text: POWER-SUM DENOMINATORS 9 5. Further properties Here we give an intermediate result that shows another formula for the values of q , which we need later on. n ≥ Theorem 3. Let q and M be deﬁned as in Theorem 1. For any ﬁxed n 0 we n n (cid:89) have q = pεp, (17) n ≤ (cid:40) p Mn where ε , which depends on n, is deﬁned for p = 2 by p − ≥ 0, if n = 2r 1 for some r 0, def ε = 2 1, otherwise, (cid:40) (cid:0) (cid:1) (cid:106) (cid:107) ≤ and for an odd prime p M by n (cid:45) | − 0, if p (n + 2) and p n+−1 for all j = 2, 3, . . . , −n 1, ε d=ef j(p 1) p 1 p 1, otherwise. For a reﬁnement step in the proof of Theorem 3, we need the following congru- ence. See Hermite [18] for the case m odd, and Bachmann [2, Eq. (116), p. 46] for the general case. For a recent elementary proof, see [23]. (cid:18) (cid:19) ≥ (cid:88) Lemma 7 (Hermite, Bachmann). If p is a prime and m 1, then m ≡ − 0 (mod p). (18) j(p 1) ≤ ≤ − 1 j m−1 p 1 Proof of Theorem 3. By Theorem 1 we know that q is the product of certain ≤ n primes p M . Thus, to show (17), we have to determine the exponents ε . n ≥ p If p = 2, then the value of ε is given by Theorem 1 part (iii) for n 1. Since 2 M = 1, the case n = 0 does not occur here. So we are done in case p = 2. 0 ≤ ≤ ≤ We now consider the case of an odd prime p M . Since M 2 if n 3, we ≥ n n may ﬁx n 4. Set m = n + 1 and recall the deﬁnition of δ in (16). As in the m proof of Theorem 1 and in relation (12), we have that (cid:45) ⇐⇒ ∈ P p q p / n m and by Lemma 3 we obtain that (cid:45) ⇐⇒ ∈ P − p q (cid:0) (cid:1) p / (k = 2, 4, . . . , m δ ). n m,k m (cid:45) ∈ P Note that p q is equivalent to(cid:22)ε = 0.(cid:23)Rec(cid:22)all from L(cid:23)emma 3 that p if and − n| (cid:45) p − − m,k only if (p 1) k and p m . As m δ and p 1 are even, we have k m − − m 1 m δ L d=ef − = − m . p 1 p 1 (cid:55)→ − (cid:18) (cid:19)− | Substituting k j(p 1) for those k with (p 1) k, we ﬁnally conclude that ⇐⇒ | m ε = 0 p − (j = 1, 2, . . . , L) . (19) p j(p 1) ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: POWER-SUM

text: DENOMINATORS

text: Further

text: properties

text: Here

text: we

text: give

text: an

text: intermediate

text: result

text: that

text: shows

text: another

text: formula

text: for

text: the

text: values

text: of

text: q

text: which

text: we

text: need

text: later

text: on.

text: n

text: Theorem

text: Let

text: q

text: and

text: M

text: be

text: deﬁned

text: as

text: in

text: Theorem

text: For

text: any

text: ﬁxed

text: n

text: we

text: n

text: n

label: equation

text: have

text: q

text: pεp,

text: n

text: p

text: Mn

text: where

text: ε

text: which

text: depends

text: on

text: n,

text: is

text: deﬁned

text: for

text: p

text: by

text: p

text: if

text: n

text: for

text: some

text: r

text: def

text: ε

text: otherwise,

text: and

text: for

text: an

text: odd

text: prime

text: p

text: M

text: by

text: n

text: if

text: p

text: (n

text: and

text: p

text: for

text: all

text: j

text: −n

text: ε

text: d=ef

text: j(p

text: p

text: p

text: otherwise.

text: For

text: a

text: reﬁnement

text: step

text: in

text: the

text: proof

text: of

text: Theorem

text: we

text: need

text: the

text: following

text: congru-

text: ence.

text: See

text: Hermite

text: for

text: the

text: case

text: m

text: odd,

text: and

text: Bachmann

text: Eq.

text: p.

text: for

text: the

text: general

text: case.

text: For

text: a

text: recent

text: elementary

text: proof,

text: see

text: Lemma

text: (Hermite,

text: Bachmann).

text: If

text: p

text: is

text: a

text: prime

text: and

text: m

text: then

text: m

text: (mod

text: p).

text: j(p

text: j

text: p

text: Proof

text: of

text: Theorem

text: By

text: Theorem

text: we

text: know

text: that

text: q

text: is

text: the

text: product

text: of

text: certain

text: n

text: primes

text: p

text: M

text: Thus,

text: to

text: show

text: we

text: have

text: to

text: determine

text: the

text: exponents

text: ε

text: n

text: p

text: If

text: p

text: then

text: the

text: value

text: of

text: ε

text: is

text: given

text: by

text: Theorem

text: part

text: (iii)

text: for

text: n

text: Since

text: M

text: the

text: case

text: n

text: does

text: not

text: occur

text: here.

text: So

text: we

text: are

text: done

text: in

text: case

text: p

text: We

text: now

text: consider

text: the

text: case

text: of

text: an

text: odd

text: prime

text: p

text: M

text: Since

text: M

text: if

text: n

text: we

text: n

text: n

text: may

text: ﬁx

text: n

text: Set

text: m

text: n

text: and

text: recall

text: the

text: deﬁnition

text: of

text: δ

text: in

text: As

text: in

text: the

text: m

text: proof

text: of

text: Theorem

text: and

text: in

text: relation

text: we

text: have

text: that

text: P

text: p

text: q

text: p

text: n

text: m

text: and

text: by

text: Lemma

text: we

text: obtain

text: that

text: P

text: p

text: q

text: p

text: (k

text: m

text: δ

text: n

text: m,k

text: m

text: P

text: Note

text: that

text: p

text: q

text: is

text: equivalent

text: from

text: that

text: p

text: if

text: and

text: n|

text: p

text: m,k

text: only

text: if

text: (p

text: k

text: and

text: p

text: m

text: As

text: m

text: δ

text: and

text: p

text: are

text: even,

text: we

text: have

text: k

text: m

text: m

text: m

text: δ

text: L

text: d=ef

text: m

text: p

text: p

text: Substituting

text: k

text: j(p

text: for

text: those

text: k

text: with

text: (p

text: k,

text: we

text: ﬁnally

text: conclude

text: that

text: m

text: ε

text: p

text: (j

text: L)

text: p

text: j(p

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Hypothesis 5.2. Consider the ﬁrst-order system of equations describing stationary solutions to (5.1) − u = D 1v x − (5.6) v = G(u) x W W (cid:62) and let s(x) and u(x) denote the stable and unstable manifolds of (u, v) = 0 associated with ∗ ∗ (cid:62) (5.6). We assume that (ϕ (x), ϕ (x)) is the unique solution, up to spatial translation, contained in the W ∩ W x intersection s(x) u(x). ∗ W ∩ W ≥ Remark 5.3. Since we assume that ϕ (x) is a pulse solution to (5.1), dim( s(x) u(x)) 1. The assumption that this dimension is exactly equal to one is generic for the following reason. We append the x W W direction so that the manifolds s(x) and u(x) are n + 1 dimensional manifolds in a 2n + 1 dimensional ambient space. Then it is a well known fact of diﬀerential topology that the dimension of a transverse intersection of two manifolds X and Z in the ambient space Y is given by ∩ − dim(X Z) = dim(X) + dim(Z) dim(Y ) (c.f. [19, pg. 30]) which in our case gives W ∩ W − dim( s(x) u(x)) = (n + 1) + (n + 1) (2n + 1) = 1. ∗ The even-symmetry of ϕ (x) is a straightforward consequence of Hypothesis 5.2 and the spatial- reversibility of (5.1). ∈ R ∗ Proposition 5.4. Assume Hypothesis 5.2. Then there exists some x so that ϕ (x) is even-symmetric 0 about x = x . 0 − ∗ Proof. Equation (5.1) is reversible; i.e. if u(x) is a solution, so is u( x). Since ϕ (x) is a solution to (5.1), so ∗ − W ∩W is ϕ ( x) and, by the deﬁnition of a pulse, both solutions are contained in the intersection s(x) u(x). ∗ ∗ − By Hypothesis 5.2, ϕ (x) and ϕ ( x) are the same up to spatial translations. This can only be true if ∗ ∗ ∗ − ∈ R ϕ (x) is even-symmetric about some point x . More precisely, if ϕ (x) = ϕ ( x + δ) for all x and ∗ ∗ − 0 ∈ R some ﬁxed δ, then ϕ (x + x) = ϕ (x x) for all x , where x = δ/2. 0 0 0 We are now ready to prove our main result. ∗ Theorem 5.5. Assume Hypothesis 5.2. Then ϕ (x) is unstable. (cid:12) ∗ (cid:12) Proof. By Proposition 5.4, ϕ (x) is even-symmetric (cid:12)about some x0; thus ∗ ∂ ϕ (x) = 0 x x=x0 and so (5.5) is satisﬁed for s = x . 0 Remark 5.6. It was assumed as the start of the section that the essential spectrum of the linearized operator was stable; this allowed us to verify Hypothesis 1.1 and thus apply Theorem 4.1 to conclude −∇ instability of the pulse. On the other hand, if 2F (0) has any negative eigenvalues, the pulse is unstable due to the essential spectrum. ∗ Remark 5.7. The proof of Theorem 5.5 only requires the even-symmetry of ϕ (which may be valid even if Hypothesis 5.2 does not hold, or cannot be veriﬁed). We choose to state the result in terms of Hypothesis 5.2, rather than its consequence, Proposition 5.4, as it is less apparent that the latter condition is generically satisﬁed. 18 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Hypothesis

label: paragraph

text: Consider

text: the

text: ﬁrst-order

text: system

text: of

text: equations

text: describing

text: stationary

text: solutions

text: to

text: u

text: D

text: x

text: v

text: G(u)

text: x

text: W

text: W

text: and

text: let

text: s(x)

text: and

text: u(x)

text: denote

text: the

text: stable

text: and

text: unstable

text: manifolds

text: of

text: (u,

text: v)

text: associated

text: with

text: We

text: assume

text: that

text: (ϕ

text: (x),

text: ϕ

text: (x))

text: is

text: the

text: unique

text: solution,

text: up

text: to

text: spatial

text: translation,

text: contained

text: in

text: the

text: W

text: W

text: x

text: intersection

text: s(x)

text: u(x).

