Robust concurrent remote entanglement between two superconducting qubits

A. Narla, S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, E. Zalys-Geller,

S. O. Mundhada, W. Pfaff, L. Frunzio, R. J. Schoelkopf, M. H. DevoretDepartment of Applied Physics, Yale University

(Dated: April 27, 2016)

Entangling two remote quantum systems which never interact directly is an essential primitive inquantum information science. In quantum optics, remote entanglement experiments[1–3] providesone approach[4] for loophole-free tests of quantum non-locality[5, 6] and form the basis for the mod-ular architecture of quantum computing[7]. In these experiments, the two qubits, Alice and Bob,are each first entangled with a traveling photon. Subsequently, the two photons paths interfere ona beam-splitter before being directed to single-photon detectors. Such concurrent remote entangle-ment protocols using discrete Fock states can be made robust to photon losses[8], unlike schemes thatrely on continuous variable states[9, 10]. This robustness arises from heralding the entanglement onthe detection of events which can be selected for their unambiguity. However, efficiently detectingsingle photons is challenging in the domain of superconducting quantum circuits because of the lowenergy of microwave quanta. Here, we report the realization of a novel microwave photon detectorimplemented in the circuit quantum electrodynamics (cQED) framework of superconducting quan-tum information, and the demonstration, with this detector, of a robust form of concurrent remoteentanglement. Our experiment opens the way for the implementation of the modular architectureof quantum computation with superconducting qubits.

The concept of a photon, the quantum of excitation ofthe electromagnetic field, was introduced by Planck andEinstein to explain black-body radiation spectrum[11]and the photoelectric effect[12]. However, experimentsthat would definitively prove the existence of optical pho-tons were only understood[13, 14] and realized[15] muchlater in the 20th century. Although there is no reasonto doubt that microwave photons would behave differ-ently than their optical counterparts, revealing and ma-nipulating them is much more challenging because theirenergies are 4 to 5 orders of magnitude lower. Cavity-QED, and later on circuit-QED, have established thereality of stationary quantum microwave excitations ofa superconducting resonator by strongly coupling themto Rydberg[16] and superconducting artificial atoms[17].The production of traveling microwave photons was thenindirectly demonstrated using linear amplifiers to mea-sure the state of the radiation[18–20]. However, whilethere have been proposals and implementations of sin-gle flying microwave photon detectors[21–23], controllingand employing the single-photon nature of microwaveradiation is still an open challenge. Here, we presentsuch a detector using a superconducting 3D transmonqubit[24], a promising candidate system for a scalablequantum computer. We operate this artificial atom in aregime that directly exploits the properties of single pho-tons and carry over to the microwave domain the remoteentanglement experiments performed in quantum optics.In the absence of single microwave photon detectors,

the only form of remote entanglement realized so farwith superconducting qubits has been through the useof continuous variable coherent states as the flying in-formation carriers[25]. While such states can be effi-ciently processed by quantum-limited linear parametricamplifiers[26, 27] readily available at microwave frequen-cies, a disadvantage of this protocol is its sensitivity to

losses in the paths of the flying states. In contrast, re-mote entanglement using flying single photons is robustto these losses because only the successful detection ofphotons is linked to the production of a pure entangledstate[8, 28]. This feature is particularly important forgenerating entanglement between two distant stationaryqubits, a crucial element of the modular architectureof quantum computation[7] and the proposed quantuminternet[29]. Furthermore, scaling up the modular archi-tecture requires a protocol with no direct connections be-tween modules, unlike previously demonstrated sequen-tial methods[25], in order to maintain a strong on/offratio. Thus, demonstrating robust remote entanglementwhich satisfies this requirement, i.e. a concurrent proto-col, is a vital step in the implementation of the modulararchitecture with superconducting qubits.The experiment, housed in a dilution refrigerator below

20 mK, consists of two superconducting transmon qubits(see Fig. 1A), referred to as Alice and Bob, in separate 3Dcavities[24]. The cavities have nearly identical resonancefrequencies ωg

