The realization of a full-scale quantum computer presents one

Transcription

1 ARICLES PUBLISED ONLINE: 7 DECEMBER 28 DOI:.38/NPYS5 Simplifying quntum logic using higher-dimensionl ilert spces Benjmin P. Lnyon *, Mrco Brieri, Mrcelo P. Almeid, homs Jennewein,2, imothy C. Rlph, Kevin J. Resch,3, Geoff J. Pryde,4, Jeremy L. O Brien,5, Alexei Gilchrist,6 nd Andrew G. White Quntum computtion promises to solve fundmentl, yet otherwise intrctle, prolems cross rnge of ctive fields of reserch. Recently, universl quntum logic-gte sets the elementl uilding locks for quntum computer hve een demonstrted in severl physicl rchitectures. A serious ostcle to full-scle implementtion is the lrge numer of these gtes required to uild even smll quntum circuits. ere, we present nd demonstrte generl technique tht hrnesses multi-level informtion crriers to significntly reduce this numer, enling the construction of key quntum circuits with existing technology. We present implementtions of two key quntum circuits: the three-quit offoli gte nd the generl two-quit controlled-unitry gte. Although our experiment is crried out in photonic rchitecture, the technique is independent of the prticulr physicl encoding of quntum informtion, nd hs the potentil for wider ppliction. he reliztion of full-scle quntum computer presents one of the most chllenging prolems fcing modern science. Even implementing smll-scle quntum lgorithms requires high level of control over multiple quntum systems. Recently, much progress hs een mde with demonstrtions of universl quntum gte sets in numer of physicl rchitectures including ion trps,2, liner optics 3 6, superconductors 7,8 nd toms 9,. In theory, these gtes cn now e put together to implement ny quntum circuit nd uild sclle quntum computer. In prctice, there re mny significnt ostcles tht will require oth theoreticl nd technologicl developments to overcome. One is the sheer numer of elementl gtes required to uild quntum logic circuits. Most pproches to quntum computing use quits the quntum version of its. A quit is two-level quntum system tht cn e represented mthemticlly y vector in two-dimensionl ilert spce. Relizing quits typiclly requires enforcing twolevel structure on systems tht re nturlly fr more complex nd which hve mny redily ccessile degrees of freedom, such s toms, ions or photons. ere, we show how hrnessing these extr levels during computtion significntly reduces the numer of elementl gtes required to uild key quntum circuits. Becuse the technique is independent of the physicl encoding of quntum informtion nd the wy in which the elementl gtes re themselves constructed, it hs the potentil to e used in conjunction with existing gte technology in wide vriety of rchitectures. Our technique extends recent proposl, nd we use it to demonstrte two key quntum logic circuits: the offoli nd controlled-unitry 2 gtes. We first outline the technique in generl context, then present n experimentl reliztion in liner optic rchitecture: without our resource-sving technique, liner optic implementtions of these gtes re infesile with current technology. Simplifying the offoli gte One of the most importnt quntum logic gtes is the offoli 2 three-quit entngling gte tht flips the logicl stte of the trget quit conditionl on the logicl stte of the two control quits. Fmously, these gtes enle universl reversile clssicl computtion, nd hve centrl role in quntum error correction 3 nd fult tolernce 4. Furthermore, the comintion of the offoli nd the one-quit dmrd offers simple universl quntum gte set 5. he simplest known decomposition of offoli when restricted to operting on quits throughout the clcultion is circuit tht requires five two-quit gtes 2. If we further restrict ourselves to controlled-z (or cnot) gtes, this numer clims to six 2 (Fig. ). A decomposition tht requires only three two-quit gtes is shown in Fig.. he incresed efficiency is chieved y hrnessing third level of the trget informtion crrier the trget is ctully qutrit with logicl sttes, nd 2. At the input nd output of the circuit, informtion is encoded only in the