text: Since P is a thread, if we add a new edge uw in [G , Σ ], the resulting graph will still be planar. i i (cid:48) (cid:48) (cid:48) · · · (cid:48) So we add such an edge and subdivide it r times to obtain the new thread P = uv v v w. (cid:48) 1 2 r Also we assign signs of the new edges in such a way that the edges uv and v w have a same (cid:48) (cid:48) (cid:48) 1 r sign, the edges v w and uv have a same sign and the edges v v and v − v − have a same r 1 i i+1 r i+1 r i sign. (cid:48) Consider a planar drawing of the graph in which P and P form a facial cycle of length 2r. (cid:48) (cid:48) − − In the face P P connect v and v by a new edge. Subdivide this new edge 2k r 2 times − − 1 r (that is, into 2k r 1 edges, we color them green), so that each of the facial cycles containing the new thread is of length 2k. Choose signs of the edges of this new path in such a way that each of the facial cycles containing the new thread is unbalanced. (cid:48) (cid:48) (cid:48) (cid:48) Let [G , Σ ] be the resulting signed graph. We ﬁrst note that [G , Σ ] is also planar bipartite i i i i of unbalanced-girth 2k. Now suppose that the vertices of P , indexed by odd (or even) numbers − are all part of a maximal clique in the graph induced by one partite set of G in [G , Σ ](2k 2). ∪ (cid:48) i i i Then the vertices of P P , indexed by odd (or even) numbers are all part of a maximal clique (cid:48) (cid:48) (cid:48) − in the graph induced by the corresponding partite set of G in [G , Σ ](2k 2). This can be proved i i i by similar logic used in the proof of Theorem 2.1. The only diﬀerence is that to prove the above claim one needs to repeat the argument based on the parity of number of negative edges instead of the parity of number of edges and use Lemma 3.2 instead of Lemma 2.2. − Now we describe our general construction. At ﬁrst we have [G , Σ ] on 4k 2 vertices and 1 1 two maximal threads. By Lemma 3.2 the vertices, indexed by numbers with the same parity, of − these two threads are parts of the unique clique of order 2k 1 in their respective components − in [G , Σ ](2k 2). We apply the previously mentioned construction on both the threads to build 1 1 − [G , Σ ] which will have four maximal threads each of length 2k 3 (we are only considering 2 2 − maximal threads that are part of a path of length 2k 2 connecting a, b or c, d, the green − − threads are not considered). There are two disjoint cliques, each of order (2k 1) + (2k 3) in − (cid:48) − − (cid:80) [G , Σ ](2k 2), that is, f(2) = f (2) = (2k 1) + (2k 3), and there are four maximal threads 2 2 − − of length 2k 3, each a part of a path of length 2k 2 either connecting a, b or connecting c, d. (cid:48) − − − − − Continuing this construction, in general, f(i) = f (i) = (2k 1) + i 1 2j 1(2k j 2) − − j=1 − and there are 2i maximal threads of length 2k i 1 which are part of a path of length 2k 2 connecting a, b or connecting c, d. Note that [G , Σ ] at each step is a planar bipartite signed graph of unbalanced-girth 2k. i i Therefore (cid:88) (cid:88) (cid:88) − − − 2k 2 2k 2 2k 2 − (cid:48) − − − − − − − − − f(2k 1) = f (2k 1) = 2k 1 + 2j 1(2k j 2) = 2k 1 + (k 1) 2j j2j 1 j=1 j=1 j=1 − − − − − − − − − − − = 2k 1 + [(k 1)(22k 1 2)] [(1 22k 1) ( 1)(2k 1)22k 2] − − − − − − − − − − − = 2k 1 + [k22k 1 2k 22k 1 + 2] [1 22k 1 + k22k 1 22k 2] − = 22k 2. This completes the proof. Corollary 3.4. If [B, Π] is a minimal signed bipartite graph of unbalanced-girth 2k to which ev- | | ≥ ery planar signed bipartite graph of unbalanced-girth 2k admits a homomorphism, then V (B) − ≥ 22k 1 and δ(B) 2k. 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Since

text: P

text: is

text: a

text: thread,

text: if

text: we

text: add

text: a

text: new

text: edge

text: uw

text: in

text: [G

text: Σ

text: the

text: resulting

text: graph

text: will

text: still

text: be

text: planar.

text: i

text: i

text: So

text: we

text: add

text: such

text: an

text: edge

text: and

text: subdivide

text: it

text: r

text: times

text: to

text: obtain

text: the

text: new

text: thread

text: P

text: uv

text: v

text: v

text: w.

text: r

text: Also

text: we

text: assign

text: signs

text: of

text: the

text: new

text: edges

text: in

text: such

text: a

text: way

text: that

text: the

text: edges

text: uv

text: and

text: v

text: w

text: have

text: a

text: same

text: r

text: sign,

text: the

text: edges

text: v

text: w

text: and

text: uv

text: have

text: a

text: same

text: sign

text: and

text: the

text: edges

text: v

text: v

text: and

text: v

text: v

text: have

text: a

text: same

text: r

text: i

text: r

text: r

text: i

text: sign.

text: Consider

text: a

text: planar

text: drawing

text: of

text: the

text: graph

text: in

text: which

text: P

text: and

text: P

text: form

text: a

text: facial

text: cycle

text: of

text: length

text: In

text: the

text: face

text: P

text: P

text: connect

text: v

text: and

text: v

text: by

text: a

text: new

text: edge.

text: Subdivide

text: this

text: new

text: edge

text: r

text: times

text: r

text: (that

text: is,

text: into

text: r

text: edges,

text: we

text: color

text: them

text: green),

text: so

text: that

text: each

text: of

text: the

text: facial

text: cycles

text: containing

text: the

text: new

text: thread

text: is

text: of

text: length

text: Choose

text: signs

text: of

text: the

text: edges

text: of

text: this

text: new

text: path

text: in

text: such

text: a

text: way

text: that

text: each

text: of

text: the

text: facial

text: cycles

text: containing

text: the

text: new

text: thread

text: is

text: unbalanced.

text: Let

text: [G

text: Σ

text: be

text: the

text: resulting

text: signed

text: graph.

text: We

text: ﬁrst

text: note

text: that

text: [G

text: Σ

text: is

text: also

text: planar

text: bipartite

text: i

text: i

text: i

text: i

text: of

text: unbalanced-girth

text: Now

text: suppose

text: that

text: the

text: vertices

text: of

text: P

text: indexed

text: by

text: odd

text: (or

text: even)

text: numbers

text: are

text: all

text: part

text: of

text: a

text: maximal

text: clique

text: in

text: the

text: graph

text: induced

text: by

text: one

text: partite

text: set

text: of

text: G

text: in

text: [G

text: Σ

text: i

text: i

text: i

text: Then

text: the

text: vertices

text: of

text: P

text: P

text: indexed

text: by

text: odd

text: (or

text: even)

text: numbers

text: are

text: all

text: part

text: of

text: a

text: maximal

text: clique

text: in

text: the

text: graph

text: induced

text: by

text: the

text: corresponding

text: partite

text: set

text: of

text: G

text: in

text: [G

text: Σ

text: This

text: can

text: be

text: proved

text: i

text: i

text: i

text: by

text: similar

text: logic

text: used

text: in

text: the

text: proof

text: of

text: Theorem

text: The

text: only

text: diﬀerence

text: is

text: that

text: to

text: prove

text: the

text: above

text: claim

text: one

text: needs

text: to

text: repeat

text: the

text: argument

text: based

text: on

text: the

text: parity

text: of

text: number

text: of

text: negative

text: edges

text: instead

text: of

text: the

text: parity

text: of

text: number

text: of

text: edges

text: and

text: use

text: Lemma

text: instead

text: of

text: Lemma

text: Now

text: we

text: describe

text: our

text: general

text: construction.

text: At

text: ﬁrst

text: we

text: have

text: [G

text: Σ

text: on

text: vertices

text: and

text: two

text: maximal

text: threads.

text: By

text: Lemma

text: the

text: vertices,

text: indexed

text: by

text: numbers

text: with

text: the

text: same

text: parity,

text: of

text: these

text: two

text: threads

text: are

text: parts

text: of

text: the

text: unique

text: clique

text: of

text: order

text: in

text: their

text: respective

text: components

text: in

text: [G

text: Σ

text: We

text: apply

text: the

text: previously

text: mentioned

text: construction

text: on

text: both

text: the

text: threads

text: to

text: build

text: [G

text: Σ

text: which

text: will

text: have

text: four

text: maximal

text: threads

text: each

text: of

text: length

text: (we

text: are

text: only

text: considering

text: maximal

text: threads

text: that

text: are

text: part

text: of

text: a

text: path

text: of

text: length

text: connecting

text: a,

text: b

text: or

text: c,

text: d,

text: the

text: green

text: threads

text: are

text: not

text: considered).