text: W

text: W

text: Remark

text: Since

text: we

text: assume

text: that

text: ϕ

text: (x)

text: is

text: a

text: pulse

text: solution

text: to

text: dim(

text: s(x)

text: u(x))

text: The

text: assumption

text: that

text: this

text: dimension

text: is

text: exactly

text: equal

text: to

text: one

text: is

text: generic

text: for

text: the

text: following

text: reason.

text: We

text: append

text: the

text: x

text: W

text: W

text: direction

text: so

text: that

text: the

text: manifolds

text: s(x)

text: and

text: u(x)

text: are

text: n

text: dimensional

text: manifolds

text: in

text: a

text: dimensional

text: ambient

text: space.

text: Then

text: it

text: is

text: a

text: well

text: known

text: fact

text: of

text: diﬀerential

text: topology

text: that

text: the

text: dimension

text: of

text: a

text: transverse

text: intersection

text: of

text: two

text: manifolds

text: X

text: and

text: Z

text: in

text: the

text: ambient

text: space

text: Y

text: is

text: given

text: by

text: dim(X

text: Z)

text: dim(X)

text: dim(Z)

text: dim(Y

text: (c.f.

text: pg.

text: which

text: in

text: our

text: case

text: gives

text: W

text: W

text: dim(

text: s(x)

text: u(x))

text: (n

text: (n

text: The

text: even-symmetry

text: of

text: ϕ

text: (x)

text: is

text: a

text: straightforward

text: consequence

text: of

text: Hypothesis

text: and

text: the

text: spatial-

text: reversibility

text: of

text: R

text: Proposition

text: Assume

text: Hypothesis

text: Then

text: there

text: exists

text: some

text: x

text: so

text: that

text: ϕ

text: (x)

text: is

text: even-symmetric

text: about

text: x

text: x

text: Proof.

text: Equation

text: is

text: reversible;

text: i.e.

text: if

text: u(x)

text: is

text: a

text: solution,

text: so

text: is

text: u(

text: x).

text: Since

text: ϕ

text: (x)

text: is

text: a

text: solution

text: to

text: so

text: W

text: ∩W

text: is

text: ϕ

text: x)

text: and,

text: by

text: the

text: deﬁnition

text: of

text: a

text: pulse,

text: both

text: solutions

text: are

text: contained

text: in

text: the

text: intersection

text: s(x)

text: u(x).

text: By

text: Hypothesis

text: ϕ

text: (x)

text: and

text: ϕ

text: x)

text: are

text: the

text: same

text: up

text: to

text: spatial

text: translations.

text: This

text: can

text: only

text: be

text: true

text: if

text: R

text: ϕ

text: (x)

text: is

text: even-symmetric

text: about

text: some

text: point

text: x

text: More

text: precisely,

text: if

text: ϕ

text: (x)

text: ϕ

text: x

text: δ)

text: for

text: all

text: x

text: and

text: R

text: some

text: ﬁxed

text: δ,

text: then

text: ϕ

text: (x

text: x)

text: ϕ

text: (x

text: x)

text: for

text: all

text: x

text: where

text: x

text: We

text: are

text: now

text: ready

text: to

text: prove

text: our

text: main

text: result.

text: Theorem

text: Assume

text: Hypothesis

text: Then

text: ϕ

text: (x)

text: is

text: unstable.

text: Proof.

text: By

text: Proposition

text: ϕ

text: (x)

text: is

text: even-symmetric

text: some

text: thus

text: ϕ

text: (x)

text: x

text: and

text: so

text: is

text: satisﬁed

text: for

text: s

text: x

text: Remark

text: It

text: was

text: assumed

text: as

text: the

text: start

text: of

text: the

text: section

text: that

text: the

text: essential

text: spectrum

text: of

text: the

text: linearized

text: operator

text: was

text: stable;

text: this

text: allowed

text: us

text: to

text: verify

text: Hypothesis

text: and

text: thus

text: apply

text: Theorem

text: to

text: conclude

text: instability

text: of

text: the

text: pulse.

text: On

text: the

text: other

text: hand,

text: if

text: has

text: any

text: negative

text: eigenvalues,

text: the

text: pulse

text: is

text: unstable

text: due

text: to

text: the

text: essential

text: spectrum.

text: Remark

text: The

text: proof

text: of

text: Theorem

text: only

text: requires

text: the

text: even-symmetry

text: of

text: ϕ

text: (which

text: may

text: be

text: valid

text: even

text: if

text: Hypothesis

text: does

text: not

text: hold,

text: or

text: cannot

text: be

text: veriﬁed).

text: We

text: choose

text: to

text: state

text: the

text: result

text: in

text: terms

text: of

text: Hypothesis

text: rather

text: than

text: its

text: consequence,

text: Proposition

text: as

text: it

text: is

text: less

text: apparent

text: that

text: the

text: latter

text: condition

text: is

text: generically

text: satisﬁed.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:12) (cid:12) (cid:12) d ∈ ∩ QM(v, w) = ω(v, φ(t)w) for v, w Φ(t ) V, (2.1) 0 dt t=t0 is called the crossing form. We denote by n (QM) and n−(QM) the number of positive and negative + − eigenvalues of QM, counted with multiplicity, so that sign QM = n (QM) n−(QM) is its signature. + Since QM is a symmetric matrix its eigenvalues are purely real; thus the notion of positive and negative (cid:12) eigenvalues is well de(cid:12)ﬁned. A crossing is called regular if the crossing form is nondegenerate. T Deﬁnition 2.1. If s is the only regular crossing of the path Φ with (V ) in a segment [a , b ], then the 0 0 0 Maslov index Mas(Φ , V ) is deﬁned as [a0,b0] (cid:40) (cid:12) (cid:12) − n−(QM) if s = a − 0 ∈ 0 Mas(Φ , V ) := n (QM) n−(QM) if s (a , b ) (2.2) [a0,b0] n+(QM) if s0 = b 0 0 + 0 0 The Maslov index of any regular, piecewise C1 path can be determined by computing the Maslov index on each segment and summing. The important features of the Maslov index for this work are summarized below. Theorem 2.2. [27] (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 1. (Additivity) For a < c < b, Mas(Φ , V ) = Mas(Φ , V ) + Mas(Φ , V ). [a,b] [a,c] [c,b] → 2. (Homotopy invariance) Two paths Φ , Φ : [a, b] Λ(n), with Φ (a) = Φ (a) and Φ (b) = Φ (b), are 0 1 0 1 0 1 homotopic with ﬁxed endpoints if and only if they have the same Maslov index. (cid:54) Remark 2.3. When deﬁning the Maslov index for a curve with Φ(a) = Φ(b), one needs to be careful ∈ about crossings at the endpoints, to ensure that the additivity property holds. That is, if c (a, b) is | | a crossing, should it contribute towards Mas(Φ , V ) or Mas(Φ , V )? Our (arbitrary) convention is [a,c] [c,b] that the positive part of the crossing is assigned to [a, c], whereas the negative part is assigned to [c, b]. We could just as well have chosen the opposite sign convention, or could have assigned 1sign QM to each 2 segment. For a closed curve these are all equivalent, so our choice ultimately does not matter, but it will aﬀect some intermediate results; see in particular Theorem 3.1 and Remark 3.2. In our application to the spectral theory of diﬀerential operators, a special role is played by paths for which all crossings are sign deﬁnite (and have the same sign). Remark 2.4. A crossing at t is said to be positive if the crossing form is positive deﬁnite; in this case 0 ∩ → the local contribution to the Maslov index is just n (QM) = dim(Φ(t ) V ). A curve Φ: [a, b] Λ(n) is (cid:88)+ 0 said to be positive if all of its crossings are positive, in which case ∩ Mas(Φ, V ) = dim(Φ(t) V ). ≤ a<t b (cid:88) Similarly, the Maslov index of a negative curve is given by ∩ Mas(Φ, V ) = dim(Φ(t) V ). ≤ a t<b Both sums are necessarily ﬁnite because regular crossings are isolated. 7 ##LTLine## ##LTLine##

text: d

text: QM(v,

text: w)

text: ω(v,

text: φ(t)w)

text: for

text: v,

text: w

text: Φ(t

text: V,

text: dt

text: is

text: called

text: the

text: crossing

text: form.

text: We

text: denote

text: by

text: n

text: (QM)

text: and

text: n−(QM)

text: the

text: number

text: of

text: positive

text: and

text: negative

text: eigenvalues

text: of

text: QM,

text: counted

text: with

text: multiplicity,

text: so

text: that

text: sign

text: QM

text: n

text: (QM)

text: n−(QM)

text: is

text: its

text: signature.

text: Since

text: QM

text: is

text: a

text: symmetric

text: matrix

text: its

text: eigenvalues

text: are

text: purely

text: real;

text: thus

text: the

text: notion

text: of

text: positive

text: and

text: negative

text: eigenvalues

text: is

text: well

text: A

text: crossing

text: is

text: called

text: regular

text: if

text: the

text: crossing

text: form

text: is

text: nondegenerate.

text: T

text: Deﬁnition

text: If

text: s

text: is

text: the

text: only

text: regular

text: crossing

text: of

text: the

text: path

text: Φ

text: with

text: (V

text: in

text: a

text: segment

text: [a

text: b

text: then

text: the

text: Maslov

text: index

text: Mas(Φ

text: V

text: is

text: deﬁned

text: as

text: n−(QM)

text: if

text: s

text: a

text: Mas(Φ

text: V

text: n

text: (QM)

text: n−(QM)

text: if

text: s

text: (a

text: b

text: n+(QM)

text: if

text: b

text: The

text: Maslov

text: index

text: of

text: any

text: regular,

text: piecewise

text: path

text: can

text: be

text: determined

text: by

text: computing

text: the

text: Maslov

text: index

text: on

text: each

text: segment

text: and

text: summing.

