text: 8 I.D. Roberts et al. 1.5 1.0 Low-M (r =0.25) s High-M (r =0.50) s 0.8 Total (r =0.42) 1.0 s 0.6 0.5 0.4 0.2 0.0 w vs. A2 0.0 0.00 0.01 0.02 0.03 3 2 1 0 Normalization log A phot Figure 7. Same as Fig. 6 however for the centroid shift instead Figure 8. Estimated probability of a cluster being non-Gaussian of photon asymmetry. as a function of photon asymmetry. Photon asymmetry data points are shown for Gaussian (0, green) and non-Gaussian (1, ± purple) clusters and the black line shows the best-ﬁt logistic curve. have a slightly la±rger median Aphot (0.08 0.03 ) than low- Shaded vertical lines show the median asymmetry and 1σ stan- mass halos (0.04 0.01 ). dard error for Gaussian and non-Gaussian groups. 4.2.2 Centroid shift In Fig. 7 we now show the power-law ﬁt results to the − w A2 relationship for the two halo mass subsamples. We 1.0 ﬁnd qualitatively similar results when considering centroid shift instead of photon asymmetry, with the high-mass ha- 0.8 los displaying a clear correlation (slope: 0.86+−0.27 , Spearman 0.24 p-value: 0.005 ) whereas the correlation for low-mass halos 0.6 is marginal and not statistically signiﬁcant (slope: 0.24+−0.35, 0.30 Spearman p-value: 0.19 ). We also ﬁnd that the median cen- ± 0.4 troid shift is larger for high-mass halos (0.015 0.002 ) than ± low-mass halos (0.007 0.001). 0.2 4.3 The discrete case: X-ray asymmetry of Gaussian and non-Gaussian groups 0.0 Thus far we have treated the shape of the velocity distri- 3.0 2.5 2.0 1.5 1.0 bution in a continuous fashion with the AD statistic, though log w it is commonplace in the literature to deﬁne a dichotomy between “Gaussian” and “non-Gaussian” clusters (Hou et al. Figure 9. Same as Fig. 8 however for the centroid shift instead 2009; Ribeiro et al. 2013; Roberts & Parker 2017; de Car- of photon asymmetry. valho et al. 2017). We use the AD test and choose a critical p-value of 0.10 to deﬁne G and NG groups – where G groups ≥ have pAD 0.10 and NG groups have pAD < 0.10 (though our results are not sensitive to the precise p-value chosen over a where β0 and β1 are parameters of the ﬁt, and for this work reasonable range). we have pˆ = pˆ(NG) and x = logAphot or logw. To quantify the relationship between photon asymme- 4.3.1 Photon asymmetry try and whether a cluster is classiﬁed as G or NG we employ In Fig. 8 we show the photon asymmetry for G (0, green) the method of logistic regression (e.g. Cox 1958). Logistic and NG (1, purple) clusters along with the best-ﬁtting lo- regression is a classiﬁcation tool used to estimate the prob- gistic curve (black line, equation 9) describing the proba- ability of a binary response as a function of one (or many) independent variables, which may be numeric or categorical. bility of being classiﬁed as NG as a function of Aphot. It is clear from Fig. 8 that the probability of a cluster being For this application, a galaxy cluster is classiﬁed as either NG increases with photon asymmetry, we obtain a best-ﬁt G or NG (the boolean, dependent variable) and we are in- ± terested in the probability of a galaxy cluster being NG as coeﬃcient of β1 = 2.1 0.8 indicating a signiﬁcant correla- a function of photon asymmetry or centroid shift (the nu- tion at 2.6σ. According to our logistic model, the Aphot value where the probability of being a NG cluster reaches 50 per meric, independent variable). The estimated probability is then cent is Aphot = 0.14 and the asymmetry where the probability reaches 75 per cent is Aphot = 0.46. Additionally in Fig. 8 we show the median photon asymmetry and the 1σ standard pˆ = eβ1x+β0 , (9) error for G and NG clus±ters, NG clusters have a±larger me- 1+eβ1x+β0 dian asymmetry of 0.13 0.03 compared to 0.05 0.02 for G clusters. 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text: 10 I.D. Roberts et al. massive systems. This result hints that low-mass and high- these relationships for correlations leaves the Spearman cor- mass clusters may follow slightly diﬀerent scaling relations relation coeﬃcient virtually unchanged from Sections and when it comes to Aphot or w versus A2, though a larger sample , suggesting that ﬁbre collision galaxies are not biasing the is necessary to build up the statistics required to conclude results. this with high conﬁdence. In principle, this diﬀerence could In Section we ﬁnd no evidence for a correlation between be explained through simple hierarchical growth where low- Aphot or w and δRMMG, which suggests that the MMG oﬀset is mass halos are on average more virialized than higher-mass not a reliable tracer of cluster relaxation. One caveat which is clusters at the present day. High-mass clusters will be more important to consider is the assumptions made to justify the recently formed through mergers and accretion which can in use of M2/M1 and δRMMG as relaxation proxies, in particu- turn increase Aphot and w (e.g. Cassano et al. 2010; Nurgaliev lar that for relaxed systems the MMG (or brightest galaxy) et al. 2017). From the optical perspective, we also ﬁnd that resides at rest at the centre of the dark matter potential high-mass clusters have velocity distributions which are less – the so-called central galaxy paradigm (CGP). For exam- Gaussian than low-mass systems, in agreement with previ- ple, if the MMG is instead a satellite galaxy then the use of ous studies (Roberts & Parker 2017; de Carvalho et al. 2017). M2/M1 as a relaxation measure may not be valid as it is pred- Although it is important to note that it is easier to sta- icated on the MMG being the central and growing through tistically identify departures from normality for high-mass accretion and mergers at the centre of the potential well. systems with many members. Similarly, if the MMG is a satellite then its oﬀset from the luminosity-weighted centre would not be expected to trace 5.2 Interpreting MMG-based relaxation cluster relaxation. Many recent studies have called into ques- parameters tion the ubiquity of the CGP by highlighting the fact that a The second optical relaxation proxy that shows a signif- substantial fraction of brightest cluster galaxies (BCGs) are icant correlation with X-ray relaxation proxies is the stellar signiﬁcantly oﬀset from the cluster centroid, both in terms of mass gap between the two most-massive cluster galaxies (see projected distance and velocity (van den Bosch et al. 2005; Section and Section ). The correlations between X-ray re- Coziol et al. 2009; Skibba et al. 2011; Sehgal et al. 2013; laxation proxies and M2/M1 are found to be weaker than for Lauer et al. 2014; Hoshino et al. 2015). In particular, Skibba A2 (especially as measured by the Spearman correlation), et al. (2011) ﬁnd that the fraction of halos where the bright- ∼ pusoetfeunlt)iatlrlaycesrugogfedsitﬀinugsethXa-traMy2m/Mor1pihsoalogpyo.oTrehris(tmhoauygbhesetxil-l ecestntgainlaxloywi-sminasfsahctalaossa(t1e0l1li2the−(1fB≤NCM)(cid:46)ra2ng×e1s0f1r3ohm−1M2(cid:12)5)pteor pected given that satellite galaxies (i.e. the velocity distri- ∼ 40 per cent in high-mass halos (M (cid:38) 5×1013h−1M(cid:12)). Fur- ∼ − bution) and the diﬀuse hot gas proﬁle should both trace thermore, Hoshino et al. (2015) ﬁnd fBNC 20 30 per cent the larger-scale cluster potential relatively directly, whereas for galaxies in redMaPPer clusters, and in terms of velocity central galaxy growth is governed more by dynamical inter- Coziol et al. (2009) show that the median peculiar velocity ∼ actions and gas accretion at the cluster centre. It is also pos- for BCGs in a sample of Abell clusters is one third of the − − sible that the Aphot M2/M1 and w M2/M1 trends are being cluster velocity dispersion. It is plausible that systems where aﬀected by selection eﬀects related to the diﬃculty indenti- the CGP is not valid are diluting stronger trends between fying the true MMG (and second most massive galaxy) in Aphot or w and M2/M1, or perhaps masking trends between these clusters. A particular concern regarding the SDSS is Aphot or w and δRMMG. Unfortunately, identifying systems the impact of ﬁbre collisions in the dense inner regions of where the CGP is violated is diﬃcult on a case-by-case ba- clusters, as it has been estimated that up to 30 per cent of sis, limited by observing in projection, and is generally done clusters may be missing a spectra for the true BCG (Von Der in the statistical sense for large samples (ie. thousands) of Linden et al. 2007). In an attempt to mitigate the eﬀect of ﬁ- groups and clusters (e.g. van den Bosch et al. 2005; Skibba bre collisions we use the systems from sample III in the Yang et al. 2011). Therefore we continue to argue that the AD test group catalogue which attaches redshifts to galaxies that (or some other measure of the velocity distribution shape, lack spectra due to ﬁbre collisions by assigning these galax- see e.g. Ribeiro et al. 2013) is a better optical relaxation ies the redshift of the galaxy it “collided” with. While this proxy as it is not complicated by CGP assumptions. procedure allows the group ﬁnder to include galaxies which 6 SUMMARY otherwise would be missed due to ﬁbre collisions, the trade- oﬀ is uncertainty regarding whether the added galaxies are In this paper we present a comparison between diﬀuse ∼ true group members. 60 per cent of ﬁbre collision galaxies X-ray morphology and cluster relaxation proxies based on − have redshifts within 500kms 1 of the estimated value (Ze- optical measures. Using the Yang et al. (2007) SDSS grou≥p havi et al. 2002), though this still leaves a signiﬁcant number catalogue we match optically identiﬁed clusters with N of ﬁbre collision galaxies which may have true redshifts that 10 members to X-ray observations from both the Chandra diﬀer substantially from the assigned value. To ensure that and XMM-Newton X-ray observatories. With a sample of 58 our results are not being aﬀected by the inclusion of these ﬁ- X-ray matched clusters we compare X-ray asymmetry and − − bre collisi−on galaxies w−e re-test the Aphot M2/M1, w M2/M1 centroid shift to three diﬀerent optical relaxation probes: and Aphot δRMMG, w δRMMG relationships for correlations, the Anderson-Darling statistic, the stellar mass gap, and now removing any systems where the MMG (and in the case the MMG oﬀset. The main conclusions of this work are as of M2/M1, the second-most-massive galaxy as well) is a ﬁbre follows: collision galaxy. 26 per cent of the clusters in the sample 1. We detect a signiﬁcant positive correlation between have an MMG which is a ﬁbre collision galaxy and 39 per X-ray relaxation proxies (photon asymmetry, centroid shift) ∼ − cent of the sample have either the MMG or the second- and Anderson-Darling statistic at 3 4σ as measured by most-massive galaxy as a ﬁbre collision galaxy. Re-testing both a power-law ﬁt and by the Spearman correlation test, MNRAS 000, 1–12 (2017)

