text: is

text: critical

text: to

text: our

text: error

text: analysis

text: of

text: the

text: method.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:18) (cid:19) (cid:88) 11 (cid:48) ≈ † h(k) ∆(k) − (cid:48) H Ψ † − − Ψ f + ε , (B26) (cid:80) k ∆ (k) hT ( k) k 0 k (cid:48) (cid:88) where only covers half 1BZ(covering the whole 1BZ if counting its inversion partner), k 1 (cid:48) ∗ (cid:48) f = V (k, k )b (k)b (k ), (cid:80) 2 αβγδ βα γδ (cid:48) k,k ,α,β,γ,δ (cid:48) (cid:48) − and ε = Tr[h( k)]. 0 k (cid:88) Plugging Eq.B26 into deﬁnition of b (k) and keeping ∆ to ﬁrst order on the right-hand side, we obtain γδ 1 b (k) = [G (k, iω )∆(k)G (k, iω )] + O(∆2). (B27) γδ β e n h n δγ ωn (cid:88) (cid:88) Combining the above equation with Eq.B24, the self-consistent linearized gap equation is derived as 1 (cid:48) (cid:48) (cid:48) (cid:48) ∆ (k) = V (k, k )[G (k , iω )∆(k )G (k , iω )] , (B28) αβ β αβγδ e n h n δγ (cid:48) k ,ωn γδ − − − − where G (k, iω ) = [iω h(k)] 1 is the normal state Green function, G (k, iω ) = [iω + hT ( k)] 1 and ω = e n n h n n n (2n+1)π/β is the fermionic Matsubara frequency with n being integers. The superconducting transition temperature can be solved from Eq.(B28). 4. s-Wave Singlet and d-Wave Quintet Mixing in linearized gap equation In this section, we will show the singlet-quintet mixing is allowed in the above linearized gap equation for Luttinger Hamiltonian in the isotropic case and the symmetric SOC term h deﬁned in the main text plays a central role in SOC this pairing mixing mechanism. If we choose the gap funct(cid:90)ion on the left hand side of the gap equation to be s-wave singlet pairing, the gap equation (cid:88) (cid:88) (cid:88) will take the form 1 dkˆ † (cid:48) (cid:48) (cid:48) (cid:48) ∆00(k) = [N00(kˆ)γ] V (k, k )[G (k , iω )∆(k )G (k , iω )] . 00 β 4π 00 βα αβγδ e n h n δγ (cid:48) αβ k ,ωn γδ (cid:88) With the interaction form in Eq.(B18), we have ∆00(k) = 1 V˜ (|k|, |k(cid:48)|)Tr{G (k(cid:48), iω )∆(k(cid:48))G (k(cid:48), iω )[N00(kˆ(cid:48))γ]†} (B29) 00 β 000 e n h n 00 (cid:48) k ,ωn (cid:88) (cid:88) (cid:88) (cid:88) (cid:48) | | l+S j ⇒ ∆00(k) = ∆lS (k) 1 V˜ (|k|, |k(cid:48)|)Tr{G (k(cid:48), iω )[NlS (kˆ(cid:48))γ]G (k(cid:48), iω )[N00(kˆ(cid:48))γ]†}. 00 | − | − jmj β (cid:48) 000 e n jmj h n 00 S,l j= l S mj= j k ,ωn (B30) (cid:88) The mixing between ∆00(k) and ∆lS (k) in the above equation is determined by 00 jmj 1 V˜ (|k|, |k(cid:48)|)Tr{G (k(cid:48), iω )[NlS (kˆ(cid:48))γ]G (k(cid:48), iω )[N00(kˆ(cid:48))γ]†}. β (cid:48) 000 e n jmj h n 00 k ,ωn (cid:88) To simplify our discussion, we assume the O(3) symmetry of non-interacting Hamiltonian. In this limit, we ﬁnd Tr{G (k(cid:48), iω )[NlS (kˆ(cid:48))γ]G (k(cid:48), iω )[N00(kˆ(cid:48))γ]†} = 0 e n jmj h n 00 kˆ(cid:48),ωn (cid:54) (cid:54) for j = 0 or m = 0 or l is not even. Therefore, only the isotropic d-wave quintet pairing ∆22(k) can be mixed into j 00 ∆00(k) under the O(3) symmetry. 00 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: h(k)