text: There

text: are

text: two

text: disjoint

text: cliques,

text: each

text: of

text: order

text: in

text: [G

text: Σ

text: that

text: is,

text: f

text: and

text: there

text: are

text: four

text: maximal

text: threads

text: of

text: length

text: each

text: a

text: part

text: of

text: a

text: path

text: of

text: length

text: either

text: connecting

text: a,

text: b

text: or

text: connecting

text: c,

text: d.

text: Continuing

text: this

text: construction,

text: in

text: general,

text: f(i)

text: f

text: (i)

text: i

text: j

text: and

text: there

text: are

text: maximal

text: threads

text: of

text: length

text: i

text: which

text: are

text: part

text: of

text: a

text: path

text: of

text: length

text: connecting

text: a,

text: b

text: or

text: connecting

text: c,

text: d.

text: Note

text: that

text: [G

text: Σ

text: at

text: each

text: step

text: is

text: a

text: planar

text: bipartite

text: signed

text: graph

text: of

text: unbalanced-girth

text: i

text: i

text: Therefore

text: f

text: j

text: (k

text: [(k

text: This

text: completes

text: the

text: proof.

text: Corollary

text: If

text: [B,

text: Π]

text: is

text: a

text: minimal

text: signed

text: bipartite

text: graph

text: of

text: unbalanced-girth

text: to

text: which

text: ev-

text: ery

text: planar

text: signed

text: bipartite

text: graph

text: of

text: unbalanced-girth

text: admits

text: a

text: homomorphism,

text: then

text: V

text: (B)

text: and

text: δ(B)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 quantum walker and the qubit that is measured plays the can be used to observe localisation [31, 32], or to simu- role of the coin. If the interaction between the pointer late physical systems with non-homogenous interactions and the qubit occurs at the same time as the evolution [33]. In [12] it was shown that quantum walks with posi- of the qubit, the measured value changes during the pro- tion and time-dependent coin operations C can be also x,t cess and the pointer starts to move back and forth. This used to implement POVMs. movement leads to an interference and the interference III. POVM IMPLEMENTATION WITH pattern produced in this process can be interpreted as QUANTUM WALKS a POVM. In fact, in [12] it was shown that any POVM can be implemented in such a way, provided a necessary The probability distribution of a quantum walk, that evolution is applied to the qubit. is initially localised at the origin x = 0, depends on the | →(cid:105) | ←(cid:105) In this work we report an experimental implementa- initial coin state α + β and on the subsequent tion of the above quantum walk POVM scenario. Here, coin operations. A single application of the operator T | →(cid:105) we use the optical setup in which the qubit is encoded makes the particle to go into superposition α 1, + | − ←(cid:105) in a polarization state of a single photon and the posi- β 1, . In this case the position measurement would tion of the quantum walker is implemented on a pho- correspond to a von Neumann measurement of the coin {← →} → tonic path [25]. We construct setups realising (i) three in the basis , , where the result corresponds to ← POVM elements symmetrically distributed on an equa- ﬁnding the particle at x = 1 and to ﬁnding it at x = − torial plane of the Bloch sphere (Trine-POVM); (ii) 1. However, if one allows quantum walk to evolve for Symmetric-Informationally-Complete (SIC) POVM; and more than one step and the coin operation to change from (iii) Unambiguous Discrimination of two non-orthogonal one step to another step, then the particle can spread quantum states. over many positions and the measurement of its location II. DISCRETE-TIME QUANTUM WALKS at x may correspond to a measurement of some POVM element E . Indeed, a proper choice of coin operations x Discrete-time quantum walks are quantum counter- can lead to an arbitrary qubit POVM scenario [12]. parts of classical random walks in which a particle takes The POVM elements E (i = 1, 2, . . . , n) are nonnega- i a random step to the left or right. In the classical case tive operators obeying (cid:88) the particle spreads in a diﬀusive manner√(a standard deviation of its position is proportional to t) and after n many steps a spatial probability distribution is Gaussian. E = 11. (4) In quantum case the system is described by two degrees i | (cid:105) | (cid:105) − i of freedom ψ = x, c : the position x = . . . , 1, 0, 1, . . . aconidn ac =tw←o-,l→eve.l Tinhteerenvaolludteiognreeis oufnfirteaerdyoamndkncoonwsnistassoaf in Tthhaeyt dthiﬀeeyr dfroomnosttahnadvaerdtovobneNoeruthmoannonrmparol j(eΠctoΠrs Π=i t(wthoatsuibs-aop2er×at2iounns iUtar=y CmTat,rnixamacetlyinga uonnliytaroyn ctohien ctooisns δbie,jΠgrie,abteurt tEhaiEnjth(cid:54)=e dδiim,j)enasniodn tohfatthtehesiyrstneummbneir>jcadn. degree of freedom) (cid:18) (cid:19) The following quantum walk algorithm, proposed in [12], generates an arbitrary set of rank 1 POVM elements { } − E , . . . , E (rank 2 elements can be generated with a C = − cos θ e iβ sin θ (1) mo1diﬁed venrsion of this algorithm). eiβ sin θ cos θ 1. Initiate the quantum walk at position x = 0 with the coin state corresponding to the qubit state one and a (cid:88)conditional translation operation wants to measure. 2. Set i := 1. | →(cid:105)(cid:104) → | | − ←(cid:105)(cid:104) ← | T = x + 1, x, + x 1, x, . (2) 3. While i < n do the following: x • Apply coin operation C(1) at position x = 0 Quantum walks with uniform (position-independent) and identity elsewhere aind then apply trans- coin operation spread ballistically (standard deviation lation operator T. proportional to t) and its probability distribution diﬀers • (cid:18) (cid:19) form the classical Gaussian shape [29, 30]. On the other Apply coin operation C(2) at position x = 1, i hand, quantum walks with position-dependent coin op- eration (cid:18) (cid:19) NOT = 0 1 1 0 − Cx = −eicβoxssθixn θx e−icβoxssθinx θx (3) • athtepnosaiptipolny xtr=ansla1taionndoidpeenratittoyreTls.ewhere and i := i + 1. ##LTLine##

label: paragraph

text: quantum

text: walker

text: and

text: the

text: qubit

text: that

text: is

text: measured

text: plays

text: the

text: can

text: be

text: used

text: to

text: observe

text: localisation

text: or

text: to

text: simu-

text: role

text: of

text: the

text: coin.

text: If

text: the

text: interaction

text: between

text: the

text: pointer

text: late

text: physical

text: systems

text: with

text: non-homogenous

text: interactions

text: and

text: the

text: qubit

text: occurs

text: at

text: the

text: same

text: time

text: as

text: the

text: evolution

text: In

text: it

text: was

text: shown

text: that

text: quantum

text: walks

text: with

text: posi-

text: of

text: the

text: qubit,

text: the

text: measured

text: value

text: changes

text: during

text: the

text: pro-

text: tion

text: and

text: time-dependent

text: coin

text: operations

text: C

text: can

text: be

text: also

text: x,t

text: cess

text: and

text: the

text: pointer

text: starts

text: to

text: move

text: back

text: and

text: forth.

text: This

text: used

text: to

text: implement

text: POVMs.

text: movement

text: leads

text: to

text: an

text: interference

text: and

text: the

text: interference

text: III.

text: POVM

text: IMPLEMENTATION

text: WITH

text: pattern

text: produced

text: in

text: this

text: process

text: can

text: be

text: interpreted

text: as

text: QUANTUM

text: WALKS

text: a

text: POVM.

text: In

text: fact,

text: in

text: it

text: was

text: shown

text: that

text: any

text: POVM

text: can

text: be

text: implemented

text: in

text: such

text: a

text: way,

text: provided

text: a

text: necessary

text: The

text: probability

text: distribution

text: of

text: a

text: quantum

text: walk,

text: that

text: evolution

text: is

text: applied

text: to

text: the

text: qubit.

text: is

text: initially

text: localised

text: at

text: the

text: origin

text: x

text: depends

text: on

text: the

text: In

text: this

text: work

text: we

text: report

text: an

text: experimental

text: implementa-

text: initial

text: coin

text: state

text: α

text: β

text: and

text: on

text: the

text: subsequent

text: tion

text: of

text: the

text: above

text: quantum

text: walk

text: POVM

text: scenario.

text: Here,

text: coin

text: operations.