label: paragraph

text: I.D.

text: Roberts

text: et

text: al.

text: massive

text: systems.

text: This

text: result

text: hints

text: that

text: low-mass

text: and

text: high-

text: these

text: relationships

text: for

text: correlations

text: leaves

text: the

text: Spearman

text: cor-

text: mass

text: clusters

text: may

text: follow

text: slightly

text: diﬀerent

text: scaling

text: relations

text: relation

text: coeﬃcient

text: virtually

text: unchanged

text: from

text: Sections

text: and

text: when

text: it

text: comes

text: to

text: Aphot

text: or

text: w

text: versus

text: though

text: a

text: larger

text: sample

text: suggesting

text: that

text: ﬁbre

text: collision

text: galaxies

text: are

text: not

text: biasing

text: the

text: is

text: necessary

text: to

text: build

text: up

text: the

text: statistics

text: required

text: to

text: conclude

text: results.

text: this

text: with

text: high

text: conﬁdence.

text: In

text: principle,

text: this

text: diﬀerence

text: could

text: In

text: Section

text: we

text: ﬁnd

text: no

text: evidence

text: for

text: a

text: correlation

text: between

text: be

text: explained

text: through

text: simple

text: hierarchical

text: growth

text: where

text: low-

text: Aphot

text: or

text: w

text: and

text: δRMMG,

text: which

text: suggests

text: that

text: the

text: MMG

text: oﬀset

text: is

text: mass

text: halos

text: are

text: on

text: average

text: more

text: virialized

text: than

text: higher-mass

text: not

text: a

text: reliable

text: tracer

text: of

text: cluster

text: relaxation.

text: One

text: caveat

text: which

text: is

text: clusters

text: at

text: the

text: present

text: day.

text: High-mass

text: clusters

text: will

text: be

text: more

text: important

text: to

text: consider

text: is

text: the

text: assumptions

text: made

text: to

text: justify

text: the

text: recently

text: formed

text: through

text: mergers

text: and

text: accretion

text: which

text: can

text: in

text: use

text: of

text: and

text: δRMMG

text: as

text: relaxation

text: proxies,

text: in

text: particu-

text: turn

text: increase

text: Aphot

text: and

text: w

text: (e.g.

text: Cassano

text: et

text: al.

text: Nurgaliev

text: lar

text: that

text: for

text: relaxed

text: systems

text: the

text: MMG

text: (or

text: brightest

text: galaxy)

text: et

text: al.

text: From

text: the

text: optical

text: perspective,

text: we

text: also

text: ﬁnd

text: that

text: resides

text: at

text: rest

text: at

text: the

text: centre

text: of

text: the

text: dark

text: matter

text: potential

text: high-mass

text: clusters

text: have

text: velocity

text: distributions

text: which

text: are

text: less

text: the

text: so-called

text: central

text: galaxy

text: paradigm

text: (CGP).

text: For

text: exam-

text: Gaussian

text: than

text: low-mass

text: systems,

text: in

text: agreement

text: with

text: previ-

text: ple,

text: if

text: the

text: MMG

text: is

text: instead

text: a

text: satellite

text: galaxy

text: then

text: the

text: use

text: of

text: ous

text: studies

text: (Roberts

text: Parker

text: de

text: Carvalho

text: et

text: al.