text: The

text: important

text: features

text: of

text: the

text: Maslov

text: index

text: for

text: this

text: work

text: are

text: summarized

text: below.

text: Theorem

text: (Additivity)

text: For

text: a

text: c

text: b,

text: Mas(Φ

text: V

text: Mas(Φ

text: V

text: Mas(Φ

text: V

text: [a,b]

text: [a,c]

text: [c,b]

text: (Homotopy

text: invariance)

text: Two

text: paths

text: Φ

text: Φ

text: [a,

text: b]

text: Λ(n),

text: with

text: Φ

text: (a)

text: Φ

text: (a)

text: and

text: Φ

text: (b)

text: Φ

text: (b),

text: are

text: homotopic

text: with

text: ﬁxed

text: endpoints

text: if

text: and

text: only

text: if

text: they

text: have

text: the

text: same

text: Maslov

text: index.

text: Remark

text: When

text: deﬁning

text: the

text: Maslov

text: index

text: for

text: a

text: curve

text: with

text: Φ(a)

text: Φ(b),

text: one

text: needs

text: to

text: be

text: careful

text: about

text: crossings

text: at

text: the

text: endpoints,

text: to

text: ensure

text: that

text: the

text: additivity

text: property

text: holds.

text: That

text: is,

text: if

text: c

text: (a,

text: b)

text: is

text: a

text: crossing,

text: should

text: it

text: contribute

text: towards

text: Mas(Φ

text: V

text: or

text: Mas(Φ

text: V

text: Our

text: (arbitrary)

text: convention

text: is

text: [a,c]

text: [c,b]

text: that

text: the

text: positive

text: part

text: of

text: the

text: crossing

text: is

text: assigned

text: to

text: [a,

text: c],

text: whereas

text: the

text: negative

text: part

text: is

text: assigned

text: to

text: [c,

text: b].

text: We

text: could

text: just

text: as

text: well

text: have

text: chosen

text: the

text: opposite

text: sign

text: convention,

text: or

text: could

text: have

text: assigned

text: QM

text: to

text: each

text: segment.

text: For

text: a

text: closed

text: curve

text: these

text: are

text: all

text: equivalent,

text: so

text: our

text: choice

text: ultimately

text: does

text: not

text: matter,

text: but

text: it

text: will

text: aﬀect

text: some

text: intermediate

text: results;

text: see

text: in

text: particular

text: Theorem

text: and

text: Remark

text: In

text: our

text: application

text: to

text: the

text: spectral

text: theory

text: of

text: diﬀerential

text: operators,

text: a

text: special

text: role

text: is

text: played

text: by

text: paths

text: for

text: which

text: all

text: crossings

text: are

text: sign

text: deﬁnite

text: (and

text: have

text: the

text: same

text: sign).

text: Remark

text: A

text: crossing

text: at

text: t

text: is

text: said

text: to

text: be

text: positive

text: if

text: the

text: crossing

text: form

text: is

text: positive

text: deﬁnite;

text: in

text: this

text: case

text: the

text: local

text: contribution

text: to

text: the

text: Maslov

text: index

text: is

text: just

text: n

text: (QM)

text: dim(Φ(t

text: V

text: A

text: curve

text: Φ:

text: [a,

text: b]

text: Λ(n)

text: is

text: said

text: to

text: be

text: positive

text: if

text: all

text: of

text: its

text: crossings

text: are

text: positive,

text: in

text: which

text: case

text: Mas(Φ,

text: V

text: dim(Φ(t)

text: V

text: a<t

text: b

text: Similarly,

text: the

text: Maslov

text: index

text: of

text: a

text: negative

text: curve

text: is

text: given

text: by

text: Mas(Φ,

text: V

text: dim(Φ(t)

text: V

text: a

text: t<b

text: Both

text: sums

text: are

text: necessarily

text: ﬁnite

text: because

text: regular

text: crossings

text: are

text: isolated.

text: ##LTLine##

text: ##LTLine##


text: 5 FIG. 4. Site-projected partial density of states of solvent molecules (red) and the 1¯100 terminated Li O surface (blue) for 2 2 solvents (a) ACN (b) DME (c) THF and (d) DMSO (clockwise) with their converged DFT optimized geometries and indicated locations of frontier energy levels. ferent ways for diﬀerent solvents13, indicating the pres- oxidation. ence of undesirable parasitic processes14 in varying de- grees. Further, one can assert that the larger the devia- To test this hypothesis, we plot our chosen marker of tion, the less eﬃcient and rechargeable the cell is. − cell performance, e /OER, adopted from the work of Mc- Closkey et al. 13 against the calculated values of the dif- In an earlier work, we demonstrated that, to a ﬁrst ap- ference between the renormalized HOMO level and VBM proximation, the highest occupied molecular orbital en- energy levels, δ, for the four solvents in Fig. 5. As can be ergy level of the solvent in gas phase is a good descriptor clearly seen in Fig 5a, δ, shows a clear correlation with for oxidative stability, which is accounted for by choosing − the rechargeability metric, signaling a greater tendency e /OER as the metric of rechargeability17. However, the for electrochemical decomposition of solvents for which highest occupied molecular orbital (HOMO) and lowest this diﬀerence is smaller. The results obtained in our unoccupied molecular orbital (LUMO) levels of the sol- work point to an exponential variation of the recharge- vent molecule in its gas or even bulk liquid phase are not ability metric with respect to δ, in comparison to a more representative of its true state at the interface where it linear variation with respect to the gas phase HOMO interacts with the electrode components. To establish a levels17. descriptor for oxidative stability based on renormaliza- tion eﬀects, we refer back to the thermodynamic model of electron energy alignments developed earlier in Fig. 1, The analysis performed here leads to a reﬁned picture and that the VBM of the surface also undergoes renor- of solvent-surface interaction and subsequent renormal- malization. We surmise that the probability of an elec- ization of frontier energy levels on the 1¯100 facet of Li O 2 2 tron transfer from the solvent to hole states at the VBM during charging. A good descriptor for solvent stability on the surface will depend not on the absolute values is one which can be computed easily and requires min- of renormalized HOMO level values of solvent molecules, imal user intervention to calculate. In that sense, the but the diﬀerence in the energy levels between the renor- gas phase HOMO level for aprotic solvents in Li-O bat- 2 malized HOMO level and VBM. This diﬀerence has been teries can already be considered a very good descriptor, deﬁned earlier as δ = (VBM - ReHOMO). Further, during because it can be easily computed for a large number the process of charging, the presence of a charging over- of solvents which satisfy other requirements from an ap- potential U at the cathode surface will lower the Fermi plication point of view. When choosing a descriptor, it chg level of the surface by eU , thereby reducing the diﬀer- is also important to rule out any signiﬁcant inconsisten- chg ence between ReHOMO and VBM by the same amount cies, which in the present case may arise from the ﬁrst and further increasing the solvent’s propensity toward principles method being used for calculation. ##LTFigure##

text: FIG.

text: Site-projected

label: caption

text: partial

text: density

text: of

text: states

text: of

text: solvent

text: molecules

text: (red)

text: and

text: the

text: terminated

text: Li

text: O

text: surface

text: (blue)

text: for

text: solvents

text: (a)

text: ACN

text: (b)

text: DME

text: (c)

text: THF

text: and

text: (d)

text: DMSO

text: (clockwise)

text: with

text: their

text: converged

text: DFT

text: optimized

text: geometries

text: and

text: indicated

text: locations

text: of

text: frontier

text: energy

text: levels.

text: ferent

text: ways

text: for

text: diﬀerent

text: indicating

text: the

text: pres-

text: oxidation.

text: ence

text: of

text: undesirable

text: parasitic

text: in

text: varying

text: de-

text: grees.

text: Further,

text: one

text: can

text: assert

text: that

text: the

text: larger

text: the

text: devia-

text: To

text: test

text: this

text: hypothesis,

text: we

text: plot

text: our

text: chosen

text: marker

text: of

text: tion,

text: the

text: less

text: eﬃcient

text: and

text: rechargeable

text: the

text: cell

text: is.

text: cell

text: performance,

text: e

text: /OER,

text: adopted

text: from

text: the

text: work

text: of

text: Mc-

text: Closkey

text: et

text: al.

text: against

text: the

text: calculated

text: values

text: of

text: the

text: dif-

text: In

text: an

text: earlier

text: work,

text: we

text: demonstrated

text: that,

text: to

text: a

text: ﬁrst

text: ap-

text: ference

text: between

text: the

text: renormalized

text: HOMO

text: level

text: and

text: VBM

text: proximation,

text: the

text: highest

text: occupied

text: molecular

text: orbital

text: en-

text: energy

text: levels,

text: δ,

text: for

text: the

text: four

text: solvents

text: in

text: Fig.

text: As

text: can

text: be

text: ergy

text: level

text: of

text: the

text: solvent

text: in

text: gas

text: phase

text: is

text: a

text: good

text: descriptor

text: clearly

text: seen

text: in

text: Fig

text: δ,

text: shows

text: a

text: clear

text: correlation

text: with

text: for

text: oxidative

text: stability,

text: which

text: is

text: accounted

text: for

text: by

text: choosing

text: the

text: rechargeability

text: metric,

text: signaling

text: a

text: greater

text: tendency

text: e

text: /OER

text: as

text: the

text: metric

text: of

text: However,

text: the

text: for

text: electrochemical

text: decomposition

text: of

text: solvents

text: for

text: which

text: highest

text: occupied

text: molecular

text: orbital

text: (HOMO)

text: and

text: lowest

text: this

text: diﬀerence

text: is

text: smaller.