A/2π = 7.6314 GHz, ωgB/2π = 7.6316 GHz

and bandwidths κA/2π = 1.2 MHz, κB/2π = 0.9 MHz.Their strongly coupled output ports are connected by mi-crowave coaxial cables to the two input ports of a 180◦hybrid, the microwave equivalent of a 50/50 beamsplit-ter. One of the output ports of the hybrid is connectedto a microwave single photon detector which is realizedby a third 3D cavity also containing a transmon. Theother output port of the hybrid is terminated in a 50 Ωload. To ensure signal flow as shown by the arrows inFig. 1A, microwave isolators/circulators (not shown, seeexperimental schematic in Methods) are inserted into thelines connecting each qubit to the hybrid. These providerobust isolation between modules and connect the systemoutput to readout electronics.To entangle the remote qubits, flying microwave

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Bob

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Time

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Figure 1. | Experiment and protocol schematic for remote entanglement of transmon qubits using flying singlemicrowave photons. A) Two superconducting 3D transmon qubits, Alice and Bob, are connected by coaxial cables to thetwo input ports of the microwave equivalent of a 50/50 beam-splitter. One of the output ports of the splitter is connected to amicrowave single photon detector also realized by a 3D transmon qubit. The other port of the splitter is terminated in a cold50 Ω load. B) Quantum circuit diagram of the remote entanglement protocol, with the states of the quantum system at varioussteps. The Alice and Bob (red and blue) qubits are each prepared in the state 1√

2(|g〉+ |e〉) by a single qubit gate Ry

(π2

).

They are then entangled with flying single photons (black) via a CNOT-like operation. The states∣∣O±⟩

= 1√2(|ge〉 ± |eg〉)

represent odd Bell states of the Alice and Bob qubits while∣∣o±

⟩= 1√

2(|10〉 ± |01〉) represent odd Bell states of flying single

photons in the Alice and Bob channels respectively. The flying photons interfere on the beam-splitter whose unitary actionUBS erases their which-path information. Following a π-pulse on Alice and Bob, the CNOT-like operation and beam-splittersteps are repeated to remove contributions of the unwanted |ee〉 state. Detecting two photon clicks in a pair of consecutiverounds heralds the

∣∣O+

⟩= 1√

2(|ge〉+ |eg〉) Bell state of Alice and Bob.

single photon states are used as carriers of quantuminformation according to the protocol proposed in[8]. As outlined in Fig. 1B, the remote entanglementprotocol begins by initializing both qubit-cavity systemsin 1√

2(|g〉+ |e〉) ⊗ |0〉, the state on the equator of

the Bloch sphere with no photons in their respectivecavities. Through a controlled-NOT (CNOT)-likeoperation, whose implementation is detailed later inthe text, the qubits are now entangled with flyingsingle photons where the state of each qubit-photonpair becomes 1√

2(|g0〉+ |e1〉). The joint state of

all stationary and flying qubits can be expressed as|ψ〉1 = 1

2 (|gg〉 |00〉+ |O+〉 |o+〉+ |O−〉 |o−〉+ |ee〉 |11〉)where |O±〉 = 1√

2(|ge〉 ± |eg〉) represent the odd

Bell states of the Alice and Bob qubits and

|o±〉 = 1√2(|10〉 ± |01〉) represent the odd Bell states

of flying single photons in Alice’s and Bob’s channels,respectively. The photons interfere on the 180◦ hybridwhose action, analogous to that of a beam-splitter,

is described by the unitary UBS = e−3π(a†b−ab†)/4.This maps |o+〉 → |10〉 (|o−〉 → |01〉), taking thetwo flying odd Bell states to a single photon in theAlice or Bob branch of the system. This operationerases the which-path information of the photonsand produces Hong-Ou-Mandel interference[15]. Af-ter the hybrid, the total system state is |ψ〉2 =12

(|gg〉 |00〉+ |O+〉 |10〉+ |O−〉 |01〉+ 1√

2|ee〉 (|02〉 − |20〉)

).