text: as

text: a

text: relaxation

text: measure

text: may

text: not

text: be

text: valid

text: as

text: it

text: is

text: pred-

text: Although

text: it

text: is

text: important

text: to

text: note

text: that

text: it

text: is

text: easier

text: to

text: sta-

text: icated

text: on

text: the

text: MMG

text: being

text: the

text: central

text: and

text: growing

text: through

text: tistically

text: identify

text: departures

text: from

text: normality

text: for

text: high-mass

text: accretion

text: and

text: mergers

text: at

text: the

text: centre

text: of

text: the

text: potential

text: well.

text: systems

text: with

text: many

text: members.

text: Similarly,

text: if

text: the

text: MMG

text: is

text: a

text: satellite

text: then

text: its

text: oﬀset

text: from

text: the

text: luminosity-weighted

text: centre

text: would

text: not

text: be

text: expected

text: to

text: trace

text: Interpreting

text: MMG-based

text: relaxation

text: cluster

text: relaxation.

text: Many

text: recent

text: studies

text: have

text: called

text: into

text: ques-

text: parameters

text: tion

text: the

text: ubiquity

text: of

text: the

text: CGP

text: by

text: highlighting

text: the

text: fact

text: that

text: a

text: The

text: second

text: optical

text: relaxation

text: proxy

text: that

text: shows

text: a

text: signif-

text: substantial

text: fraction

text: of

text: brightest

text: cluster

text: galaxies

text: (BCGs)

text: are

text: icant

text: correlation

text: with

text: X-ray

text: relaxation

text: proxies

text: is

text: the

text: stellar

text: signiﬁcantly

text: oﬀset

text: from

text: the

text: cluster

text: centroid,

text: both

text: in

text: terms

text: of

text: mass

text: gap

text: between

text: the

text: two

text: most-massive

text: cluster

text: galaxies

text: (see

text: projected

text: distance

text: and

text: velocity

text: (van

text: den

text: Bosch

text: et

text: al.

text: Section

text: and

text: Section

text: The

text: correlations

text: between

text: X-ray

text: re-

text: Coziol

text: et

text: al.

text: Skibba

text: et

text: al.

text: Sehgal

text: et

text: al.

text: laxation

text: proxies

text: and

text: are

text: found

text: to

text: be

text: weaker

text: than

text: for

text: Lauer

text: et

text: al.

text: Hoshino

text: et

text: al.

text: In

text: particular,

text: Skibba

text: (especially

text: as

text: measured

text: by

text: the

text: Spearman

text: correlation),

text: et

text: al.

text: ﬁnd

text: that

text: the

text: fraction

text: of

text: halos

text: where

text: the

text: bright-

text: pected

text: given

text: that

text: satellite

text: galaxies

text: (i.e.

text: the

text: velocity

text: distri-

text: per

text: cent

text: in

text: high-mass

text: halos

text: (M

text: Fur-

text: bution)

text: and

text: the

text: diﬀuse

text: hot

text: gas

text: proﬁle

text: should

text: both

text: trace

text: thermore,

text: Hoshino

text: et

text: al.

text: ﬁnd

text: fBNC

text: per

text: cent

text: the

text: larger-scale

text: cluster

text: potential

text: relatively

text: directly,

text: whereas

text: for

text: galaxies

text: in

text: redMaPPer

text: clusters,

text: and

text: in

text: terms

text: of

text: velocity

text: central

text: galaxy

text: growth

text: is

text: governed

text: more

text: by

text: dynamical

text: inter-

text: Coziol

text: et

text: al.

text: show

text: that

text: the

text: median

text: peculiar

text: velocity

text: actions

text: and

text: gas

text: accretion

text: at

text: the

text: cluster

text: centre.

text: It

text: is

text: also

text: pos-

text: for

text: BCGs

text: in

text: a

text: sample

text: of

text: Abell

text: clusters

text: is

text: one

text: third

text: of

text: the

text: sible

text: that

text: the

text: Aphot

text: and

text: w

text: trends

text: are

text: being

text: cluster

text: velocity

text: dispersion.

text: It

text: is

text: plausible

text: that

text: systems

text: where

text: aﬀected

text: by

text: selection

text: eﬀects

text: related

text: to

text: the

text: diﬃculty

text: indenti-

text: the

text: CGP

text: is

text: not

text: valid

text: are

text: diluting

text: stronger

text: trends

text: between

text: fying

text: the

text: true

text: MMG

text: (and

text: second

text: most

text: massive

text: galaxy)

text: in

text: Aphot

text: or

text: w

text: and

text: or

text: perhaps

text: masking

text: trends

text: between

text: these

text: clusters.

text: A

text: particular

text: concern

text: regarding

text: the

text: SDSS

text: is

text: Aphot

text: or

text: w

text: and

text: δRMMG.

text: Unfortunately,

text: identifying

text: systems

text: the

text: impact

text: of

text: ﬁbre

text: collisions

text: in

text: the

text: dense

text: inner

text: regions

text: of

text: where

text: the

text: CGP

text: is

text: violated

text: is

text: diﬃcult

text: on

text: a

text: case-by-case

text: ba-

text: clusters,

text: as

text: it

text: has

text: been

text: estimated

text: that

text: up

text: to

text: per

text: cent

text: of

text: sis,

text: limited

text: by

text: observing

text: in

text: projection,

text: and

text: is

text: generally

text: done

text: clusters

text: may

text: be

text: missing

text: a

text: spectra

text: for

text: the

text: true

text: BCG

text: (Von

text: Der

text: in

text: the

text: statistical

text: sense

text: for

text: large

text: samples

text: (ie.

text: thousands)

text: of

text: Linden

text: et

text: al.

text: In

text: an

text: attempt

text: to

text: mitigate

text: the

text: eﬀect

text: of

text: ﬁ-

text: groups

text: and

text: clusters

text: (e.g.

text: van

text: den

text: Bosch

text: et

text: al.

text: Skibba

text: bre

text: collisions

text: we

text: use

text: the

text: systems

text: from

text: sample

text: III

text: in

text: the

text: Yang

text: et

text: al.

text: Therefore

text: we

text: continue

text: to

text: argue

text: that

text: the

text: AD

text: test

text: group

text: catalogue

text: which

text: attaches

text: redshifts

text: to

text: galaxies

text: that

text: (or

text: some

text: other

text: measure

text: of

text: the

text: velocity

text: distribution

text: shape,

text: lack

text: spectra

text: due

text: to

text: ﬁbre

text: collisions

text: by

text: assigning

text: these

text: galax-

text: see

text: e.g.

text: Ribeiro

text: et

text: al.

text: is

text: a

text: better

text: optical

text: relaxation

text: ies

text: the

text: redshift

text: of

text: the

text: galaxy

text: it

text: “collided”

text: with.

text: While

text: this

text: proxy

text: as

text: it

text: is

text: not

text: complicated

text: by

text: CGP

text: assumptions.

text: procedure

text: allows

text: the

text: group

text: ﬁnder

text: to

text: include

text: galaxies

text: which

text: SUMMARY

text: otherwise

text: would

text: be

text: missed

text: due

text: to

text: ﬁbre

text: collisions,

text: the

text: trade-

text: oﬀ

text: is

text: uncertainty

text: regarding

text: whether

text: the

text: added

text: galaxies

text: are

text: In

text: this

text: paper

text: we

text: present

text: a

text: comparison

text: between

text: diﬀuse

text: true

text: group

text: members.