text: The

text: results

text: obtained

text: in

text: our

text: unoccupied

text: molecular

text: orbital

text: (LUMO)

text: levels

text: of

text: the

text: sol-

text: work

text: point

text: to

text: an

text: exponential

text: variation

text: of

text: the

text: recharge-

text: vent

text: molecule

text: in

text: its

text: gas

text: or

text: even

text: bulk

text: liquid

text: phase

text: are

text: not

text: ability

text: metric

text: with

text: respect

text: to

text: δ,

text: in

text: comparison

text: to

text: a

text: more

text: representative

text: of

text: its

text: true

text: state

text: at

text: the

text: interface

text: where

text: it

text: linear

text: variation

text: with

text: respect

text: to

text: the

text: gas

text: phase

text: HOMO

text: interacts

text: with

text: the

text: electrode

text: components.

text: To

text: establish

text: a

text: descriptor

text: for

text: oxidative

text: stability

text: based

text: on

text: renormaliza-

text: tion

text: eﬀects,

text: we

text: refer

text: back

text: to

text: the

text: thermodynamic

text: model

text: of

text: electron

text: energy

text: alignments

text: developed

text: earlier

text: in

text: Fig.

text: The

text: analysis

text: performed

text: here

text: leads

text: to

text: a

text: reﬁned

text: picture

text: and

text: that

text: the

text: VBM

text: of

text: the

text: surface

text: also

text: undergoes

text: renor-

text: of

text: solvent-surface

text: interaction

text: and

text: subsequent

text: renormal-

text: malization.

text: We

text: surmise

text: that

text: the

text: probability

text: of

text: an

text: elec-

text: ization

text: of

text: frontier

text: energy

text: levels

text: on

text: the

text: facet

text: of

text: Li

text: O

text: tron

text: transfer

text: from

text: the

text: solvent

text: to

text: hole

text: states

text: at

text: the

text: VBM

text: during

text: charging.

text: A

text: good

text: descriptor

text: for

text: solvent

text: stability

text: on

text: the

text: surface

text: will

text: depend

text: not

text: on

text: the

text: absolute

text: values

text: is

text: one

text: which

text: can

text: be

text: computed

text: easily

text: and

text: requires

text: min-

text: of

text: renormalized

text: HOMO

text: level

text: values

text: of

text: solvent

text: molecules,

text: imal

text: user

text: intervention

text: to

text: calculate.

text: In

text: that

text: sense,

text: the

text: but

text: the

text: diﬀerence

text: in

text: the

text: energy

text: levels

text: between

text: the

text: renor-

text: gas

text: phase

text: HOMO

text: level

text: for

text: aprotic

text: solvents

text: in

text: Li-O

text: bat-

text: malized

text: HOMO

text: level

text: and

text: VBM.

text: This

text: diﬀerence

text: has

text: been

text: teries

text: can

text: already

text: be

text: considered

text: a

text: very

text: good

text: descriptor,

text: deﬁned

text: earlier

text: as

text: δ

text: (VBM

text: ReHOMO).

text: Further,

text: during

text: because

text: it

text: can

text: be

text: easily

text: computed

text: for

text: a

text: large

text: number

text: the

text: process

text: of

text: charging,

text: the

text: presence

text: of

text: a

text: charging

text: over-

text: of

text: solvents

text: which

text: satisfy

text: other

text: requirements

text: from

text: an

text: ap-

text: potential

text: U

text: at

text: the

text: cathode

text: surface

text: will

text: lower

text: the

text: Fermi

text: plication

text: point

text: of

text: view.

text: When

text: choosing

text: a

text: descriptor,

text: it

text: chg

text: level

text: of

text: the

text: surface

text: by

text: eU

text: thereby

text: reducing

text: the

text: diﬀer-

text: is

text: also

text: important

text: to

text: rule

text: out

text: any

text: signiﬁcant

text: inconsisten-

text: chg

text: ence

text: between

text: ReHOMO

text: and

text: VBM

text: by

text: the

text: same

text: amount

text: cies,

text: which

text: in

text: the

text: present

text: case

text: may

text: arise

text: from

text: the

text: ﬁrst

text: and

text: further

text: increasing

text: the

text: solvent’s

text: propensity

text: toward

text: principles

text: method

text: being

text: used

text: for

text: calculation.

text: ##LTFigure##


text: 11 APPENDIX C EXPONENT FOR REPETITION In Section , we perform an asymptotic analysis of the exponent of the probability of failure to meet a deadline in the limit of the deadline diverging to inﬁnity. Recall from Section that an upper bound for the leading coefﬁcient of the failure exponent for Coded Convolution is given by − − (P K + 1)µα (cid:15)(s) = (28) ((2Cs) log (2s))α Here K is the number of processors one needs to wait for in the worst case for Coded Convolution. We also showed that for uncoded strategy, this upper bound is actually exactly equal to the leading co-efﬁcient of the failure exponent (putting worst case K = P) as the deadline diverges to inﬁnity. Now, we show that even for repetition strategy this upper bound is actually exactly equal to the upper bound using worst case K as the deadline diverges to inﬁnity. Theorem 4: For a (P, r) repetition strategy, the leading coefﬁcient of the exponent in the probability of failure to meet a deadline converges as − log(Pf(t)) − (P K + 1)µα lim s = (cid:15)(s) = →∞ t tα ((2Cs) log (2s))α − where K = P r + 1 is the number of processors to wait for in the worst case for a (P, r) repetition strategy. Proof: Recall that in a (P, r) repetition strategy, we basically divide the operation of convolution into P/r equal tasks, and each task has r repetitions, so as to use all the given P processors. Let us introduce the notation T to represent the i,j random variable corresponding to the different computation times on different processors. Here i is the task index which varies from 1 to P/r for the P/r different parts of the convolution task. Also let j be the replica index for each task which varies from 1 to r for r replicas of each task. Thus T represents the computation time of the j-th copy of the i-th task. i,j (cid:48) Consider any task with index i. Let T denote the completion time of the particular task. The task ﬁnishes when any one i (cid:48) of its r replicas ﬁnish. Thus, the distribution of T is given by i (cid:48) { } T = min T i i,j j (cid:113) (cid:48) Now let us try to ﬁnd the distribution of T . Recall that each T is a random variable denoting the time required to perform i i,j a convolution of length s at a processor where s = N1N2r. Its computational complexity is given by 2Cs log(2s). From P shifted Weibull model assumptions, the c.d.f. of each T is given by i,j (cid:0) (cid:1) Pr(Ti,j ≤ t) = Fs(t) = 1 − e− µ(2Cslotg(2s)−1) α ∀ t ≥ 2Cs log (2s) (29) { } Since the T s are i.i.d., we thus have i,j (cid:0) (cid:1) Pr(Ti(cid:48) < t) = 1 − Πrj=1 Pr(Ti,j ≥ t) = 1 − e−r µ(2Cslotg(2s)−1) α ∀ t ≥ 2Cs log (2s) (30) (cid:48) Let us denote F˜ (t) = Pr(T < t). Now, for repetition the probability of failure to meet a deadline occurs when at least s i one of the P/r tasks have not completed. Thus the failure probability can be exactly computed as: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: APPENDIX

label: section

text: C

text: EXPONENT

text: FOR

text: REPETITION

text: In

text: Section

text: we

text: perform

text: an

text: asymptotic

text: analysis

text: of

text: the

text: exponent

text: of

text: the

text: probability

text: of

text: failure

text: to

text: meet

text: a

text: deadline

text: in

text: the

text: limit

text: of

text: the

text: deadline

text: diverging

text: to

text: inﬁnity.

text: Recall

text: from

text: Section

text: that

text: an

text: upper

text: bound

text: for

text: the

text: leading

text: coefﬁcient

text: of

text: the

text: failure

text: exponent

text: for

text: Coded

text: Convolution

text: is

text: given

text: by

text: (P

text: K

text: 1)µα

text: log

text: Here

text: K

text: is

text: the

text: number

text: of

text: processors

text: one

text: needs

text: to

text: wait

text: for

text: in

text: the

text: worst

text: case

text: for

text: Coded

text: Convolution.

text: We

text: also

text: showed

text: that

text: for

text: uncoded

text: strategy,

text: this

text: upper

text: bound

text: is

text: actually

text: exactly

text: equal

text: to

text: the

text: leading

text: co-efﬁcient

text: of

text: the

text: failure

text: exponent

text: (putting

text: worst

text: case

text: K

text: P)

text: as

text: the

text: deadline

text: diverges

text: to

text: inﬁnity.

text: Now,

text: we

text: show

text: that

text: even

text: for

text: repetition

text: strategy

text: this

text: upper

text: bound

text: is

text: actually

text: exactly

text: equal

text: to

text: the

text: upper

text: bound

text: using

text: worst

text: case

text: K

text: as

text: the

text: deadline

text: diverges

text: to

text: inﬁnity.

text: Theorem

text: For

text: a

text: (P,

text: r)

text: repetition

text: strategy,

text: the

text: leading

text: coefﬁcient

text: of

text: the

text: exponent

text: in

text: the

text: probability

text: of

text: failure

text: to

text: meet

text: a

text: deadline

text: converges

text: as

text: log(Pf(t))

text: (P

text: K

text: 1)µα

text: lim

text: s

text: t

text: tα

text: log

text: where

text: K

text: P

text: r

text: is

text: the

text: number

text: of

text: processors

text: to

text: wait

text: for

text: in

text: the

text: worst

text: case

text: for

text: a

text: (P,

text: r)

text: repetition

text: strategy.

text: Proof:

text: Recall

text: that

text: in

text: a

text: (P,

text: r)

text: repetition

text: strategy,

text: we

text: basically

text: divide

text: the

text: operation

text: of

text: convolution

text: into

text: P/r

text: equal

text: tasks,

text: and

text: each

text: task

text: has

text: r

text: repetitions,

text: so

text: as

text: to

text: use

text: all

text: the

text: given

text: P

text: processors.

text: Let

text: us

text: introduce

text: the

text: notation

text: T

text: to

text: represent

text: the

text: i,j

text: random

text: variable

text: corresponding

text: to

text: the

text: different

text: computation

text: times

text: on

text: different

text: processors.

text: Here

text: i

text: is

text: the

text: task

text: index

text: which

text: varies

text: from

text: to

text: P/r

text: for

text: the

text: P/r

text: different

text: parts

text: of

text: the

text: convolution

text: task.