text: A

text: single

text: application

text: of

text: the

text: operator

text: T

text: we

text: use

text: the

text: optical

text: setup

text: in

text: which

text: the

text: qubit

text: is

text: encoded

text: makes

text: the

text: particle

text: to

text: go

text: into

text: superposition

text: α

text: in

text: a

text: polarization

text: state

text: of

text: a

text: single

text: photon

text: and

text: the

text: posi-

text: β

text: In

text: this

text: case

text: the

text: position

text: measurement

text: would

text: tion

text: of

text: the

text: quantum

text: walker

text: is

text: implemented

text: on

text: a

text: pho-

text: correspond

text: to

text: a

text: von

text: Neumann

text: measurement

text: of

text: the

text: coin

text: tonic

text: path

text: We

text: construct

text: setups

text: realising

text: (i)

text: three

text: in

text: the

text: basis

text: where

text: the

text: result

text: corresponds

text: to

text: POVM

text: elements

text: symmetrically

text: distributed

text: on

text: an

text: equa-

text: ﬁnding

text: the

text: particle

text: at

text: x

text: and

text: to

text: ﬁnding

text: it

text: at

text: x

text: torial

text: plane

text: of

text: the

text: Bloch

text: sphere

text: (Trine-POVM);

text: (ii)

text: However,

text: if

text: one

text: allows

text: quantum

text: walk

text: to

text: evolve

text: for

text: Symmetric-Informationally-Complete

text: (SIC)

text: POVM;

text: and

text: more

text: than

text: one

text: step

text: and

text: the

text: coin

text: operation

text: to

text: change

text: from

text: (iii)

text: Unambiguous

text: Discrimination

text: of

text: two

text: non-orthogonal

text: one

text: step

text: to

text: another

text: step,

text: then

text: the

text: particle

text: can

text: spread

text: quantum

text: states.

text: over

text: many

text: positions

text: and

text: the

text: measurement

text: of

text: its

text: location

text: II.

text: DISCRETE-TIME

text: QUANTUM

text: WALKS

text: at

text: x

text: may

text: correspond

text: to

text: a

text: measurement

text: of

text: some

text: POVM

text: element

text: E

text: Indeed,

text: a

text: proper

text: choice

text: of

text: coin

text: operations

text: x

text: Discrete-time

text: quantum

text: walks

text: are

text: quantum

text: counter-

text: can

text: lead

text: to

text: an

text: arbitrary

text: qubit

text: POVM

text: scenario

text: parts

text: of

text: classical

text: random

text: walks

text: in

text: which

text: a

text: particle

text: takes

text: The

text: POVM

text: elements

text: E

text: (i

text: n)

text: are

text: nonnega-

text: i

text: a

text: random

text: step

text: to

text: the

text: left

text: or

text: right.

text: In

text: the

text: classical

text: case

text: tive

text: operators

text: obeying

text: the

text: particle

text: spreads

text: in

text: a

text: diﬀusive

text: manner√(a

text: standard

text: deviation

text: of

text: its

text: position

text: is

text: proportional

text: to

text: t)

text: and

text: after

text: n

text: many

text: steps

text: a

text: spatial

text: probability

text: distribution

text: is

text: Gaussian.

text: E

text: In

text: quantum

text: case

text: the

text: system

text: is

text: described

text: by

text: two

text: degrees

text: i

text: i

text: of

text: freedom

text: ψ

text: x,

text: c

text: the

text: position

text: x

text: aconidn

text: ac

text: =tw←o-,l→eve.l

text: Tinhteerenvaolludteiognreeis

text: oufnfirteaerdyoamndkncoonwsnistassoaf

text: in

text: Tthhaeyt

text: dthiﬀeeyr

text: j(eΠctoΠrs

text: Π=i

text: iUtar=y

text: CmTat,rnixamacetlyinga

text: uonnliytaroyn

text: ctohien

text: ctooisns

text: δbie,jΠgrie,abteurt

text: dδiim,j)enasniodn

text: tohfatthtehesiyrstneummbneir>jcadn.

text: degree

text: of

text: freedom)

text: The

text: following

text: quantum

text: walk

text: algorithm,

text: proposed

text: in

text: generates

text: an

text: arbitrary

text: set

text: of

text: rank

text: POVM

text: elements

text: E

text: E

text: (rank

text: elements

text: can

text: be

text: generated

text: with

text: a

text: C

text: cos

text: θ

text: e

text: iβ

text: sin

text: θ

text: venrsion

text: of

text: this

text: algorithm).

text: eiβ

text: sin

text: θ

text: cos

text: θ

text: Initiate

text: the

text: quantum

text: walk

text: at

text: position

text: x

text: with

text: the

text: coin

text: state

text: corresponding

text: to

text: the

text: qubit

text: state

text: one

text: and

text: a

text: translation

text: operation

text: wants

text: to

text: measure.

text: Set

text: i

text: T

text: x

text: x,

text: x

text: x,

text: While

text: i

text: n

text: do

text: the

text: following:

text: x

text: Apply

text: coin

text: operation

text: at

text: position

text: x

text: Quantum

text: walks

text: with

text: uniform

text: (position-independent)

text: and

text: identity

text: elsewhere

text: aind

text: then

text: apply

text: trans-

text: coin

text: operation

text: spread

text: ballistically

text: (standard

text: deviation

text: lation

text: operator

text: T.

text: proportional

text: to

text: t)

text: and

text: its

text: probability

text: distribution

text: diﬀers

text: form

text: the

text: classical

text: Gaussian

text: shape

text: On

text: the

text: other

text: Apply

text: coin

text: operation

text: at

text: position

text: x

text: i

text: hand,

text: quantum

text: walks

text: with

text: position-dependent

text: coin

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text: eration

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text: θx

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text: and

text: i

text: i

text: ##LTLine##


text: 3 Since in this work we are only interested in the measure- tion operator as given by Eq.(2) is realized by the de- ment statistics, and not in the post-measurement states, signed BD, that does not displace the vertical polarized | (cid:105) → | − (cid:105) we simpliﬁed the algorithm in [12] and omitted the last photons ( x, V x 1, V ) but makes the horizon- step. The POVM elements that are generated depend tal polarized ones undergo a 4 mm lateral displacement | (cid:105) → | (cid:105) solely on the form of operators C(1) and C(2). ( x, H x + 1, H ). i i IV. EXPERIMENT A. Trine POVM In our experiment, frequency-doubled femtosecond The experimental setup in Fig. 1(a) is used to con- | (cid:105)(cid:104) | pulse (390 nm, 76 MHz repetition rate, 80 mw aver- struct the trine POVMs, 2 ψi ψi (i = 1, 2, 3), age power) from a mode-locked Ti:sapphire laser pump a 3 3 3 × × type-I beta-barium-borate (BBO, 6.0 6.0 2.0 mm3, θ= | (cid:105) | (cid:105) 29.9) crystal to produces the degenerate photon pairs. ψ1 = H 3 √ After being redirected by the mirrors (M1 and M2, as | (cid:105) −1 | (cid:105) − | (cid:105) in the Fig. 1(a)) and the interference ﬁlters (IF, (cid:52)λ=3 ψ32 = 2( H √3 V ) (5) nm, λ=780 nm), the photon pairs generated in the spon- | (cid:105) −1 | (cid:105) | (cid:105) taneous parametric down-conversion (SPDC) process are ψ3 = ( H + 3 V ). 3 2 coupled into single-mode ﬁbers separately. Single photon state is prepared by triggering on one of these two pho- According to the settings of the coin operators, the tons, and the coincidence counting rate collected by the optical axes of BD1 and BD2 must be aligned, in other × avalanche photo-diodes (APD) are about 4 104 in one words, they form an interferometer. When rotating minute. ◦ HWP1 and HWP3 by 22.5 , we observed that the in- terference visibility of the interferometer was about 99.8 and the system was stable over 2.5 hours of timescale. IIFF HHWWPP QQWWPP BBBBOO PPBBSS BBDD AAPPDD After aligning, we begin to set the corresponding coin ((aa)) ttrriiggggeerr HHWWPP11 HHWWPP22 004422 operators in each step. For (cid:114)the T(cid:18)rine POVM(cid:19)s, we have √ HHWWPP33 NNOOTT MM22 BBDD11 NNOOTT BBDD22 BBDD33 BBDD44 (cid:114) (cid:18) (cid:19)1 2 √1 ppuummpp MM11 C(1) = 1, C(2) = − , 1 1 3 1 2 (6) CCCoooiiinnnccciiidddeeennnccceee DDDeeecccttteeeccctttiiiooonnn 1 1 1 ((bb)) C(1) = − , C(2) = 1, 2 2 1 1 2 66 44 HHWWPP22 HHWWPP44 HHWWPP55 0022 where C1(2) and C◦2(1) are rea◦lized by rotating HWP2 HHWWPP11 and HWP3 by 17.32 and 22.5 respectively. The initial QQWWPP22 NNOOTT | (cid:105) QQWWPP11 BBDD11NNOOTT BBDD22HHWWPP33 BBDD33NNOOTT BBDD44 BBDD55 BBDD66 tbryin0e◦,co−in30s◦taatneds 3ψ03i◦, awrheicleontshteruacntteid-trbiynerosttaattiensg|ψH¯iW(cid:105) aPr1e ◦ ◦ − ◦ 3 constructed by rotating it by 45 , 15 and 15 . At last, every output port’s detection eﬃciency are calibrated so FIG. 1: Experimental setup. (a) Experimental setup for con- | (cid:105) that the diﬀerences among them are below 5%. structing the trine POVMs corresponding to ψ3i and realiz- Fig. 2 shows that the results in our experiment agree ing the unambiguous state discrimination. (b) Optical net- work for constructing SIC-POVMs. Initial coin states are with theoretical predictions. |Th(cid:105)e r|ati(cid:105)os 2/3 :(cid:104)1/6| : 1(cid:105)/6 prepared by passing the single photons through a polarizing (0 : 1/2 : 1/2) for the cases of ψi ( ψ¯i , where ψi ψ¯i = 3 3 3 3 beam splitter (PBS), a half-wave plate (HWP) HWP1 and a 0) are given in theory and the detailed numerical results quarter-wave plate (QWP) QWP1 in a speciﬁc conﬁguration. of the probability distributions can be found in Table II The conditional position shifts are implemented by Beam Dis- and III. To visualize that the setup has constructed the placers (BDs) and the coin operators in diﬀerent positions are Trine POVMs, it is important to demonstrate that we | (cid:105) | (cid:105) realized by wave plates with diﬀerent angles (Table 1). The cannot ﬁnd states ψ¯1 in position 4, ψ¯2 in position 0 indices in the ﬁgure denote the position of walker. and |ψ¯3(cid:105) in position 23. Fig. 2(b) and Tab3le III show that 3 the probabilities of these events are indeed very close to One-dimensional discrete time quantum walk system zero, with an average value of 0.0085. In addition, the | (cid:105) has two degrees of freedom, x and c, x is the position results for states ψi also indicate the coeﬃcients of the 3 of the particle and c is the state of the coin. In our POVM we constructed is 2, see Fig. 2(a). The errors 3 experiment they are encoded in the longitudinal spa- in our experiment mainly stem from the imperfect wave | (cid:105) | (cid:105) tial modes and polarizations H , V of the single pho- plates and the interferometers and the counting statistics tons respectively. In this case, the conditional transla- of the photons. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Since