text: per

text: cent

text: of

text: ﬁbre

text: collision

text: galaxies

text: X-ray

text: morphology

text: and

text: cluster

text: relaxation

text: proxies

text: based

text: on

text: have

text: redshifts

text: within

text: of

text: the

text: estimated

text: value

text: (Ze-

text: optical

text: measures.

text: Using

text: the

text: Yang

text: et

text: al.

text: SDSS

text: grou≥p

text: havi

text: et

text: al.

text: though

text: this

text: still

text: leaves

text: a

text: signiﬁcant

text: number

text: catalogue

text: we

text: match

text: optically

text: identiﬁed

text: clusters

text: with

text: N

text: of

text: ﬁbre

text: collision

text: galaxies

text: which

text: may

text: have

text: true

text: redshifts

text: that

text: members

text: to

text: X-ray

text: observations

text: from

text: both

text: the

text: Chandra

text: diﬀer

text: substantially

text: from

text: the

text: assigned

text: value.

text: To

text: ensure

text: that

text: and

text: XMM-Newton

text: X-ray

text: observatories.

text: With

text: a

text: sample

text: of

text: our

text: results

text: are

text: not

text: being

text: aﬀected

text: by

text: the

text: inclusion

text: of

text: these

text: ﬁ-

text: X-ray

text: matched

text: clusters

text: we

text: compare

text: X-ray

text: asymmetry

text: and

text: bre

text: collisi−on

text: galaxies

text: w−e

text: re-test

text: the

text: Aphot

text: w

text: centroid

text: shift

text: to

text: three

text: diﬀerent

text: optical

text: relaxation

text: probes:

text: and

text: Aphot

text: δRMMG,

text: w

text: δRMMG

text: relationships

text: for

text: correlations,

text: the

text: Anderson-Darling

text: statistic,

text: the

text: stellar

text: mass

text: gap,

text: and

text: now

text: removing

text: any

text: systems

text: where

text: the

text: MMG

text: (and

text: in

text: the

text: case

text: the

text: MMG

text: oﬀset.

text: The

text: main

text: conclusions

text: of

text: this

text: work

text: are

text: as

text: of

text: the

text: second-most-massive

text: galaxy

text: as

text: well)

text: is

text: a

text: ﬁbre

text: follows:

text: collision

text: galaxy.

text: per

text: cent

text: of

text: the

text: clusters

text: in

text: the

text: sample

text: We

text: detect

text: a

text: signiﬁcant

text: positive

text: correlation

text: between

text: have

text: an

text: MMG

text: which

text: is

text: a

text: ﬁbre

text: collision

text: galaxy

text: and

text: per

text: X-ray

text: relaxation

text: proxies

text: (photon

text: asymmetry,

text: centroid

text: shift)

text: cent

text: of

text: the

text: sample

text: have

text: either

text: the

text: MMG

text: or

text: the

text: second-

text: and

text: Anderson-Darling

text: statistic

text: at

text: 4σ

text: as

text: measured

text: by

text: most-massive

text: galaxy

text: as

text: a

text: ﬁbre

text: collision

text: galaxy.

