text: 20 PERIODIC WAVES FOR THE LOG-KDV EQUATION | | − Figure 4.6. Left: Phase space for A > 2eω/2 1, A < 0. Right: Phase | | − space for A > 2eω/2 1, and A > 0. The orbits in blue are those for which φ is positive. (ω,A) | | − Remark 4.7. It is clear if A = 2eω/2 1 in Case 2, we also obtain periodic solutions that do not change sign. However, in this situation (ω, A) does not belong to an open set of R 2. Remark 4.8. Having disposed Theorem 4.6, a few words of explanation are in order. Here, contrary to the case where A = 0, we have no control of how large the period of φ may be. There is the possibility that solutions, even those turning around the (ω,A) √ critical center points, change its sign. Recall in the case A = 0, we obtain L-periodic solutions that do not change sign for any L > 2π. Now we can construct the smooth family of periodic solutions we need. Theorem 4.9. Fix (ω , A ) according to Theorem 4.6 and let L be the period of φ . 0 0 O 0 (ω0,A0) Then there are an open neighborhood of (ω , A ) and a family, 0 0 ∈ O (cid:55)→ ∈ (ω, A) ψ H2 ([0, L ]), (ω,A) per,e 0 ∈ O of L -periodic solutions of (3.1), which depends smoothly on (ω, A) . 0 Proof. To calculate the value of θ in (3.12) and, consequently, to apply Corollary 3.10, we (cid:48) need to obtain the value of y¯ (L ) by solving the linear equation (3.11) with g(ω , A , φ) = − 0 0 0 ω φ log(φ2)φ + A . To ﬁx ideas, let us consider ω = 1 and A = 1. In this case, one 0 ∈ P 0 0 · 0 − − has (ω , A ) and the zeros of the function g(ω , A , ) are r = 1, r = 0.28 and 0 0 1 0 0 0 1 r = 2.09. To obtain strictly positive solutions with a local maximum at x = 0, the initial 2 condition of φ must satisfy 2.09 < φ (0) < 3.51. Collecting these informations and (1,1)  (1,1) taking φ (0) = 3, we are enable to see that φ = φ satisﬁes (1,1) (1,1)  − (cid:48)(cid:48) − φ + φ φ log φ2 + 1 = 0, φ(0) = 3, (4.11) (cid:48) φ (0) = 0. ##LTFigure## ##LTFigure## ##LTLine##

text: PERIODIC

text: WAVES

text: FOR

text: THE

text: LOG-KDV

text: EQUATION

text: Figure

text: Left:

text: Phase

text: space

text: for

text: A

text: A

text: Right:

text: Phase

text: space

text: for

text: A

text: and

text: A

text: The

text: orbits

text: in

text: blue

text: are

text: those

text: for

text: which

text: φ

text: is

text: positive.

text: (ω,A)

text: Remark

text: It

text: is

text: clear

text: if

text: A

text: in

text: Case

text: we

text: also

text: obtain

text: periodic

text: solutions

text: that

text: do

text: not

text: change

text: sign.

text: However,

text: in

text: this

text: situation

text: (ω,

text: A)

text: does

text: not

text: belong

text: to

text: an

text: open

text: set

text: of

text: R

text: Remark

text: Having

text: disposed

text: Theorem

text: a

text: few

text: words

text: of

text: explanation

text: are

text: in

text: order.

text: Here,

text: contrary

text: to

text: the

text: case

text: where

text: A

text: we

text: have

text: no

text: control

text: of

text: how

text: large

text: the

text: period

text: of

text: φ

text: may

text: be.

text: There

text: is

text: the

text: possibility

text: that

text: solutions,

text: even

text: those

text: turning

text: around

text: the

text: (ω,A)

text: critical

text: center

text: points,

text: change

text: its

text: sign.

text: Recall

text: in

text: the

text: case

text: A

text: we

text: obtain

text: L-periodic

text: solutions

text: that

text: do

text: not

text: change

text: sign

text: for

text: any

text: L

text: Now

text: we

text: can

text: construct

text: the

text: smooth

text: family

text: of

text: periodic

text: solutions

text: we

text: need.

text: Theorem

text: Fix

text: (ω

text: A

text: according

text: to

text: Theorem

text: and

text: let

text: L

text: be

text: the

text: period

text: of

text: φ

text: O

text: Then

text: there

text: are

text: an

text: open

text: neighborhood

text: of

text: (ω

text: A

text: and

text: a

text: family,

text: O

text: (ω,

text: A)

text: ψ

text: L

text: (ω,A)

text: per,e

text: O

text: of

text: L

text: -periodic

text: solutions

text: of

text: which

text: depends

text: smoothly

text: on

text: (ω,

text: A)

text: Proof.

text: To

text: calculate

text: the

text: value

text: of

text: θ

text: in

text: and,

text: consequently,

text: to

text: apply

text: Corollary

text: we

text: need

text: to

text: obtain

text: the

text: value

text: of

text: y¯

text: (L

text: by

text: solving

text: the

text: linear

text: equation

text: with

text: g(ω

text: A

text: φ)

text: ω

text: φ

text: A

text: To

text: ﬁx

text: ideas,

text: let

text: us

text: consider

text: ω

text: and

text: A

text: In

text: this

text: case,

text: one

text: P

text: has

text: (ω

text: A

text: and

text: the

text: zeros

text: of

text: the

text: function

text: g(ω

text: A

text: are

text: r

text: r

text: and

text: r

text: To

text: obtain

text: strictly

text: positive

text: solutions

text: with

text: a

text: local

text: maximum

text: at

text: x

text: the

text: initial

text: condition

text: of

text: φ

text: must

text: satisfy

text: φ

text: Collecting

text: these

text: informations

text: and

text: taking

text: φ

text: we

text: are

text: enable

text: to

text: see

text: that

text: φ

text: φ

text: satisﬁes

text: φ

text: φ

text: φ

text: log

text: φ2

text: φ

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: 4 PERIODIC WAVES FOR THE LOG-KDV EQUATION of L-periodic travelling waves. Some arguments in our approach were borrowed from [25], where, following close the arguments in [9] and [20], the author have established a general criterion to obtain the orbital stability of periodic waves associated with the generalized Korteweg-de Vries equation (gKdV henceforth), u + f(u) + u = 0, (1.13) t x xxx R → R where f : is a smooth real-valued function, One may note that the functional E is not smooth at the origin. Nevertheless, as we will see below, our periodic waves are strictly positive or negative. Thus, at least in a neighborhood of such waves, E is smooth and this allows us to use the abstract theories mentioned above. As is well known, there are two key ingredients in the nonlinear stability theory. The O ⊂ R ﬁrst one is, for a ﬁxed L > 0, the existence of an open set n, and a smooth branch ∈ O (cid:55)→ ∈ O µ ψ , such that ψ is L-periodic and solves (1.2), for all µ . In our case, we µ µ R will see that L belongs to a convenient open interval contained in and µ is either ω (in (cid:54) the case A = 0) or (ω, A) (in the case A = 0). The second ingredient is the knowledge of the non-positive spectrum of the linearized operator around the periodic traveling wave in question. Usually, this turns out to be a Hill’s operator as L − (cid:48) = ∂2 + g (µ, ψ ). (1.14) x µ Here, we use the recent theory developed in [28], where the authors presented a new method based on the classical Floquet theorem to establish a characterization of the ﬁrst L three eigenvalues of by knowing one of its eigenfunctions. The main point is that it is not necessary to know an explicit solution of a general nonlinear diﬀerential equation of the form − (cid:48)(cid:48) ψ + g(µ, ψ ) = 0. (1.15) µ µ In addition, it is possible to decide that the eigenvalue zero is simple also without L knowing an explicit solution of (1.15). In particular, we show the operator appearing in our context has only one negative eigenvalue which is simple and zero is a simple eigenvalue with L { (cid:48) } ker( ) = span ψ . µ Moreover, the remainder of the spectrum is discrete and bounded away from zero. For the precise statements we refer the reader to Section 4. This paper is organized as follows: In Section 2 is proved the well-posedness and the existence of conservation laws related to the model (1.1). Existence of periodic waves for an ODE like (1.15) is treated in Section 3. In Section 4 we apply the method developed in Section 3 to study the existence of periodic travelling waves for (1.2). The orbital stability of such waves is then established. 2. Well Posedness Results - Verbatim of [11] In this section we sketch the proof of the local well-posedness theory by using the leading arguments in [11] and [12] (see also [13] and [15]). The main diﬀerent point here is that instead of proving the well-posedness in a class similar to that in (1.11), we establish our result in the whole energy space X. Here and throughout this section, L > 0 will be a

label: paragraph

text: PERIODIC

text: WAVES

text: FOR

text: THE

text: LOG-KDV

text: EQUATION

text: of

text: L-periodic

text: travelling

text: waves.

text: Some

text: arguments

text: in

text: our

text: approach

text: were

text: borrowed

text: from

text: where,

text: following

text: close

text: the

text: arguments

text: in

text: and

text: the

text: author

text: have

text: established

text: a

text: general

text: criterion

text: to

text: obtain

text: the

text: orbital

text: stability

text: of

text: periodic

text: waves

text: associated

text: with

text: the

text: generalized

text: Korteweg-de

text: Vries

text: equation

text: (gKdV

text: henceforth),

text: u

text: f(u)

text: u

text: t

text: x

text: xxx

text: R

text: R

text: where

text: f

text: is

text: a

text: smooth

text: real-valued

text: function,

text: One

text: may

text: note

text: that

text: the

text: functional

text: E

text: is

text: not

text: smooth

text: at

text: the

text: origin.

text: Nevertheless,

text: as

text: we

text: will

text: see

text: below,

text: our

text: periodic

text: waves

text: are

text: strictly

text: positive

text: or

text: negative.

text: Thus,

text: at

text: least

text: in

text: a

text: neighborhood

text: of

text: such

text: waves,

text: E

text: is

text: smooth

text: and

text: this

text: allows

text: us

text: to

text: use

text: the

text: abstract

text: theories

text: mentioned

text: above.

text: As

text: is

text: well

text: known,

text: there

text: are

text: two

text: key

text: ingredients

text: in

text: the

text: nonlinear

text: stability

text: theory.

text: The

text: O

text: R

text: ﬁrst

text: one

text: is,

text: for

text: a

text: ﬁxed

text: L

text: the

text: existence

text: of

text: an

text: open

text: set

text: n,

text: and

text: a

text: smooth

text: branch

text: O

text: O

text: µ

text: ψ

text: such

text: that

text: ψ

text: is

text: L-periodic

text: and

text: solves

text: for

text: all

text: µ

text: In

text: our

text: case,

text: we

text: µ

text: µ

text: R

text: will

text: see

text: that

text: L

text: belongs

text: to

text: a

text: convenient

text: open

text: interval

text: contained

text: in

text: and

text: µ

text: is

text: either

text: ω

text: (in

text: the

text: case

text: A

text: or

text: (ω,

text: A)

text: (in

text: the

text: case

text: A

text: The

text: second

text: ingredient

text: is

text: the

text: knowledge

text: of

text: the

text: non-positive

text: spectrum

text: of

text: the

text: linearized

text: operator

text: around

text: the

text: periodic

text: traveling

text: wave

text: in

text: question.

text: Usually,

text: this

text: turns

text: out

text: to

text: be

text: a

text: Hill’s

text: operator

text: as

text: L

text: ∂2

text: g

text: (µ,

text: ψ

text: x

text: µ

text: Here,

text: we

text: use

text: the

text: recent

text: theory

text: developed

text: in

text: where

text: the

text: authors

text: presented

text: a

text: new

text: method

text: based

text: on

text: the

text: classical

text: Floquet

text: theorem

text: to

text: establish

text: a

text: characterization

text: of

text: the

text: ﬁrst

text: L

text: three

text: eigenvalues

text: of

text: by

text: knowing

text: one

text: of

text: its

text: eigenfunctions.