text: in

text: this

text: work

text: we

text: are

text: only

text: interested

text: in

text: the

text: measure-

text: tion

text: operator

text: as

text: given

text: by

text: is

text: realized

text: by

text: the

text: de-

text: ment

text: statistics,

text: and

text: not

text: in

text: the

text: post-measurement

text: states,

text: signed

text: BD,

text: that

text: does

text: not

text: displace

text: the

text: vertical

text: polarized

text: we

text: simpliﬁed

text: the

text: algorithm

text: in

text: and

text: omitted

text: the

text: last

text: photons

text: x,

text: V

text: x

text: V

text: but

text: makes

text: the

text: horizon-

text: step.

text: The

text: POVM

text: elements

text: that

text: are

text: generated

text: depend

text: tal

text: polarized

text: ones

text: undergo

text: a

text: mm

text: lateral

text: displacement

text: solely

text: on

text: the

text: form

text: of

text: operators

text: and

text: x,

text: H

text: x

text: H

text: i

text: i

text: IV.

text: EXPERIMENT

text: A.

text: Trine

text: POVM

text: In

text: our

text: experiment,

text: frequency-doubled

text: femtosecond

text: The

text: experimental

text: setup

text: in

text: Fig.

text: is

text: used

text: to

text: con-

text: pulse

text: nm,

text: MHz

text: repetition

text: rate,

text: mw

text: aver-

text: struct

text: the

text: trine

text: POVMs,

text: ψi

text: ψi

text: (i

text: age

text: power)

text: from

text: a

text: mode-locked

text: Ti:sapphire

text: laser

text: pump

text: a

text: type-I

text: beta-barium-borate

text: (BBO,

text: θ=

text: crystal

text: to

text: produces

text: the

text: degenerate

text: photon

text: pairs.

text: ψ1

text: H

text: After

text: being

text: redirected

text: by

text: the

text: mirrors

text: and

text: as

text: −1

text: in

text: the

text: Fig.

text: and

text: the

text: interference

text: ﬁlters

text: (IF,

text: H

text: √3

text: V

text: nm,

text: nm),

text: the

text: photon

text: pairs

text: generated

text: in

text: the

text: spon-

text: −1

text: taneous

text: parametric

text: down-conversion

text: (SPDC)

text: process

text: are

text: ψ3

text: H

text: V

text: coupled

text: into

text: single-mode

text: ﬁbers

text: separately.

text: Single

text: photon

text: state

text: is

text: prepared

text: by

text: triggering

text: on

text: one

text: of

text: these

text: two

text: pho-

text: According

text: to

text: the

text: settings

text: of

text: the

text: coin

text: operators,

text: the

text: tons,

text: and

text: the

text: coincidence

text: counting

text: rate

text: collected

text: by

text: the

text: optical

text: axes

text: of

text: and

text: must

text: be

text: aligned,

text: in

text: other

text: avalanche

text: photo-diodes

text: (APD)

text: are

text: about

text: in

text: one

text: words,

text: they

text: form

text: an

text: interferometer.

text: When

text: rotating

text: minute.

text: and

text: by

text: we

text: observed

text: that

text: the

text: in-

text: terference

text: visibility

text: of

text: the

text: interferometer

text: was

text: about

text: and

text: the

text: system

text: was

text: stable

text: over

text: hours

text: of

text: timescale.

text: IIFF

text: HHWWPP

text: QQWWPP

text: BBBBOO

text: PPBBSS

text: BBDD

text: AAPPDD

text: After

text: aligning,

text: we

text: begin

text: to

text: set

text: the

text: corresponding

text: coin

text: ((aa))

text: ttrriiggggeerr

text: operators

text: in

text: each

text: step.

text: For

text: we

text: have

text: NNOOTT

text: NNOOTT

text: √1

text: ppuummpp

text: CCCoooiiinnnccciiidddeeennnccceee

text: DDDeeecccttteeeccctttiiiooonnn

text: ((bb))

text: where

text: and

text: are

text: rea◦lized

text: by

text: rotating

text: and

text: by

text: and

text: respectively.

text: The

text: initial

text: NNOOTT

text: constructed

text: by

text: rotating

text: it

text: by

text: and

text: At

text: last,

text: every

text: output

text: port’s

text: detection

text: eﬃciency

text: are

text: calibrated

text: so

text: FIG.

text: Experimental

text: setup.

text: (a)

text: Experimental

text: setup

text: for

text: con-

text: that

text: the

text: diﬀerences

text: among

text: them

text: are

text: below

text: structing

text: the

text: trine

text: POVMs

text: corresponding

text: to

text: and

text: realiz-

text: Fig.

text: shows

text: that

text: the

text: results

text: in

text: our

text: experiment

text: agree

text: ing

text: the

text: unambiguous

text: state

text: discrimination.

text: (b)

text: Optical

text: net-

text: work

text: for

text: constructing

text: SIC-POVMs.

text: Initial

text: coin

text: states

text: are

text: with

text: theoretical

text: predictions.

text: prepared

text: by

text: passing

text: the

text: single

text: photons

text: through

text: a

text: polarizing

text: for

text: the

text: cases

text: of

text: ψi

text: ψ¯i

text: where

text: ψi

text: ψ¯i

text: beam

text: splitter

text: (PBS),

text: a

text: half-wave

text: plate

text: (HWP)

text: and

text: a

text: are

text: given

text: in

text: theory

text: and

text: the

text: detailed

text: numerical

text: results

text: quarter-wave

text: plate

text: (QWP)

text: in

text: a

text: speciﬁc

text: conﬁguration.

text: of

text: the

text: probability

text: distributions

text: can

text: be

text: found

text: in

text: Table

text: II

text: The

text: conditional

text: position

text: shifts

text: are

text: implemented

text: by

text: Beam

text: Dis-

text: and

text: III.