text: Re-testing

text: both

text: a

text: power-law

text: ﬁt

text: and

text: by

text: the

text: Spearman

text: correlation

text: test,

text: MNRAS


text: 3 Powder X-ray diﬀraction measurements show that the cerium. The magnetic anisotropy decreases by roughly → ≈ ≈ ThCr Si -type structure persists across entire the Pd a factor of 10 to approach χ⊥/χ(cid:107) 20 near x 0.35, 2 2 Ni substitution series, while the tetragonal lattice con- after which it recovers to nearly the x = 0 value at x ≈ stants (a and c) and the unit cell volume (V ) decrease 0.6 and ﬁnally decreases to become isotropic at x = with increasing x (Fig. 1). These trends are consis- 1 (Fig. 2). The Curie-Weiss temperature θ is positive tent with Vegard’s law, where the lattice contraction is at x = 0, consistent with a ferromagnetic exchange in- due to the smaller size of Ni by comparison to Pd and teraction. It subsequently changes sign from positive to ≈ suggest that the room temperature Ce valence remains negative near x 0.66 (Fig. 2c), after which its magni- nearly trivalent with increasing x. The volume contrac- tude grows rapidly. A large and negative θ is a common tion results in a chemical pressure which is estimated to feature in Ce-based materials with strong hybridization be near P = 13.6 GPa for x = 0.955. This value is between the f- and conduction electrons and indicates ch calculated using the Birch-Murnaghan equation P = an enhanced Kondo interaction at large x. ch B ∆V (x)/V (0), where the value of the bulk modulus for 0 CeCu2Si2 (B0 = 110 GPa) is used.[31] C e ( P d N i ) P H c 2 . 0 4 . 2 5 1 0 . 0 6 1 - x x 2 2 // 0 C e ( P d N i ) P 1 . 0 H = 0 . 1 T 1 . 5 0 .1 5 0 .3 5 4 . 2 0 1 - x x 2 2 0 . 8 0 .6 0 4 0 0 1 . 0 0 .6 6 0 . 6 0 .6 9 3 0 0 0 .7 5 4 . 1 5 0 . 4 a 9 . 8 4 2 0 0 0 . 5 00 ..89 31 b 0 .9 6 0 . 2 1 0 0 4 . 1 0 0 . 0 0 . 00 . 0 0 . 2 0 . 4 x 0 . 6 0 . 8 1 . 0 0 0 5 0 1 0 0 T 1 5(K0 ) 2 0 0 2 5 0 3 0 0 4 . 0 5 n o m 9 . 6 - 0 . 5 2 1 0 0 - 1 . 0 4 . 0 0 b 1 0 c - 1 . 5 3 . 9 5 9 . 4 10 .0 0 .2 0 .4 0 .6 0 .8 a1 .0 d 1 7 0 0 x - 2 . 0 1 2 0 1 0 2 0 3 0 4 0 5 0 6 0 - 6 - 3 0 3 6 T ( K ) H ( T ) 1 6 5 1 6 0 6 FIG. 2. (a) Magnetic susceptibility χ = M/H vs temper- 1 5 5 (cid:107) c ature T for H = 0.1 T applied parallel to the c-axis for 1 5 0 0 Ce(P−d1−xNix)2P2. (b) The inverse of Magn(cid:107)etic susceptibil- 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 ity χ 1(T) for representative curves with H c. The dotted x lines are Curie-Weiss ﬁts to the data for 150 K < T < 300 K. (c) The magnetic anisotropy χ⊥/χ(cid:107) vs x at T = 2 K. (d) Magnetization ﬁeld dependence for diﬀerent x-concentrations measured at 1.8K. FIG. 1. (a) Comparison between the measured phospho- rous concentration x and the nominal concentration x , act nom where x was determined using energy dispersive spectrom- → act To further evaluate the eﬀect of Pd Ni substitution eter analysis. Throughout the rest of the manuscript we use on the Ce f-electron state, Ce L -edge X-ray absorp- x = x unless otherwise speciﬁed. (b) The lattice constants, 3 act tion spectra (XAS) were obtained from single crystals a(x) (left axis) and c(x) (right axis). (c) the unit cell volume V (x) (left axis) and chemical pressure Pch(right axis), calcu- of CeNi2P2 and CePd2P2, the end members of the se- lated using the Birch-Murnaghan equation as described in the ries. Typically, CeIII provides a single absorption peak text with the bulk modulus B = 110 GPa. at ca. 5723-5725 eV, whereas CeIV exhibits a “double- 0 white line” feature with maxima at ca. 5724-5728 and 5736-5739 eV. As shown in Fig. 3, the Ce L -edge spec- 3 The magnetic susceptibility χ = M/H vs temperature trum from CePd P was typical of CeIII, containing a (cid:107) 2 2 and magnetization M vs H for H c data are shown single pronounced absorption peak with a maximum at in Fig. 2, where the high T behavior provides insight 5725.2 eV (FWHM = 6.60 eV). Although substituting into the x-evolution of the f-electron state, magnetocrys- Ni for Pd had essentially no impact on the main absorp- talline anisotropy, and strength of the Kondo hybridiza- tion peak energy (maximum = 5725.1 eV), subtle spec- tion. χ(T) for CePd P follows a Curie Weiss (CW) tem- tral changes emerged; most notably, the main absorption 2 2 perature dependence for 150 K < T < 300 K (Fig. 2b) peak broadened by 1.24 eV (FWHM = 7.84 eV) and a (cid:107) with large anisotropy between the H c and in-plane di- minor post-edge feature emerged near 5735 eV. Addition- rections where the (001)-direction is the easy axis. [24, 25] ally, spectra obtained from CeNi P and CePd P were 2 2 2 2 The CW behavior persists across the entire substitution unchanged as a function of temperature between 85 and series and ﬁts to the data reveal eﬀective magnetic mo- 10 K. ≈ − ments µ 2.5 2.6 µ /Ce, as expected for trivalent To characterize the origin of these changes, the Ce L - eﬀ B 3 ) Å ( a ) 3 Å ( V x t c a ) Å ( c ) a P G ( P m e h c ) l o m / 3 m c ( (cid:1) )3 m c/l o m(1 -(cid:1) K 8. 1 = T| //(cid:1)/ ^(cid:1) ) . U . 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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 (cid:12) (cid:11) (cid:12) Protocol 2: Secured blind simulation gates such as the T gate by choosing the right evo- lution time t. Similarly, by changing the magnetic Input: the n bath spins in a state ψ(0) . ∈ { } ﬁeld in Eq. (1) to the x axis, Alice can simulate the 1 for k 1, . . . , m do Hadamard gate H. To form a universal gate set, we 2 Alice sends Bob the classical parameters still need a two-qubit gate such as the following U tk and γj(k). gate [51]: XY | (cid:105)   3 Alice sets the central spin to αk and   sends to Bob.    1 0 0 0  4 Bob simulates evolution of the n+1 spins ≡ 0 √1 √i 0 under Hc(k) for time t2k . UXY 0 √i2 √12 0 . (10) 2 2 5 Bob returns the central spin back to Alice. 0 0 0 1 6 Alice applies either a π pulse, a π/2 pulse or identity to rotate the central spin to | (cid:105) To engineer a U gate, for example, between the β and sends to Bob. XY k ﬁrst and the second qubits, we turn all γ oﬀ except j 7 Bob simulates evolution of the n+1 spins for γ = γ = γ. The eﬀective time evolution of the under Hc(k) for time t2k . bath1s(cid:26)pins2(cid:18)is (cid:19) (cid:27) 8 Bob sends the central spin back to Alice. 9 Alice| ab(cid:105)orts if the returned central spin is − γ2 γ (cid:12) n(cid:11)ot β . exp i (X X + Y Y ) + (Z + Z ) t , (cid:12) k 8 1 2 1 2 2 1 2 10 end 11 return the bath spins in the ﬁnal state (11) ψ(m) . where X, Y are respective Pauli gates. With γ = ± ∓ 27c for some large integer c and t = 7γπ2 , the above if αk = for all k. With this restriction±, it is unitary reduces to the UXY gate. Since the UXY straightforward to verify that whenever αk = , the gate and single-qubit gates form a universal gate net unitary applied on the bath spins in the kth iter- set [51], Alice can eﬀectively perform universal quan- ation is equivalent to identity. Such iterations there- tum computation on the bath spins. In the following U fore only play the role of ﬂagging malicious measure- discussion, we refer to this universal gate set as . ment attempts and do not contribute to the overall ∈ { } Protocol 2 can be further adapted to this situa- simulation. Note that when β = α 0, 1 m, Pro- tion to guarantee the security of the computation tocol 2 reduces to Protocol 1. by using honeypots to detect measurement attacks. Universal blind computation– So far we have A desired circuit U of length m on n qubits can shown that Alice can request a general simulation m0 0 be embedded into a much larger circuit U with U given by Eq. (8) on n bath spins without re- (cid:29) m m m m that consists mostly of honeypots (Fig. 2). 0 vealing her data. But how general is U ? In other m These honeypots perform trivial operations on the words, what type of quantum computation Alice qubits and serve only as detectors of malicious be- can achieve by simulating U ? We now show that m haviors. Mean while in each non-honeypot itera- by choosing the right parameters γ(k), t in each (cid:110) (cid:111) ∈ { } j k tion of the protocol, i.e. αk 0, 1 , Alice can iteration, Alice can simulate any gate in a universal choose a Hamiltonian H↑(k) and a time t such that gate set and therefore is able to perform a blind sim- − k ulation of an arbitrary quantum circuit. Indeed, by Gk = exp iH↑(k)tk is a gate in the universal gate U turning oﬀ γj for all except j = 1, the time evolution set . Note that such a gate (or its inverse) is only unitary is a local rotation of the ﬁrst spin about the implemented on the bath spins at the end of the ⊕ z axis, (cid:18) (cid:19) iteration if β α = 0. On the other hand, if ⊕ k k β α = 1, the two evolutions in the kth iteration k k cancel each other out and therefore only a trivial e−iαZ1t = e−i(cid:126)2γ γ1t 10 ei(cid:126)0γ1t , (9) gate is implemented. Denote by ω(αk, βk) a func- tion of the characters α , β such that ω(α , β ) = 1 ⊕ k k k k if α β = 0 and ω(α , β ) = 0 otherwise. The k k k k where Z is the Pauli-Z matrix on the ﬁrst qubit. gate sequence generated by the protocol can then 1 Using this Hamiltonian, we can obtain phase-shift be summarized by the following equation: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Protocol

text: Secured

label: caption

text: blind

text: simulation

text: gates

text: such

text: as

text: the

text: T

text: gate

text: by

text: choosing

text: the

text: right

text: evo-

text: lution

text: time

text: t.