At this point, the photons in the Alice channel enter thedetector which distinguishes between detecting a photon,a ‘click’, or detecting nothing, called ‘no click’. Ideally,

3

by heralding on only single photon detection events, the|O+〉 is selected out from all the other states. However,losses in the system between the qubits and the detectorand the inability of the detector to distinguish betweenthe Fock states |1〉 and |2〉 instead result in the mixeddensity matrix ρclick3 = N |O+〉 〈O+| + (1−N) |ee〉 〈ee|when the detector goes click. Here, the normalizationconstant N depends on loss in the system and thecharacteristics of the detector. In particular, it dependson the probabilities with which it maps the input flyingphoton states, |1〉 and |2〉, to an outcome of click.Another crucial assumption in ρclick3 is that the detectorhas no dark counts, i.e. it never clicks when it receives|0〉. A fuller version of ρclick3 including dark counts isgiven in the Methods. Thus, at this stage, the qubitsare in the state |O+〉 with probability N and we wouldlike to remove the undesired |ee〉 state.To achieve this, a Ry(π) pulse is applied on both Alice

and Bob followed by a second round of entangling thequbits with flying photons, interfering them on the hy-brid and detecting them. The π-pulse takes |ee〉 → |gg〉;consequently, in the second round, the unwanted state ismapped onto |gg〉 |00〉, and thus it can be selected out bydetecting a photon. On the other hand, |O+〉 is mappedonto a mixture of |O+〉 |10〉 and |O−〉 |01〉. Conditioningon measuring clicks in two consecutive rounds of the pro-

tocol results in the odd Bell state |ψ〉click, click6 = |O+〉.

A result of this dual conditioning is that loss in the sys-tem only reduces the success probability of the protocoland not the fidelity of the generated entangled state. Re-placing the cold 50 Ω load with a second detector wouldincrease the success probability by a factor of 4 and al-lows for the generation of both the |O+〉 and |O−〉 statesdepending on whether the same or different detectors goclick on each round, respectively. Since it does not im-prove the fidelity of entanglement, we omitted the seconddetector to reduce the hardware in the dilution fridge andsimplify the microwave control electronics.Successfully realizing this protocol required simulta-

neously: (1) implementing the generation of single pho-ton Fock states which are entangled with the stationaryqubits and (2) detecting the subsequent single photonstates. Furthermore, the frequencies and temporal en-velopes of the photons arising from each cavity had to becontrolled to ensure that the detector cannot distinguishbetween them.The first ingredient, previously termed a CNOT-like

operation, actually maps an arbitrary qubit state α |g0〉+β |e0〉, where α and β are arbitrary complex coeffi-cients, onto the joint qubit-flying photon state α |g0〉 +β |e1〉 (this operation is not a unitary in the manifold{|g0〉 , |g1〉 , |e0〉 , |e1〉} because it takes |e1〉 to |f1〉; how-ever, this has no effects on the protocol since the cavityalways starts in |0〉). This is done by exploiting |f〉, thesecond excited state of the transmon qubit[30], as well asthe two-photon transition |f0〉 ↔ |e1〉[31, 32]. As shownin Fig. 2A, starting with the qubit in α |g〉 + β |e〉, theoperation is realized by first applying a π-pulse at ω0

ef ,

taking the qubit to α |g〉+ β |f〉, and then applying a π-pulse on the |f0〉 ↔ |e1〉 with two sideband tones (ωQSB ,ωCSB). This maps the qubit state onto the joint qubit-intra-cavity state, α |g0〉 + β |e1〉. Finally, the photonstate leaks out of the cavity, becoming a flying state thatis entangled with the qubit. As a result, the travelingphoton has the frequency ωe

A (ωeB) and a decaying expo-

nential temporal waveform with the decay constant κA

(κB). The indistinguishability of the photons, then, wasachieved in this experiment by the nearly identical fre-quencies and bandwidths of the Alice and Bob cavities(as given above and in Methods). Note that although allthe photons need to be at the same frequency, there isno requirement here for the qubits to be identical.The second ingredient, microwave single photon detec-

tion, is the novel technical component of our demonstra-tion. Physically, this detector is another transmon-3Dcavity system like Alice and Bob. The strongly cou-pled port of the cavity is the detector input port. Inthe strong dispersive regime where the qubit is oper-ated (χD/2π = 3 MHz, κD/2π = 1 MHz), we can se-lectively π-pulse the qubit conditioned on the presenceof one intra-cavity photon[17], mapping the flying pho-ton onto the state of the detector qubit. To operate thissystem as a detector of single flying photons, we tunedthe cavity frequency ωg