text: To

text: visualize

text: that

text: the

text: setup

text: has

text: constructed

text: the

text: placers

text: (BDs)

text: and

text: the

text: coin

text: operators

text: in

text: diﬀerent

text: positions

text: are

text: Trine

text: POVMs,

text: it

text: is

text: important

text: to

text: demonstrate

text: that

text: we

text: realized

text: by

text: wave

text: plates

text: with

text: diﬀerent

text: angles

text: (Table

text: The

text: cannot

text: ﬁnd

text: states

text: ψ¯1

text: in

text: position

text: ψ¯2

text: in

text: position

text: indices

text: in

text: the

text: ﬁgure

text: denote

text: the

text: position

text: of

text: walker.

text: and

text: in

text: position

text: Fig.

text: and

text: III

text: show

text: that

text: the

text: probabilities

text: of

text: these

text: events

text: are

text: indeed

text: very

text: close

text: to

text: One-dimensional

text: discrete

text: time

text: quantum

text: walk

text: system

text: zero,

text: with

text: an

text: average

text: value

text: of

text: In

text: addition,

text: the

text: has

text: two

text: degrees

text: of

text: freedom,

text: x

text: and

text: c,

text: x

text: is

text: the

text: position

text: results

text: for

text: states

text: ψi

text: also

text: indicate

text: the

text: coeﬃcients

text: of

text: the

text: of

text: the

text: particle

text: and

text: c

text: is

text: the

text: state

text: of

text: the

text: coin.

text: In

text: our

text: POVM

text: we

text: constructed

text: is

text: see

text: Fig.

text: The

text: errors

text: experiment

text: they

text: are

text: encoded

text: in

text: the

text: longitudinal

text: spa-

text: in

text: our

text: experiment

text: mainly

text: stem

text: from

text: the

text: imperfect

text: wave

text: tial

text: modes

text: and

text: polarizations

text: H

text: V

text: of

text: the

text: single

text: pho-

text: plates

text: and

text: the

text: interferometers

text: and

text: the

text: counting

text: statistics

text: tons

text: respectively.

text: In

text: this

text: case,

text: the

text: conditional

text: transla-

text: of

text: the

text: photons.

text: ##LTFigure##

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text: 6 θ θ C(2) P P 1/2 1 theory e state P0 P2 P4 P6 ◦ (cid:48) ◦ (cid:48) π/20 2 15 2 15 0.0123 0.013(06) ψ41 0.1662(19) 0.1654(19) 0.1934(20) 0.4749(26) ◦ (cid:48) ◦ (cid:48) π/10 4 30 4 34 0.0489 0.050(10) ψ42 0.1585(18) 0.1571(18) 0.5220(27) 0.1625(20) ◦ (cid:48) ◦ (cid:48) 3π/20 6 45 6 57 0.0109 0.103(15) ψ43 0.5015(26) 0.1676(19) 0.1695(19) 0.1614(19) ◦ ◦ (cid:48) π/5 9 9 29 0.191 0.181(19) ψ44 0.1885(20) 0.4843(25) 0.1623(19) 0.1649(19) ◦ (cid:48) ◦ (cid:48) π/4 11 15 12 14 0.293 0.285(22) ◦ (cid:48) ◦ (cid:48) 3π/10 13 30 15 19 0.412 0.402(23) TABLE IV: The nomarized probabilities of counting rates for state ◦ (cid:48) ◦ (cid:48) ψi in position 0, 2, 4 and 6. 7π/20 15 45 18 54 0.546 0.531(24) 4 ◦ ◦ (cid:48) 2π/5 18 23 18 0.691 0.673(22) ◦ (cid:48) ◦ (cid:48) 9π/20 20 15 29 21 0.844 0.832(18) ◦ (cid:48) ◦ π/2 22 30 45 1.000 0.996(03) state P P P P 0 2 4 6 ψ¯1 0.3341(24) 0.3367(25) 0.3289(24) 0.0003(01) 4 TABLE VIII: θ, the angle related to the input state ψ+ which ψ¯42 0.3182(23) 0.3283(24) 0.0051(04) 0.3485(24) is prepared by HWP1 rotated to θ1/2; C12, the angle of HWP2 to ψ¯3 0.0040(03) 0.3209(24) 0.3671(25) 0.3080(23) realize the operator C12. Ptheory and Pe represent the theoretical 4 and the experimental successful probability. ψ¯4 0.3152(24) 0.0005(01) 0.3647(24) 0.3196(24) 4 TABLE V: The nomarized probabilities of counting rates for state θ θ C(2) P P 1/2 1 theory e ψ¯i in position 0, 2, 4 and 6. − − ◦ (cid:48) ◦ (cid:48) 4 π/20 2 15 2 15 0.0123 0.0127(05) − − ◦ (cid:48) ◦ (cid:48) π/10 4 30 4 34 0.0489 0.0511(11) − − ◦ (cid:48) ◦ (cid:48) ψ1 ψ2 ψ3 ψ4 3π/20 6 45 6 57 0.0109 0.1096(15) ◦4 − ◦4 (cid:48) ◦4 (cid:48) 4◦ −π/5 −9◦ 9◦29(cid:48) 0.191 0.1889(19) HWP1 0 27 22 17 38 45 ◦ ◦ (cid:48) − ◦ (cid:48) − ◦ (cid:48) −π/4 −11◦15(cid:48) 12◦14(cid:48) 0.293 0.2883(23) QWP1 0 35 16 27 22 27 22 − − ◦ (cid:48) ◦ (cid:48) 3π/10 13 30 15 19 0.412 0.4092(24) TABLE VI: The angles of HWP1 and QWP1 used to prepare − − ◦ (cid:48) ◦ (cid:48) 7π/20 15 45 18 54 0.546 0.5340(24) the states ψ4i. − − ◦ ◦ (cid:48) 2π/5 18 23 18 0.691 0.6812(22) − − ◦ (cid:48) ◦ (cid:48) 9π/20 20 15 29 21 0.844 0.8367(17) − − ◦ (cid:48) ◦ ψ¯1 ψ¯2 ψ¯3 ψ¯4 π/2 22 30 45 1.000 0.9951(03) 4 4 4 4 ◦ ◦ (cid:48) − ◦ (cid:48) ◦ HWP1 45 17 38 27 22 0 ◦ ◦ (cid:48) − ◦ (cid:48) − ◦ (cid:48) TABLE IX: θ, the angle related to the input states ψ− which is QWP1 0 35 16 27 22 27 22 prepared by HWP1 rotated to θ ; C2, the angle of HWP2 to 1/2 1 realise the operator C12. Ptheory and Pe represent the theoretical TABLE VII: The angles of HWP1 and QWP1 used to prepare and the experimental successful probability. the states ψ¯i. 4 [1] G. Zauner, Ph.D. thesis, Universitat Wien, (1999). [7] H. Yang, W. Wei, W. Chun-Wang, D. Hong-Yi, and L. [2] M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Cheng-Zu, Chin. Phys. Lett. 25, 4195 (2008). Caves, J. Math. Phys. (N.Y.) 45, 2171 (2004). [8] A. C. Dada, E. Andersson, M. L. Jones, V. M. Kendon, [3] S. M. Barnett and S. Croke, Adv. Opt. Photonics 1, 238 and M. S. Everitt, Phys. Rev. A 83, 042339 (2011). (2009). [9] G. Waldherr, A.C. Dada, P. Neumann, F. Jelezko, E. An- [4] M. E. Deconinck and B. M. Terhal, Phys. Rev. A 81, dersson, and J. Wrachtrup, Phys. Rev. Lett. 109, 180501 062304 (2010). (2012). [5] R. B. M. Clarke, A. Cheﬂes, S. M. Barnett, and E. Riis, [10] Roger B. M. Clarke, Vivien M. Kendon, Anthony Cheﬂes, Phys. Rev. A 63, 040305(R) (2001). Stephen M. Barnett, Erling Riis, and Masahide Sasaki, [6] P. J. Mosley, S. Croke, I. A. Walmsley, and S. M. Barnett, Phys. Rev. A 64, 012303 (2001). Phys. Rev. Lett. 97, 193601 (2006). [11] Roger B. M. Clarke, Anthony Cheﬂes, Stephen M. 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text: We report an order-N approach to compute the Kubo Hall conductivity for disorderd two- dimensional systems reaching tens of millions of orbitals, and realistic values of the applied external magnetic ﬁelds (as low as a few Tesla). A time-evolution scheme is employed to evaluate the Hall conductivity σ using a wavepacket propagation method and a continued fraction expansion for the xy computation of diagonal and oﬀ-diagonal matrix elements of the Green functions. The validity of the method is demonstrated by comparison of results with brute-force diagonalization of the Kubo formula, using (disordered) graphene as system of study. This approach to mesoscopic system sizes is opening an unprecedented perspective for so-called reverse engineering in which the available experimental transport data are used to get a deeper understanding of the microscopic structure of the samples. Besides, this will not only allow addressing subtle issues in terms of resistance standardization of large scale materials (such as wafer scale polycrystalline graphene), but will also enable the discovery of new quantum transport phenomena in complex two-dimensional materials, out of reach with classical methods. PACS numbers: 73.43.-f,72.80.Vp, 73.22.Pr I. INTRODUCTION and the QHE features, with very robust behavior to large disorder19, also present additional peculiarities de- pending on the symmetry breaking aspects conveyed The Hall eﬀect is one of the central phenomenon in by defects.20–24 Recently, we have studied the case of Condensed Matter physics with great importance for oxidized graphene, and have shown the formation of measuring carrier density and mobility in semiconduc- disorder-induced resonant critical states appearing in the tors. The long history started almost 150 years ago when zero-energy Landau level with ﬁnite σ and suggesting E. H. Hall observed a transverse voltage drop over a con- xx ducting bar driven by a longitudinal current1 and it expe- a zero-energy σxy quantized plateau25. The conﬁrmation of such a plateau will demand to revise the description of rienced an important turn in 1980 when K. von Klitzing topological invariant in disordered graphene, an exciting observed a quantized version of this eﬀect, the so-called Quantum Hall Eﬀect (QHE)2. This had an immediate direction of work. However a real space calculation of σ is needed and demand for the development of new impact and triggered huge amount of work which led to xy the deﬁnition of a resistance standard.3 methodology. The eﬀect of disorder and related localization phe- Indeed, the importance of an eﬃcient computational nomenon is central to understand the integer QHE from a methodology for the Hall conductivity σ stands in xy bulk perspective, where σ plateaus develop by varying sharp contrast to the limited possibilities to simulate this xy the charge density, while the plateau region coincide with quantity with usual approaches. One of them starts with a region of vanishing σ .4 This is explained by Anderson the Kubo formula and requires the knowledge of the full xx localization of the wave functions in the Landau levels in- eigenstates of a given system.5. Such formulation allows duced by disorder. The most standard method to treat for advanced analysis26,27 but suﬀers from unfavourable bulk conductivities microscopically is the linear-response cubic scaling with system size which restricts to small theory, or the Kubo formula, which was ﬁrst applied by systems because diagonalization is necessary. All eigen- Aoki and Ando5 to the QHE problem4. Such quantity states are needed and have to be combined to calculate can be connected to measurements in the Corbino geom- σ . This computational limitation makes the analysis xy etry for which electrodes are attached to the inner and of realistic models of disordered samples a priori im- outer perimeters of an annular sample, which allow to possible. Additionally, such approach is restricted to investigate bulk transport physics in the high magnetic unreasonably strong magnetic ﬁelds that exceed by far ﬁeld regime. what is achievable experimentally, and forces the mag- Recently, the QHE has played a crucial role for the netic length scale to dominate over all other length scales ﬁrst unambigous proof of the existence of single atomic of the problem (mean free path, average distance between layers of graphene6–8. Indeed, in graphene, the pecu- impurities,...) limiting the exploration of complex mag- liar Berry phase related to the pseudospin quantum de- netotransport in intermediate or even low-magnetic ﬁeld gree of freedom results in a unique spectrum with four- regimes, thus calling for approaches with improved nu- fold degenerate Landau levels (due to the spin and valley merical performance.28–30 degeneracies), and half-integer QHE with Hall plateaus In this paper, we present a highly eﬃcient order-N emerging at σ = 4e2 (n + 1/2), with integers n. QHE algorithm for the computation of the Kubo Hall con- xy h related research in graphene is a vivid ﬁeld, study- ductivity that allows us to circumvent aforementioned ing electron-electron interaction9,10, superlattices11–14, limitations, and oﬀer fascinating perspectives for explor- graphene functionalization and topological currents. Ad- ing transverse transport phenomena in disordered two- ditional transverse transport phenomena are being stud- dimensional materials with almost arbitrary complexity. ied such as (quantum-) anomaleous Hall eﬀect,15,16, spin A validation of the method on several models of clean or Hall eﬀect17 and quantum spin Hall eﬀect.18 disordered graphene is made by comparing this new time The role of disorder in graphene is also debated evolution method with brute force diagonalization tech- ##LTLine##