text: Similarly,

text: by

text: changing

text: the

text: magnetic

text: Input:

text: the

text: n

text: bath

text: spins

text: in

text: a

text: state

text: ﬁeld

text: in

text: Eq.

text: to

text: the

text: x

text: axis,

text: Alice

text: can

text: simulate

text: the

text: for

text: k

text: m

text: do

text: Hadamard

text: gate

text: H.

text: To

text: form

text: a

text: universal

text: gate

text: set,

text: we

text: Alice

text: sends

text: Bob

text: the

text: classical

text: parameters

text: still

text: need

text: a

text: two-qubit

text: gate

text: such

text: as

text: the

text: following

text: U

text: tk

text: and

text: γj(k).

text: gate

text: XY

label: equation

text: Alice

text: sets

text: the

text: central

text: spin

text: to

text: αk

text: and

text: sends

text: to

text: Bob.

text: Bob

text: simulates

text: evolution

text: of

text: the

text: spins

text: √1

text: √i

text: under

text: Hc(k)

text: for

text: time

text: UXY

text: Bob

text: returns

text: the

text: central

text: spin

text: back

text: to

text: Alice.

text: Alice

text: applies

text: either

text: a

text: π

text: pulse,

text: a

text: pulse

text: or

text: identity

text: to

text: rotate

text: the

text: central

text: spin

text: to

text: To

text: engineer

text: a

text: U

text: gate,

text: for

text: example,

text: between

text: the

text: β

text: and

text: sends

text: to

text: Bob.

text: XY

text: k

text: ﬁrst

text: and

text: the

text: second

text: qubits,

text: we

text: turn

text: all

text: γ

text: oﬀ

text: except

text: j

text: Bob

text: simulates

text: evolution

text: of

text: the

text: spins

text: for

text: γ

text: γ

text: γ.

text: The

text: eﬀective

text: time

text: evolution

text: of

text: the

text: under

text: Hc(k)

text: for

text: time

text: Bob

text: sends

text: the

text: central

text: spin

text: back

text: to

text: Alice.

text: Alice|

text: if

text: the

text: returned

text: central

text: spin

text: is

text: γ2

text: γ

text: β

text: exp

text: i

text: (X

text: X

text: Y

text: Y

text: (Z

text: Z

text: t

text: k

text: end

text: return

text: the

text: bath

text: spins

text: in

text: the

text: ﬁnal

text: state

text: ψ(m)

text: where

text: X,

text: Y

text: are

text: respective

text: Pauli

text: gates.

text: With

text: γ

text: for

text: some

text: large

text: integer

text: c

text: and

text: t

text: 7γπ2

text: the

text: above

text: if

text: αk

text: for

text: all

text: k.

text: With

text: this

text: restriction±,

text: it

text: is

text: unitary

text: reduces

text: to

text: the

text: UXY

text: gate.

text: Since

text: the

text: UXY

text: straightforward

text: to

text: verify

text: that

text: whenever

text: αk

text: the

text: gate

text: and

text: single-qubit

text: gates

text: form

text: a

text: universal

text: gate

text: net

text: unitary

text: applied

text: on

text: the

text: bath

text: spins

text: in

text: the

text: kth

text: iter-

text: set

text: Alice

text: can

text: eﬀectively

text: perform

text: universal

text: quan-

text: ation

text: is

text: equivalent

text: to

text: identity.

text: Such

text: iterations

text: there-

text: tum

text: computation

text: on

text: the

text: bath

text: spins.

text: In

text: the

text: following

text: U

text: fore

text: only

text: play

text: the

text: role

text: of

text: ﬂagging

text: malicious

text: measure-

text: discussion,

text: we

text: refer

text: to

text: this

text: universal

text: gate

text: set

text: as

text: ment

text: attempts

text: and

text: do

text: not

text: contribute

text: to

text: the

text: overall

text: Protocol

text: can

text: be

text: further

text: adapted

text: to

text: this

text: situa-

text: simulation.

text: Note

text: that

text: when

text: β

text: α

text: m,

text: Pro-

text: tion

text: to

text: guarantee

text: the

text: security

text: of

text: the

text: computation

text: tocol

text: reduces

text: to

text: Protocol

text: by

text: using

text: honeypots

text: to

text: detect

text: measurement

text: attacks.

text: Universal

text: blind

text: computation–

text: So

text: far

text: we

text: have

text: A

text: desired

text: circuit

text: U

text: of

text: length

text: m

text: on

text: n

text: qubits

text: can

text: shown

text: that

text: Alice

text: can

text: request

text: a

text: general

text: simulation

text: be

text: embedded

text: into

text: a

text: much

text: larger

text: circuit

text: U

text: with

text: U

text: given

text: by

text: Eq.

text: on

text: n

text: bath

text: spins

text: without

text: re-

text: m

text: m

text: m

text: m

text: that

text: consists

text: mostly

text: of

text: honeypots

text: (Fig.

text: vealing

text: her

text: data.

text: But

text: how

text: general

text: is

text: U

text: In

text: other

text: m

text: These

text: honeypots

text: perform

text: trivial

text: operations

text: on

text: the

text: words,

text: what

text: type

text: of

text: quantum

text: computation

text: Alice

text: qubits

text: and

text: serve

text: only

text: as

text: detectors

text: of

text: malicious

text: be-

text: can

text: achieve

text: by

text: simulating

text: U

text: We

text: now

text: show

text: that

text: m

text: haviors.

text: Mean

text: while

text: in

text: each

text: non-honeypot

text: itera-

text: by

text: choosing

text: the

text: right

text: parameters

text: γ(k),

text: t

text: in

text: each

text: j

text: k

text: tion

text: of

text: the

text: protocol,

text: i.e.

text: αk

text: Alice

text: can

text: iteration,

text: Alice

text: can

text: simulate

text: any

text: gate

text: in

text: a

text: universal

text: choose

text: a

text: Hamiltonian

text: H↑(k)

text: and

text: a

text: time

text: t

text: such

text: that

text: gate

text: set

text: and

text: therefore

text: is

text: able

text: to

text: perform

text: a

text: blind

text: sim-

text: k

text: ulation

text: of

text: an

text: arbitrary

text: quantum

text: circuit.

text: Indeed,

text: by

text: Gk

text: exp

text: iH↑(k)tk

text: is

text: a

text: gate

text: in

text: the

text: universal

text: gate

text: U

text: turning

text: oﬀ

text: γj

text: for

text: all

text: except

text: j

text: the

text: time

text: evolution

text: set

text: Note

text: that

text: such

text: a

text: gate

text: (or

text: its

text: inverse)

text: is

text: only

text: unitary

text: is

text: a

text: local

text: rotation

text: of

text: the

text: ﬁrst

text: spin

text: about

text: the

text: implemented

text: on

text: the

text: bath

text: spins

text: at

text: the

text: end

text: of

text: the

text: z

text: axis,

text: iteration

text: if

text: β

text: α

text: On

text: the

text: other

text: hand,

text: if

text: k

text: k

text: β

text: α

text: the

text: two

text: evolutions

text: in

text: the

text: kth

text: iteration

text: k

text: k

text: cancel

text: each

text: other

text: out

text: and

text: therefore

text: only

text: a

text: trivial

text: gate

text: is

text: implemented.