D/2π = 7.6222 GHz close to ωeA

and ωeB and matched the linewidths of all three cavi-

ties. This condition ensured that the detector efficiencyis maximized. The incident single photons from Aliceand Bob will excite the detector cavity ∼ 50% of the time(see Methods) since their decaying exponential temporalwaveforms are not mode-matched to the cavity. Thus,the selective π-pulse excites the qubit only if a photonwas received, with the length and timing of this pulsedetermining the detector efficiency (see Methods). Oncethe photon leaks back out, a conventional cQED disper-sive readout of the qubit state[33] completes the quantumnon-demolition (QND) photon detection process. Mea-suring the qubit in the excited state corresponds to aphoton detection event (click). Finally, the detector isreset by returning the qubit to |g〉 with an un-selectiveπ-pulse.

This microwave photon detector satisfies three impor-tant criteria in an architecture that is easily integratedwith current state-of-the art cQED experiments. First,the detector has a reasonable efficiency, η ≈ 0.5, sinceabout half of all incident photons enter the detector. Sec-ond, this detector has low dark counts (the probability ofthe detector reporting a click even when no photon en-tered the detector) Pd < 0.01, limited by the frequencyselectivity of the π-pulse. Finally, it has a short re-armtime of 450 ns determined by how long it takes to emptythe cavity and reset the qubit. We discuss avenues tofurther improving this detector in the Methods section ofthe paper. Nevertheless, as we show below, the detectorperformance is sufficient for generating remote entangle-ment.As a preliminary step towards the realization of the full

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Figure 2. | Signatures of qubit/flying photon entanglement. A) Frequency spectra of the Alice and detector qubit-cavity systems (left) and experimental pulse sequence (right). The colors denote transitions which are driven to perform theCNOT-like operation and flying single photon detection. The Alice qubit is prepared in an arbitrary initial state by the pulseRge

φ (θ) at ωge. The CNOT-like operation consists of a Ry (π) pulse at ωef followed by a pair of sideband pulses. The sidebandpulses are applied at ωQSB , detuned by Δ from ωef , and ωCSB , detuned by Δ from ωe

A. To detect flying photons, a frequencyselective π-pulse is applied to the detector qubit at ω1

ge followed by a measurement of the qubit state. B) Color plots of theprobability, Pclick, of the detector qubit ending in |e〉(left) and the Alice qubit polarization, 〈ZA〉 (right), as a function of thesideband pulse length TSB and θ (for φ = π/2). The dashed line at TSB = 254 ns corresponds to a transfer |f0〉 → |e1〉, i.e.a CNOT-like operation. C) Detector click probability, Pclick, and Alice equatorial Bloch vector components, 〈XA〉 and 〈YA〉,as a function of φ for θ = π/2 when the CNOT-like operation is either performed (bottom) or not (top). Open circles areexperimental data and lines are fits.

remote entanglement protocol, we demonstrate in Fig. 2Band C signatures of entanglement between the Alice qubitand its corresponding traveling photon state by show-ing that the CNOT-like operation maps α |g0〉 + β |e0〉to α |g0〉 + β |e1〉 (for data on the Bob qubit and sim-ulations, see Methods). We first show that the relativeweights of |g〉 and |e〉 were correctly mapped by initial-izing the qubit in cos (θ/2) |g〉+ sin (θ/2) |e〉, followed by

a π pulse on the ω0A,ef and sideband pulses for a varying

time TSB (see Fig. 2A, right). The selective π-pulse onthe detector was a 480 ns Gaussian pulse (σ = 120 ns)and was timed such that the center of the Gaussian co-incides with the end of the sideband pulse. Finally, wemeasured the probability of detecting a photon in thedetector, Pclick, and the Alice polarization, 〈ZA〉. Asshown in Fig. 2B (black dashed line), a π-pulse on the

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Mea

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A B

ZZ XX YY XY YX ZZ XX YY XY YX

Figure 3. | Two-qubit remote entanglement. Measured amplitudes of the relevant two-qubit Pauli vector components asa function of qubit preparation. After the remote entanglement protocol described in Fig. 1B, joint tomography was performedon the qubits conditioned on the detector reporting a click for each round. A) With Bob always initialized in 1√