text: We

label: abstract

text: report

text: an

text: order-N

text: approach

text: to

text: compute

text: the

text: Kubo

text: Hall

text: conductivity

text: for

text: disorderd

text: two-

text: dimensional

text: systems

text: reaching

text: tens

text: of

text: millions

text: of

text: orbitals,

text: and

text: realistic

text: values

text: of

text: the

text: applied

text: external

text: magnetic

text: ﬁelds

text: (as

text: low

text: as

text: a

text: few

text: Tesla).

text: A

text: time-evolution

text: scheme

text: is

text: employed

text: to

text: evaluate

text: the

text: Hall

text: conductivity

text: σ

text: using

text: a

text: wavepacket

text: propagation

text: method

text: and

text: a

text: continued

text: fraction

text: expansion

text: for

text: the

text: xy

text: computation

text: of

text: diagonal

text: and

text: oﬀ-diagonal

text: matrix

text: elements

text: of

text: the

text: Green

text: functions.

text: The

text: validity

text: of

text: the

text: method

text: is

text: demonstrated

text: by

text: comparison

text: of

text: results

text: with

text: brute-force

text: diagonalization

text: of

text: the

text: Kubo

text: formula,

text: using

text: (disordered)

text: graphene

text: as

text: system

text: of

text: study.

text: This

text: approach

text: to

text: mesoscopic

text: system

text: sizes

text: is

text: opening

text: an

text: unprecedented

text: perspective

text: for

text: so-called

text: reverse

text: engineering

text: in

text: which

text: the

text: available

text: experimental

text: transport

text: data

text: are

text: used

text: to

text: get

text: a

text: deeper

text: understanding

text: of

text: the

text: microscopic

text: structure

text: of

text: the

text: samples.

text: Besides,

text: this

text: will

text: not

text: only

text: allow

text: addressing

text: subtle

text: issues

text: in

text: terms

text: of

text: resistance

text: standardization

text: of

text: large

text: scale

text: materials

text: (such

text: as

text: wafer

text: scale

text: polycrystalline

text: graphene),

text: but

text: will

text: also

text: enable

text: the

text: discovery

text: of

text: new

text: quantum

text: transport

text: phenomena

text: in

text: complex

text: two-dimensional

text: materials,

text: out

text: of

text: reach

text: with

text: classical

text: methods.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: and

text: the

text: QHE

text: features,

text: with

text: very

text: robust

text: behavior

text: to

text: large

text: also

text: present

text: additional

text: peculiarities

text: de-

text: pending

text: on

text: the

text: symmetry

text: breaking

text: aspects

text: conveyed

text: The

text: Hall

text: eﬀect

text: is

text: one

text: of

text: the

text: central

text: phenomenon

text: in

text: by

text: Recently,

text: we

text: have

text: studied

text: the

text: case

text: of

text: Condensed

text: Matter

text: physics

text: with

text: great

text: importance

text: for

text: oxidized

text: graphene,

text: and

text: have

text: shown

text: the

text: formation

text: of

text: measuring

text: carrier

text: density

text: and

text: mobility

text: in

text: semiconduc-

text: disorder-induced

text: resonant

text: critical

text: states

text: appearing

text: in

text: the

text: tors.

text: The

text: long

text: history

text: started

text: almost

text: years

text: ago

text: when

text: zero-energy

text: Landau

text: level

text: with

text: ﬁnite

text: σ

text: and

text: suggesting

text: E.

text: H.

text: Hall

text: observed

text: a

text: transverse

text: voltage

text: drop

text: over

text: a

text: con-

text: xx

text: ducting

text: bar

text: driven

text: by

text: a

text: longitudinal

text: and

text: it

text: expe-

text: a

text: zero-energy

text: σxy

text: quantized

text: The

text: conﬁrmation

text: of

text: such

text: a

text: plateau

text: will

text: demand

text: to

text: revise

text: the

text: description

text: of

text: rienced

text: an

text: important

text: turn

text: in

text: when

text: K.

text: von

text: Klitzing

text: topological

text: invariant

text: in

text: disordered

text: graphene,

text: an

text: exciting

text: observed

text: a

text: quantized

text: version

text: of

text: this

text: eﬀect,

text: the

text: so-called

text: Quantum

text: Hall

text: Eﬀect

text: This

text: had

text: an

text: immediate

text: direction

text: of

text: work.

text: However

text: a

text: real

text: space

text: calculation

text: of

text: σ

text: is

text: needed

text: and

text: demand

text: for

text: the

text: development

text: of

text: new

text: impact

text: and

text: triggered

text: huge

text: amount

text: of

text: work

text: which

text: led

text: to

text: xy

text: the

text: deﬁnition

text: of

text: a

text: resistance

text: methodology.