text: Denote

text: by

text: ω(αk,

text: βk)

text: a

text: func-

text: tion

text: of

text: the

text: characters

text: α

text: β

text: such

text: that

text: ω(α

text: β

text: k

text: k

text: k

text: k

text: if

text: α

text: β

text: and

text: ω(α

text: β

text: otherwise.

text: The

text: k

text: k

text: k

text: k

text: where

text: Z

text: is

text: the

text: Pauli-Z

text: matrix

text: on

text: the

text: ﬁrst

text: qubit.

text: gate

text: sequence

text: generated

text: by

text: the

text: protocol

text: can

text: then

text: Using

text: this

text: Hamiltonian,

text: we

text: can

text: obtain

text: phase-shift

text: be

text: summarized

text: by

text: the

text: following

text: equation:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 (cid:89) ing a quantum circuit that is known to solve a prob- lem faster than classical algorithms. For example, in m { } → Um = Gωk(αk,βk). (12) t{h0e, 1S}inmoisn’psropmroibseledmto[56sa],tiasfyfunthctaitonf(fx): =0,f1(yn) if k=1 ⊕ ∈ { } and only if x = y or x y = s for all x, y 0, 1 n ∈ { } and a ﬁxed string s 0, 1 n. To ﬁnd s, classi- Therefore the keys α, β eﬀectively encrypt the cir- cal algorithms require at least Ω(2n/2) queries to cuit Alice implements and make the quantum com- the function f while the quantum Simon’s algorithm putation blind. Our protocol for universal blind can solve the problem using only O (n) queries. Us- quantum computation is be summarized in Proto- ing Protocol 3, Alice can simulate a quantum circuit col 3 below. corresponding to a secret string s. She then asks Bob to measure the output and announce the measured Protocol 3: Universal blind computation string. If Bob is able to answer correctly what s is for large enough n, Alice can be conﬁdent that the 1 Alice has a circuit Um0 of length m0 to be per- server Bob has access to at least a BQP machine. formed on n qubits. Outlook– Here we have shown how to implement 2 Alice embeds Um0 into a much ∈lar{ger ci±rc}uit an arbitrary circuit controlled by a single spin. This U by choosing two keys α, β 0, 1, m, m (cid:29) enables us to deﬁne several blind computing proto- each of length m m such that U in 0 m Eq. (12) reduces to U . cols that can be a powerful test of computing power m0 in quantum simulators. However, we have not yet 3 Alice and Bo|b (cid:105)perform Protocol 2 to imple- developed natural observables whose, e.g., distribu- ment U on ψ . m tion function is distinctly diﬀerent given classical versus quantum computational power. We consider Quantum veriﬁcation– Blindness allows the client this an intriguing direction for future research. Alice to not only hide the computation from the We thank S. -H Hung, B. Lackey, R. Matthew, server Bob but also to verify if Bob performs the cor- and Y. Wang for helpful discussions. This research rect computation. Indeed, Alice can verify Bob by was supported in part by the NSF funded Physics simply requesting quantum circuits that have out- Frontier Center at the Joint Quantum Institute and comes that can be classically veriﬁed. By the deﬁni- the Army Research Laboratory’s CDQI. tion of blind computation used in this work, Bob has no information about what circuits are being imple- mented, and the only way he can return the correct output to Alice is to perform the exact simulation ∗ sequence as instructed. For example, Alice can ask Bob to initialize the n bath spins in a product state [1] R. P. Feynman, International Journal of Theoretical Physics 21, 467 (1982). such that some of the spins are “up” and some are [2] A. A. Houck, H. E. Tu¨reci, and J. Koch, Nature “down”, and use Protocol 3 to simulate a sequence of Physics 8, 292 (2012). SWAP gates between the bath spins known only to [3] X. Ma, B. Daki´c, S. Kropatschek, W. Naylor, Alice. The ﬁnal state is a permutation of the initial Y. Chan, Z. Gong, L. Duan, A. Zeilinger, and state and is known only to Alice. Bob has to ﬁnd P. Walther, Scientiﬁc Reports 4, 3583 (2014). the ﬁnal state and the only way he can pass with [4] P. J. J. O’Malley, R. Babbush, I. D. Kivlichan, certainty is to correctly perform the computation. J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Since both the initial state and the ﬁnal state are Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. fully separable, the permutation circuit is essentially Fowler, E. Jeﬀrey, E. Lucero, A. Megrant, J. Y. Mu- classical. Stabilizer circuits [52, 53], on the other tus, M. Neeley, C. Neill, C. Quintana, D. Sank, hand, can perform nontrivial quantum operations, A. Vainsencher, J. Wenner, T. C. White, P. V. such as quantum teleportation [54] and preparation Coveney, P. J. Love, H. Neven, A. Aspuru-Guzik, of highly entangled states. They consist of only Clif- and J. M. Martinis, Phys. Rev. X 6, 031007 (2016). ford gates and can be simulated eﬃciently on a clas- [5] T. Hensgens, T. Fujita, L. Janssen, X. Li, C. J. Van Diepen, C. Reichl, W. Wegscheider, sical computer [55]. Therefore by requesting simula- S. Das Sarma, and L. M. K. Vandersypen, Nature tion of an arbitrary stabilizer circuits, Alice can also 548, 70 (2017). eﬃciently verify quantumness of the server. [6] J. C. Loredo, M. P. Almeida, R. Di Candia, J. S. Alice can also take a step further to verify even the Pedernales, J. Casanova, E. Solano, and A. G. quantum computing power of the server by request- White, Phys. Rev. Lett. 116, 070503 (2016). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ing

text: a

text: quantum

text: circuit

text: that

text: is

text: known

text: to

text: solve

text: a

text: prob-

text: lem

text: faster

text: than

text: classical

text: algorithms.

text: For

text: example,

text: in

text: m

text: Um

text: Gωk(αk,βk).

text: if

text: and

text: only

text: if

text: x

text: y

text: or

text: x

text: y

text: s

text: for

text: all

text: x,

text: y

text: n

text: and

text: a

text: ﬁxed

text: string

text: s

text: n.

text: To

text: ﬁnd

text: s,

text: classi-

text: Therefore

text: the

text: keys

text: α,

text: β

text: eﬀectively

text: encrypt

text: the

text: cir-

text: cal

text: algorithms

text: require

text: at

text: least

text: queries

text: to

text: cuit

text: Alice

text: implements

text: and

text: make

text: the

text: quantum

text: com-

text: the

text: function

text: f

text: while

text: the

text: quantum

text: Simon’s

text: algorithm

text: putation

text: blind.

text: Our

text: protocol

text: for

text: universal

text: blind

text: can

text: solve

text: the

text: problem

text: using

text: only

text: O

text: (n)

text: queries.

text: Us-

text: quantum

text: computation

text: is

text: be

text: summarized

text: in

text: Proto-

text: ing

text: Protocol

text: Alice

text: can

text: simulate

text: a

text: quantum

text: circuit

text: col

text: below.

text: corresponding

text: to

text: a

text: secret

text: string

text: s.