text: ∆(k)

text: H

text: Ψ

text: Ψ

text: f

text: ε

text: k

text: (k)

text: hT

text: k)

text: k

text: k

text: where

text: only

text: covers

text: half

text: the

text: whole

text: if

text: counting

text: its

text: inversion

text: partner),

text: k

text: f

text: V

text: (k,

text: k

text: )b

text: (k)b

text: (k

text: αβγδ

text: βα

text: γδ

text: k,k

text: ,α,β,γ,δ

text: and

text: ε

text: Tr[h(

text: k)].

text: k

text: Plugging

text: into

text: deﬁnition

text: of

text: b

text: (k)

text: and

text: keeping

text: to

text: ﬁrst

text: order

text: on

text: the

text: right-hand

text: side,

text: we

text: obtain

text: γδ

text: b

text: (k)

text: [G

text: (k,

text: iω

text: )∆(k)G

text: (k,

text: iω

text: γδ

text: β

text: e

text: n

text: h

text: n

text: δγ

text: ωn

text: Combining

text: the

text: above

text: equation

text: with

text: the

text: self-consistent

text: linearized

text: gap

text: equation

text: is

text: derived

text: as

text: (k)

text: V

text: (k,

text: k

text: )[G

text: (k

text: iω

text: )∆(k

text: )G

text: (k

text: iω

text: αβ

text: β

text: αβγδ

text: e

text: n

text: h

text: n

text: δγ

text: k

text: ,ωn

text: γδ

text: where

text: G

text: (k,

text: iω

text: [iω

text: h(k)]

text: is

text: the

text: normal

text: state

text: Green

text: function,

text: G

text: (k,

text: iω

text: [iω

text: hT

text: k)]

text: and

text: ω

text: e

text: n

text: n

text: h

text: n

text: n

text: n

text: is

text: the

text: fermionic

text: Matsubara

text: frequency

text: with

text: n

text: being

text: integers.

text: The

text: superconducting

text: transition

text: temperature

text: can

text: be

text: solved

text: from

text: s-Wave

text: Singlet

text: and

text: d-Wave

text: Quintet

text: Mixing

text: in

text: linearized

text: gap

text: equation

text: In

text: this

text: section,

text: we

text: will

text: show

text: the

text: singlet-quintet

text: mixing

text: is

text: allowed

text: in

text: the

text: above

text: linearized

text: gap

text: equation

text: for

text: Luttinger

text: Hamiltonian

text: in

text: the

text: isotropic

text: case

text: and

text: the

text: symmetric

text: SOC

text: term

text: h

text: deﬁned

text: in

text: the

text: main

text: text

text: plays

text: a

text: central

text: role

text: in

text: SOC

text: this

text: pairing

text: mixing

text: mechanism.

text: If

text: we

text: choose

text: the

text: gap

text: on

text: the

text: left

text: hand

text: side

text: of

text: the

text: gap

text: equation

text: to

text: be

text: s-wave

text: singlet

text: pairing,

text: the

text: gap

text: equation

text: will

text: take

text: the

text: form

text: dkˆ

text: V

text: (k,

text: k

text: )[G

text: (k

text: iω

text: )∆(k

text: )G

text: (k

text: iω

text: β

text: 4π

text: βα

text: αβγδ

text: e

text: n

text: h

text: n

text: δγ

text: αβ

text: k

text: ,ωn

text: γδ

text: With

text: the

text: interaction

text: form

text: in

text: we

text: have

text: V˜

text: (|k|,

text: iω

text: iω

text: β

text: e

text: n

text: h

text: n

text: k

text: ,ωn

text: l+S

text: j

text: ∆lS

text: (k)