ottom two (quit) levels of the trget. he ction of the first X gte is to move informtion from the logicl stte of the trget into the third level ( 2 ), which then ypsses the susequent two-quit gtes. he finl X gte then coherently rings this informtion ck into the stte, reconstructing the logicl quit. By temporrily storing prt of the informtion in this third level, we re effectively removing it from the clcultion enling the susequent two-quit gtes to operte on suspce of the trget. his enles n implementtion of the offoli with significntly reduced numer of gtes. Note tht only stndrd two-quit gtes re necessry, with the extr requirement tht they ct only trivilly on (tht is, pply the identity to) level 2 of the qutrit. As such, it is not necessry to develop universl set of gtes for qutrits. his technique cn e redily generlized to implement higher-order n-control-quit offoli gtes ( n t) y hrnessing single (n+)-level informtion crrier during computtion nd Deprtment of Physics nd Centre for Quntum Computer echnology, University of Queenslnd, Brisne 472, Austrli, 2 Institute for Quntum Optics nd Quntum Informtion, Austrin Acdemy of Sciences, Boltzmnng. 3, A-9 Vienn, Austri, 3 Institute for Quntum Computing nd Deprtment of Physics & Astronomy, University of Wterloo, N2L 3G, Cnd, 4 Centre for Quntum Dynmics, Griffith University, Brisne 4, Austrli, 5 Centre for Quntum Photonics,.. Wills Physics Lortory nd Deprtment of Electricl nd Electronic Engineering, University of Bristol, Merchnt Venturers Building, Woodlnd Rod, Bristol BS8 UB, UK, 6 Physics Deprtment, Mcqurie University, Sydney 29, Austrli. *e-mil: lnyon@physics.uq.edu.u. 34 NAURE PYSICS VOL 5 FEBRUARY 29

2 NAURE PYSICS DOI:.38/NPYS5 ARICLES C 3 C Z π C X Quit Quit Qutrit X Quit opertors 2 Qutrit opertor X X = e iπ/4 Figure Simplifying the offoli gte., Most efficient known decomposition into the universl gte set CNO +ritrry one-quit gte, when restricted to operting on quits 2., Our decomposition requiring only three two-quit gtes. ere, the trget is three-level qutrit with logicl sttes, nd 2. Initilly nd finlly, ll of the quntum informtion is encoded in the nd levels of ech informtion crrier. he ction of the X gtes is to swp informtion etween the logicl nd 2 sttes of the trget. he trget undergoes sign shift only for the input term,c, =,,. his opertion is equivlent to the offoli under the ction of only three one-quit gtes, s shown. he second gte in the decomposition is CZ nd is equivlent to CNO under the ction of two one-quit dmrd () gtes. only 2n stndrd two-quit gtes ; tht is, with ech extr control quit we need n extr level in the trget crrier (see Fig. 2). Compre this with the previous est known scheme, which requires 2n two-quit gtes nd n extr overhed of n extr ncill quits 2. When restrined from using ncill, this scheme requires of the order of n 2 two-quit gtes. In either cse, we chieve significnt resource reduction, y hrnessing only higher levels of existing informtion crriers. For exmple, the simplest known decomposition of the 5 t requires 5 two-quit gtes nd four ncill quits, when restricted to operting on quits 2. Our technique requires only nine two-quit gtes nd no ncillry informtion crriers. Extension to more generl quntum circuits Figure 3 shows n extension to simplify the construction of nother key quntum circuit: the n-control-quit unitry gte (c n u), which pplies n ritrry one-quit gte (u) to trget conditionl on the stte of n control quits. hese circuits hve centrl role in quntum computing, prticulrly in the phse-estimtion lgorithm 2. Phse estimtion underpins mny importnt pplictions of quntum computing including quntum simultion 6 nd Shor s fmous lgorithm for fctoring 7. Furthermore, the set of c u gtes lone is sufficient for universl quntum computing; c u cn implement cnot nd induce ny single-quit rottion t the expense of n ncill quit. Our technique cn implement c n u using n (n+)-level trget nd only 2n two-quit gtes. his is similr improvement, over schemes limited to quits, to tht chieved for the offoli 2. Figure 4 shows further generliztion to efficiently