text: The

text: main

text: point

text: is

text: that

text: it

text: is

text: not

text: necessary

text: to

text: know

text: an

text: explicit

text: solution

text: of

text: a

text: general

text: nonlinear

text: diﬀerential

text: equation

text: of

text: the

text: form

text: ψ

text: g(µ,

text: ψ

text: µ

text: µ

text: In

text: addition,

text: it

text: is

text: possible

text: to

text: decide

text: that

text: the

text: eigenvalue

text: zero

text: is

text: simple

text: also

text: without

text: L

text: knowing

text: an

text: explicit

text: solution

text: of

text: In

text: particular,

text: we

text: show

text: the

text: operator

text: appearing

text: in

text: our

text: context

text: has

text: only

text: one

text: negative

text: eigenvalue

text: which

text: is

text: simple

text: and

text: zero

text: is

text: a

text: simple

text: eigenvalue

text: with

text: L

text: ker(

text: span

text: ψ

text: µ

text: Moreover,

text: the

text: remainder

text: of

text: the

text: spectrum

text: is

text: discrete

text: and

text: bounded

text: away

text: from

text: zero.

text: For

text: the

text: precise

text: statements

text: we

text: refer

text: the

text: reader

text: to

text: Section

text: This

text: paper

text: is

text: organized

text: as

text: follows:

text: In

text: Section

text: is

text: proved

text: the

text: well-posedness

text: and

text: the

text: existence

text: of

text: conservation

text: laws

text: related

text: to

text: the

text: model

text: Existence

text: of

text: periodic

text: waves

text: for

text: an

text: ODE

text: like

text: is

text: treated

text: in

text: Section

text: In

text: Section

text: we

text: apply

text: the

text: method

text: developed

text: in

text: Section

text: to

text: study

text: the

text: existence

text: of

text: periodic

text: travelling

text: waves

text: for

text: The

text: orbital

text: stability

text: of

text: such

text: waves

text: is

text: then

text: established.

text: Well

text: Posedness

text: Results

text: Verbatim

text: of

text: In

text: this

text: section

text: we

text: sketch

text: the

text: proof

text: of

text: the

text: local

text: well-posedness

text: theory

text: by

text: using

text: the

text: leading

text: arguments

text: in

text: and

text: (see

text: also

text: and

text: The

text: main

text: diﬀerent

text: point

text: here

text: is

text: that

text: instead

text: of

text: proving

text: the

text: well-posedness

text: in

text: a

text: class

text: similar

text: to

text: that

text: in

text: we

text: establish

text: our

text: result

text: in

text: the

text: whole

text: energy

text: space

text: X.

text: Here

text: and

text: throughout

text: this

text: section,

text: L

text: will

text: be

text: a


text: 10 PERIODIC WAVES FOR THE LOG-KDV EQUATION Proof. Here we let H2 ([0, L ]) denote the subspace of H2 ([0, L ]) constituted by the per,e 0F P × → per 0 even periodic functions. Let : H2 ([0, L ]) L2 ([0, L ]) be the operator per,e 0 per,e 0 deﬁned as F − (cid:48)(cid:48) (µ, ψ) = ψ + g(µ, ψ). (3.10) F F Since φ is an even periodic solution of (3.1), it is clear that (µ , φ ) = 0. Also, µ0 0 µ0 F is Fr´echet-diﬀerentiable with respect to ψ and, in particular, the derivative ∂ (µ , φ ) L (cid:48) ∂ψ 0 µ0 is exactly the operator . By noting that φ is an odd function, the assumption L (cid:48) µ0 L µ0→ ker( ) = [φ ] imply that : H2 ([0, L ]) L2 ([0, L ]) is invertible and its µ0 µ0 µ0 per,e 0 per,e 0 inverse is bounded. Therefore, the conclusion follows from the Implicit Function Theorem in Banach spaces (see e.g., Theorem 15.1 and Corollary 15.1 in [16]) and the Sobolev (cid:3) embedding. As we already discussed in the introduction, the existence of the smooth family of L - 0 periodic solutions in Theorem 3.5 is a ﬁrst step to study the orbital stability of the traveling L wave φ . As a second step, the non-positive spectrum of plays a fundamental role µ0 ψµ and, in connection, it deliveries the major diﬃculty in the theory. In our case, we study L L the non-positive spectrum of by studying the inertial index in( ), which we now ψµ ψµ introduce. To simplify notation and avoid too many technicalities, we restrict ourselves to the study of the operators treated in our paper. However, it is possible to obtain similar results for a large class of self-adjoint operators and we strongly recommend the reader to [30] and [31] for additional informations. L Deﬁnition 3.6. Let Q be an L-periodic function. Let be the Hill’s operator deﬁned in L L2 ([0, L]) with domain D( ) = H2 ([0, L]) by per per L − = ∂2 + Q(x). x L L The inertial index of , denoted by in( ), is the pair (n, z), where n denotes the dimension L L of the negative subspace of and z denotes the dimension of ker( ). ∈ P ∈ O (cid:55)→ Deﬁnition 3.7. Fix µ and let µ ψ be the smooth family of L -periodic 0 µ L − 0 (cid:48) solutions obtained in Theorem 3.5. The family of linear operators := ∂2 + g (µ, ψ ), ∈ O L L ∈ O ψµ x µ µ , is said to be isoinertial if in( ) = in( ), for all µ . ψµ µ0 The next results in this section are based on [30] and [31]. The ﬁrst one concerns the ∈ O invariance of the inertial index with respect to the parameter µ . ∈ P ∈ O (cid:55)→ Theorem 3.8. Fix µ and let µ ψ be the smooth family of L -periodic 0 µ L − 0 (cid:48) solutions obtained in Theorem 3.5. Then the family of operators := ∂2 + g (µ, ψ ), ∈ O ψµ x µ µ , is isoinertial. ∈ P (cid:48) L Proof. Since, for every µ , φ is an eigenfunction of associated with the eigenvalue µ ψµ (cid:3) λ = 0, the result follows from Theorem 3.1 in [30]. ##LTLine##

text: PERIODIC

text: WAVES

text: FOR

text: THE

text: LOG-KDV

text: EQUATION

text: Proof.

text: Here

text: we

text: let

text: L

text: denote

text: the

text: subspace

text: of

text: L

text: constituted

text: by

text: the

text: per,e

text: P

text: per

text: even

text: periodic

text: functions.

text: Let

text: L

text: L

text: be

text: the

text: operator

text: per,e

text: per,e

text: deﬁned

text: as

text: F

text: (µ,

text: ψ)

text: ψ

text: g(µ,

text: ψ).

text: F

text: F

text: Since

text: φ

text: is

text: an

text: even

text: periodic

text: solution

text: of

text: it

text: is

text: clear

text: that

text: (µ

text: φ

text: Also,

text: µ0

text: µ0

text: F

text: is

text: Fr´echet-diﬀerentiable

text: with

text: respect

text: to

text: ψ

text: and,

text: in

text: particular,

text: the

text: derivative

text: (µ

text: φ

text: L

text: ∂ψ

text: µ0

text: is

text: exactly

text: the

text: operator

text: By

text: noting

text: that

text: φ

text: is

text: an

text: odd

text: function,

text: the

text: assumption

text: L

text: µ0

text: L

text: µ0→

text: ker(

text: [φ

text: imply

text: that

text: L

text: L

text: is

text: invertible

text: and

text: its

text: µ0

text: µ0

text: µ0

text: per,e

text: per,e

text: inverse

text: is

text: bounded.

text: Therefore,

text: the

text: conclusion

text: follows

text: from

text: the

text: Implicit

text: Function

text: Theorem

text: in

text: Banach

text: spaces

text: (see

text: e.g.,

text: Theorem

text: and

text: Corollary

text: in

text: and

text: the

text: Sobolev

text: embedding.

text: As

text: we

text: already

text: discussed

text: in

text: the

text: introduction,

text: the

text: existence

text: of

text: the

text: smooth

text: family

text: of

text: L

text: periodic

text: solutions

text: in

text: Theorem

text: is

text: a

text: ﬁrst

text: step

text: to

text: study

text: the

text: orbital

text: stability

text: of

text: the

text: traveling

text: L

text: wave

text: φ

text: As

text: a

text: second

text: step,

text: the

text: non-positive

text: spectrum

text: of

text: plays

text: a

text: fundamental

text: role

text: µ0

text: ψµ

text: and,

text: in

text: connection,

text: it

text: deliveries

text: the

text: major

text: diﬃculty

text: in

text: the

text: theory.

text: In

text: our

text: case,

text: we

text: study

text: L

text: L

text: the

text: non-positive

text: spectrum

text: of

text: by

text: studying

text: the

text: inertial

text: index

text: in(

text: which

text: we

text: now

text: ψµ

text: ψµ

text: introduce.

text: To

text: simplify

text: notation

text: and

text: avoid

text: too

text: many

text: technicalities,

text: we

text: restrict

text: ourselves

text: to

text: the

text: study

text: of

text: the

text: operators

text: treated

text: in

text: our

text: paper.

text: However,

text: it

text: is

text: possible

text: to

text: obtain

text: similar

text: results

text: for

text: a

text: large

text: class

text: of

text: self-adjoint

text: operators

text: and

text: we

text: strongly

text: recommend

text: the

text: reader

text: to

text: and

text: for

text: additional

text: informations.

text: L

text: Deﬁnition

text: Let

text: Q

text: be

text: an

text: L-periodic

text: function.

text: Let

text: be

text: the

text: Hill’s

text: operator

text: deﬁned

text: in

text: L

text: L])

text: with

text: domain

text: D(

text: L])

text: by

text: per

text: per

text: L

text: ∂2

text: Q(x).

text: x

text: L

text: L

text: The

text: inertial

text: index

text: of

text: denoted

text: by

text: in(

text: is

text: the

text: pair

text: (n,

text: z),

text: where

text: n

text: denotes

text: the

text: dimension

text: L

text: L

text: of

text: the

text: negative

text: subspace

text: of

text: and

text: z

text: denotes

text: the

text: dimension

text: of

text: ker(

text: P

text: O

text: Deﬁnition

text: Fix

text: µ

text: and

text: let

text: µ

text: ψ

text: be

text: the

text: smooth

text: family

text: of

text: L

text: -periodic

text: µ

text: L

text: solutions

text: obtained

text: in

text: Theorem

text: The

text: family

text: of

text: linear

text: operators

text: ∂2

text: g

text: (µ,

text: ψ

text: O

text: L

text: L

text: O

text: ψµ

text: x

text: µ

text: µ

text: is

text: said

text: to

text: be

text: isoinertial

text: if

text: in(

text: in(

text: for

text: all

text: µ

text: ψµ

text: µ0

text: The

text: next

text: results

text: in

text: this

text: section

text: are

text: based

text: on

text: and

text: The

text: ﬁrst

text: one

text: concerns

text: the

text: O

text: invariance

text: of

text: the

text: inertial

text: index

text: with

text: respect

text: to

text: the

text: parameter

text: µ

text: P

text: O

text: Theorem

text: Fix

text: µ

text: and

text: let

text: µ

text: ψ

text: be

text: the

text: smooth

text: family

text: of

text: L

text: -periodic

text: µ

text: L

text: solutions

text: obtained

text: in

text: Theorem

text: Then

text: the

text: family

text: of

text: operators

text: ∂2

text: g

text: (µ,

text: ψ

text: O

text: ψµ

text: x

text: µ

text: µ

text: is

text: isoinertial.

text: P

text: L

text: Proof.