text: V˜

text: (|k|,

text: iω

text: )[NlS

text: iω

text: jmj

text: β

text: e

text: n

text: jmj

text: h

text: n

text: S,l

text: j=

text: l

text: S

text: mj=

text: j

text: k

text: ,ωn

text: The

text: mixing

text: between

text: and

text: ∆lS

text: (k)

text: in

text: the

text: above

text: equation

text: is

text: determined

text: by

text: jmj

text: V˜

text: (|k|,

text: iω

text: )[NlS

text: iω

text: β

text: e

text: n

text: jmj

text: h

text: n

text: k

text: ,ωn

text: To

text: simplify

text: our

text: discussion,

text: we

text: assume

text: the

text: symmetry

text: of

text: non-interacting

text: Hamiltonian.

text: In

text: this

text: limit,

text: we

text: ﬁnd

text: Tr{G

text: iω

text: )[NlS

text: iω

text: e

text: n

text: jmj

text: h

text: n

text: for

text: j

text: or

text: m

text: or

text: l

text: is

text: not

text: even.

text: Therefore,

text: only

text: the

text: isotropic

text: d-wave

text: quintet

text: pairing

text: can

text: be

text: mixed

text: into

text: j

text: under

text: the

text: symmetry.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:88) (cid:88) 15 · (cid:48) (cid:48) Γ0 a2g Γ ∆ (k) = V (k, k )b (k ) = ∆ ( γ) + ∆ ( k γ) , (B56) αβ αβγδ γδ 0 2 αβ 1 2 αβ (cid:48) k γδ (cid:88) (cid:88) where V Γ0 † (cid:48) ∆ = V0 [ γ] b (k ) (B57) 0 2 γδ γδ (cid:48) k γδ (cid:88) (cid:88) and · V a2g (cid:48) Γ † (cid:48) ∆ = V1 [ k γ] b (k ). (B58) 1 2 γδ γδ (cid:48) k γδ With the linearized gap equation (B28), we ﬁnd the coupled linearized gap equations in the singlet and qunitet channels take the form (cid:20) (cid:21) (cid:88) · ∆ =(cid:88)VV0 (cid:20) Tr G (k, iω )(∆ Γ0 γ + ∆ a2gk Γγ)G (k, iω )(Γ0 γ)(cid:21)† (B59) 0 β e n 0 2 1 2 h n 2 k,ωn · · V Γ0 a2g Γ a2g Γ † ∆ = V1 Tr G (k, iω )(∆ γ + ∆ k γ)G (k, iω )( k γ) . (B60) 1 β e n 0 2 1 2 h n 2 k,ωn (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:88) With the Green functions in Eq.B3 and Eq.B4, the coupled gap equations are re-written as − ∆ 1 V (k4Q2 + ξ2 + ω2) V 2k4a2(c Q2 + c Q2)ξ ∆ ∆01 = βVb(k, iωn) V12k04a2(c1cQ21 + c2Qn22)ξ −V1k4a4[(k4Q2c +0ξ2 + ωn2(cid:112))1−12k4(Q2 2c 2− (c1Q21 +(cid:112)c2Q22)2)] ∆01 , (cid:112) k,ωn − ± where b(k, iω ) = (ξ−2 +ω2)(ξ2 +ω2) , ξ± = k2 µ, m± = m/(1 2mQ ) ,Q = c2Q2 + c2Q2, Q = gˆ2 + gˆ2 + gˆ2, n n + n 2m± c c 1 1 2 2 1 1 2 3 Q = gˆ2 + gˆ2 and gˆ = g /k2. It is easy to see that the mixing is zero if c = c = 0, which means the symmetric 2 4 5 i i 1 2 SOC is essential. The above equations would be easier to deal with if expressed in terms of the projection operators. With Eq.B5 and Eq.B6, we have (cid:88) (cid:88)    − V 1 ∆ ∆ a2k2 c Q2 + c Q2 1 ∆ − ∆ a2k2 c Q2 + c Q2 ∆ = V0 ( 0 + 1 1 1 2 2 ) + ( 0 1 1 1 2 2 ) 0 β ω2 + ξ2 2 2 Q ω2 + ξ−2 2 2 Q k,ωn n +(cid:88) c k,ωn n (cid:18) (cid:19) c  − V 1 ∆ a2k2 c Q2 + c Q2 ∆ a4k4 c Q2 + c Q2 2 ∆ = V1 ( 0 1 1 2 2 + 1 1 1 2 2 ) 1 β ω2 + ξ2 2 Q 2 Q (cid:88) k,ωn n + (cid:18) c (cid:19) (cid:88) c  − −  1 ∆ a2k2 c Q2 + c Q2 ∆ a4k4 c Q2 + c Q2 2 − 1 (c c )2Q2Q2k4a4∆ + ( 0 1 1 2 2 + 1 1 1 2 2 ) − 1 2 1 2 1 . ω2 + ξ−2 2 Q 2 Q (iω ξ )(iω + ξ−) Q2 k,ωn n c c k,ωn n + n c (cid:28) | | (cid:28) We consider the limit (cid:15) /2Q k2 1 and (cid:15) /µ 1, where (cid:15) labels the energy range for the momentum summation c c F c c around the chemical potential µ and is an energy sc(cid:115)ale much smaller than SOC stren(cid:115)gth and chemical potential. In (cid:90) (cid:90) (cid:90) (cid:90) the continuous limit, the(cid:88)momentum summation can thus be written as 1 dΩ (cid:15)c ξ− dΩ (cid:15)c ξ V = N−(0) dξ− 1 + + N (0) dξ 1 + + , 4π − µ 4π + − + µ k √ (cid:15)c (cid:15)c ≡ − | | where N±(0) (2π) 34π m± Re[ 2m±µ] is the density of states for ξ± bands at Fermi energy without spin degen- eracy. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Γ0