dd control quits to n ritrry controlled-unitry tht opertes on k quits. Potentil for ppliction he technique tht we descrie is independent of the prticulr physicl system used to encode quntum informtion nd the X e i θ C X X X X Figure 2 Simplifying higher-order offoli gtes. hree-control-quit offoli. he X gte swps informtion etween the logicl nd 2 sttes of the trget. he X gte flips informtion etween the logicl nd 3 stte of the trget. hus, we require ccess to four-level trget informtion crrier: two levels in the originl ril nd one in ech of the dshed rils. he trget undergoes sign shift only for the input term C 3,,C, =,,,. his opertion is equivlent to the offoli under the ction of only two one-quit gtes, s shown. See Fig. for gte opertions. C V U V C V U V X X X X X Figure 3 Simplifying controlled-unitry gtes., One control quit (we implement simplified version, see Fig. 5): the control opertion occurs if C =., wo control quits: the control opertion occurs if,c =,. V V is the spectrl decomposition of U, up to glol phse fctor. See Fig. for gte opertions. wy in which the elementl gtes re relized. Consequently, it hs the potentil for ppliction in mny rchitectures, yielding the sme resource svings. he only physicl requirements re ccess to multi-level systems nd the ility to coherently swp informtion etween these levels, tht is, implement the generlized X gtes (Fig. 2). Fortuntely, most of the cndidte systems for encoding quntum informtion nturlly offer multi-level structures tht re redily ccessile. For exmple, the photon hs lrge numer of degrees of freedom including polriztion, trnsverse sptil mode, rrivl time, photon numer nd frequency. Coherent control over nd etween mny of these dimensions hs lredy een demonstrted nd shown to offer significnt dvntges in rnge of pplictions such s quntum communiction nd mesurement 8,9. rpped ions lso offer redily ccessile levels including multiple electronic nd virtionl modes. Indeed, oth liner optic 2 nd trpped-ion 2,22 quntum computing rchitectures lredy routinely use multi-level systems to implement two-quit gtes nd relize universl gte sets. Clerly the tools re ville to exploit our technique, the enefits of which lie t the next level of construction uilding lrge quntum circuits. An immedite enefit of significnt reduction in the numer of two-quit gtes required for quntum circuits is n eqully significnt speed-up in processing time. his hs prticulr dvntges in the mny cses where short coherence times re n ostcle in the pth to sclility. Furthermore, s we illustrte in the next section, our technique rings whole rnge of logic circuits X NAURE PYSICS VOL 5 FEBRUARY

3 ARICLES K C 3 U k C X X X X Figure 4 Efficiently dding control quits to n ritrry controlled circuit. Circuit for three-control-quit unitry cting on k quits, c 3 u k. Given the ility to crry out single instnce of c u k, n extr control quits cn e dded t cost of n extr 2n two-quit gtes nd n extr n levels in C. he X j perform s descried in the cption of Fig. 2. he control opertion occurs if C 3,,C =,,. within rech of current technology, enling the implementtion nd explortion of new circuits in the lortory. Demonstrtion in liner opticl rchitecture ere, we present n implementtion of the offoli nd the c u, using photons to encode informtion nd liner optics to construct the component quntum logic gtes (see the Methods section). We cknowledge previous demonstrtions of offoli gte in liquid stte nmr, which do not exploit our resource-sving U k NAURE PYSICS DOI:.38/NPYS5 technique 3, Our demonstrtion uses two-quit gtes, the successful opertion of which is indicted y detection of one photon in ech of the sptil output modes 3,27 3. Such gtes re high performing, well chrcterized, offer fst gte speeds nd hve severl known pths to sclle quntum computing 2,3 33. We note tht our resource-sving technique is fundmentlly different from nd potentilly complementry to the numerous liner optics schemes for reducing the overhed ssocited with generting universl resource ; here, we re concerned with reducing the mount of tht resource required to