text: Since,

text: for

text: every

text: µ

text: φ

text: is

text: an

text: eigenfunction

text: of

text: associated

text: with

text: the

text: eigenvalue

text: µ

text: ψµ

text: λ

text: the

text: result

text: follows

text: from

text: Theorem

text: in

text: ##LTLine##


text: T. Alexander, P. Natarajan, Science, 7 August 2014 (10.1126/science.1251053). ∼ (cid:112) (cid:39) M• M (Figure ??). At these early times, the growth can be approximated as linear, M(t) eq M (1+t/t∞). It then follows that the captured mass grows as ∆M = (M /t∞)t, while the angu- i i lar momentum in the ﬂow grows in a random walk fashion as ∆J = [j (M /t∞)t ] t/t i i vRR vRR on times longer than vector RR timescale, t . The speciﬁc angular momentum accumulated vRR(cid:114) in the accretion ﬂow therefore falls with time as ∆J(t) ∼ t j (t) = j vRR , (t > t ) , (S12) a i vRR ∆M(t) t or in terms of the BH mass, (cid:112) √ j M j (M•) = i− , (M• > − i ) , (S13) a N 1 M /M• 1 t /t∞ vRR i vRR wh√ere N = t∞/t is the number of RR angular momentum steps over the divergence vRR vRR time ( N = 7.6 for the cluster model of table ??). Figure ?? shows that early suppression vRR by RR decreases j rapidly below j , thereby enabling prompt radial accretion until dynamical a ISO decoupling becomes effective. This suppression of angular momentum is further aided by the angular momentum minimum near Q , 0 References 44. H. Dejonghe, MNRAS 224, 13 (1987). 45. P. Chatterjee, L. Hernquist, A. Loeb, ApJ 572, 371 (2002). 46. M. Giersz, D. C. Heggie, MNRAS 268, 257 (1994). 47. B. W. Murphy, H. N. Cohn, R. H. Durisen, ApJ 370, 60 (1991). 48. D. Heggie, P. Hut, The Gravitational Million-Body Problem: A Multidisciplinary Approach to Star Cluster Dynamics (Cambridge University Press, 2003, 372 pp., 2003). 49. M. A. Gu¨rkan, C. Hopman, MNRAS 379, 1083 (2007). 50. E. Eilon, G. Kupi, T. Alexander, ApJ 698, 641 (2009). 51. J. Frank, A. King, D. J. Raine, Accretion Power in Astrophysics (Cambridge University Press, 2002), third edn. 52. A. F. Illarionov, A. M. Beloborodov, MNRAS 323, 159 (2001). 53. R. E. Davies, J. E. Pringle, MNRAS 191, 599 (1980). 54. M. Ruffert, A&A 317, 793 (1997). 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: T.

text: Alexander,

text: P.

text: Natarajan,

text: Science,

text: August

text: M•

text: M

text: (Figure

text: At

text: these

text: early

text: times,

text: the

text: growth

text: can

text: be

text: approximated

text: as

text: linear,

text: M(t)

text: eq

text: M

text: It

text: then

text: follows

text: that

text: the

text: captured

text: mass

text: grows

text: as

text: ∆M

text: (M

text: /t∞)t,

text: while

text: the

text: angu-

text: i

text: i

text: lar

text: momentum

text: in

text: the

text: ﬂow

text: grows

text: in

text: a

text: random

text: walk

text: fashion

text: as

text: ∆J

text: [j

text: (M

text: /t∞)t

text: t/t

text: i

text: i

text: vRR

text: vRR

text: on

text: times

text: longer

text: than

text: vector

text: RR

text: timescale,

text: t

text: The

text: speciﬁc

text: angular

text: momentum

text: accumulated

label: equation

text: in

text: the

text: accretion

text: ﬂow

text: therefore

text: falls

text: with

text: time

text: as

text: ∆J(t)

text: t

text: j

text: (t)

text: j

text: vRR

text: (t

text: t

text: a

text: i

text: vRR

text: ∆M(t)

text: t

text: or

text: in

text: terms

text: of

text: the

text: BH

text: mass,

text: j

text: M

text: j

text: (M•)

text: i−

text: (M•

text: i

text: a

text: N

text: M

text: /M•

text: t

text: /t∞

text: vRR

text: i

text: vRR

text: wh√ere

text: N

text: t∞/t

text: is

text: the

text: number

text: of

text: RR

text: angular

text: momentum

text: steps

text: over

text: the

text: divergence

text: vRR

text: vRR

text: time

text: N

text: for

text: the

text: cluster

text: model

text: of

text: table

text: Figure

text: shows

text: that

text: early

text: suppression

text: vRR

text: by

text: RR

text: decreases

text: j

text: rapidly

text: below

text: j

text: thereby

text: enabling

text: prompt

text: radial

text: accretion

text: until

text: dynamical

text: a

text: ISO

text: decoupling

text: becomes

text: effective.

text: This

text: suppression

text: of

text: angular

text: momentum

text: is

text: further

text: aided

text: by

text: the

text: angular

text: momentum

text: minimum

text: near

text: Q

text: References

text: H.

text: Dejonghe,

text: MNRAS

text: P.

text: Chatterjee,

text: L.

text: Hernquist,

text: A.

text: Loeb,

text: ApJ

text: M.

text: Giersz,

text: D.

text: C.

text: Heggie,

text: MNRAS

text: B.

text: W.

text: Murphy,

text: H.

text: N.

text: Cohn,

text: R.

text: H.

text: Durisen,

text: ApJ

text: D.

text: Heggie,

text: P.

text: Hut,

text: The

text: Gravitational

text: Million-Body

text: Problem:

text: A

text: Multidisciplinary

text: Approach

text: to

text: Star

text: Cluster

text: Dynamics

text: (Cambridge

text: University

text: Press,

text: pp.,

text: M.

text: A.

text: Gu¨rkan,

text: C.

text: Hopman,

text: MNRAS

text: E.

text: Eilon,

text: G.

text: Kupi,

text: T.

text: Alexander,

text: ApJ

text: J.

text: Frank,

text: A.

text: King,

text: D.

text: J.

text: Raine,

text: Accretion

text: Power

text: in

text: Astrophysics

text: (Cambridge

text: University

text: Press,

text: third

text: edn.

text: A.

text: F.

text: Illarionov,

text: A.

text: M.

text: Beloborodov,

text: MNRAS

text: R.

text: E.

text: Davies,

text: J.

text: E.

text: Pringle,

text: MNRAS

text: M.

text: Ruffert,

text: A&A

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: T. Alexander, P. Natarajan, Science, 7 August 2014 (10.1126/science.1251053). (cid:39) a reduction of the accretion rate by 0.42 1/6. The supra-exponential divergence of spherical accretion is therefore M (cid:39) 3 c3∞ M•(t) = − i , t∞ 6t = . (1) 1 t/t∞ B 21/2π G2M ρ∞ i (cid:38) The typical lifetime of cold ﬂow streams seen in simulations is 107 years (?), which is ∼ long enough for the 105 M(cid:12) of gas in the marginally unstable reservoir on the few parsec scale to accrete onto the growing BH, and yet short enough to be relevant for forming z > 7 quasars. As a demonstration of concept, we adopt the gas properties found in these simulations, ∼ × and match a divergence time of t∞ few 107 years to the mean physical parameters of the × 4 104 M(cid:12) of gas on the 0.25-pc scale. We further assume that half of that mass is in a star cluster with 1 M(cid:12) stars in a nonsingular distribution (a Plummer law), containing a 10 M(cid:12) stellar BH, and that the escaped radiation from r > r does not substantially affect the cold ﬂow γ (cid:29) on larger scales. This cluster, while dense, is dynamically stable on timescales t∞. Table ?? lists the input physical parameters of the model, the derived gas and cluster properties, and the predicted accretion properties. Simulations (?) ﬁnd that the cold ﬂow is nearly pressure-supported, and thus has little angu- lar momentum. It is also plausible that there is little net rotation between the gas and the stars that were formed from it. However, unavoidable gravitational interactions of the low-mass BH ∼ with cluster stars accelerate it at orbital frequencies of Ω(r•) Ω , which induce a velocity gra- 0 dient across the capture radius. Gas captured by the BH then acquires speciﬁc angular momen- tum relative to it, j , due to the opposite velocity and density gradients and the velocity-skewed a ∼ ∼ capture cross-section (j Ωr2 and j (d logρ/d log r) Ωr2) (?). Gas with j > j a,v a a,ρ r• a a ISO cannot plunge directly into the BH but rather circularizes at a radius r = j2/GM•, and accre- c a tion then proceeds on the slow viscous timescale rather than the fast, near-free fall timescale. Analytic and numeric results on the capture efﬁciency of angular momentum by inhomogeneous wind accretion are currently available only in the ballistic (supersonic) limit (?, ?). We adapt these results to provide a rough estimate of the angular momentum in the accretion ﬂow on the BH seed in the subsonic regime considered here (see details in the Supplementary Materials). ∝ ∼ The formal divergence of j /j M• in the test particle limit, where Ω Ω , is reversed a ISO 0 by the accretion itself. The nonrotating gas accreted by the BH from the cluster exerts a drag − on it, r¨• = 2γ r˙•, where γ = M˙ /2M• = M•(t)/2M t∞. This provides additional damping a a i ∝ beyond dissipation by dynamical friction against the gas and stars, γ [M•(t)/M ]Ω (?), df c 0 which is balanced by the two-body ﬂuctuations. Accretion damping drives the BH to sube- quipartition energy and angular momentum. Two-body interactions with the cluster stars can ∼ × reestablish equipartition on the central relaxation timescale t 2 107 years only as long as r0 − ∼ the BH growth rate 2γ is slower than the relaxation rate, up to time t /t∞ = 1 t /t∞ 0.6 a ∼ eq (cid:39) r0 (Eq. ??), when the BH has grown by a factor of only t∞/t 2.5 to M 25 M(cid:12). At later r0 eq times, j is expected to fall below the extrapolated test particle limit, because both the BH wan- a dering radius ∆• and the orbital frequency Ω are increasingly damped by dynamical friction 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: T.

text: Alexander,

text: P.

text: Natarajan,

text: Science,

text: August

text: a

text: reduction

text: of

text: the

text: accretion

text: rate

text: by

text: The

text: supra-exponential

text: divergence

text: of

text: spherical

text: accretion

text: is

text: therefore

text: M

text: M•(t)

text: i

text: t∞

text: t/t∞

text: B

text: ρ∞

text: i

text: The

text: typical

text: lifetime

text: of

text: cold

text: ﬂow

text: streams

text: seen

text: in

text: simulations

text: is

text: years

text: which

text: is

text: long

text: enough

text: for

text: the

text: of

text: gas

text: in

text: the

text: marginally

text: unstable

text: reservoir

text: on

text: the

text: few

text: parsec

text: scale

text: to

text: accrete

text: onto

text: the

text: growing

text: BH,

text: and

text: yet

text: short

text: enough

text: to

text: be

text: relevant

text: for

text: forming

text: z

text: quasars.

text: As

text: a

text: demonstration

text: of

text: concept,

text: we

text: adopt

text: the

text: gas

text: properties

text: found

text: in

text: these

text: simulations,

text: and

text: match

text: a

text: divergence

text: time

text: of

text: t∞

text: few

text: years

text: to

text: the

text: mean

text: physical

text: parameters

text: of

text: the

text: of

text: gas

text: on

text: the

text: scale.

text: We

text: further

text: assume

text: that

text: half

text: of

text: that

text: mass

text: is

text: in

text: a

text: star

text: cluster

text: with

text: stars

text: in

text: a

text: nonsingular

text: distribution

text: (a

text: Plummer

text: law),

text: containing

text: a

text: stellar

text: BH,

text: and

text: that

text: the

text: escaped

text: radiation

text: from

text: r

text: r

text: does

text: not

text: substantially

text: affect

text: the

text: cold

text: ﬂow

text: γ

text: on

text: larger

text: scales.

text: This

text: cluster,

text: while

text: dense,

text: is

text: dynamically

text: stable

text: on

text: timescales

text: t∞.

text: Table

text: lists

text: the

text: input

text: physical

text: parameters

text: of

text: the

text: model,

text: the

text: derived

text: gas

text: and

text: cluster

text: properties,

text: and

text: the

text: predicted

text: accretion

text: properties.

text: Simulations

text: ﬁnd

text: that

text: the

text: cold

text: ﬂow

text: is

text: nearly

text: pressure-supported,

text: and

text: thus

text: has

text: little

text: angu-

text: lar

text: momentum.

text: It

text: is

text: also

text: plausible

text: that

text: there

text: is

text: little

text: net

text: rotation

text: between

text: the

text: gas

text: and

text: the

text: stars

text: that

text: were

text: formed

text: from

text: it.