2(|g〉+ |e〉),

Alice was prepared in the variable state cos (θ/2) |g〉+ sin (θ/2) |e〉. Data (points) and fits (lines) confirm that entanglement ismaximized when θ = π/2 (dotted line). B) With Alice always initialized in 1√

2(|g〉+ |e〉), Bob was prepared in the variable

state 1√2

(|g〉+ eiφ |e〉). The components of the Pauli vector oscillate with φ sinusoidally as expected. The complete density

matrix for φ given by the dotted line is shown in Fig. 4 (left) in the Pauli basis.

|f0〉 ↔ |e1〉 transition occurs for TSB = 254 ns when theprobability of detecting a photon, Pclick is maximized.On the other hand, for shorter sideband pulse lengths,no photons are generated and Pclick = 0. Moreover, theobserved increase in Pclick with θ confirms that the rel-ative weight of the superposition state between |g〉 and|e〉 is mapped onto |g0〉 and |e1〉 (Fig. 2B, left). We alsoconfirm that this process does not destroy the qubit stateby observing that the final value of 〈ZA〉 agrees with theinitial preparation angle θ (Fig. 2B, right).Furthermore, in Fig. 2C, we show that this operation

also maps the phase of α |g〉 + β |e〉 onto α |g0〉 + β |e1〉.Directly measuring the phase of |e1〉 relative to |g0〉 isnot possible in this experiment since the detector onlydetects the presence or absence of a photon. Instead,the Alice qubit was first prepared on the equator of theBloch sphere in 1√

2

(|g〉+ eiφ |e〉), the CNOT-like opera-

tion was either performed or not and finally both Pclick,and the qubit equatorial Bloch vector components, 〈XA〉and 〈YA〉, were measured. When no photon is generated,Pclick = 0 as expected and 〈XA〉 and 〈YA〉 oscillate withthe preparation phase φ (Fig. 2C, top). However, whenthe operation is performed, a photon is generated andthus Pclick is now non-zero. Since, the preparation phaseφ is now mapped onto the entangled state, the measure-ment of the photon, either by the detector or some otherloss in the system, results in the unconditional dephasingof the qubit, 〈XA〉 , 〈YA〉 = 0 (Fig. 2B, bottom).Having demonstrated qubit-photon entanglement, we

next perform the full remote entanglement protocol. Thefinal two-qubit density matrix was measured in the Paulibasis with joint tomography (see Methods) conditionedon detecting two clicks. For an arbitrary Bell state, the

only non-zero Pauli components are 〈ZZ〉, 〈XX〉, 〈Y Y 〉,〈XY 〉 and 〈Y Z〉, which are displayed in Fig. 3. We firstdemonstrate that the protocol entangles the qubits onlywhen they start in the correct state. With Bob initializedin 1√

2(|g〉+ |e〉), Alice was prepared in cos (θ/2) |g〉 +

sin (θ/2) |e〉. Entanglement is maximized for θ = π/2(see Fig. 3A dotted line), with extremal values for 〈XX〉,〈Y Y 〉, 〈XY 〉 and 〈Y X〉, and with the expected negative〈ZZ〉 indicating a state of odd parity. On the other hand,for θ = 0 (θ = π), the final two-qubit state should bethe separable state |eg〉 (|ge〉) as indicated by 〈XX〉 =〈Y Y 〉 = 〈XY 〉 = 〈Y X〉 = 0 and 〈ZZ〉 < 0. We attributethe deviation of 〈ZZ〉 from −1 to the dark counts in thedetector and the finite T1’s of the two qubits.Next, we show that when both qubits are ini-

tialized along the equator of the Bloch sphere, re-mote entanglement is always generated. Alice wasnow prepared in 1√

2(|g〉+ |e〉) with Bob prepared in

1√2

(|g〉+ eiφ |e〉). In this case, the final state should be1√2

(|ge〉+ ei(φ+φoff) |eg〉), where φoff is an arbitrary off-

set phase included to account for frequency offsets andpath length differences between the two flying photons.This Bell state is witnessed by the tomography resultsin Fig. 3B, where 〈ZZ〉 is constant and negative whilethe other four displayed Pauli components follow the ex-pected sinusoidal behavior. From the fits to the data, weextract φoff = 3π/10.The complete density matrix, ρmeas, is shown in Fig. 4