text: Also

text: let

text: j

text: be

text: the

text: replica

text: index

text: for

text: each

text: task

text: which

text: varies

text: from

text: to

text: r

text: for

text: r

text: replicas

text: of

text: each

text: task.

text: Thus

text: T

text: represents

text: the

text: computation

text: time

text: of

text: the

text: j-th

text: copy

text: of

text: the

text: i-th

text: task.

text: i,j

text: Consider

text: any

text: task

text: with

text: index

text: i.

text: Let

text: T

text: denote

text: the

text: completion

text: time

text: of

text: the

text: particular

text: task.

text: The

text: task

text: ﬁnishes

text: when

text: any

text: one

text: i

text: of

text: its

text: r

text: replicas

text: ﬁnish.

text: Thus,

text: the

text: distribution

text: of

text: T

text: is

text: given

text: by

text: i

text: T

text: min

text: T

text: i

text: i,j

text: j

text: Now

text: let

text: us

text: try

text: to

text: ﬁnd

text: the

text: distribution

text: of

text: T

text: Recall

text: that

text: each

text: T

text: is

text: a

text: random

text: variable

text: denoting

text: the

text: time

text: required

text: to

text: perform

text: i

text: i,j

text: a

text: convolution

text: of

text: length

text: s

text: at

text: a

text: processor

text: where

text: s

text: Its

text: computational

text: complexity

text: is

text: given

text: by

text: From

text: P

text: shifted

text: Weibull

text: model

text: assumptions,

text: the

text: c.d.f.

text: of

text: each

text: T

text: is

text: given

text: by

text: i,j

text: Pr(Ti,j

text: t)

text: Fs(t)

text: e−

text: α

text: t

text: log

text: Since

text: the

text: T

text: s

text: are

text: i.i.d.,

text: we

text: thus

text: have

text: i,j

text: t)

text: Pr(Ti,j

text: t)

text: e−r

text: α

text: t

text: log

text: Let

text: us

text: denote

text: F˜

text: (t)

text: Pr(T

text: t).

text: Now,

text: for

text: repetition

text: the

text: probability

text: of

text: failure

text: to

text: meet

text: a

text: deadline

text: occurs

text: when

text: at

text: least

text: s

text: i

text: one

text: of

text: the

text: P/r

text: tasks

text: have

text: not

text: completed.

text: Thus

text: the

text: failure

text: probability

text: can

text: be

text: exactly

text: computed

text: as:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 (cid:113) √ Fanigd. f5o.r cCooddeidngwoeuctpheorofsoermss=un√c4oNde,ds:trCichtloyosgereNat1er=th4anNu,nNco2d=ed.NThaendplPots=showlogloNg.(−N(cid:15)o(tse)t)hafot rsU=ncodNed1P(Ns2==√2√2NNP) caonrdreCspoodnedds(sto=th√4eNun)csotdraetdegsitersataesgya, function of P. P P P ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Fanigd.

text: function

text: of

text: P.

text: P

text: P

text: P

text: ##LTFigure##

label: figure

font: default

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Proof. Executing each phase of part III of the algorithm can be done in 8 + 8 = 8( + 1) rounds, i i where is the size of the blocks at the start of the phase. Recall that after each phase, the size i of the blocks grows from to 4 + 3. Note also that 1 log phases are enough to grow the block ‘ ‘ 2 n size so large that the algorithm has to either accept or reject. It follows that 8(1 + 1) + 8(7 + 1) + 8(31 + 1) + · · · + 8(22(12 log n)−1)) ≤ 8n is an upper bound for the running time. On the other hand, Ω( ) rounds are clearly necessary to solve ThueMorse: node at the end n of the path has to receive information from the other end of the path to be able to verify that the instance is actually a yes-instance. This concludes the proof of Theorem 5. 5 Conclusions We identiﬁed a model of distributed computing and a class of graphs, where the question on the existence of constant-space, non-constant-time algorithms is interesting and non-trivial. The answer turned out to be positive. This opens up the way to study constant space further—we can ask, for example, what other time complexities besides Θ(log ) and Θ( ) can possibly be n n found and in which graph classes. Acknowledgements We have discussed this work with numerous people, including at least Juho Hirvonen, Janne H. Korhonen, Antti Kuusisto, Joel Rybicki and Przemysław Uznański. Many thanks to all of them, as well as to others that we may have forgotten. References [1] Yehuda Afek, Noga Alon, Ziv Bar-Joseph, Alejandro Cornejo, Bernhard Haeupler, and Fabian Kuhn. Beeping a maximal independent set. In Proc. 25th International Symposium on Distributed Computing (DISC 2011), volume 6950 of Lecture Notes in Computer Science, pages 32–50. Springer, 2011. doi:10.1007/978-3-642-24100-0_3. [2] Dana Angluin. Local and global properties in networks of processors. In Proc. 12th Annual ACM Symposium on Theory of Computing (STOC 1980), pages 82–93. ACM Press, 1980. doi:10.1145/800141.804655. [3] Sebastian Brandt, Orr Fischer, Juho Hirvonen, Barbara Keller, Tuomo Lempiäinen, Joel Rybicki, Jukka Suomela, and Jara Uitto. A lower bound for the distributed Lovász local lemma. In Proc. 48th Annual ACM Symposium on Theory of Computing (STOC 2016), pages 479–488. ACM Press, 2016. doi:10.1145/2897518.2897570. arXiv:1511.00900. [4] Sebastian Brandt, Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, Patric R. J. Östergård, Christopher Purcell, Joel Rybicki, Jukka Suomela, and Przemysław Uznański. LCL problems on grids. In Proc. 36th Annual ACM Symposium on Principles of Distributed Computing (PODC 2017), 2017. To appear. arXiv:1702.05456. [5] Yi-Jun Chang, Tsvi Kopelowitz, and Seth Pettie. An exponential separation between randomized and deterministic complexity in the LOCAL model. In Proc. 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2016), pages 615–624. IEEE Computer Society Press, 2016. doi:10.1109/FOCS.2016.72. arXiv:1602.08166. [6] Yi-Jun Chang and Seth Pettie. A time hierarchy theorem for the LOCAL model, 2017. arXiv:1704.06297. 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: Executing

text: each

text: phase

text: of

text: part

text: III

text: of

text: the

text: algorithm

text: can

text: be

text: done

text: in

text: rounds,

text: i

text: i

text: where

text: is

text: the

text: size

text: of

text: the

text: blocks

text: at

text: the

text: start

text: of

text: the

text: phase.

text: Recall

text: that

text: after

text: each

text: phase,

text: the

text: size

text: i

text: of

text: the

text: blocks

text: grows

text: from

text: to

text: Note

text: also

text: that

text: log

text: phases

text: are

text: enough

text: to

text: grow

text: the

text: block

text: n

text: size

text: so

text: large

text: that

text: the

text: algorithm

text: has

text: to

text: either

text: accept

text: or

text: reject.

text: It

text: follows

text: that

text: log

text: is

text: an

text: upper

text: bound

text: for

text: the

text: running

text: time.

text: On

text: the

text: other

text: hand,

text: Ω(

text: rounds

text: are

text: clearly

text: necessary

text: to

text: solve

text: ThueMorse:

text: node

text: at

text: the

text: end

text: n

text: of

text: the

text: path

text: has

text: to

text: receive

text: information

text: from

text: the

text: other

text: end

text: of

text: the

text: path

text: to

text: be

text: able

text: to

text: verify

text: that

text: the

text: instance

text: is

text: actually

text: a

text: yes-instance.

text: This

text: concludes

text: the

text: proof

text: of

text: Theorem

text: Conclusions

text: We

text: identiﬁed

text: a

text: model

text: of

text: distributed

text: computing

text: and

text: a

text: class

text: of

text: graphs,

text: where

text: the

text: question

text: on

text: the

text: existence

text: of

text: constant-space,

text: non-constant-time

text: algorithms

text: is

text: interesting

text: and

text: non-trivial.

text: The

text: answer

text: turned

text: out

text: to

text: be

text: positive.

text: This

text: opens

text: up

text: the

text: way

text: to

text: study

text: constant

text: space

text: further—we

text: can

text: ask,

text: for

text: example,

text: what

text: other

text: time

text: complexities

text: besides

text: Θ(log

text: and

text: Θ(

text: can

text: possibly

text: be

text: n

text: n

text: found

text: and

text: in

text: which

text: graph

text: classes.

text: Acknowledgements

text: We

text: have

text: discussed

text: this

text: work

text: with

text: numerous

text: people,

text: including

text: at

text: least

text: Juho

text: Hirvonen,

text: Janne

text: H.

text: Korhonen,

text: Antti

text: Kuusisto,

text: Joel

text: Rybicki

text: and

text: Przemysław

text: Uznański.

text: Many

text: thanks

text: to

text: all

text: of

text: them,

text: as

text: well

text: as

text: to

text: others

text: that

text: we

text: may

text: have

text: forgotten.

text: References

text: Yehuda

text: Afek,

text: Noga

text: Alon,

text: Ziv

text: Bar-Joseph,

text: Alejandro

text: Cornejo,

text: Bernhard

text: Haeupler,

text: and

text: Fabian

text: Kuhn.

text: Beeping

text: a

text: maximal

text: independent

text: set.

text: In

text: Proc.

text: International

text: Symposium

text: on

text: Distributed

text: Computing

text: (DISC

text: volume

text: of

text: Lecture

text: Notes

text: in

text: Computer

text: Science,

text: pages

text: Springer,

text: Dana

text: Angluin.

text: Local

text: and

text: global

text: properties

text: in

text: networks

text: of

text: processors.

text: In

text: Proc.

text: Annual

text: ACM

text: Symposium

text: on

text: Theory

text: of

text: Computing

text: (STOC

text: pages

text: ACM

text: Press,

text: Sebastian

text: Brandt,

text: Orr

text: Fischer,

text: Juho

text: Hirvonen,

text: Barbara

text: Keller,

text: Tuomo

text: Lempiäinen,

text: Joel

text: Rybicki,

text: Jukka

text: Suomela,

text: and

text: Jara

text: Uitto.