text: However,

text: unavoidable

text: gravitational

text: interactions

text: of

text: the

text: low-mass

text: BH

text: with

text: cluster

text: stars

text: accelerate

text: it

text: at

text: orbital

text: frequencies

text: of

text: Ω(r•)

text: Ω

text: which

text: induce

text: a

text: velocity

text: gra-

text: dient

text: across

text: the

text: capture

text: radius.

text: Gas

text: captured

text: by

text: the

text: BH

text: then

text: acquires

text: speciﬁc

text: angular

text: momen-

text: tum

text: relative

text: to

text: it,

text: j

text: due

text: to

text: the

text: opposite

text: velocity

text: and

text: density

text: gradients

text: and

text: the

text: velocity-skewed

text: a

text: capture

text: cross-section

text: (j

text: and

text: j

text: (d

text: logρ/d

text: log

text: r)

text: Gas

text: with

text: j

text: j

text: a,v

text: a

text: a,ρ

text: r•

text: a

text: a

text: ISO

text: cannot

text: plunge

text: directly

text: into

text: the

text: BH

text: but

text: rather

text: circularizes

text: at

text: a

text: radius

text: r

text: and

text: accre-

text: c

text: a

text: tion

text: then

text: proceeds

text: on

text: the

text: slow

text: viscous

text: timescale

text: rather

text: than

text: the

text: fast,

text: near-free

text: fall

text: timescale.

text: Analytic

text: and

text: numeric

text: results

text: on

text: the

text: capture

text: efﬁciency

text: of

text: angular

text: momentum

text: by

text: inhomogeneous

text: wind

text: accretion

text: are

text: currently

text: available

text: only

text: in

text: the

text: ballistic

text: (supersonic)

text: limit

text: We

text: adapt

text: these

text: results

text: to

text: provide

text: a

text: rough

text: estimate

text: of

text: the

text: angular

text: momentum

text: in

text: the

text: accretion

text: ﬂow

text: on

text: the

text: BH

text: seed

text: in

text: the

text: subsonic

text: regime

text: considered

text: here

text: (see

text: details

text: in

text: the

text: Supplementary

text: Materials).

text: The

text: formal

text: divergence

text: of

text: j

text: /j

text: M•

text: in

text: the

text: test

text: particle

text: limit,

text: where

text: Ω

text: Ω

text: is

text: reversed

text: a

text: ISO

text: by

text: the

text: accretion

text: itself.

text: The

text: nonrotating

text: gas

text: accreted

text: by

text: the

text: BH

text: from

text: the

text: cluster

text: exerts

text: a

text: drag

text: on

text: it,

text: r¨•

text: 2γ

text: r˙•,

text: where

text: γ

text: M˙

text: t∞.

text: This

text: provides

text: additional

text: damping

text: a

text: a

text: i

text: beyond

text: dissipation

text: by

text: dynamical

text: friction

text: against

text: the

text: gas

text: and

text: stars,

text: γ

text: [M•(t)/M

text: ]Ω

text: df

text: c

text: which

text: is

text: balanced

text: by

text: the

text: two-body

text: ﬂuctuations.

text: Accretion

text: damping

text: drives

text: the

text: BH

text: to

text: sube-

text: quipartition

text: energy

text: and

text: angular

text: momentum.

text: Two-body

text: interactions

text: with

text: the

text: cluster

text: stars

text: can

text: reestablish

text: equipartition

text: on

text: the

text: central

text: relaxation

text: timescale

text: t

text: years

text: only

text: as

text: long

text: as

text: the

text: BH

text: growth

text: rate

text: 2γ

text: is

text: slower

text: than

text: the

text: relaxation

text: rate,

text: up

text: to

text: time

text: t

text: /t∞

text: t

text: /t∞

text: a

text: eq

text: (Eq.

text: when

text: the

text: BH

text: has

text: grown

text: by

text: a

text: factor

text: of

text: only

text: t∞/t

text: to

text: M

text: At

text: later

text: eq

text: times,

text: j

text: is

text: expected

text: to

text: fall

text: below

text: the

text: extrapolated

text: test

text: particle

text: limit,

text: because

text: both

text: the

text: BH

text: wan-

text: a

text: dering

text: radius

text: and

text: the

text: orbital

text: frequency

text: Ω

text: are

text: increasingly

text: damped

text: by

text: dynamical

text: friction

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: T. Alexander, P. Natarajan, Science, 7 August 2014 (10.1126/science.1251053). t/t ∞ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100 t > M/(dM/dt) r,0 10 O S jI 1 / | a |j 0.1 Analytic ja (ηj=1/3) no RR no damping Monte Carlo ja (ηj=1/3) no RR no damping M = 25.1 Mo RR-suppressed j eq a 0.01 10 100 M (M ) bh o Figure 2: The speciﬁc angular momentum ratio, j /j , in gas captured by the BH, as function a ISO of the BH mass M• (and the corresponding t/t∞ for Bondi accretion). The evolution of j /j a ISO during the initial stages of the BH growth is calculated in the ballistic wind accretion limit by a ﬁrst-order analytic estimate with an angular momentum capture efﬁciency of η = 1/3 (red j line), which is validated against results from a Monte Carlo integration over the exact capture (cid:39) cross-section (circles). j falls to zero at M 20 M(cid:12), where the density and velocity gradients a (cid:39)0 cancel each other. The vertical line at M 25 M(cid:12) marks the transition to a dynamical regime eq where two-body relaxation can no longer establish equipartition between the growing BH and the stars, and the acceleration frequency Ω is damped. The early dynamical suppression of the angular momentum down to j < j by resonant relaxation of the BH orbit (thick blue a ISO line) is approximated by conservatively assuming a constant j (t) = j (0) (thin blue line). a a At M• > M , where the BH dynamics are sub-equipartition, the actual value of j /j is eq a ISO expected to progressively drop below the extrapolated one (dashed lines). (See the detailed discussion in the Supplementary Materials). 6 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: T.

text: Alexander,

text: P.

text: Natarajan,

text: Science,

text: August

text: t/t

font: default

text: t

text: M/(dM/dt)

text: O

text: S

text: jI

text: a

text: |j

text: Analytic

text: ja

text: no

text: RR

text: no

text: damping

text: Monte

text: Carlo

text: ja

text: no

text: RR

text: no

text: damping

text: M

text: Mo

text: RR-suppressed

text: j

text: eq

text: a

text: M

text: (M

text: bh

text: o

text: Figure

text: The

text: speciﬁc

text: angular

text: momentum

text: ratio,

text: j

text: /j

text: in

text: gas

text: captured

text: by

text: the

text: BH,

text: as

text: function

text: a

text: ISO

text: of

text: the

text: BH

text: mass

text: M•

text: (and

text: the

text: corresponding

text: t/t∞

text: for

text: Bondi

text: accretion).

text: The

text: evolution

text: of

text: j

text: /j

text: a

text: ISO

text: during

text: the

text: initial

text: stages

text: of

text: the

text: BH

text: growth

text: is

text: calculated

text: in

text: the

text: ballistic

text: wind

text: accretion

text: limit

text: by

text: a

text: ﬁrst-order

text: analytic

text: estimate

text: with

text: an

text: angular

text: momentum

text: capture

text: efﬁciency

text: of

text: η

text: (red

text: j

text: line),

text: which

text: is

text: validated

text: against

text: results

text: from

text: a

text: Monte

text: Carlo

text: integration

text: over

text: the

text: exact

text: capture

text: cross-section

text: (circles).

text: j

text: falls

text: to

text: zero

text: at

text: M

text: where

text: the

text: density

text: and

text: velocity

text: gradients

text: a

text: cancel

text: each

text: other.

text: The

text: vertical

text: line

text: at

text: M

text: marks

text: the

text: transition

text: to

text: a

text: dynamical

text: regime

text: eq

text: where

text: two-body

text: relaxation

text: can

text: no

text: longer

text: establish

text: equipartition

text: between

text: the

text: growing

text: BH

text: and

text: the

text: stars,

text: and

text: the

text: acceleration

text: frequency

text: Ω

text: is

text: damped.

text: The

text: early

text: dynamical

text: suppression

text: of

text: the

text: angular

text: momentum

text: down

text: to

text: j

text: j

text: by

text: resonant

text: relaxation

text: of

text: the

text: BH

text: orbit

text: (thick

text: blue

text: a

text: ISO

text: line)

text: is

text: approximated

text: by

text: conservatively

text: assuming

text: a

text: constant

text: j

text: (t)

text: j

text: (thin

text: blue

text: line).

text: a

text: a

text: At

text: M•

text: M

text: where

text: the

text: BH

text: dynamics

text: are

text: sub-equipartition,

text: the

text: actual

text: value

text: of

text: j

text: /j

text: is

text: eq

text: a

text: ISO

text: expected

text: to

text: progressively

text: drop

text: below

text: the

text: extrapolated

text: one

text: (dashed

text: lines).

text: (See

text: the

text: detailed

text: discussion

text: in

text: the

text: Supplementary

text: Materials).

text: ##LTFigure##

text: ##LTFigure##

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text: REFERENCES AND NOTES REFERENCES AND NOTES Grant PHY11-2591. T.A. is grateful for the warm hospitality of Angel Milla´n and Lucy Ark- wright of La Posada San Marcos, Ala´jar, Spain, who hosted the Ala´jar Workshop where this work was continued. 10 ##LTLine##

text: REFERENCES

text: AND

text: NOTES

text: REFERENCES

text: AND

text: NOTES

text: Grant

text: T.A.

text: is

text: grateful

text: for

text: the

text: warm

text: hospitality

text: of

text: Angel

text: Milla´n

text: and

text: Lucy

text: Ark-

text: wright

text: of

text: La

text: Posada

text: San

text: Marcos,

text: Ala´jar,

text: Spain,

text: who

text: hosted

text: the

text: Ala´jar

text: Workshop

text: where

text: this

text: work

text: was

text: continued.