text: Γ

text: (k)

text: V

text: (k,

text: k

text: )b

text: (k

text: γ)

text: k

text: γ)

text: αβ

text: αβγδ

text: γδ

text: αβ

text: αβ

text: k

text: γδ

text: where

text: V

text: Γ0

text: γ]

text: b

text: (k

text: γδ

text: γδ

text: k

text: γδ

text: and

text: V

text: Γ

text: k

text: γ]

text: b

text: (k

text: γδ

text: γδ

text: k

text: γδ

text: With

text: the

text: linearized

text: gap

text: equation

text: we

text: ﬁnd

text: the

text: coupled

text: linearized

text: gap

text: equations

text: in

text: the

text: singlet

text: and

text: qunitet

text: channels

text: take

text: the

text: form

text: Tr

text: G

text: (k,

text: iω

text: Γ0

text: γ

text: Γγ)G

text: (k,

text: iω

text: β

text: e

text: n

text: h

text: n

text: k,ωn

text: V

text: Γ0

text: Γ

text: Γ

text: Tr

text: G

text: (k,

text: iω

text: γ

text: k

text: γ)G

text: (k,

text: iω

text: k

text: γ)

text: β

text: e

text: n

text: h

text: n

text: k,ωn

text: With

text: the

text: Green

text: functions

text: in

text: and

text: the

text: coupled

text: gap

text: equations

text: are

text: re-written

text: as

text: V

text: ξ2

text: V

text: c

text: βVb(k,

text: iωn)

text: 2−

text: k,ωn

text: where

text: b(k,

text: iω

text: (ξ−2

text: ξ±

text: µ,

text: m±

text: ,Q

text: Q

text: n

text: n

text: n

text: c

text: c

text: Q

text: and

text: gˆ

text: g

text: It

text: is

text: easy

text: to

text: see

text: that

text: the

text: mixing

text: is

text: zero

text: if

text: c

text: c

text: which

text: means

text: the

text: symmetric

text: i

text: i

text: SOC

text: is

text: essential.