uild circuits. Figure 5 shows schemtic digrms of our experiment (see the Methods section). Key steps re the expnsion of the ilert spce of the trget informtion crrier (), effected y the first polrizing emsplitter (PBS), nd contrction ck into the originl spce, effected y the components in the dshed rectngle. Before PBS, we hve two-level system in the trget ril with logicl sttes = nd V = (horizontl nd verticl photon polriztion). PBS then moves informtion encoded in the logicl stte into seprte sptil mode, which ypsses the susequent two-quit gtes. After PBS, we hve ccess to four-level system; two levels in the top ril (t) nd two in the ottom ril (), with logicl sis sttes,t, V,t,, nd V,, respectively. Although we need to use only one of the extr levels in the ottom ril to enct our technique, we use oth in our experiment simply to lnce opticl pth lengths. he contrction ck into the originl two-level polriztion quit is crried out non-deterministiclly, tht is, given deterministic two-quit gtes, mesurement of single photon t D herlds successful run of the gte. his enles demonstrtion of the offoli nd c u without the lst cnot in c D4 C t R D Lser ±2.5 nm V PBS PBS2 SG PDC Non-deterministic recomintion of C ±2.5 nm L L2 ±.5 nm D2 D3 t PPBS PPBS2 '' L3 ±.5 nm D Bem displcer (PBS / X) CNO CZ R Bem displcer (PBS2 / X ) Interference filter Loss element C R = R V = 2/3 R = /2 R V = /2 SPCM Lens Fire coupler Anlysis/prep λ /2 λ /4 Figure 5 offoli nd controlled-unitry experimentl lyout., Conceptul logic circuit. A polrizing em splitter temporrily expnds the ilert spce of the trget informtion crrier, from polriztion-encoded photonic quit to multi-level system distriuted cross polriztion nd longitudinl sptil mode. Informtion in the ottom ril () ypsses the two-quit gtes. Detection of photon t D herlds successful implementtion. R = I (the identity) implements offoli. R = (see Fig. ) implements C U etween C nd (in this cse, no photon is injected into ).,c, Experimentl circuit nd opticl source (see the Methods section). We use n inherently stle polriztion interferometer using two clcite em displcers 3. PPBS, prtilly polrizing em splitter; SPCM, single-photon counting module; PDC, prmetric downconversion; SG, second-hrmonic genertion. 36 NAURE PYSICS VOL 5 FEBRUARY 29

4 NAURE PYSICS DOI:.38/NPYS Figure 6 Experimentlly constructed offoli logicl truth tle. he lels on the x nd y xes identify the stte,c,. Idelly, flip of the logicl stte of the trget quit () occurs only when oth control quits ( nd C ) re in the logicl stte. he idel cse is shown s wire grid nd the overlp is I =.8±.3 (see the Methods section). Error rs re shown representing one stndrd devition, clculted from Poissonin photon-counting sttistics. he tle required four dys of mesurement. Figs nd 3, therey mking n implementtion fesile with recent developments in liner optic quntum gtes 37,38. For our implementtion of the offoli, we require four photons. We oserve fourfold coincidence rte t the output of our circuit of pproximtely mz when running t full pump lser power. Although this is not sufficient to crry out full process tomogrphy 27 of the gte over prcticl time period, we re le to demonstrte ll of the key spects of its ehviour. he first step in our chrcteriztion is to test the clssicl ction of the gte, tht ARICLES is, the ility to pply the correct opertion to ll eight logicl input sttes. Figure 6 shows the experimentlly reconstructed logicl truth tle. In the idel cse of our implementtion, the trget () undergoes logicl flip if, nd only if, oth control quits re in the logicl stte. We mesure good overlp etween the idel nd mesured truth tles 39 of I=.8±.3, compred with.84 nd.85 chieved for the originl opticl implementtions of two-quit gtes 3,3. his is comprehensive test of the clssicl ction of our gte. he next step is to test the quntum ction of the gte, tht is, the ility to pply the correct opertion to input superposition sttes. At our count rtes, we re not le to test tomogrphiclly complete set required for full process chrcteriztion, over prcticl time period. Our concession is