text: The

text: eﬀect

text: of

text: disorder

text: and

text: related

text: localization

text: phe-

text: Indeed,

text: the

text: importance

text: of

text: an

text: eﬃcient

text: computational

text: nomenon

text: is

text: central

text: to

text: understand

text: the

text: integer

text: QHE

text: from

text: a

text: methodology

text: for

text: the

text: Hall

text: conductivity

text: σ

text: stands

text: in

text: xy

text: bulk

text: perspective,

text: where

text: σ

text: plateaus

text: develop

text: by

text: varying

text: sharp

text: contrast

text: to

text: the

text: limited

text: possibilities

text: to

text: simulate

text: this

text: xy

text: the

text: charge

text: density,

text: while

text: the

text: plateau

text: region

text: coincide

text: with

text: quantity

text: with

text: usual

text: approaches.

text: One

text: of

text: them

text: starts

text: with

text: a

text: region

text: of

text: vanishing

text: σ

text: This

text: is

text: explained

text: by

text: Anderson

text: the

text: Kubo

text: formula

text: and

text: requires

text: the

text: knowledge

text: of

text: the

text: full

text: xx

text: localization

text: of

text: the

text: wave

text: functions

text: in

text: the

text: Landau

text: levels

text: in-

text: eigenstates

text: of

text: a

text: given

text: Such

text: formulation

text: allows

text: duced

text: by

text: disorder.

text: The

text: most

text: standard

text: method

text: to

text: treat

text: for

text: advanced

text: but

text: suﬀers

text: from

text: unfavourable

text: bulk

text: conductivities

text: microscopically

text: is

text: the

text: linear-response

text: cubic

text: scaling

text: with

text: system

text: size

text: which

text: restricts

text: to

text: small

text: theory,

text: or

text: the

text: Kubo

text: formula,

text: which

text: was

text: ﬁrst

text: applied

text: by

text: systems

text: because

text: diagonalization

text: is

text: necessary.

text: All

text: eigen-

text: Aoki

text: and

text: to

text: the

text: QHE

text: Such

text: quantity

text: states

text: are

text: needed

text: and

text: have

text: to

text: be

text: combined

text: to

text: calculate

text: can

text: be

text: connected

text: to

text: measurements

text: in

text: the

text: Corbino

text: geom-

text: σ

text: This

text: computational

text: limitation

text: makes

text: the

text: analysis

text: xy

text: etry

text: for

text: which

text: electrodes

text: are

text: attached

text: to

text: the

text: inner

text: and

text: of

text: realistic

text: models

text: of

text: disordered

text: samples

text: a

text: priori

text: im-

text: outer

text: perimeters

text: of

text: an

text: annular

text: sample,

text: which

text: allow

text: to

text: possible.

text: Additionally,

text: such

text: approach

text: is

text: restricted

text: to

text: investigate

text: bulk

text: transport

text: physics

text: in

text: the

text: high

text: magnetic

text: unreasonably

text: strong

text: magnetic

text: ﬁelds

text: that

text: exceed

text: by

text: far

text: ﬁeld

text: regime.

text: what

text: is

text: achievable

text: experimentally,

text: and

text: forces

text: the

text: mag-

text: Recently,

text: the

text: QHE

text: has

text: played

text: a

text: crucial

text: role

text: for

text: the

text: netic

text: length

text: scale

text: to

text: dominate

text: over

text: all

text: other

text: length

text: scales

text: ﬁrst

text: unambigous

text: proof

text: of

text: the

text: existence

text: of

text: single

text: atomic

text: of

text: the

text: problem

text: (mean

text: free

text: path,

text: average

text: distance

text: between

text: layers

text: of

text: Indeed,

text: in

text: graphene,

text: the

text: pecu-

text: impurities,...)

text: limiting

text: the

text: exploration

text: of

text: complex

text: mag-

text: liar

text: Berry

text: phase

text: related

text: to

text: the

text: pseudospin

text: quantum

text: de-

text: netotransport

text: in

text: intermediate

text: or

text: even

text: low-magnetic

text: ﬁeld

text: gree

text: of

text: freedom

text: results

text: in

text: a

text: unique

text: spectrum

text: with

text: four-

text: regimes,

text: thus

text: calling

text: for

text: approaches

text: with

text: improved

text: nu-

text: fold

text: degenerate

text: Landau

text: levels

text: (due

text: to

text: the

text: spin

text: and

text: valley

text: merical

text: degeneracies),

text: and

text: half-integer

text: QHE

text: with

text: Hall

text: plateaus

text: In

text: this

text: paper,

text: we

text: present

text: a

text: highly

text: eﬃcient

text: order-N

text: emerging

text: at

text: σ

text: (n

text: with

text: integers

text: n.

text: QHE

text: algorithm

text: for

text: the

text: computation

text: of

text: the

text: Kubo

text: Hall

text: con-

text: xy

text: h

text: related

text: research

text: in

text: graphene

text: is

text: a

text: vivid

text: ﬁeld,

text: study-

text: ductivity

text: that

text: allows

text: us

text: to

text: circumvent

text: aforementioned

text: ing

text: electron-electron

text: limitations,

text: and

text: oﬀer

text: fascinating

text: perspectives

text: for

text: explor-

text: graphene

text: functionalization

text: and

text: topological

text: currents.

text: Ad-

text: ing

text: transverse

text: transport

text: phenomena

text: in

text: disordered

text: two-

text: ditional

text: transverse

text: transport

text: phenomena

text: are

text: being

text: stud-

text: dimensional

text: materials

text: with

text: almost

text: arbitrary

text: complexity.

text: ied

text: such

text: as

text: (quantum-)

text: anomaleous

text: Hall

text: spin

text: A

text: validation

text: of

text: the

text: method

text: on

text: several

text: models

text: of

text: clean

text: or

text: Hall

text: and

text: quantum

text: spin

text: Hall

text: disordered

text: graphene

text: is

text: made

text: by

text: comparing

text: this

text: new

text: time

text: The

text: role

text: of

text: disorder

text: in

text: graphene

text: is

text: also

text: debated

text: evolution

text: method

text: with

text: brute

text: force

text: diagonalization

text: tech-

text: ##LTLine##


text: plotroc( sim.obj, bdgraph.obj, bdgraph.obj2 = NULL, cut.num = 20, smooth = FALSE ) provides a ROC plot for visualization comparison based on the estimated pos- terior link inclusion probabilities. 4.5 Data simulation The function bdgraph.sim is designed to simulate diﬀerent types of datasets with various graph structures. The function bdgraph.sim( n = 2, p = 10, type = "Gaussian", graph = "random", prob = 0.2, size = NULL, mean = 0, class = NULL, cut = 4, b = 3, D = diag(p), K = NULL, sigma = NULL, vis = FALSE ) can simulate multivariate Gaussian, non-Gaussian, discrete and mixed data with diﬀerent undirected graph structures, including “random”, “cluster”, “hub”, “scale-free”, “circle” and “ﬁxed” graphs. Users can specify the type of multi- variate data by option type and the graph structure by option graph. They can determine the sparsity level of the obtained graph by using option prob. With this function users can generate mixed data from “count”, “ordinal”, “binary”, “Gaussian” and “non-Gaussian” distributions. bdgraph.sim returns an object of the S3 class type “sim”. Functions plot and print work with this object type. 4.6 An example on simulated data We illustrate the user interface of the BDgraph package by use of a simple simulation; for extensive simulation study see Section . We perform all the com- putations on an Intel(R) Core(TM) i5 CPU 2.67GHz processor. By using the function bdgraph.sim we simulate 70 observations (n = 70) from a multivariate Gaussian distribution with 8 variables (p = 8) and “scale-free” graph structure, as below. R> data.sim <- bdgraph.sim( n = 70, p = 8, type = "Gaussian", graph = "scale-free" ) R> round( head( data.sim $ data, 4 ), 2 ) [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [1,] -0.74 1.10 0.47 -0.89 -0.40 0.10 -0.30 -0.49 [2,] -0.23 -0.37 -0.38 -0.53 -0.05 0.36 -0.53 -1.02 [3,] 0.17 -0.72 -0.20 -0.01 -0.45 -0.12 -0.47 -0.28 [4,] 0.83 -2.02 0.24 1.32 0.32 -0.08 -0.07 -0.88 Since the generated data are Gaussian, we run the BDMCMC algorithm which is based on Gaussian graphical models. For this we choose method = "ggm", as follows: R> sample.bdmcmc <- bdgraph( data = data.sim, method = "ggm", iter = 5000, save.all = TRUE ) We choose option “save.all = TRUE” to save the samples in order to check convergence of the algorithm. Running this function takes around 3 seconds, as the computational intensive tasks are performed in C++ and interfaced with R. Since the function bdgraph returns an object of class S3, users can see the summary result as follows 13

text: plotroc(

text: sim.obj,

text: bdgraph.obj,

text: NULL,

text: cut.num

text: smooth

text: FALSE

text: provides

text: a

text: ROC

text: plot

text: for

text: visualization

text: comparison

text: based

text: on

text: the

text: estimated

text: pos-

text: terior

text: link

text: inclusion

text: probabilities.