text: The

text: above

text: equations

text: would

text: be

text: easier

text: to

text: deal

text: with

text: if

text: expressed

text: in

text: terms

text: of

text: the

text: projection

text: operators.

text: With

text: and

text: we

text: have

text: V

text: c

text: c

text: c

text: c

text: β

text: ω2

text: ξ2

text: Q

text: ω2

text: ξ−2

text: Q

text: k,ωn

text: n

text: c

text: k,ωn

text: n

text: c

text: V

text: c

text: c

text: c

text: c

text: β

text: ω2

text: ξ2

text: Q

text: Q

text: k,ωn

text: n

text: c

text: c

text: c

text: c

text: c

text: c

text: (c

text: c

text: ω2

text: ξ−2

text: Q

text: Q

text: (iω

text: ξ

text: )(iω

text: ξ−)

text: k,ωn

text: n

text: c

text: c

text: k,ωn

text: n

text: n

text: c

text: We

text: consider

text: the

text: limit

text: and

text: /µ

text: where

text: labels

text: the

text: energy

text: range

text: for

text: the

text: momentum

text: summation

text: c

text: c

text: F

text: c

text: c

text: around

text: the

text: chemical

text: potential

text: µ

text: and

text: is

text: an

text: energy

text: much

text: smaller

text: than

text: SOC

text: and

text: chemical

text: potential.