text: ##LTLine##


text: 12 Lemma 9. P = C · Pauli and P = Cliﬀord for n ≥ 2. cal quantum circuit with the spread s . The proof pro- 1 n n U ceeds by induction. Assuming m = 0, the dressed logical Proof. Let us ﬁrst show that P = C · Pauli. Suppose 1 operator U is supported on a bare-cleanable region and that U ∈ P . By hypothesis, the group commutator 1 by deﬁnition 4 must be a trivial logical operator in P . 0 of U with any Pauli operator P ∈ P is trivial up to a phase UPU†P† = eiθ. This phase must be ±1, since it Let us now assume that our statement is true up to m is an eigenvalue for the rank one superoperator resulting and prove it for m + 1. Consider a unitary U with the from conjugation by a Pauli operator PU†P† = eiθU†. spread s such that U Conversely, we may consider the rank one superoperator U · U† for which the Pauli operators constitute a full set of eigenoperators with eigenvalues ±1. This uniquely de- m(cid:91)+1 termines U to be equivalent to a Pauli operator itself up supp(U) ⊆ R . (B1) j to a global phase. Here, we have crucially used that the j=0 Pauli operators linearly span the full operator algebra. We will now prove that U ∈ P ⇔ U ∈ Cliﬀord by n n induction for n ≥ 2. The proof relies on the observa- Any logical Pauli operator [P] has a dressed incarna- (cid:0) (cid:1) L tion that all the levels of the usual Cliﬀord hierarchy tion P fully supported on R+ c. Observe that are closed under right multiplication by Pauli operators 1 Cliﬀord = Cliﬀord · Pauli which can be proven induc- n n   tively. m(cid:91)+1 (cid:0) (cid:1) thaStupUpPoUse†UP†∈∈PPn+1a.nHdecnocnes,eqfourenatnlyy PUP∈UP†au∈liCwlieﬀohradve. supp(UPU†P†) ⊆  Rj ∩ B( R1+ c , sU) (B2) n n j=0 This implies that U ∈ Cliﬀord . The converse can be n+1 m(cid:91)+1 proven identically. ⊆ R ∪ R . (B3) 0 j The hierarchy is composed of increasingly larger sets j=2 of gates, where Cliﬀord ⊂ Cliﬀord . These sets are n n+1 closed under group multiplication only for n ≤ 2. Fur- Furthermore, we have that s † † ≤ 2s . Hence, by thermore, Cliﬀordn/C are ﬁnite sets. For n > 2, Cliﬀordn inductive hypothesis, [UPU†P†U]P,UwPhich is aUlso a dressed generate a dense subset of the full unitary group. A full L logical operator, must belong to P when restricted to characterization of subgroups included in P remains an m n the codespace. Thus, by deﬁnition of the Cliﬀord hierar- interesting open problem. chy, [U] ∈ P . L m+1 Appendix B: Constant depth local circuits (proof of lemma 8) Proof. Let us assume that the unitary U preserves the codespace and is implementable by a constant depth lo- [1] P. W. Shor, in Proceedings of the 37th Annual Symposium 97, 180501 (2006). on Foundations of Computer Science (FOCS) (IEEE [11] H. Bombin, arXiv:1311.0879. Computer Society, Los Alamitos, CA, 1996), p. 56. [12] S. Bravyi, D. Poulin, and B. Terhal, Phys. Rev. Lett. [2] J. Preskill, Proc. Roy. Soc. Lond. 454, 385 (1998). 104, 050503 (2010). [3] B. Eastin and E. Knill, Phys. Rev. Lett. 102, 110502 [13] S. Bravyi, Phys. Rev. A 83, 012320 (2011). (2009). [14] D. Gottesman, Phys. Rev. A 54, 1862 (1996). [4] S. Bravyi and R. Ko¨nig, Phys. Rev. Lett. 110, 170503 [15] H. Bombin and M. A. Martin-Delgado, Journal of Physics (2013). A: Mathematical and General 42, 095302 (2009). [5] D. Gottesman and I. L. Chuang, Nature 402, 390 (1999), [16] D. Poulin, Phys. Rev. Lett. 95, 230504 (2005). 0906.1579v1. [17] D. W. Kribs, R. Laﬂamme, D. Poulin, and M. Lesosky, [6] M. A. Nielsen and I. L. Chuang, Quantum Computation Quant. Inf. Comp. 6, 383 (2006). and Quantum Information (Cambridge University Press, [18] M. Nielsen and D. Poulin, Phys. Rev. A 75, 064304 Cambridge, 2000). (2007). [7] D. Gottesman, Phys. Rev. A 57, 127 (1998). [19] D. Bacon, Phys. Rev. A 73, 012340 (2006). [8] S. Bravyi, Phys. Rev. A 73, 042313 (2006). [20] B. Yoshida and I. L. Chuang, Phys. Rev. A 81, 052302 [9] S. Bravyi and B. Terhal, New. J. Phys. 11, 043029 (2009). (2010). [10] H. Bombin and M. A. Martin-Delgado, Phys. Rev. Lett. [21] T. M. Stace, S. D. Barrett, and A. C. Doherty, Phys. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lemma

text: P

text: C

font: unknown

text: Pauli

text: and

text: P

text: Cliﬀord

text: for

text: n

text: cal

text: quantum

text: circuit

text: with

text: the

text: spread

text: s

text: The

text: proof

text: pro-

text: n

text: n

text: U

text: ceeds

text: by

text: induction.

text: Assuming

text: m

text: the

text: dressed

text: logical

text: Proof.

text: Let

text: us

text: ﬁrst

text: show

text: that

text: P

text: C

text: Pauli.

text: Suppose

text: operator

text: U

text: is

text: supported

text: on

text: a

text: bare-cleanable

text: region

text: and

text: that

text: U

text: P

text: By

text: hypothesis,

text: the

text: group

text: commutator

text: by

text: deﬁnition

text: must

text: be

text: a

text: trivial

text: logical

text: operator

text: in

text: P

text: of

text: U

text: with

text: any

text: Pauli

text: operator

text: P

text: P

text: is

text: trivial

text: up

text: to

text: a

text: phase

text: UPU†P†

text: eiθ.

text: This

text: phase

text: must

text: be

text: since

text: it

text: Let

text: us

text: now

text: assume

text: that

text: our

text: statement

text: is

text: true

text: up

text: to

text: m

text: is

text: an

text: eigenvalue

text: for

text: the

text: rank

text: one

text: superoperator

text: resulting

text: and

text: prove

text: it

text: for

text: m

text: Consider

text: a

text: unitary

text: U

text: with

text: the

text: from

text: conjugation

text: by

text: a

text: Pauli

text: operator

text: PU†P†

text: eiθU†.

text: spread

text: s

text: such

text: that

text: U

text: Conversely,

text: we

text: may

text: consider

text: the

text: rank

text: one

text: superoperator

text: U

text: U†

text: for

text: which

text: the

text: Pauli

text: operators

text: constitute

text: a

text: full

text: set

text: of

text: eigenoperators

text: with

text: eigenvalues

text: This

text: uniquely

text: de-

text: termines

text: U

text: to

text: be

text: equivalent

text: to

text: a

text: Pauli

text: operator

text: itself

text: up

text: supp(U)

text: R

text: j

text: to

text: a

text: global

text: phase.

text: Here,

text: we

text: have

text: crucially

text: used

text: that

text: the

text: Pauli

text: operators

text: linearly

text: span

text: the

text: full

text: operator

text: algebra.

text: We

text: will

text: now

text: prove

text: that

text: U

text: P

text: U

text: Cliﬀord

text: by

text: n

text: n

text: induction

text: for

text: n

text: The

text: proof

text: relies

text: on

text: the

text: observa-

text: Any

text: logical

text: Pauli

text: operator

text: [P]

text: has

text: a

text: dressed

text: incarna-

text: L

text: tion

text: that

text: all

text: the

text: levels

text: of

text: the

text: usual

text: Cliﬀord

text: hierarchy

text: tion

text: P

text: fully

text: supported

text: on

text: R+

text: c.

text: Observe

text: that

text: are

text: closed

text: under

text: right

text: multiplication

text: by

text: Pauli

text: operators

text: Cliﬀord

text: Cliﬀord

text: Pauli

text: which

text: can

text: be

text: proven

text: induc-

text: n

text: n

text: tively.

text: PUP∈UP†au∈liCwlieﬀohradve.

text: supp(UPU†P†)

text: Rj

text: B(

text: c

text: sU)

text: n

text: n

text: This

text: implies

text: that

text: U

text: Cliﬀord

text: The

text: converse

text: can

text: be

text: proven

text: identically.

text: R

text: R

text: j

text: The

text: hierarchy

text: is

text: composed

text: of

text: increasingly

text: larger

text: sets

text: of

text: gates,

text: where

text: Cliﬀord

text: Cliﬀord

text: These

text: sets

text: are

text: n

text: closed

text: under

text: group

text: multiplication

text: only

text: for

text: n

text: Fur-

text: Furthermore,

text: we

text: have

text: that

text: s

text: Hence,

text: by

text: thermore,

text: Cliﬀordn/C

text: are

text: ﬁnite

text: sets.

text: For

text: n

text: Cliﬀordn

text: inductive

text: hypothesis,

text: [UPU†P†U]P,UwPhich

text: is

text: aUlso

text: a

text: dressed

text: generate

text: a

text: dense

text: subset

text: of

text: the

text: full

text: unitary

text: group.

text: A

text: full

text: L

text: logical

text: operator,

text: must

text: belong

text: to

text: P

text: when

text: restricted

text: to

text: characterization

text: of

text: subgroups

text: included

text: in

text: P

text: remains

text: an

text: m

text: n

text: the

text: codespace.

text: Thus,

text: by

text: deﬁnition

text: of

text: the

text: Cliﬀord

text: hierar-

text: interesting

text: open

text: problem.

text: chy,

text: [U]

text: P

text: L

text: Appendix

text: B:

text: Constant

text: depth

text: local

text: circuits

text: (proof

text: of

text: lemma

text: Proof.

text: Let

text: us

text: assume

text: that

text: the

text: unitary

text: U

text: preserves

text: the

text: codespace

text: and

text: is

text: implementable

text: by

text: a

text: constant

text: depth

text: lo-

text: P.

text: W.

text: Shor,

text: in

text: Proceedings

text: of

text: the

text: Annual

text: Symposium

text: on

text: Foundations

text: of

text: Computer

text: Science

text: (FOCS)

text: (IEEE

text: H.

text: Bombin,

text: Computer

text: Society,

text: Los

text: Alamitos,

text: CA,

text: p.

text: S.

text: Bravyi,

text: D.

text: Poulin,

text: and

text: B.

text: Terhal,

text: Phys.

text: Rev.

text: Lett.

text: J.

text: Preskill,

text: Proc.

text: Roy.

text: Soc.

text: Lond.

text: B.

text: Eastin

text: and

text: E.

text: Knill,

text: Phys.

text: Rev.

text: Lett.

text: S.

text: Bravyi,

text: Phys.

text: Rev.

text: A

text: D.

text: Gottesman,

text: Phys.

text: Rev.

text: A

text: S.

text: Bravyi

text: and

text: R.

text: Ko¨nig,

text: Phys.

text: Rev.

text: Lett.

text: H.

text: Bombin

text: and

text: M.

text: A.

text: Martin-Delgado,

text: Journal

text: of

text: Physics

text: A:

text: Mathematical

text: and

text: General

text: D.

text: Gottesman

text: and

text: I.

text: L.

text: Chuang,

text: Nature

text: D.

text: Poulin,

text: Phys.

text: Rev.

text: Lett.

text: D.

text: W.

text: Kribs,

text: R.

text: Laﬂamme,

text: D.

text: Poulin,

text: and

text: M.

text: Lesosky,

text: M.

text: A.

text: Nielsen

text: and

text: I.

text: L.

text: Chuang,

text: Quantum

text: Computation

text: Quant.

text: Inf.

text: Comp.

text: and

text: Quantum

text: Information

text: (Cambridge

text: University

text: Press,

text: M.

text: Nielsen

text: and

text: D.

text: Poulin,

text: Phys.

text: Rev.

text: A

text: Cambridge,

text: D.

text: Gottesman,

text: Phys.

text: Rev.

text: A

text: D.

text: Bacon,

text: Phys.

text: Rev.

text: A

text: S.

text: Bravyi,

text: Phys.

text: Rev.

text: A

text: B.

text: Yoshida

text: and

text: I.

text: L.

text: Chuang,

text: Phys.

text: Rev.

text: A

text: S.

text: Bravyi

text: and

text: B.

text: Terhal,

text: New.

text: J.

text: Phys.

text: H.

text: Bombin

text: and

text: M.

text: A.

text: Martin-Delgado,

text: Phys.

text: Rev.

text: Lett.

text: T.

text: M.

text: Stace,

text: S.

text: D.

text: Barrett,

text: and

text: A.

text: C.

text: Doherty,

text: Phys.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


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text: Rev.

text: Lett.

text: S.

text: A.

text: Ocko,

text: X.

text: Chen,

text: B.

text: Zeng,

text: B.

text: Yoshida,

text: Z.

text: Ji,

text: M.

text: B.

text: J.

text: Dorier,

text: F.

text: Becca,

text: and

text: F.

text: Mila,

text: Physical

text: Review

text: B

text: Ruskai,

text: and

text: I.

text: L.

text: Chuang,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: A.

text: Kovalev

text: and

text: L.

text: P.

text: Pryadko,

text: Phys.

text: Rev.

text: A

text: If

label: footer

text: two

text: copies

text: of

text: a

text: CSS

text: code

text: are

text: stacked

text: such

text: that

text: corre-

text: sponding

text: qubits

text: are

text: geometrically

text: close,

text: performing

text: pair-

text: F.

text: Pastawski,

text: A.

text: Kay,

text: N.

text: Schuch,

text: and

text: I.

text: Cirac,

text: Quantum

text: wise

text: CNOT

text: on

text: all

text: physical

text: qubits

text: implements

text: a

text: CNOT

text: Inf.

text: Comput.

text: gates

text: on

text: all

text: pairs

text: of

text: encoded

text: qubits

text: in

text: the

text: two

text: copies.

text: M.

text: B.

text: Hastings,

text: Phys.

text: Rev.

text: Lett.

text: Formally,

text: erasure

text: errors

text: are

text: modeled

text: by

text: extending

text: the

text: E.

text: Dennis,

text: A.

text: Kitaev,

text: A.

text: Landahl,

text: and

text: J.