text: A

text: lower

text: bound

text: for

text: the

text: distributed

text: Lovász

text: local

text: lemma.

text: In

text: Proc.

text: Annual

text: ACM

text: Symposium

text: on

text: Theory

text: of

text: Computing

text: (STOC

text: pages

text: ACM

text: Press,

text: Sebastian

text: Brandt,

text: Juho

text: Hirvonen,

text: Janne

text: H.

text: Korhonen,

text: Tuomo

text: Lempiäinen,

text: Patric

text: R.

text: J.

text: Östergård,

text: Christopher

text: Purcell,

text: Joel

text: Rybicki,

text: Jukka

text: Suomela,

text: and

text: Przemysław

text: Uznański.

text: LCL

text: problems

text: on

text: grids.

text: In

text: Proc.

text: Annual

text: ACM

text: Symposium

text: on

text: Principles

text: of

text: Distributed

text: Computing

text: (PODC

text: To

text: appear.

text: Yi-Jun

text: Chang,

text: Tsvi

text: Kopelowitz,

text: and

text: Seth

text: Pettie.

text: An

text: exponential

text: separation

text: between

text: randomized

text: and

text: deterministic

text: complexity

text: in

text: the

text: LOCAL

text: model.

text: In

text: Proc.

text: Annual

text: IEEE

text: Symposium

text: on

text: Foundations

text: of

text: Computer

text: Science

text: (FOCS

text: pages

text: IEEE

text: Computer

text: Society

text: Press,

text: Yi-Jun

text: Chang

text: and

text: Seth

text: Pettie.

text: A

text: time

text: hierarchy

text: theorem

text: for

text: the

text: LOCAL

text: model,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: V Reordering and Frbeaqnuden1cy SVD ForeggernoeurnadtioSnignal Interpolation Background Signal generation HOA data MDCT Frbeaqnudenicy SVD ForeggernoeurnadtioSnignal CoFnGcaStiegnnaatlisng QEunatnrotipzyatCioondianngd HOA data U,Σ Foreground Signal Quantization and SVD generation MDCT Entropy Coding Background Signal Concatenating Quantization and generation BG Signals Entropy Coding Frbeaqnudenncy SVD ForeggernoeurnadtioSnignal Background Signal Quantization and Background Signal generation MDCT Entropy Coding generation Figure 2: Overview of proposed method encoder Figure 1: Overview of MPEG-H encoder and further improve perceptual quality of the rendered HOA blocking artifacts if blindly concatenated foreground compo- data. nents are fed to the core codec. While there are several ap- 2. MPEG-H APPROACH FOR COMPRESSION OF proaches to reorder and match components with the previous HOA DATA frame, we employ the magnitude of correlation between col- umn vectors of Vˆ and Vˆ − as the criterion in an Hungarian f f 1 The MPEG HOA encoder [8] processes the input HOA matching algorithm [9], which we found to be effective. data over frames of length 2L (L = 1024) with 50% overlap. Even with matched components, simple concatenation Let the number of HOA channels be M = (N+1)2, where N across frames would introduce noticeable artifacts as a small is the ambisonics order. The encoder operates on the current change in the basis vector causes some mismatch at the frame × frame data X , which is an 2L M matrix, and performs boundary. Hence the encoder interpolates the column vec- f standard singular value decomposition (SVD), tors of Vˆ between current frame and previous frame to en- sure continuity over time. Speciﬁcally, a different transform matrix is used for each sample of the current frame, whose X = U Σ V T , (1) f f f f column vectors are obtained as, × × − where Uf is an 2L 2L unitary matrix, Σf is a 2L M v¯fi (l) = (1 w−(l))vˆfi−1 + w(l−)vˆfi , (3) rectangular diagonal matrix×with non-zero elements on the l = 0 to L 1, i = 0 to r 1, diagonal and V is an M M unitary matrix. The SVD f construction ensures that predominant components, corre- where vˆi , vˆi− are the ith matched column vectors of V˜ f f 1 sponding to the largest r singular values have as basis vectors for current and previous frames, v¯i (l) is ith column vector the ﬁrst r columns of Vf. Let Vf be truncated to the ﬁrst r for sample l in current frame and wf(l) is a window function, columns, and further be independently or deferentially quan- which may be the triangular or Hanning window. The in- tized to Vˆf and sent to the decoder as side information for terpolation should also account for the fact that the vectors each frame, so as to enable it to transform back the predomi- might get negated from one frame to next frame by perform- nant components to the ambisonics domain. To keep encoder ing a sign correction when needed. and decoder in sync, the quantized Vˆf is used to generate the An approximation of the HOA data, X˜ , is generated f predominant components Y˜ (now an approximation of ﬁrst by transforming the foreground components back to the am- f r columns of U Σ ), as, bisonics domain, which is then subtracted from the original f f data to produce the ambient (or background) HOA data. The − Y˜ = X Vˆ (Vˆ T Vˆ ) 1. (2) foreground components are coded using separate instances f f f f f of the core audio codec. The order of background HOA data is then reduced (from N to some t) and this lower order HOA Note that the inverse term is for renormalization of the data are also coded using the core audio codec. An illustra- quantized basis vectors (to maintain unitarity). The next tion of the MPEG-H approach is shown in Figure 1. step is to code the predominant or foreground components, 3. FREQUENCY DOMAIN SVD FOR HOA DATA each corresponding to a column of Y˜f, using separate in- COMPRESSION stances of the core audio codec. This requires concatenating components across frames. However, since SVD arranges Clearly, the MPEG-H approach performs an elaborate the basis vectors based on the singular value magnitudes, process of matching and interpolating transform basis vec- the same foreground component might change position in Y˜ tors of consecutive frames to improve their continuity over from frame to frame depending on the magnitude of its sin- time and to mitigate the artifacts stemming from blockwise gular value relative to others. This can result in noticeable SVD application. We propose to circumvent this underly- ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: V

text: Reordering

text: and

text: SVD

text: ForeggernoeurnadtioSnignal

text: Interpolation

text: Background

text: Signal

text: generation

text: HOA

text: data

text: MDCT

text: Frbeaqnudenicy

text: SVD

text: ForeggernoeurnadtioSnignal

text: CoFnGcaStiegnnaatlisng

text: QEunatnrotipzyatCioondianngd

text: HOA

text: data

text: U,Σ

text: Foreground

text: Signal

text: Quantization

text: and

text: SVD

text: generation

text: MDCT

text: Entropy

text: Coding

text: Background

text: Signal

text: Concatenating

text: Quantization

text: and

text: generation

text: BG

text: Signals

text: Entropy

text: Coding

text: Frbeaqnudenncy

text: SVD

text: ForeggernoeurnadtioSnignal

text: Background

text: Signal

text: Quantization

text: and

text: Background

text: Signal

text: generation

text: MDCT

text: Entropy

text: Coding

text: generation

text: Figure

text: Overview

text: of

text: proposed

text: method

text: encoder

text: Figure

text: Overview

text: of

text: MPEG-H

text: encoder

text: and

text: further

text: improve

text: perceptual

text: quality

text: of

text: the

text: rendered

text: HOA

text: blocking

text: artifacts

text: if

text: blindly

text: concatenated

text: foreground

text: compo-

text: data.

text: nents

text: are

text: fed

text: to

text: the

text: core

text: codec.

text: While

text: there

text: are

text: several

text: ap-

text: MPEG-H

text: APPROACH

text: FOR

text: COMPRESSION

text: OF

text: proaches

text: to

text: reorder

text: and

text: match

text: components

text: with

text: the

text: previous

text: HOA

text: DATA

text: frame,

text: we

text: employ

text: the

text: magnitude

text: of

text: correlation

text: between

text: col-

text: umn

text: vectors

text: of

text: Vˆ

text: and

text: Vˆ

text: as

text: the

text: criterion

text: in

text: an

text: Hungarian

text: f

text: f

text: The

text: MPEG

text: HOA

text: encoder

text: processes

text: the

text: input

text: HOA

text: matching

text: algorithm

text: which

text: we

text: found

text: to

text: be

text: effective.

text: data

text: over

text: frames

text: of

text: length

text: (L

text: with

text: overlap.

text: Even

text: with

text: matched

text: components,

text: simple

text: concatenation

text: Let

text: the

text: number

text: of

text: HOA

text: channels

text: be

text: M

text: where

text: N

text: across

text: frames

text: would

text: introduce

text: noticeable

text: artifacts

text: as

text: a

text: small

text: is

text: the

text: ambisonics

text: order.

text: The

text: encoder

text: operates

text: on

text: the

text: current

text: change

text: in

text: the

text: basis

text: vector

text: causes

text: some

text: mismatch

text: at

text: the

text: frame

text: frame

text: data

text: X

text: which

text: is

text: an

text: M

text: matrix,

text: and

text: performs

text: boundary.

text: Hence

text: the

text: encoder

text: interpolates

text: the

text: column

text: vec-

text: f

text: standard

text: singular

text: value

text: decomposition

text: (SVD),

text: tors

text: of

text: Vˆ

text: between

text: current

text: frame

text: and

text: previous

text: frame

text: to

text: en-

text: sure

text: continuity

text: over

text: time.

text: Speciﬁcally,

text: a

text: different

text: transform

text: matrix

text: is

text: used

text: for

text: each

text: sample

text: of

text: the

text: current

text: frame,

text: whose

text: X

text: U

text: Σ

text: V

text: T

text: f

text: f

text: f

text: f

text: column

text: vectors

text: are

text: obtained

text: as,

text: where

text: Uf

text: is

text: an

text: unitary

text: matrix,

text: Σf

text: is

text: a

text: M

text: v¯fi

text: (l)

text: w(l−)vˆfi

text: rectangular

text: diagonal

text: matrix×with

text: non-zero

text: elements

text: on

text: the

text: l

text: to

text: L

text: i

text: to

text: r

text: diagonal

text: and

text: V

text: is

text: an

text: M

text: M

text: unitary

text: matrix.