text: In

text: the

text: continuous

text: limit,

text: summation

text: can

text: thus

text: be

text: written

text: as

text: dΩ

text: ξ−

text: dΩ

text: ξ

text: V

text: dξ−

text: N

text: dξ

text: 4π

text: µ

text: 4π

text: µ

text: k

text: where

text: m±

text: Re[

text: is

text: the

text: density

text: of

text: states

text: for

text: ξ±

text: bands

text: at

text: Fermi

text: energy

text: without

text: spin

text: degen-

text: eracy.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 17 where T0 = 2eγ¯(cid:15)c/(πkB). The correspo(cid:113)nding eigen-vector gives rise to the ratio of order parameters ∆∆˜˜01 in diﬀerent channels, which reads − − ∆˜ 0 = (λ0y1 λ˜1y3)2 + 4λ0λ˜1y22 + λ0y1 λ˜1y3 . (B68) ∆˜ 2λ˜ y 1 1 2 We notice that the singlet-quintet mixing can enhance the critical temperature T . To see that, we can neglect the c oﬀ-diagonal term in the gap equation (B61) or equivalently choose y = 0. In this case, the critical temperatures 2 in the singlet and quintet channels can be determined by 1/x = λ y /2 and 1/x = λ˜ y /2, respectively, where cs 0 1 cq 1 3 x = ln(T /T ) and x = ln(T /T ). Since cs 0 cs cq 0 cq 1 1 1 > max( , ) (B69) x x x cs cq with x = ln(T /T ), we conclude that the T in Eq. (B67) is always larger than T and T . 0 c c cs cq 7. Kink Structure of ∆˜ /∆˜ in Regime III 1 0 This section is devoted to the understanding of three kinks in Fig.1b of the main text, whose positions are shown in Fig.3a by gray dashed lines. Let us ﬁrst discuss the band structure and the momentum cut-oﬀ. In regime III, the ξ− bands always bend down, − − while the ξ band bends up along Γ X and down along Γ L or vice verse, as depicted in Fig.1a(iii) in the main + text. Therefore, a saddle point exists at Γ for the ξ bands, and leads to hyperbolic Fermi surface with divergent + density of states. Such hyperbolic Fermi surface is due to the limitation of the Luttinger model, which is only valid in a small momentum region around Γ. More importantly, it will cause the divergence of the functions y . To avoid 1,2,3 this pr√oblem, we introduce a momentum cut-oﬀ Λ, wh√ich can be implemented by inserting a Heaviside step function − − − − − θ(Λ 2m−µ) = θ(1 2mQ 2mµ/Λ2) and θ(Λ 2m µ) = θ(1 + 2mQ 2mµ/Λ2) into the integral for the ξ− c(cid:90) (cid:20) + c (cid:21) and ξ bands, respectively. As a result, the functions y are re-deﬁned as + 1,2,3 − − − ≡ dΩ θ(1 2mQ 2mµ/Λ2) θ(1 + 2mQ 2mµ/Λ2) y − c + c , (cid:90) 1 (cid:20) 4π (1 2mQc)3/2 (1 + 2mQc)3/(cid:21)2 − − − | | | | ≡ dΩ −θ(1 2mQ 2mµ/Λ2) θ(1 + 2mQ 2mµ/Λ2) c Q2 + c Q2 y − c + c 1 1 2 2 2 4π (1 2mQ )5/2 (1 + 2mQ )5/2 Q c c c (cid:90) (cid:20) (cid:21) (cid:18) (cid:19) and − − − | | | | ≡ dΩ θ(1 2mQ 2mµ/Λ2) θ(1 + 2mQ 2mµ/Λ2) Q2 c + Q2 c 2 y − c + c 1 1 2 2 . 3 4π (1 2mQ )7/2 (1 + 2mQ )7/2 Q c c c √ The Fermi surface shape of the ξ band plays an important role in determining the values of y , and consequently + 1,2,3∞ the kink structures. We choose the momentum cut-oﬀ as Λ = 3 2mµ for the red line and Λ = for the blue line in Fig. 3 (a). For a small SOC parameter c , the momentum cut-oﬀ is not important and thus the red line coincides 1 | | with the blue line in Fig. 3 (a). With increasing the SOC to 2mc = 4/9, the Fermi momentum of the ξ band 1 + starts becoming larger than the cut-oﬀ Λ along certain angles, and thus the integrals y are limited by Λ (see |1,2,3 | Fig.3b), leading to the appearance of the ﬁrst kink in ∆˜ /∆˜ . With further increasing 2mc , we ﬁnd y show a 1 0 1 2,3 peak behavior in Fig.3b due to the shrink of the Fermi surface range for the ξ bands, giving rise to the second kink. | | + When the SOC reaches 2mc = 8/9, the ξ Fermi surface moves away from the momentum range within the cut-oﬀ 1 + Λ. This yields a signiﬁcant decreasing of y , as well as a dramatic drop of ∆˜ /∆˜ . The quintet mixing is negligible | | 1,2,3 1 0 in the regime III when 2mc > 8/9. 1 C. Bogoliubov-de Gennes Hamiltonian and Topological Nodal-line superconductivity We will study the energy dispersion of the Bogoliubov-de Gennes (BdG) Hamiltonian with the singlet-quintet mixing and extract the phase diagram for the topological nodal-line superconducting phase. 1. BdG Hamiltonian ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: The

text: eigen-vector

text: gives

text: rise

text: to

text: the

text: ratio

text: of

text: order

text: parameters

text: ∆∆˜˜01

text: in

text: diﬀerent

text: channels,

text: which

text: reads

text: 2λ˜

text: y

text: We

text: notice

text: that

text: the

text: singlet-quintet

text: mixing

text: can

text: enhance

text: the

text: critical

text: temperature

text: T

text: To

text: see

text: that,

text: we

text: can

text: neglect

text: the

text: c

text: oﬀ-diagonal

text: term

text: in

text: the

text: gap

text: equation

text: or

text: equivalently

text: choose

text: y

text: In

text: this

text: case,

text: the

text: critical

text: temperatures

text: in

text: the

text: singlet

text: and

text: quintet

text: channels

text: can

text: be

text: determined

text: by

text: λ

text: y

text: and

text: λ˜

text: y

text: respectively,

text: where

text: cs

text: cq

text: x

text: ln(T

text: /T

text: and

text: x

text: ln(T

text: /T

text: Since

text: cs

text: cs

text: cq

text: cq

text: max(

text: x

text: x

text: x

text: cs

text: cq

text: with

text: x

text: ln(T

text: /T

text: we

text: conclude

text: that

text: the

text: T

text: in

text: Eq.