text: Preskill,

text: J.

text: space

text: associated

text: to

text: each

text: physical

text: particle

text: with

text: one

text: addi-

text: Math.

text: Phys.

text: tional

text: state

text: which

text: indicates

text: loss

text: of

text: the

text: corresponding

text: B.

text: Yoshida,

text: Ann.

text: Phys.

text: particle.

text: An

text: error-correcting

text: recovery

text: map

text: for

text: loss

text: errors

text: J.

text: Haah

text: and

text: J.

text: Preskill,

text: may

text: mimic

text: the

text: one

text: for

text: depolarizing

text: noise

text: by

text: mapping

text: all

text: O.

text: Landon-Cardinal

text: and

text: D.

text: Poulin,

text: Phys.

text: Rev.

text: Lett.

text: particles

text: marked

text: as

text: to

text: the

text: ﬁxed-point

text: of

text: corresponding

text: depolarizing

text: channel.

text: Hence,

text: the

text: loss

text: threshold

text: p

text: must

text: l

text: J.

text: Haah,

text: Phys.

text: Rev.

text: A

text: necessarily

text: be

text: no

text: smaller

text: than

text: any

text: depolarization

text: thresh-

text: S.

text: Bravyi

text: and

text: J.

text: Haah,

text: Phys.

text: Rev.

text: Lett.

text: old

text: p

text: p

text: l

text: e

text: Models

text: proposed

text: in

text: have

text: a

text: macroscopic

text: energy

text: I.

text: H.

text: Kim,

text: barrier,

text: yet

text: quantum

text: memory

text: time

text: is

text: upper

text: bounded

text: by

text: B.

text: Yoshida,

text: Phys.

text: Rev.

text: B

text: constant,

text: which

text: is

text: perhaps

text: due

text: to

text: topological

text: transition

text: K.

text: Michnicki,

text: temperature

text: being

text: zero.

text: Finite

text: transition

text: temperature

text: is

text: H.

text: Bombin,

text: R.

text: W.

text: Chhajlany,

text: M.

text: Horodecki,

text: and

text: M.

text: A.

text: not

text: suﬃcient

text: to

text: guarantee

text: exponentially

text: growing

text: quan-

text: Martin-Delgado,

text: New.

text: J.

text: Phys.

text: tum

text: memory

text: time

text: F.

text: Pastawski,

text: L.

text: Clemente,

text: and

text: J.

text: I.

text: Cirac,

text: Phys.

text: Rev.

text: A

text: The

text: absence

text: of

text: string-like

text: logical

text: operators

text: does

text: not

text: nec-

text: essarily

text: imply

text: the

text: presence

text: of

text: macroscopic

text: energy

text: bar-

text: X.

text: Chen,

text: Z.-C.

text: Gu,

text: and

text: X.-G.

text: Wen,

text: Phys.

text: Rev.

text: B

text: rier

text: Due

text: to

text: the

text: non-commutativity

text: of

text: G

text: a

text: ground

text: state

text: j

text: S.

text: Bravyi

text: and

text: A.

text: Kitaev,

text: Phys.

text: Rev.

text: A

text: of

text: the

text: Hamiltonian

text: H

text: α

text: G

text: is

text: not

text: necessarily

text: inside

text: j

text: j

text: j

text: A.

text: Paetznick

text: and

text: B.

text: W.

text: Reichardt,

text: Phys.

text: Rev.

text: Lett.

text: the

text: stabilized

text: subspace.

text: Indeed

text: it

text: is

text: analytically

text: and

text: com-

text: putationally

text: diﬃcult

text: to

text: ﬁnd

text: values

text: of

text: S

text: in

text: the

text: ground

text: j

text: T.

text: Jochym-O’Connor

text: and

text: R.

text: Laﬂamme,

text: Phys.

text: Rev.

text: Lett.

text: space.

text: For

text: CSS

text: subsystem

text: codes,

text: the

text: ground

text: space

text: of

text: H

text: G

text: is

text: guaranteed

text: to

text: be

text: in

text: the

text: stabilized

text: sub-

text: j

text: j

text: B.

text: Yoshida,

text: space

text: deﬁned

text: by

text: S

text: due

text: to

text: the

text: Perron-Frobenius

text: theo-

text: S.

text: Bravyi,

text: B.

text: Leemhuis,

text: and

text: B.

text: M.

text: Terhal,

text: Ann.

text: Phys.

text: rem


text: 8 procedures to be small. Namely, the failure probability gions R and {R } . If R is bare-cleanable and 0 j j∈[1,m] 0 is lower bounded by pfail ≥ O(pd) where p is the physical each Rj+ := B(Rj, 2j−1sU) is dressed-cleanable for j > 0, error rate. In order for pfail to converge to zero for suﬃ- then the logical unitary implemented by U belongs to Pm. ciently small but ﬁnite p, the distance d must be macro- scopic. It is highly desirable to have a code with exponen- This means that when dealing with locality-preserving tially vanishing p . Polynomially decaying p may be circuits which implement logical unitary gates, it is suﬃ- fail fail still acceptable, which imposes that the distance d must cient to use extended correctable regions such that they grow at least logarithmically. However, sub-polynomially overlap in a boundary of width 2m−1sU, where m is the vanishing p would be impractical. As such we may as- number of regions to be used. As such, much of discus- fail sume that a code distance grows at least logarithmically sion dealing with transversal gates applies to ﬁnite depth in a fault-tolerant code. circuits. The proof is presented in appendix . Macroscopic code distance is necessary, but not suﬃ- With an assumption of macroscopic code distance cient for a ﬁnite error threshold in the code. For a sta- alone, one is able to obtain the following statement for bilizer code with low-weight generators, a macroscopic topological subsystem codes. code distance guarantees a ﬁnite error threshold against Corollary 1. Consider a family of subsystem codes with depolarization. Namely, if the code distance grows loga- increasing code distance deﬁned by geometrically local rithmically with n, a ﬁnite error threshold is guaranteed gauge generators of diameter bounded by ξ in D spatial as proven by Kovalev and Pryadko [23]. dimensions. Then the set of dressed logical unitary gates Theorem 3. [Error threshold [23]] Consider a family implementable by constant depth circuits is included in of stabilizer codes whose stabilizer group generators have P . constant weight. If the code distance grows logarithmi- D+1 cally in the system size n, a ﬁnite error threshold always Proof. Since gauge generators are geometrically local exists such that the recovery failure probability pfail ap- with diameter bounded by ξ, the union lemma (lemma 7) proaches to zero as n increases. applies to dressed cleanable regions that are separated by a distance ξ or larger. Furthermore, any region with vol- Yet this theorem does not apply to all subsystem ume smaller than d is dressed-cleanable by the deﬁnition codes, and applies only to subsystem codes with low- of a code distance. Let s be the spread of the circuit U. weight stabilizer generators. Indeed, two and three- U One has d > (2Ds +ξ)D for suﬃciently large n since the dimensional quantum compass models have a zero er- U code has a macroscopic distance. Then the lattice may ror threshold, and thus are not scalable quantum error- be partitioned into D + 1 disjoint regions {R } correcting codes [24]. To our knowledge, there is no j j∈[1,D+1] known relation between the existence (or absence) of a such that Rj+ := B(Rj, 2j−1sU) is dressed-cleanable for ﬁnite error threshold and locality (of non-locality) of sta- all j > 0. For instance, we construct a D-dimensional bilizer generators. It seems plausible that such a relation hyper-cubic tiling and fatten m-dimensional objects to could exist. obtain Rm+1 for m = 0, . . . , D. By taking R0 to be an BK’s derivation relies on a macroscopic code distance, empty set ∅, we conclude that the logical action of U is which is required for a ﬁnite error threshold. In the included in PD+1. present work, we use the fault-tolerance itself as the guid- Note that R are dressed-cleanable, but not necessarily ing principle. Namely, we will assume that (i) the code j bare-cleanable since the union lemma does not hold for distance grows at least logarithmically, and (ii) the code bare-cleanable regions. Thus, we needed to take R to be has a ﬁnite (loss) error threshold. 0 an empty set, which result in increasing the level of the implementable Cliﬀord hierarchy by one with respect to BK’s result for topological stabilizer codes. An interest- C. Bravyi-K¨onig for subsystem code ing open problem is to ﬁnd subsystem codes with growing distance which achieve the bound stated in corollary 1. If Let us now proceed to generalization of BK’s result such subsystem codes exist, we believe that they would to a topological subsystem code. The distance d(, ) be- be highly artiﬁcial and would possess highly non-local tween particles on the lattice will be used to deﬁne the stabilizer generators. r-neighborhood B(R, r) of a region R which includes re- We now further assume that the family of codes has gion R and all particles within distance r to it. Fur- a non-zero loss threshold p > 0 and that a code dis- l thermore, we deﬁne the spread s of a unitary as the U tance d grows at least logarithmically with the number smallest possible distance such that ∀A : supp(UAU†) ⊆ of particles n. Under these reasonable and perhaps in- B(supp(A), s ). In particular, if U is implemented by a U dispensable assumptions for fault-tolerance of the code, constant depth circuits composed of geometrically local we obtain the same thesis as BK’s result for topological gates, the spread s will also be bounded by a constant. U subsystem codes. A version of lemma 5 involving the lattice geometry can now be stated. Theorem 4. Consider a family of subsystem codes with Lemma 8. Let U be a dressed logical unitary operator geometrically local gauge generators in D spatial dimen- supported on the union of mutually non-intersecting re- sion with i) a loss threshold p > 0 and ii) a code distance l ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: procedures

text: to

text: be

text: small.

text: Namely,

text: the

text: failure

text: probability

text: gions

text: R

text: and

text: {R

text: If

text: R

text: is

text: bare-cleanable

text: and

text: j

text: is

text: lower

text: bounded

text: by

text: pfail

text: O(pd)

text: where

text: p

text: is

text: the

text: physical

text: each

text: Rj+

text: B(Rj,

text: is

text: dressed-cleanable

text: for

text: j

text: error

text: rate.

text: In

text: order

text: for

text: pfail

text: to

text: converge

text: to

text: zero

text: for

text: suﬃ-

text: then

text: the

text: logical

text: unitary

text: implemented

text: by

text: U

text: belongs

text: to

text: Pm.

text: ciently

text: small

text: but

text: ﬁnite

text: p,

text: the

text: distance

text: d

text: must

text: be

text: macro-

text: scopic.

text: It

text: is

text: highly

text: desirable

text: to

text: have

text: a

text: code

text: with

text: exponen-

text: This

text: means

text: that

text: when

text: dealing

text: with

text: locality-preserving

text: tially

text: vanishing

text: p

text: Polynomially

text: decaying

text: p

text: may

text: be

text: circuits

text: which

text: implement

text: logical

text: unitary

text: gates,

text: it

text: is

text: suﬃ-

text: fail

text: fail

text: still

text: acceptable,

text: which

text: imposes

text: that

text: the

text: distance

text: d

text: must

text: cient

text: to

text: use

text: extended

text: correctable

text: regions

text: such

text: that

text: they

text: grow

text: at

text: least

text: logarithmically.

text: However,

text: sub-polynomially

text: overlap

text: in

text: a

text: boundary

text: of

text: width

text: where

text: m

text: is

text: the

text: vanishing

text: p

text: would

text: be

text: impractical.

text: As

text: such

text: we

text: may

text: as-

text: number

text: of

text: regions

text: to

text: be

text: used.

text: As

text: such,

text: much

text: of

text: discus-

text: fail

text: sume

text: that

text: a

text: code

text: distance

text: grows

text: at

text: least

text: logarithmically

text: sion

text: dealing

text: with

text: transversal

text: gates

text: applies

text: to

text: ﬁnite

text: depth

text: in

text: a

text: fault-tolerant

text: code.

text: circuits.

text: The

text: proof

text: is

text: presented

text: in

text: appendix

text: Macroscopic

text: code

text: distance

text: is

text: necessary,

text: but

text: not

text: suﬃ-

text: With

text: an

text: assumption

text: of

text: macroscopic

text: code

text: distance

text: cient

text: for

text: a

text: ﬁnite

text: error

text: threshold

text: in

text: the

text: code.