to test the most experimentlly chllenging nd functionlly importnt cses. hey re chllenging ecuse they require coherent interction etween ll three quits nd, in two cses, idelly generte mximlly entngled Bell sttes 2. hey re functionlly importnt ecuse they demonstrte the gtes ility to generte nd control lrge mount of entnglement. his is of fundmentl importnce to the dvntges offered y universl quntum computer 4 nd is stndrd figure of merit 3 6. In the idel cse: with n input stte of,(+), / 2, our offoli will produce the entngled stte,ψ +, where Ψ + is the mximlly entngled Bell stte 2 (, +, )/ 2; with n input stte of,c, =,(+), / 2, it will produce the seprle output stte,(+), / 2. In the former (ltter) cse, the entngling opertion etween C nd is coherently turned on (off) y. We then swp the roles of the control quits nd repet the test. We crry out over-complete full stte tomogrphy to reconstruct the density mtrix of two-quit output sttes, while projecting the remining quit into its input stte (see the Methods section). Figure 7 shows the experimentlly reconstructed density mtrices representing the stte of control nd trget quit, t c Figure 7 Experimentlly reconstructed offoli output density mtrices., Mesured output sttes of quits C nd for offoli gte inputs;,(+), / 2; nd,(+), / 2. We oserve fidelities with the idel sttes, liner entropies nd tngles 39 of {.9±.4,.2±.8,.68±.} nd {.75±.6,.47±,.4±.6}, respectively., As for, ut where the roles of C nd hve een swpped. We now oserve {.8±.2,.39±.5,.53±.7} nd {.8±.3,.4±.5,.±.}. he decrese in tngle in the cses reflects the difference etween dependent nd independent photon interference, s discussed in the text. c, Idel density mtrices. Note, in ll cses only rel prts re shown; imginry prts re smll. Ech density mtrix requires 36 seprte mesurements 28 nd tkes pproximtely three dys to complete. NAURE PYSICS VOL 5 FEBRUARY

5 ARICLES NAURE PYSICS DOI:.38/NPYS c d Figure 8 Experimentlly reconstructed controlled-unitry gte process mtrices. d, u= nd θ=π/4 (C) (), θ=π/2 (CJ) (), θ=3π/4 (CL) (c) nd θ=π (CZ) (d). Rel nd imginry prts re shown. We oserve high process fidelities 27 with the idel {.982±.3,.977±.4,.94±.6,.956±.3} nd low verge output-stte liner entropies {.36±.4,.47±.4,.9±.5,.86±.6}, respectively. Mtrices re presented in the stndrd Puli sis 27. the output of our offoli gte. We chieve high fidelity 39 with the idel sttes nd high level of entnglement, s detiled in the figure cption. he results show tht the offoli crries out its most importnt nd experimentlly chllenging quntum opertions with high fidelity nd entnglement. o discuss sources of experimentl imperfection, we look t the detils of our liner optic implementtion. A key requirement for correct opertion of ech component two-quit gte is perfect reltive non-clssicl interference visiility (V r ) etween two photons. his in turn requires perfectly indistinguishle single photons. We mesure V r = ± % nd V r =92 ± 4% for the first nd second two-quit gtes shown in Fig. 5, respectively (where V r =V mes /V idel, V idel =8% nd results re for verticlly polrized photons). he difference cn e understood 38 NAURE PYSICS VOL 5 FEBRUARY 29

6 NAURE PYSICS DOI:.38/NPYS5 y considering tht the former opertes on dependent pir of photons generted from the sme pss of our opticl source, wheres the ltter uses independent photons from different psses (Fig. 5c). Photons generted from different psses re intrinsiclly more distinguishle 4,42. Another contriution to experimentl imperfection re the cses when more thn one pir of photons is creted simultneously in single pss of our opticl source. Although these higher-order terms occur with very low proility, nd do not significntly ffect the visiility mesurement due to higher-order interference processes, they cn introduce significnt error in the gte opertion 42. In generl, imperfections in the mesured offoli truth tle correspond to unwnted flips of the trget quit (Fig. 6). hese cn e understood with reference to the non-clssicl interferences required for correct opertion in ech cse. o etter illuminte these