text: is

text: always

text: larger

text: than

text: T

text: and

text: T

text: c

text: c

text: cs

text: cq

text: Kink

text: Structure

text: of

text: in

text: Regime

text: III

text: This

text: section

text: is

text: devoted

text: to

text: the

text: understanding

text: of

text: three

text: kinks

text: in

text: of

text: the

text: main

text: text,

text: whose

text: positions

text: are

text: shown

text: in

text: by

text: gray

text: dashed

text: lines.

text: Let

text: us

text: ﬁrst

text: discuss

text: the

text: band

text: structure

text: and

text: the

text: momentum

text: cut-oﬀ.

text: In

text: regime

text: III,

text: the

text: ξ−

text: bands

text: always

text: bend

text: down,

text: while

text: the

text: ξ

text: band

text: bends

text: up

text: along

text: Γ

text: X

text: and

text: down

text: along

text: Γ

text: L

text: or

text: vice

text: verse,

text: as

text: depicted

text: in

text: in

text: the

text: main

text: text.

text: Therefore,

text: a

text: saddle

text: point

text: exists

text: at

text: Γ

text: for

text: the

text: ξ

text: bands,

text: and

text: leads

text: to

text: hyperbolic

text: Fermi

text: surface

text: with

text: divergent

text: density

text: of

text: states.

text: Such

text: hyperbolic

text: Fermi

text: surface

text: is

text: due

text: to

text: the

text: limitation

text: of

text: the

text: Luttinger

text: model,

text: which

text: is

text: only

text: valid

text: in

text: a

text: small

text: momentum

text: region

text: around

text: Γ.

text: More

text: importantly,

text: it

text: will

text: cause

text: the

text: divergence

text: of

text: the

text: functions

text: y

text: To

text: avoid

text: this

text: pr√oblem,

text: we

text: introduce

text: a

text: momentum

text: cut-oﬀ

text: Λ,

text: wh√ich

text: can

text: be

text: implemented

text: by

text: inserting

text: a

text: Heaviside

text: step

text: function

text: θ(Λ

text: and

text: θ(Λ

text: µ)

text: into

text: the

text: integral

text: for

text: the

text: ξ−

text: c

text: and

text: ξ

text: bands,

text: respectively.

text: As

text: a

text: result,

text: the

text: functions

text: y

text: are

text: re-deﬁned

text: as

text: dΩ

text: y

text: c

text: c

text: 4π

text: dΩ

text: −θ(1

text: c

text: c

text: y

text: c

text: c

text: 4π

text: Q

text: c

text: c

text: c

text: and

text: dΩ

text: c

text: c

text: y

text: c

text: c

text: 4π

text: Q

text: c

text: c

text: c

text: The

text: Fermi

text: surface

text: shape

text: of

text: the

text: ξ

text: band

text: plays

text: an

text: important

text: role

text: in

text: determining

text: the

text: values

text: of

text: y

text: and

text: consequently

text: the

text: kink

text: structures.

text: We

text: choose

text: the

text: momentum

text: cut-oﬀ

text: as

text: Λ

text: for

text: the

text: red

text: line

text: and

text: Λ

text: for

text: the

text: blue

text: line

text: in

text: Fig.

text: (a).

text: For

text: a

text: small

text: SOC

text: parameter

text: c

text: the

text: momentum

text: cut-oﬀ

text: is

text: not

text: important

text: and

text: thus

text: the

text: red

text: line

text: coincides

text: with

text: the

text: blue

text: line

text: in

text: Fig.

text: (a).

text: With

text: increasing

text: the

text: SOC

text: to

text: the

text: Fermi

text: momentum

text: of

text: the

text: ξ

text: band

text: starts

text: becoming

text: larger

text: than

text: the

text: cut-oﬀ

text: Λ

text: along

text: certain

text: angles,

text: and

text: thus

text: the

text: integrals

text: y

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