label: section

text: Data

text: simulation

text: The

text: function

text: bdgraph.sim

text: is

text: designed

text: to

text: simulate

text: diﬀerent

text: types

text: of

text: datasets

text: with

text: various

text: graph

text: structures.

text: The

text: function

text: bdgraph.sim(

text: n

text: p

text: type

text: "Gaussian",

text: graph

text: "random",

text: prob

text: size

text: NULL,

text: mean

text: class

text: NULL,

text: cut

text: b

text: D

text: diag(p),

text: K

text: NULL,

text: sigma

text: NULL,

text: vis

text: FALSE

text: can

text: simulate

text: multivariate

text: Gaussian,

text: non-Gaussian,

text: discrete

text: and

text: mixed

text: data

text: with

text: diﬀerent

text: undirected

text: graph

text: structures,

text: including

text: “random”,

text: “cluster”,

text: “hub”,

text: “scale-free”,

text: “circle”

text: and

text: “ﬁxed”

text: graphs.

text: Users

text: can

text: specify

text: the

text: type

text: of

text: multi-

text: variate

text: data

text: by

text: option

text: type

text: and

text: the

text: graph

text: structure

text: by

text: option

text: graph.

text: They

text: can

text: determine

text: the

text: sparsity

text: level

text: of

text: the

text: obtained

text: graph

text: by

text: using

text: option

text: prob.

text: With

text: this

text: function

text: users

text: can

text: generate

text: mixed

text: data

text: from

text: “count”,

text: “ordinal”,

text: “binary”,

text: “Gaussian”

text: and

text: “non-Gaussian”

text: distributions.

text: bdgraph.sim

text: returns

text: an

text: object

text: of

text: the

text: class

text: type

text: “sim”.

text: Functions

text: plot

text: and

text: print

text: work

text: with

text: this

text: object

text: type.

text: An

text: example

text: on

text: simulated

text: data

text: We

text: illustrate

text: the

text: user

text: interface

text: of

text: the

text: BDgraph

text: package

text: by

text: use

text: of

text: a

text: simple

text: simulation;

text: for

text: extensive

text: simulation

text: study

text: see

text: Section

text: We

text: perform

text: all

text: the

text: com-

text: putations

text: on

text: an

text: Intel(R)

text: Core(TM)

text: CPU

text: processor.

text: By

text: using

text: the

text: function

text: bdgraph.sim

text: we

text: simulate

text: observations

text: (n

text: from

text: a

text: multivariate

text: Gaussian

text: distribution

text: with

text: variables

text: (p

text: and

text: “scale-free”

text: graph

text: structure,

text: as

text: below.

text: R>

text: data.sim

text: bdgraph.sim(

text: n

text: p

text: type

text: "Gaussian",

text: graph

text: "scale-free"

text: R>

text: round(

text: head(

text: data.sim

text: data,

text: Since

text: the

text: generated

text: data

text: are

text: Gaussian,

text: we

text: run

text: the

text: BDMCMC

text: algorithm

text: which

text: is

text: based

text: on

text: Gaussian

text: graphical

text: models.

text: For

text: this

text: we

text: choose

text: method

text: "ggm",

text: as

text: follows:

text: R>

text: sample.bdmcmc

text: bdgraph(

text: data

text: data.sim,

text: method

text: "ggm",

text: iter

text: save.all

text: TRUE

text: We

text: choose

text: option

text: “save.all

text: TRUE”

text: to

text: save

text: the

text: samples

text: in

text: order

text: to

text: check

text: convergence

text: of

text: the

text: algorithm.

text: Running

text: this

text: function

text: takes

text: around

text: seconds,

text: as

text: the

text: computational

text: intensive

text: tasks

text: are

text: performed

text: in

text: C++

text: and

text: interfaced

text: with

text: R.

text: Since

text: the

text: function

text: bdgraph

text: returns

text: an

text: object

text: of

text: class

text: users

text: can

text: see

text: the

text: summary

text: result

text: as

text: follows


text: etaR evitsoP eurT 0.1 8.0 6.0 4.0 2.0 0.0 ROC Curve BDMCMC RJMCMC 0.0 0.2 0.4 0.6 0.8 1.0 False Postive Rate Figure 4: ROC plot to compare the performance of the BDMCMC and RJMCMC algorithms for a simulated toy example. true positive 7 5 5 true negative 21 18 19 false positive 0 3 2 false negative 0 2 2 true positive rate 1 0.71 0.71 false positive rate 0 0.14 0.10 accuracy 1 0.82 0.86 balanced F-score 1 0.67 0.71 positive predictive 1 0.63 0.71 The results show that for this speciﬁc simulated example both algorithms have more or less the same performance; for comprehensive comparison see next section. In this simulation example, we run both BDMCMC and RJMCMC algo- rithms for 10, 000 iterations, 5, 000 of them as burn-in. To check whether the number of iterations is enough and to monitoring the convergence of our both algorithm, we run R> plotcoda( sample.bdmcmc ) R> plotcoda( sample.rjmcmc ) The results in Figure 5 indicate that our BDMCMC algorithm converges faster with compare with RJMCMC algorithm. 5 Extensive simulation study In this section we present a comprehensive simulation study to investigate the performance and capability of the Bayesian methods implemented in our BDgraph package and compare them with two frequentist approaches, which are available in the huge package (Zhao et al., 2014, 2012). All the simulation studies were performed in parallel on a 450 batch nodes with 12 cores and 24 GB of memory, running Linux. We consider 6 various kinds of synthetic graphical models with p nodes: 16 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: etaR

font: default

text: evitsoP

text: eurT

text: ROC

text: Curve

text: BDMCMC

text: RJMCMC

text: False

text: Postive

text: Rate

text: Figure

text: ROC

text: plot

text: to

text: compare

text: the

text: performance

text: of

text: the

text: BDMCMC

text: and

text: RJMCMC

text: algorithms

text: for

text: a

text: simulated

text: toy

text: example.

text: true

text: positive

text: true

text: negative

text: false

text: positive

text: false

text: negative

text: true

text: positive

text: rate

text: false

text: positive

text: rate

text: accuracy

text: balanced

text: F-score

text: positive

text: predictive

text: The

text: results

text: show

text: that

text: for

text: this

text: speciﬁc

text: simulated

text: example

text: both

text: algorithms

text: have

text: more

text: or

text: less

text: the

text: same

text: performance;

text: for

text: comprehensive

text: comparison

text: see

text: next

text: section.

text: In

text: this

text: simulation

text: example,

text: we

text: run

text: both

text: BDMCMC

text: and

text: RJMCMC

text: algo-

text: rithms

text: for

text: iterations,

text: of

text: them

text: as

text: burn-in.

text: To

text: check

text: whether

text: the

text: number

text: of

text: iterations

text: is

text: enough

text: and

text: to

text: monitoring

text: the

text: convergence

text: of

text: our

text: both

text: algorithm,

text: we

text: run

text: R>

text: plotcoda(

text: sample.bdmcmc

text: R>

text: plotcoda(

text: sample.rjmcmc

text: The

text: results

text: in

text: Figure

text: indicate

text: that

text: our

text: BDMCMC

text: algorithm

text: converges

text: faster

text: with

text: compare

text: with

text: RJMCMC

text: algorithm.

text: Extensive

text: simulation

text: study

text: In

text: this

text: section

text: we

text: present

text: a

text: comprehensive

text: simulation

text: study

text: to

text: investigate

text: the

text: performance

text: and

text: capability

text: of

text: the

text: Bayesian

text: methods

text: implemented

text: in

text: our

text: BDgraph

text: package

text: and

text: compare

text: them

text: with

text: two

text: frequentist

text: approaches,

text: which

text: are

text: available

text: in

text: the

text: huge

text: package

text: (Zhao

text: et

text: al.,

text: All

text: the

text: simulation

text: studies

text: were

text: performed

text: in

text: parallel

text: on

text: a

text: batch

text: nodes

text: with

text: cores

text: and

text: GB

text: of

text: memory,

text: running

text: Linux.

text: We

text: consider

text: various

text: kinds

text: of

text: synthetic

text: graphical

text: models

text: with

text: p

text: nodes:

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