text: The

text: SVD

text: f

text: construction

text: ensures

text: that

text: predominant

text: components,

text: corre-

text: where

text: vˆi

text: vˆi−

text: are

text: the

text: ith

text: matched

text: column

text: vectors

text: of

text: V˜

text: f

text: f

text: sponding

text: to

text: the

text: largest

text: r

text: singular

text: values

text: have

text: as

text: basis

text: vectors

text: for

text: current

text: and

text: previous

text: frames,

text: v¯i

text: (l)

text: is

text: ith

text: column

text: vector

text: the

text: ﬁrst

text: r

text: columns

text: of

text: Vf.

text: Let

text: Vf

text: be

text: truncated

text: to

text: the

text: ﬁrst

text: r

text: for

text: sample

text: l

text: in

text: current

text: frame

text: and

text: wf(l)

text: is

text: a

text: window

text: function,

text: columns,

text: and

text: further

text: be

text: independently

text: or

text: deferentially

text: quan-

text: which

text: may

text: be

text: the

text: triangular

text: or

text: Hanning

text: window.

text: The

text: in-

text: tized

text: to

text: Vˆf

text: and

text: sent

text: to

text: the

text: decoder

text: as

text: side

text: information

text: for

text: terpolation

text: should

text: also

text: account

text: for

text: the

text: fact

text: that

text: the

text: vectors

text: each

text: frame,

text: so

text: as

text: to

text: enable

text: it

text: to

text: transform

text: back

text: the

text: predomi-

text: might

text: get

text: negated

text: from

text: one

text: frame

text: to

text: next

text: frame

text: by

text: perform-

text: nant

text: components

text: to

text: the

text: ambisonics

text: domain.

text: To

text: keep

text: encoder

text: ing

text: a

text: sign

text: correction

text: when

text: needed.

text: and

text: decoder

text: in

text: sync,

text: the

text: quantized

text: Vˆf

text: is

text: used

text: to

text: generate

text: the

text: An

text: approximation

text: of

text: the

text: HOA

text: data,

text: X˜

text: is

text: generated

text: f

text: predominant

text: components

text: Y˜

text: (now

text: an

text: approximation

text: of

text: ﬁrst

text: by

text: transforming

text: the

text: foreground

text: components

text: back

text: to

text: the

text: am-

text: f

text: r

text: columns

text: of

text: U

text: Σ

text: as,

text: bisonics

text: domain,

text: which

text: is

text: then

text: subtracted

text: from

text: the

text: original

text: f

text: f

text: data

text: to

text: produce

text: the

text: ambient

text: (or

text: background)

text: HOA

text: data.

text: The

text: Y˜

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text: 3 − ∈ To stabilize the qubit state, we process the homodyne to Γ = (2τ η) 1, where η [0, 1] is the quantum eﬃ- m m signal I(t) and feed it back into the coherent qubit evo- ciency of the microwave detection circuitry [6, 7]. lution as a coherent Rabi drive [31–36, 41, 62]. For con- When the energy decay rate of the resonator κ is suﬃ- venience, we ﬁrst rescale the signal so that it directly ciently fast compared to both the measurement rate Γ m corresponds to σˆ . The rescaled readout has the form and the dynamics of the qubit (known as the “bad cav- ≡ − z r(t) 2(I(t) I¯)/∆I, where I¯ = (I + I )/2 is the mean ity regime”), the coherently pumped resonator eﬀectively −0 1 homodyne current, and ∆I = I I is the homodyne remains at steady-state. We may then approximately ig- 1 0 signal contrast. The rescaled readout r(t) then has a nore the resonator and phenomenologically describe the ± (cid:82) temporal average of 1 for deﬁnite qubit energy states, qubit evolution alone [6], provided that we coarse-grain { } in agreement with the eigenvalues of σˆ . Before being the collected readout r(t) into discrete segments r¯ z ≡ (cid:48) (cid:48) k fed back into the controller, this rescaled readout is nec- that are temporal averages r¯ dt tk+dt r(t )dt over essarily frequency-ﬁltered by the feedback circuitry [38], − k tk (cid:55)→ time increments dt > κ 1 longer than the relaxation r r˜, which also delays the ﬁltered readout by a dura- − timescale of the resonator ﬁeld [7]. This coarse-grained tion T . This ﬁltered and delayed readout r˜(t T ) is d d measurement evolution is then Markovian in the dis- then used to modulate a coherent Rabi drive described cretized time intervals, and can be eﬃciently described by the control Hamiltonian, using the quantum Bayesian method [6, 7], which also permits eﬃcient numerical simulation. Hˆ = (cid:126) [∆ + ∆ r˜(t − T )] σˆφ , (1) The total evolution of the qubit can be approximately c 0 1 d 2 decomposed into three parts: pure measurement back- action, coherent unitary evolution, and environmental | (cid:105)(cid:104) | | (cid:105)(cid:104) | where σˆ = cos φ σˆ +sin φ σˆ , with σˆ = 1 0 + 0 1 dissipation. These three contributions may be treated φ− | (cid:105)(cid:104) | x | (cid:105)(cid:104) | y x and σˆ = i 1 0 + i 0 1 . The choice of angle φ ﬁxes separately provided that the duration dt is smaller than y the plane of the qubit Bloch sphere to which the qubit the characteristic timescales of each evolution. We now will become stabilized. For simplicity of discussion, we describe each contributing evolution, then combine them. now ﬁx φ = π to choose clockwise oscillations in the yz- First, the form of the measurement backaction fol- plane. This feedback Hamiltonian induces oscillations at lows from Bayes’ theorem and the Born rule [6]. Given | a constant Rabi frequency ∆ that is modulated by the empirically measured conditional probabilities P(r¯ 0) − 0 | linear feedback term ∆ r˜(t T ). As we will show, and in and P(r¯ 1) for obtaining a result r¯ averaged over dt 1 d agreement with ﬂuorescence-based feedback stabilization given a deﬁnite qubit energy state (0 or 1), the qubit protocols [35, 36, 41, 62], the two parameters ∆ and ∆ energy probabilities must update according to Bayes’ 0 1 (cid:55)→ | | (cid:55)→ completely control the eﬀect of the feedback by jointly theorem P(0) P(0 r¯) = P(r¯ 0)P(0)/P(r¯) and P(1) | | determining the attraction region in state space for sta- P(1 r¯) = P(r¯ 1)P(1)/P(r¯), where the total probabil- | | bilization. In contrast to the ﬂuorescence-based protocols ity for collecting r¯ is P(r¯) = P(r¯ 0)P(0) + P(r¯ 1)P(1). that have a bias toward the ground state, we will show This evolution is equivalent to the density operator up- (cid:55)→ † that dispersive measurements can stabilize nearly any an- date rule ρˆ Mˆ ρˆMˆ /P(r¯) with probability P(r¯) = † r¯ r¯ gle within the chosen plane with high purity (i.e., nearly Tr(Mˆr¯Mˆr¯ρˆ), de(cid:112)termined entirely(cid:112)by the Kraus operator any pure state may be targeted with high ﬁdelity). We now describe the qubit evolution produced by col- | | (cid:105)(cid:104) | | | (cid:105)(cid:104) | lecting the dispersive readout r(t) and applying the feed- Mˆ = P(r¯ 0) 0 0 + P(r¯ 1) 1 1 . (2) r¯ back Hamiltonian in Eq. (1). We ﬁrst consider a time- discrete quantum Bayesian model [6, 7], from which we can numerically simulate the feedback protocol includ- For a maximally informative choice of ampliﬁed quadra- ing ineﬃciencies and delay. We then consider a time- ture, there is no additional phase-backaction on the qubit continuous stochastic master equation model [5] that in- [6, 7], so this minimal Kraus operator is suﬃcient. terpolates the Ma→rkovian feedback dynamics in the limit Because of the|central limi|t theorem, the conditional of zero delay Td 0 (and no signal ﬁltering) with a ﬁc- probabilities±P(r¯ 0) and P(r¯ 1) approximate Gaussians titious time-continuous stochastic process [31–33, 35, 36, centered at 1. For simplicity, we also assume approx- 41], and which is convenient for analytic derivations of imately equal variances, which may be justiﬁed empir- the required feedback parameters. ically. To preserve proper scaling of temporal averages A. Time-discrete model as the discrete time step dt is varied, the variances of each time step must be inversely proportional to dt. By The strength of the qubit measurement is determined deﬁnition of the unit signal-to-noise timescale τ , the m by the timescale τm needed to obtain a unit signal-to- variances must (cid:104)then have the form τm/dt, yieldin(cid:105)g noise ratio for distinguishing the qubit states. A qubit initially in a superposition of energy states will typically − | (cid:105)(cid:104) | | (cid:105)(cid:104) | collapse to a deﬁnite energy state within a few τm. This Mˆr¯ = C(r¯) e r¯dt/2τm 0 0 + er¯dt/2τm 1 1 (3) collapse timescale τ is related to the ensemble dephas- m ing rate Γ of the qubit due to measurement according m ##LTLine## ##LTLine## ##LTLine##

text: To

text: stabilize

text: the

text: qubit

text: state,

text: we

text: process

text: the

text: homodyne

text: to

text: Γ

text: η)

text: where

text: η

text: is

text: the

text: quantum

text: eﬃ-

text: m

text: m

text: signal

text: I(t)

text: and

text: feed

text: it

text: back

text: into

text: the

text: coherent

text: qubit

text: evo-

text: ciency

text: of

text: the

text: microwave

text: detection

text: circuitry

text: lution

text: as

text: a

text: coherent

text: Rabi

text: drive

text: For

text: con-

text: When

text: the

text: energy

text: decay

text: rate

text: of

text: the

text: resonator

text: κ

text: is

text: suﬃ-

text: venience,

text: we

text: ﬁrst

text: rescale

text: the

text: signal

text: so

text: that

text: it

text: directly

text: ciently

text: fast

text: compared

text: to

text: both

text: the

text: measurement

text: rate

text: Γ

text: m

text: corresponds

text: to

text: σˆ

text: The

text: rescaled

text: readout

text: has

text: the

text: form

text: and

text: the

text: dynamics

text: of

text: the

text: qubit

text: (known

text: as

text: the

text: “bad

text: cav-

text: z

text: r(t)

text: I¯)/∆I,

text: where

text: I¯

text: (I

text: I

text: is

text: the

text: mean

text: ity

text: regime”),

text: the

text: coherently

text: pumped

text: resonator

text: eﬀectively

text: −0

text: homodyne

text: current,

text: and

text: ∆I

text: I

text: I

text: is

text: the

text: homodyne

text: remains

text: at

text: steady-state.