text: She

text: then

text: asks

text: Bob

text: to

text: measure

text: the

text: output

text: and

text: announce

text: the

text: measured

text: Protocol

text: Universal

text: blind

text: computation

text: string.

text: If

text: Bob

text: is

text: able

text: to

text: answer

text: correctly

text: what

text: s

text: is

text: for

text: large

text: enough

text: n,

text: Alice

text: can

text: be

text: conﬁdent

text: that

text: the

text: Alice

text: has

text: a

text: circuit

text: of

text: length

text: to

text: be

text: per-

text: server

text: Bob

text: has

text: access

text: to

text: at

text: least

text: a

text: BQP

text: machine.

text: formed

text: on

text: n

text: qubits.

text: Outlook–

text: Here

text: we

text: have

text: shown

text: how

text: to

text: implement

text: Alice

text: embeds

text: into

text: a

text: much

text: ∈lar{ger

text: ci±rc}uit

text: an

text: arbitrary

text: circuit

text: controlled

text: by

text: a

text: single

text: spin.

text: This

text: U

text: by

text: choosing

text: two

text: keys

text: α,

text: β

text: m,

text: m

text: enables

text: us

text: to

text: deﬁne

text: several

text: blind

text: computing

text: proto-

text: each

text: of

text: length

text: m

text: m

text: such

text: that

text: U

text: in

text: m

text: Eq.

text: reduces

text: to

text: U

text: cols

text: that

text: can

text: be

text: a

text: powerful

text: test

text: of

text: computing

text: power

text: in

text: quantum

text: simulators.

text: However,

text: we

text: have

text: not

text: yet

text: Alice

text: and

text: Bo|b

text: Protocol

text: to

text: imple-

text: developed

text: natural

text: observables

text: whose,

text: e.g.,

text: distribu-

text: ment

text: U

text: on

text: ψ

text: m

text: tion

text: function

text: is

text: distinctly

text: diﬀerent

text: given

text: classical

text: versus

text: quantum

text: computational

text: power.

text: We

text: consider

text: Quantum

text: veriﬁcation–

text: Blindness

text: allows

text: the

text: client

text: this

text: an

text: intriguing

text: direction

text: for

text: future

text: research.

text: Alice

text: to

text: not

text: only

text: hide

text: the

text: computation

text: from

text: the

text: We

text: thank

text: S.

text: -H

text: Hung,

text: B.

text: Lackey,

text: R.

text: Matthew,

text: server

text: Bob

text: but

text: also

text: to

text: verify

text: if

text: Bob

text: performs

text: the

text: cor-

text: and

text: Y.

text: Wang

text: for

text: helpful

text: discussions.

text: This

text: research

text: rect

text: computation.

text: Indeed,

text: Alice

text: can

text: verify

text: Bob

text: by

text: was

text: supported

text: in

text: part

text: by

text: the

text: NSF

text: funded

text: Physics

text: simply

text: requesting

text: quantum

text: circuits

text: that

text: have

text: out-

text: Frontier

text: Center

text: at

text: the

text: Joint

text: Quantum

text: Institute

text: and

text: comes

text: that

text: can

text: be

text: classically

text: veriﬁed.

text: By

text: the

text: deﬁni-

text: the

text: Army

text: Research

text: Laboratory’s

text: CDQI.

text: tion

text: of

text: blind

text: computation

text: used

text: in

text: this

text: work,

text: Bob

text: has

text: no

text: information

text: about

text: what

text: circuits

text: are

text: being

text: imple-

text: mented,

text: and

text: the

text: only

text: way

text: he

text: can

text: return

text: the

text: correct

text: output

text: to

text: Alice

text: is

text: to

text: perform

text: the

text: exact

text: simulation

text: sequence

text: as

text: instructed.

text: For

text: example,

text: Alice

text: can

text: ask

text: Bob

text: to

text: initialize

text: the

text: n

text: bath

text: spins

text: in

text: a

text: product

text: state

text: R.

text: P.

text: Feynman,

text: International

text: Journal

text: of

text: Theoretical

text: Physics

text: such

text: that

text: some

text: of

text: the

text: spins

text: are

text: “up”

text: and

text: some

text: are

text: A.

text: A.

text: Houck,

text: H.

text: E.

text: Tu¨reci,

text: and

text: J.

text: Koch,

text: Nature

text: “down”,

text: and

text: use

text: Protocol

text: to

text: simulate

text: a

text: sequence

text: of

text: Physics

text: SWAP

text: gates

text: between

text: the

text: bath

text: spins

text: known

text: only

text: to

text: X.

text: Ma,

text: B.

text: Daki´c,

text: S.

text: Kropatschek,

text: W.

text: Naylor,

text: Alice.

text: The

text: ﬁnal

text: state

text: is

text: a

text: permutation

text: of

text: the

text: initial

text: Y.

text: Chan,

text: Z.

text: Gong,

text: L.

text: Duan,

text: A.

text: Zeilinger,

text: and

text: state

text: and

text: is

text: known

text: only

text: to

text: Alice.

text: Bob

text: has

text: to

text: ﬁnd

text: P.

text: Walther,

text: Scientiﬁc

text: Reports

text: the

text: ﬁnal

text: state

text: and

text: the

text: only

text: way

text: he

text: can

text: pass

text: with

text: P.

text: J.

text: J.

text: O’Malley,

text: R.

text: Babbush,

text: I.

text: D.

text: Kivlichan,

text: certainty

text: is

text: to

text: correctly

text: perform

text: the

text: computation.

text: J.

text: Romero,

text: J.

text: R.

text: McClean,

text: R.

text: Barends,

text: J.

text: Kelly,

text: P.

text: Roushan,

text: A.

text: Tranter,

text: N.

text: Ding,

text: B.

text: Campbell,

text: Since

text: both

text: the

text: initial

text: state

text: and

text: the

text: ﬁnal

text: state

text: are

text: Y.

text: Chen,

text: Z.

text: Chen,

text: B.

text: Chiaro,

text: A.

text: Dunsworth,

text: A.

text: G.

text: fully

text: separable,

text: the

text: permutation

text: circuit

text: is

text: essentially

text: Fowler,

text: E.

text: Jeﬀrey,

text: E.

text: Lucero,

text: A.

text: Megrant,

text: J.

text: Y.

text: Mu-

text: classical.

text: Stabilizer

text: circuits

text: on

text: the

text: other

text: tus,

text: M.

text: Neeley,

text: C.

text: Neill,

text: C.

text: Quintana,

text: D.

text: Sank,

text: hand,

text: can

text: perform

text: nontrivial

text: quantum

text: operations,

text: A.

text: Vainsencher,

text: J.

text: Wenner,

text: T.

text: C.

text: White,

text: P.

text: V.

text: such

text: as

text: quantum

text: teleportation

text: and

text: preparation

text: Coveney,

text: P.

text: J.

text: Love,

text: H.

text: Neven,

text: A.

text: Aspuru-Guzik,

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text: highly

text: entangled

text: states.

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text: consist

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text: ford

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text: can

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text: R.

text: Di

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text: J.

text: S.

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text: also

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text: a

text: step

text: further

text: to

text: verify

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text: and

text: A.

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text: server

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