text: For

text: a

text: sta-

text: alone,

text: one

text: is

text: able

text: to

text: obtain

text: the

text: following

text: statement

text: for

text: bilizer

text: code

text: with

text: low-weight

text: generators,

text: a

text: macroscopic

text: topological

text: subsystem

text: codes.

text: code

text: distance

text: guarantees

text: a

text: ﬁnite

text: error

text: threshold

text: against

text: Corollary

text: Consider

text: a

text: family

text: of

text: subsystem

text: codes

text: with

text: depolarization.

text: Namely,

text: if

text: the

text: code

text: distance

text: grows

text: loga-

text: increasing

text: code

text: distance

text: deﬁned

text: by

text: geometrically

text: local

text: rithmically

text: with

text: n,

text: a

text: ﬁnite

text: error

text: threshold

text: is

text: guaranteed

text: gauge

text: generators

text: of

text: diameter

text: bounded

text: by

text: ξ

text: in

text: D

text: spatial

text: as

text: proven

text: by

text: Kovalev

text: and

text: Pryadko

text: dimensions.

text: Then

text: the

text: set

text: of

text: dressed

text: logical

text: unitary

text: gates

text: Theorem

text: [Error

text: threshold

text: Consider

text: a

text: family

text: implementable

text: by

text: constant

text: depth

text: circuits

text: is

text: included

text: in

text: of

text: stabilizer

text: codes

text: whose

text: stabilizer

text: group

text: generators

text: have

text: P

text: constant

text: weight.

text: If

text: the

text: code

text: distance

text: grows

text: logarithmi-

text: cally

text: in

text: the

text: system

text: size

text: n,

text: a

text: ﬁnite

text: error

text: threshold

text: always

text: Proof.

text: Since

text: gauge

text: generators

text: are

text: geometrically

text: local

text: exists

text: such

text: that

text: the

text: recovery

text: failure

text: probability

text: pfail

text: ap-

text: with

text: diameter

text: bounded

text: by

text: ξ,

text: the

text: union

text: lemma

text: (lemma

text: proaches

text: to

text: zero

text: as

text: n

text: increases.

text: applies

text: to

text: dressed

text: cleanable

text: regions

text: that

text: are

text: separated

text: by

text: a

text: distance

text: ξ

text: or

text: larger.

text: Furthermore,

text: any

text: region

text: with

text: vol-

text: Yet

text: this

text: theorem

text: does

text: not

text: apply

text: to

text: all

text: subsystem

text: ume

text: smaller

text: than

text: d

text: is

text: dressed-cleanable

text: by

text: the

text: deﬁnition

text: codes,

text: and

text: applies

text: only

text: to

text: subsystem

text: codes

text: with

text: low-

text: of

text: a

text: code

text: distance.

text: Let

text: s

text: be

text: the

text: spread

text: of

text: the

text: circuit

text: U.

text: weight

text: stabilizer

text: generators.

text: Indeed,

text: two

text: and

text: three-

text: U

text: One

text: has

text: d

text: +ξ)D

text: for

text: suﬃciently

text: large

text: n

text: since

text: the

text: dimensional

text: quantum

text: compass

text: models

text: have

text: a

text: zero

text: er-

text: U

text: code

text: has

text: a

text: macroscopic

text: distance.

text: Then

text: the

text: lattice

text: may

text: ror

text: threshold,

text: and

text: thus

text: are

text: not

text: scalable

text: quantum

text: error-

text: be

text: partitioned

text: into

text: D

text: disjoint

text: regions

text: {R

text: correcting

text: codes

text: To

text: our

text: knowledge,

text: there

text: is

text: no

text: j

text: known

text: relation

text: between

text: the

text: existence

text: (or

text: absence)

text: of

text: a

text: such

text: that

text: Rj+

text: B(Rj,

text: is

text: dressed-cleanable

text: for

text: ﬁnite

text: error

text: threshold

text: and

text: locality

text: (of

text: non-locality)

text: of

text: sta-

text: all

text: j

text: For

text: instance,

text: we

text: construct

text: a

text: D-dimensional

text: bilizer

text: generators.

text: It

text: seems

text: plausible

text: that

text: such

text: a

text: relation

text: hyper-cubic

text: tiling

text: and

text: fatten

text: m-dimensional

text: objects

text: to

text: could

text: exist.

text: obtain

text: for

text: m

text: D.

text: By

text: taking

text: to

text: be

text: an

text: BK’s

text: derivation

text: relies

text: on

text: a

text: macroscopic

text: code

text: distance,

text: empty

text: set

text: we

text: conclude

text: that

text: the

text: logical

text: action

text: of

text: U

text: is

text: which

text: is

text: required

text: for

text: a

text: ﬁnite

text: error

text: threshold.

text: In

text: the

text: included

text: in

text: present

text: work,

text: we

text: use

text: the

text: fault-tolerance

text: itself

text: as

text: the

text: guid-

text: Note

text: that

text: R

text: are

text: dressed-cleanable,

text: but

text: not

text: necessarily

text: ing

text: principle.

text: Namely,

text: we

text: will

text: assume

text: that

text: (i)

text: the

text: code

text: j

text: bare-cleanable

text: since

text: the

text: union

text: lemma

text: does

text: not

text: hold

text: for

text: distance

text: grows

text: at

text: least

text: logarithmically,

text: and

text: (ii)

text: the

text: code

text: bare-cleanable

text: regions.

text: Thus,

text: we

text: needed

text: to

text: take

text: R

text: to

text: be

text: has

text: a

text: ﬁnite

text: (loss)

text: error

text: threshold.

text: an

text: empty

text: set,

text: which

text: result

text: in

text: increasing

text: the

text: level

text: of

text: the

text: implementable

text: Cliﬀord

text: hierarchy

text: by

text: one

text: with

text: respect

text: to

text: BK’s

text: result

text: for

text: topological

text: stabilizer

text: codes.

text: An

text: interest-

text: C.

text: Bravyi-K¨onig

text: for

text: subsystem

text: code

text: ing

text: open

text: problem

text: is

text: to

text: ﬁnd

text: subsystem

text: codes

text: with

text: growing

text: distance

text: which

text: achieve

text: the

text: bound

text: stated

text: in

text: corollary

text: If

text: Let

text: us

text: now

text: proceed

text: to

text: generalization

text: of

text: BK’s

text: result

text: such

text: subsystem

text: codes

text: exist,

text: we

text: believe

text: that

text: they

text: would

text: to

text: a

text: topological

text: subsystem

text: code.

text: The

text: distance

text: d(,

text: be-

text: be

text: highly

text: artiﬁcial

text: and

text: would

text: possess

text: highly

text: non-local

text: tween

text: particles

text: on

text: the

text: lattice

text: will

text: be

text: used

text: to

text: deﬁne

text: the

text: stabilizer

text: generators.

text: r-neighborhood

text: B(R,

text: r)

text: of

text: a

text: region

text: R

text: which

text: includes

text: re-

text: We

text: now

text: further

text: assume

text: that

text: the

text: family

text: of

text: codes

text: has

text: gion

text: R

text: and

text: all

text: particles

text: within

text: distance

text: r

text: to

text: it.

text: Fur-

text: a

text: non-zero

text: loss

text: threshold

text: p

text: and

text: that

text: a

text: code

text: dis-

text: l

text: thermore,

text: we

text: deﬁne

text: the

text: spread

text: s

text: of

text: a

text: unitary

text: as

text: the

text: U

text: tance

text: d

text: grows

text: at

text: least

text: logarithmically

text: with

text: the

text: number

text: smallest

text: possible

text: distance

text: such

text: that

text: ∀A

text: supp(UAU†)

text: of

text: particles

text: n.

text: Under

text: these

text: reasonable

text: and

text: perhaps

text: in-

text: B(supp(A),

text: s

text: In

text: particular,

text: if

text: U

text: is

text: implemented

text: by

text: a

text: U

text: dispensable

text: assumptions

text: for

text: fault-tolerance

text: of

text: the

text: code,

text: constant

text: depth

text: circuits

text: composed

text: of

text: geometrically

text: local

text: we

text: obtain

text: the

text: same

text: thesis

text: as

text: BK’s

text: result

text: for

text: topological

text: gates,

text: the

text: spread

text: s

text: will

text: also

text: be

text: bounded

text: by

text: a

text: constant.

text: U

text: subsystem

text: codes.

text: A

text: version

text: of

text: lemma

text: involving

text: the

text: lattice

text: geometry

text: can

text: now

text: be

text: stated.

text: Theorem

text: Consider

text: a

text: family

text: of

text: subsystem

text: codes

text: with

text: Lemma

text: Let

text: U

text: be

text: a

text: dressed

text: logical

text: unitary

text: operator

text: geometrically

text: local

text: gauge

text: generators

text: in

text: D

text: spatial

text: dimen-

text: supported

text: on

text: the

text: union

text: of

text: mutually

text: non-intersecting

text: re-

text: sion

text: with

text: i)

text: a

text: loss

text: threshold

text: p

text: and

text: ii)

text: a

text: code

text: distance

text: l

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: TABLE OF CONTENTS 1. INTRODUCTION results are presented in Section . 2. ENSTROPHY-CONSERVING SHALLOW This paper continues our cycle of publications on WATER EQUATION SOLVER the optimization of HPC applications for computing systems enabled with Intel Xeon Phi coprocessors fea- The physical model of shallow water ﬂow assumes turing the Many Integrated Core (MIC) architecture1. a perfect incompressible ﬂuid subject to the gravita- The application studied in this work is a computational tional force. This model can be used to describe, for ﬂuid dynamics code (CFD), a shallow water equation example, the movement of the ocean water on Earth or solver. The source code of the application is freely certain atmospheric phenomena. available at [1]. The code is based on the numerical The quantities that describe shallow water ﬂow are method developed by Robert Sadourny [3] and initially the pressure P (x, y) and the ﬂuid velocity V(x, y). The ≡ written by Paul Schwarztrauber of National Center for components of vector V are denoted as V u and ≡ x Atmospheric Research (NCAR). Thereafter, the code V v. These quantities can be related to the height y was modernized and adapted for hybrid parallelism in H(x, y) of the water above the equilibrium level as ≡ the OpenMP and MPI frameworks by Cliff Addison H = P + V 2/2, where V 2 u2 + v2. The evolu- (University of Liverpool). This solver is a good candi- tion of shallow water ﬂow can be expressed with the set date for acceleration on the Intel MIC architecture be- of equations (1)–(2): (cid:18) (cid:19) cause it submits to parallelization and operates in the memory bandwidth-bound regime. ∂V × ∇ 1 In our solver, just like in any MPI application de- + ηN (P V) + P + V 2 = 0, (1) signed for CPU-based clusters, initialization and data ∂t 2 trafﬁc were already implemented prior to porting to the ∂P ∇ · + (P V) = 0. (2) Intel MIC architecture. Therefore, the porting proce- ∂t dure can be limited to recompilation of the code [4]. ≡ − Furthermore, the solver is already optimized for com- Here η [∂v/∂x ∂u/∂y] /P is the potential vor- pute nodes based on multi-core CPUs. It is expressed ticity, and N is the unit vector normal to the plain do- in the Fortran language, with OpenMP and array no- main S. The system of units is chosen so that the free tation used to express parallelism. As a consequence, fall acceleration is reduced to a dimensionless value the node-level performance of the solver is also very g = 1, and the ﬂuid density ρ = 1. efﬁcient when the code is compiled for and run on an The numerical method for solving these equations (cid:90) Intel Xeon Phi coprocessor. This means that the op- proposed by Sadourny [3] conserves enstrophy timization process does not need to involve any code modiﬁcation, and only the execution environment and 1 Z = η2P dS, parameters need to be tuned. 2 S Optimization methods described here can be ap- which is a desirable stability property. Enstrophy con- plied to other bandwidth-bound stencil codes on mul- servation avoids nonlinear instability that leads to an en- tidimensional Cartesian grids. ergy catastrophe in the simulation. The time update step In Section we outline the nature of the calculation in this method is expressed as (cid:10) (cid:11) and the methodology of work distribution across com- puting nodes in the cluster. Section discusses the opti- ∂u − (cid:104) (cid:105) (cid:10)(cid:104) (cid:105) (cid:11) mization of the MPI environment for executing the code η V + δ H = 0, (3) ∂t x x y x on a single CPU-based node or a single Xeon Phi copro- ∂v (cid:104) (cid:105) (cid:104) (cid:105) cessor. Cluster-level tuning is discussed in Section , and + η U + δ H = 0, (4) ∂t y y x y 1See, e.g., [2] for information ∂P + δ U + δ V = 0. (5) x y ∂t (cid:13) c Colfax International, 2014 — http://research.colfaxinternational.com/ 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: OF

text: CONTENTS

label: section

text: INTRODUCTION

text: results

text: are

text: presented

text: in

text: Section

text: ENSTROPHY-CONSERVING

text: SHALLOW

text: This

text: paper

text: continues

text: our

text: cycle

text: of

text: publications

text: on

text: WATER

text: EQUATION

text: SOLVER

text: the

text: optimization

text: of

text: HPC

text: applications

text: for

text: computing

text: systems

text: enabled

text: with

text: Intel

text: Xeon

text: Phi

text: coprocessors

text: fea-

text: The

text: physical

text: model

text: of

text: shallow

text: water

text: ﬂow

text: assumes

text: turing

text: the

text: Many

text: Integrated

text: Core

text: (MIC)

text: a

text: perfect

text: incompressible

text: ﬂuid

text: subject

text: to

text: the

text: gravita-

text: The

text: application

text: studied

text: in

text: this

text: work

text: is

text: a

text: computational

text: tional

text: force.