effects, we define stndrd contrst C (see the Methods section), which guges our gte s ility to pply the correct opertion to suset of logicl input sttes. For inputs,c =,, no non-clssicl interference is required for correct opertion nd we mesure C=.99±., verged over oth trget logicl input sttes. Inputs,C =, require perfect non-clssicl interference etween dependent photons C nd, for idel opertion. We chieve ner-perfect interference visiility etween verticl photons in this cse. owever, the full process suffers from the higher-order photon terms. his is reflected in n verge of C=.95±.2. Inputs,C =, require perfect non-clssicl interference etween independent photons nd, for idel opertion, reflected in n verge of C=.8±.2. Inputs,C =, require perfect non-clssicl interference etween oth dependent nd independent photons, nd re therefore the most chllenging cses. ere, we oserve n verge of C=.73±.5. It is strightforwrd to show tht the rtio of single to doule photon-pir emission is proportionl to the pump power. hus, reducing the power y fctor of four should reduce these unwnted higher-order contriutions from our source y fctor of four from ech pss. Under these conditions, we oserve fourfold rte t the output of the offoli gte of only mz nd repet mesurement of the verge contrst for the most chllenging logicl input,c =,, over period of five dys. We oserve cler improvement from C=.73±.5 to C=.83±.4. he effects of photon distinguishility nd higher-order terms lso cuse the imperfections in the stte tomogrphies of Fig. 7. For exmple, the entngling process required to chieve Fig. 7 relies on interference etween dependent photons. he process required to chieve Fig. 7 relies on oth dependent nd independent photon interference. his leds to the reduced fidelity oserved in the ltter cse. We conclude tht the dominnt source of experimentl error lies in our imperfect photon source. Our implementtion of the c u requires the genertion of two photons (Fig. 5). Even when running t /4 power, we oserve pproximtely z, which is sufficient to crry out full process tomogrphy 27 in 2 h. As demonstrtion, we report the implementtion of four distinct c u gtes tht pply z θ rottions (Fig. ) of π/4 (ct), π/2 (cj), 3π/4 (cl) nd π (cz) to the trget () conditionl on the control (C ), respectively. We fully chrcterize these gtes through quntum process tomogrphy 27 : Fig. 8 shows the experimentlly reconstructed process mtrices. We chieve exceptionlly high process fidelities, s detiled in the figure cption. We ttriute the smll devitions from idel opertion to residul higher-order emissions, imperfect mode mtching nd mnufctured optics 4,42. Outlook A cler impliction of our work is tht using multi-level quntum systems to encode informtion, rther thn enforcing two-level ARICLES structure, cn offer significnt prcticl dvntges for quntum logic. Although our demonstrtion enled new photonic quntum circuits, the resource-sving technique hs the potentil for ppliction in mny other rchitectures, ringing new circuits within rech of experimentl reliztion. An importnt pth for further reserch is to look for other prcticl simplifictions to quntum logic tht my e possile y enling simple steps outside the quit ilert spce. he overriding sources of error in our demonstrtions lie in our imperfect photon source: oth the effects of photon distinguishility nd the presence of unwnted higher-order emissions from prmetric downconversion. Current developments in source technology promise significnt improvements in the ner future. he comintion of this with recently developed photon-numer resolving detectors offers pths to deterministic nd sclle implementtions of our gtes. A key result is tht it is possile to overcome inherent non-determinism using only polynomil overhed in resources 2. Other importnt next steps re to use our circuits to explore smll-scle quntum lgorithms, generte new sttes nd test error-correction schemes. During the preprtion of this mnuscript, we ecme wre of demonstrtion of the offoli gte with trpped ions 43. Methods Source. Forwrd nd ckwrd photons pirs re produced through