text: We

text: may

text: then

text: approximately

text: ig-

text: signal

text: contrast.

text: The

text: rescaled

text: readout

text: r(t)

text: then

text: has

text: a

text: nore

text: the

text: resonator

text: and

text: phenomenologically

text: describe

text: the

text: temporal

text: average

text: of

text: for

text: deﬁnite

text: qubit

text: energy

text: states,

text: qubit

text: evolution

text: alone

text: provided

text: that

text: we

text: coarse-grain

text: in

text: agreement

text: with

text: the

text: eigenvalues

text: of

text: σˆ

text: Before

text: being

text: the

text: collected

text: readout

text: r(t)

text: into

text: discrete

text: segments

text: r¯

text: z

text: k

text: fed

text: back

text: into

text: the

text: controller,

text: this

text: rescaled

text: readout

text: is

text: nec-

text: that

text: are

text: temporal

text: averages

text: r¯

text: dt

text: tk+dt

text: r(t

text: )dt

text: over

text: essarily

text: frequency-ﬁltered

text: by

text: the

text: feedback

text: circuitry

text: k

text: tk

text: time

text: increments

text: dt

text: κ

text: longer

text: than

text: the

text: relaxation

text: r

text: r˜,

text: which

text: also

text: delays

text: the

text: ﬁltered

text: readout

text: by

text: a

text: dura-

text: timescale

text: of

text: the

text: resonator

text: ﬁeld

text: This

text: coarse-grained

text: tion

text: T

text: This

text: ﬁltered

text: and

text: delayed

text: readout

text: r˜(t

text: T

text: is

text: d

text: d

text: measurement

text: evolution

text: is

text: then

text: Markovian

text: in

text: the

text: dis-

text: then

text: used

text: to

text: modulate

text: a

text: coherent

text: Rabi

text: drive

text: described

text: cretized

text: time

text: intervals,

text: and

text: can

text: be

text: eﬃciently

text: described

text: by

text: the

text: control

text: Hamiltonian,

text: using

text: the

text: quantum

text: Bayesian

text: method

text: which

text: also

text: permits

text: eﬃcient

text: numerical

text: simulation.

text: Hˆ

text: r˜(t

text: T

text: σˆφ

text: The

text: total

text: evolution

text: of

text: the

text: qubit

text: can

text: be

text: approximately

text: c

text: d

text: decomposed

text: into

text: three

text: parts:

text: pure

text: measurement

text: back-

text: action,

text: coherent

text: unitary

text: evolution,

text: and

text: environmental

text: where

text: σˆ

text: cos

text: φ

text: σˆ

text: +sin

text: φ

text: σˆ

text: with

text: σˆ

text: dissipation.

text: These

text: three

text: contributions

text: may

text: be

text: treated

text: φ−

text: x

text: y

text: x

text: and

text: σˆ

text: i

text: i

text: The

text: choice

text: of

text: angle

text: φ

text: ﬁxes

text: separately

text: provided

text: that

text: the

text: duration

text: dt

text: is

text: smaller

text: than

text: y

text: the

text: plane

text: of

text: the

text: qubit

text: Bloch

text: sphere

text: to

text: which

text: the

text: qubit

text: the

text: characteristic

text: timescales

text: of

text: each

text: evolution.

text: We

text: now

text: will

text: become

text: stabilized.

text: For

text: simplicity

text: of

text: discussion,

text: we

text: describe

text: each

text: contributing

text: evolution,

text: then

text: combine

text: them.

text: now

text: ﬁx

text: φ

text: π

text: to

text: choose

text: clockwise

text: oscillations

text: in

text: the

text: yz-

text: First,

text: the

text: form

text: of

text: the

text: measurement

text: backaction

text: fol-

text: plane.

text: This

text: feedback

text: Hamiltonian

text: induces

text: oscillations

text: at

text: lows

text: from

text: Bayes’

text: theorem

text: and

text: the

text: Born

text: rule

text: Given

text: a

text: constant

text: Rabi

text: frequency

text: that

text: is

text: modulated

text: by

text: the

text: empirically

text: measured

text: conditional

text: probabilities

text: P(r¯

text: linear

text: feedback

text: term

text: r˜(t

text: T

text: As

text: we

text: will

text: show,

text: and

text: in

text: and

text: P(r¯

text: for

text: obtaining

text: a

text: result

text: r¯

text: averaged

text: over

text: dt

text: d

text: agreement

text: with

text: ﬂuorescence-based

text: feedback

text: stabilization

text: given

text: a

text: deﬁnite

text: qubit

text: energy

text: state

text: or

text: the

text: qubit

text: protocols

text: the

text: two

text: parameters

text: and

text: energy

text: probabilities

text: must

text: update

text: according

text: to

text: Bayes’

text: completely

text: control

text: the

text: eﬀect

text: of

text: the

text: feedback

text: by

text: jointly

text: theorem

text: r¯)

text: P(r¯

text: and

text: determining

text: the

text: attraction

text: region

text: in

text: state

text: space

text: for

text: sta-

text: r¯)

text: P(r¯

text: where

text: the

text: total

text: probabil-

text: bilization.

text: In

text: contrast

text: to

text: the

text: ﬂuorescence-based

text: protocols

text: ity

text: for

text: collecting

text: r¯

text: is

text: P(r¯)

text: P(r¯

text: P(r¯

text: that

text: have

text: a

text: bias

text: toward

text: the

text: ground

text: state,

text: we

text: will

text: show

text: This

text: evolution

text: is

text: equivalent

text: to

text: the

text: density

text: operator

text: up-

text: that

text: dispersive

text: measurements

text: can

text: stabilize

text: nearly

text: any

text: an-

text: date

text: rule

text: ρˆ

text: Mˆ

text: ρˆMˆ

text: /P(r¯)

text: with

text: probability

text: P(r¯)

text: r¯

text: r¯

text: gle

text: within

text: the

text: chosen

text: plane

text: with

text: high

text: purity

text: (i.e.,

text: nearly

text: Tr(Mˆr¯Mˆr¯ρˆ),

text: the

text: Kraus

text: operator

text: any

text: pure

text: state

text: may

text: be

text: targeted

text: with

text: high

text: ﬁdelity).

text: We

text: now

text: describe

text: the

text: qubit

text: evolution

text: produced

text: by

text: col-

text: lecting

text: the

text: dispersive

text: readout

text: r(t)

text: and

text: applying

text: the

text: feed-

text: Mˆ

text: P(r¯

text: P(r¯

text: r¯

text: back

text: Hamiltonian

text: in

text: Eq.

text: We

text: ﬁrst

text: consider

text: a

text: time-

text: discrete

text: quantum

text: Bayesian

text: model

text: from

text: which

text: we

text: can

text: numerically

text: simulate

text: the

text: feedback

text: protocol

text: includ-

text: For

text: a

text: maximally

text: informative

text: choice

text: of

text: ampliﬁed

text: quadra-

text: ing

text: ineﬃciencies

text: and

text: delay.

text: We

text: then

text: consider

text: a

text: time-

text: ture,

text: there

text: is

text: no

text: additional

text: phase-backaction

text: on

text: the

text: qubit

text: continuous

text: stochastic

text: master

text: equation

text: model

text: that

text: in-

text: so

text: this

text: minimal

text: Kraus

text: operator

text: is

text: suﬃcient.

text: terpolates

text: the

text: Ma→rkovian

text: feedback

text: dynamics

text: in

text: the

text: limit

text: Because

text: of

text: the|central

text: limi|t

text: theorem,

text: the

text: conditional

text: of

text: zero

text: delay

text: Td

text: (and

text: no

text: signal

text: ﬁltering)

text: with

text: a

text: ﬁc-

text: probabilities±P(r¯

text: and

text: P(r¯

text: approximate

text: Gaussians

text: titious

text: time-continuous

text: stochastic

text: process

text: centered

text: at

text: For

text: simplicity,

text: we

text: also

text: assume

text: approx-

text: and

text: which

text: is

text: convenient

text: for

text: analytic

text: derivations

text: of

text: imately

text: equal

text: variances,

text: which

text: may

text: be

text: justiﬁed

text: empir-

text: the

text: required

text: feedback

text: parameters.

text: ically.

text: To

text: preserve

text: proper

text: scaling

text: of

text: temporal

text: averages

text: A.

text: Time-discrete

text: model

text: as

text: the

text: discrete

text: time

text: step

text: dt

text: is

text: varied,

text: the

text: variances

text: of

text: each

text: time

text: step

text: must

text: be

text: inversely

text: proportional

text: to

text: dt.

text: By

text: The

text: strength

text: of

text: the

text: qubit

text: measurement

text: is

text: determined

text: deﬁnition

text: of

text: the

text: unit

text: signal-to-noise

text: timescale

text: τ

text: the

text: m

text: by

text: the

text: timescale

text: τm

text: needed

text: to

text: obtain

text: a

text: unit

text: signal-to-

text: variances

text: must

text: have

text: the

text: form

text: τm/dt,

text: noise

text: ratio

text: for

text: distinguishing

text: the

text: qubit

text: states.

text: A

text: qubit

text: initially

text: in

text: a

text: superposition

text: of

text: energy

text: states

text: will

text: typically

text: collapse

text: to

text: a

text: deﬁnite

text: energy

text: state

text: within

text: a

text: few

text: τm.

text: This

text: Mˆr¯

text: C(r¯)

text: e

text: collapse

text: timescale

text: τ

text: is

text: related

text: to

text: the

text: ensemble

text: dephas-

text: m

text: ing

text: rate

text: Γ

text: of

text: the

text: qubit

text: due

text: to

text: measurement

text: according

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##