text: This

text: model

text: can

text: be

text: used

text: to

text: describe,

text: for

text: ﬂuid

text: dynamics

text: code

text: (CFD),

text: a

text: shallow

text: water

text: equation

text: example,

text: the

text: movement

text: of

text: the

text: ocean

text: water

text: on

text: Earth

text: or

text: solver.

text: The

text: source

text: code

text: of

text: the

text: application

text: is

text: freely

text: certain

text: atmospheric

text: phenomena.

text: available

text: at

text: The

text: code

text: is

text: based

text: on

text: the

text: numerical

text: The

text: quantities

text: that

text: describe

text: shallow

text: water

text: ﬂow

text: are

text: method

text: developed

text: by

text: Robert

text: Sadourny

text: and

text: initially

text: the

text: pressure

text: P

text: (x,

text: y)

text: and

text: the

text: ﬂuid

text: velocity

text: V(x,

text: y).

text: The

text: written

text: by

text: Paul

text: Schwarztrauber

text: of

text: National

text: Center

text: for

text: components

text: of

text: vector

text: V

text: are

text: denoted

text: as

text: V

text: u

text: and

text: x

text: Atmospheric

text: Research

text: (NCAR).

text: Thereafter,

text: the

text: code

text: V

text: v.

text: These

text: quantities

text: can

text: be

text: related

text: to

text: the

text: height

text: y

text: was

text: modernized

text: and

text: adapted

text: for

text: hybrid

text: parallelism

text: in

text: H(x,

text: y)

text: of

text: the

text: water

text: above

text: the

text: equilibrium

text: level

text: as

text: the

text: OpenMP

text: and

text: MPI

text: frameworks

text: by

text: Cliff

text: Addison

text: H

text: P

text: V

text: where

text: V

text: The

text: evolu-

text: (University

text: of

text: Liverpool).

text: This

text: solver

text: is

text: a

text: good

text: candi-

text: tion

text: of

text: shallow

text: water

text: ﬂow

text: can

text: be

text: expressed

text: with

text: the

text: set

text: date

text: for

text: acceleration

text: on

text: the

text: Intel

text: MIC

text: architecture

text: be-

text: of

text: equations

text: cause

text: it

text: submits

text: to

text: parallelization

text: and

text: operates

text: in

text: the

text: memory

text: bandwidth-bound

text: regime.

text: ∂V

text: In

text: our

text: solver,

text: just

text: like

text: in

text: any

text: MPI

text: application

text: de-

text: ηN

text: (P

text: V)

text: P

text: V

text: signed

text: for

text: CPU-based

text: clusters,

text: initialization

text: and

text: data

text: ∂t

text: trafﬁc

text: were

text: already

text: implemented

text: prior

text: to

text: porting

text: to

text: the

text: ∂P

text: (P

text: V)

text: Intel

text: MIC

text: architecture.

text: Therefore,

text: the

text: porting

text: proce-

text: ∂t

text: dure

text: can

text: be

text: limited

text: to

text: recompilation

text: of

text: the

text: code

text: Furthermore,

text: the

text: solver

text: is

text: already

text: optimized

text: for

text: com-

text: Here

text: η

text: [∂v/∂x

text: ∂u/∂y]

text: /P

text: is

text: the

text: potential

text: vor-

text: pute

text: nodes

text: based

text: on

text: multi-core

text: CPUs.

text: It

text: is

text: expressed

text: ticity,

text: and

text: N

text: is

text: the

text: unit

text: vector

text: normal

text: to

text: the

text: plain

text: do-

text: in

text: the

text: Fortran

text: language,

text: with

text: OpenMP

text: and

text: array

text: no-

text: main

text: S.

text: The

text: system

text: of

text: units

text: is

text: chosen

text: so

text: that

text: the

text: free

text: tation

text: used

text: to

text: express

text: parallelism.

text: As

text: a

text: consequence,

text: fall

text: acceleration

text: is

text: reduced

text: to

text: a

text: dimensionless

text: value

text: the

text: node-level

text: performance

text: of

text: the

text: solver

text: is

text: also

text: very

text: g

text: and

text: the

text: ﬂuid

text: density

text: ρ

text: efﬁcient

text: when

text: the

text: code

text: is

text: compiled

text: for

text: and

text: run

text: on

text: an

text: The

text: numerical

text: method

text: for

text: solving

text: these

text: equations

text: Intel

text: Xeon

text: Phi

text: coprocessor.

text: This

text: means

text: that

text: the

text: op-

text: proposed

text: by

text: Sadourny

text: conserves

text: enstrophy

text: timization

text: process

text: does

text: not

text: need

text: to

text: involve

text: any

text: code

text: modiﬁcation,

text: and

text: only

text: the

text: execution

text: environment

text: and

text: Z

text: dS,

text: parameters

text: need

text: to

text: be

text: tuned.

text: S

text: Optimization

text: methods

text: described

text: here

text: can

text: be

text: ap-

text: which

text: is

text: a

text: desirable

text: stability

text: property.

text: Enstrophy

text: con-

text: plied

text: to

text: other

text: bandwidth-bound

text: stencil

text: codes

text: on

text: mul-

text: servation

text: avoids

text: nonlinear

text: instability

text: that

text: leads

text: to

text: an

text: en-

text: tidimensional

text: Cartesian

text: grids.

text: ergy

text: catastrophe

text: in

text: the

text: simulation.

text: The

text: time

text: update

text: step

text: In

text: Section

text: we

text: outline

text: the

text: nature

text: of

text: the

text: calculation

text: in

text: this

text: method

text: is

text: expressed

text: as

text: and

text: the

text: methodology

text: of

text: work

text: distribution

text: across

text: com-

text: puting

text: nodes

text: in

text: the

text: cluster.

text: Section

text: discusses

text: the

text: opti-

text: ∂u

text: mization

text: of

text: the

text: MPI

text: environment

text: for

text: executing

text: the

text: code

text: η

text: V

text: δ

text: H

text: ∂t

text: x

text: x

text: y

text: x

text: on

text: a

text: single

text: CPU-based

text: node

text: or

text: a

text: single

text: Xeon

text: Phi

text: copro-

text: ∂v

text: cessor.

text: Cluster-level

text: tuning

text: is

text: discussed

text: in

text: Section

text: and

text: η

text: U

text: δ

text: H

text: ∂t

text: y

text: y

text: x

text: y

text: e.g.,

text: for

text: information

text: ∂P

text: δ

text: U

text: δ

text: V

text: x

text: y

text: ∂t

text: c

text: Colfax

text: International,

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text: REFERENCES REFERENCES [11234121212].......... Landing page for this paper ”Cluster-Level Tuning...”. http://research.colfaxinternational.com/post/2014/05/12/Shallow-Water.aspx. [2] Primer on Computing with Intel Xeon Phi Coprocessors. Slides from a presentation, with links to additional resources. http://research.colfaxinternational.com/post/2014/03/06/Geant4-Tutorial.aspx. [3] Robert Sadourny. The Dynamics of Finite-Difference Models of Shallow-Water Equations. Journal of Atmospheric Sci- ences, 32:680–689, 1974. [4] Colfax International. Parallel Programming and Optimization with Intel Xeon Phi Coprocessors. ISBN: 978-0-9885234- 1-8. Colfax International, 2013. http://www.colfax-intl.com/xeonphi/book.html. [5] Andrey Vladimirov. Terabyte RAM Servers: Memory Bandwidth Benchmark and How to Boost RAM Bandwidth by 20% with a Single Command . http://research.colfaxinternational.com/post/2012/01/04/Terabyte-RAM-Servers-Memory-Bandwidth-Benchmark.aspx. [6] Intel MPI Library Reference Manual for Linux* OS. http://software.intel.com/sites/products/documentation/hpc/ics/impi/41/lin/Reference Manual/index.htm. [7] Andrey Vladimirov. Multithreaded Transposition of Square Matrices with Common Code for Intel Xeon Processors and Intel Xeon Phi Coprocessors. http://research.colfaxinternational.com/post/2013/08/12/Trans-7110.aspx. [8] Vadim Karpusenko and Andrey Vladimirov. Conﬁguration and Benchmarks of Peer-to-Peer Communication over Gigabit Ethernet and InﬁniBand in a Cluster with Intel Xeon Phi Coprocessors. http://research.colfaxinternational.com/post/2014/03/11/InﬁniBand-for-MIC.aspx. (cid:13) c Colfax International, 2014 — http://research.colfaxinternational.com/ 12 ##LTLine## ##LTLine##

text: REFERENCES

text: REFERENCES

label: reference

text: Landing

text: page

text: for

text: this

text: paper

text: ”Cluster-Level

text: Tuning...”.

text: Primer

text: on

text: Computing

text: with

text: Intel

text: Xeon

text: Phi

text: Coprocessors.

text: Slides

text: from

text: a

text: presentation,

text: with

text: links

text: to

text: additional

text: resources.

text: Robert

text: Sadourny.

text: The

text: Dynamics

text: of

text: Finite-Difference

text: Models

text: of

text: Shallow-Water

text: Equations.

text: Journal

text: of

text: Atmospheric

text: Sci-

text: ences,

text: Colfax

text: International.

text: Parallel

text: Programming

text: and

text: Optimization

text: with

text: Intel

text: Xeon

text: Phi

text: Coprocessors.

text: ISBN:

text: Colfax

text: International,

text: Andrey

text: Vladimirov.

text: Terabyte

text: RAM

text: Servers:

text: Memory

text: Bandwidth

text: Benchmark

text: and

text: How

text: to

text: Boost

text: RAM

text: Bandwidth

text: by

text: with

text: a

text: Single

text: Command

text: Intel

text: MPI

text: Library

text: Reference

text: Manual

text: for

text: Linux*

text: OS.

text: Manual/index.htm.

text: Andrey

text: Vladimirov.

text: Multithreaded

text: Transposition

text: of

text: Square

text: Matrices

text: with

text: Common

text: Code

text: for

text: Intel

text: Xeon

text: Processors

text: and

text: Intel

text: Xeon

text: Phi

text: Coprocessors.

text: Vadim

text: Karpusenko

text: and

text: Andrey

text: Vladimirov.

text: Conﬁguration

text: and

text: Benchmarks

text: of

text: Peer-to-Peer

text: Communication

text: over

text: Gigabit

text: Ethernet

text: and

text: InﬁniBand

text: in

text: a

text: Cluster

text: with

text: Intel

text: Xeon

text: Phi

text: Coprocessors.

text: c

text: Colfax

text: International,

text: ##LTLine##

text: ##LTLine##