spontneous prmetric downconversion of frequency-douled mode-locked i:spphire lser (82 nm 4 nm, τ = 8 fs t 82 Mz repetition rte) doule pssed through type-i 2 mm BiB 3 O 6 crystl (Fig. 5). Photons re collected into four single-mode opticl fires nd detected using fire-coupled non-numer-resolving photon-counting modules. We spectrlly filter using unlocked interference filters centred t 82±.5 nm. Circuit. Photons re injected from single-mode opticl fires into free spce nd coupled into single-mode fires t the outputs (Fig. 5). One-quit gtes re relized deterministiclly using irefringent wve pltes. wo-quit gtes re relized non-deterministiclly using n estlished technique sed on non-clssicl interference t prtilly polrizing emsplitters in comintion with coincident mesurement Rther thn directly chining the two-quit gtes required for the offoli (Fig. 5), we use recently developed three-quit quntum logic gte 37,38. In liner optics implementtions of two-quit quntum gtes, stte-dependent loss is used to relnce mplitudes When incorporting loss elements L 3 (L), the offoli (c u) opertes with success proility of /72 (/8) (Fig. 5). Alterntively, to comt low count rtes, we chieve correct lnce y removing extr loss elements nd pre-ising the input polriztion sttes during gte chrcteriztion For the offoli, we use ll four outputs from spontneous prmetric downconversion fourfold coincident mesurement etween detectors D 4 signls successful run. We mesure fourfold coincidence rte of pproximtely mz when running t full pump lser power nd mz t /4 power. For the c u, we use only outputs C nd. In this cse, twofold coincident mesurement etween detectors D 2 signls successful run. We mesure twofold coincidence rte of pproximtely z when running t /4 pump lser power. Our imperfectly mnufctured emsplitters imprt systemtic unitry opertions on the opticl modes. For simplicity, we corrected for these effects numericlly. Alterntively, such unitries could e corrected with stndrd wve pltes. Qulity mesures nd sttistics. All error nlysis is crried out using Poissonin distriution to descrie the uncertinty in non-numer-resolving photon counting. Our stte nd process tomogrphy uses mximum likelihood estimtion to reconstruct physicl sttes nd Monte Crlo simultion for error nlysis 27,39,44. Mesurements sets re tken itertively, wherey multiple sets ech tking round h to complete re recorded. his reduces the effect of opticl source power fluctutions. he overlp etween two truth tles or inquisition (I) is defined s the verge logicl stte fidelity of truth tle I = r(m exp m idel )/d, where m exp nd m idel re the mesured nd idel truth tles, nd d is the tle dimension 39. he stndrd fidelity etween mixed (mesured) mtrix, ρ, nd the pure (idel) mtrix (either two sttes or two processes) is f= Ψ ρ Ψ ; liner entropy is sl d( r[ρ 2 ])/(d ), where d is the stte dimension 39. For the purposes of our error nlysis, we define the contrst C=/2{+(p idel p flip )/(p idel +p flip )}, where p idel is the proility of otining the idel output stte nd p flip is the proility of otining the output stte where the idel trget quit output stte hs een flipped. We clculte this property directly from the mesured truth tle. NAURE PYSICS VOL 5 FEBRUARY

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Quntum Interferometry (VC, 996). 42. Weinhold,. J. et l. Understnding photonic quntum-logic gtes: the rod to fult tolernce. Preprint t < (28). 43. Monz,. et l. Reliztion of the quntum offoli gte with trpped ions. Preprint t < (28). 44. Jmes, D. F. V., Kwit, P. G., Munro, W. J. & White, A. G. Mesurement of quits. Phys. Rev. A 64, 5232 (2). Acknowledgements We cknowledge discussions with W. Munro nd D. Kielpinski, nd finncil support from the Austrlin Reserch Council Discovery nd Federtion Fellow progrmmes, the DES Endevour Europe nd Interntionl Linkge progrmmes, nd n IARPA-funded US Army Reserch Office Contrct. Additionl informtion Reprints nd permissions informtion is ville online t reprintsndpermissions. Correspondence nd requests for mterils should e ddressed to B.P.L. 4 NAURE PYSICS VOL 5 FEBRUARY 29