(left) in the Pauli basis for φ given by the dotted linein Fig. 3B, where the fidelity F = Tr (ρmeas |O+〉 〈O+|)is maximum. The theoretically calculated density ma-trix, (Fig. 4, right), includes the effects of the coherence

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ZI XI YI IZ IX IY ZZ ZX ZY XZ XX XY YZ YX YYZI XI YI IZ IX IY ZZ ZX ZY XZ XX XY YZ YX YY

Experiment (uncorrected) Theory (with imperfections)

Alice Bob JointAlice Bob JointMea

sure

d Pa

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nt A

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Alice JointBob

Figure 4. | Entanglement characterization. Left: Experimentally measured Pauli vector components of the two-qubitentangled state confirming that the final state is the odd Bell state 1√

2(|ge〉+ |eg〉) with raw fidelity F = 0.53. Right: The

theoretically expected Pauli components accounting for qubit decoherence, detector dark counts and tomography infidelity.

times of the Alice and Bob qubits, T2Bell, the imperfec-tions of the detector and the imperfections in the jointtomography (see Methods). As expected, most of thestate information lies in the two-qubit Pauli componentsrather than the single-qubit ones. The measured fidelityF = 0.53±0.01 and concurrence C = 0.1±0.01[34] exceedthe entanglement threshold. When accounting for sys-tematic errors in tomography (see Methods), we obtainthe corrected fidelity Fcorr = 0.57 ± 0.01. This fidelitycan be understood as a result of various imperfections inthe entanglement generation protocol: (1) decoherenceof the two qubits which limits the fidelity to FT2Bell

and(2) imperfections of the detector which are characterizedby Fdet. From the measured value of T2Bell = 6 μs andthe protocol time, Tseq = 2.5 μs, we expect FT2Bell

∼= 0.8.The infidelity associated with the imperfect detector ischaracterized by the dark count ratio Pd/Pclick, which isthe fraction of detection events that are reported as clickseven though no actual photon was sent. In this exper-iment, Pd/Pclick = 0.05, primarily limited by the finiteselectivity of the detection pulse and the imperfect read-out of the detector qubit, which results in Fdet

∼= 0.9. Atheoretical model incorporating these two imperfectionswas used to calculate an expected fidelity Fthy = 0.76(see Methods). The remaining infidelity is a result ofsources that are harder to characterize and will need tobe explored in further work, like, for instance, the im-perfections of the CNOT-like operation and the distin-guishability of the photons. Nevertheless, the currentresults clearly establish the viability of this protocol and,by extension, the modular architecture for superconduct-ing qubits.Another figure of merit for this experiment is the en-

tanglement generation rate which is determined by therepetition rate, Trep = 21 μs, and the success prob-ability of the experiment. The latter is determinedby the product of state initialization via post-selection

(57%) and the detector click probability in the first(8%) and second (9%) rounds respectively leading toan overall success probability of 0.4%. The correspond-ing generation rate of about 200 s−1 is orders of mag-nitude faster than similar experiments performed withnitrogen-vacancy centers in diamond (2 × 10−3 s−1)[3],neutral atoms (9 × 10−3 s−1)[2] or trapped ion systems(4.5 s−1)[35]. We note, however, that our generation ratedoes not yet cross the threshold for fault tolerance[7, 35]though there are many prospects for enhancement.Improvements in generation rate and fidelity are possi-

ble with readily available upgrades to the hardware andsoftware of our experiment. Firstly, a factor of 4 increasein success can be achieved by installing the omitted sec-ond detector. Secondly, shaping the generated photonsand detection pulse to mode match the flying photonsto the detector would increase the detection efficiencyby at least 50% and hence multiply the generation rateby at least a factor of 2. Moreover, this would reduceboth the dark count fraction and the distinguishabilityof the traveling photons which would directly benefit theentanglement fidelity by bringing Fdet closer to unity.Thirdly, an order of magnitude better coherence timesfor the two qubits have been demonstrated in similar 3Dqubit-cavity systems[36], which should readily carry overto this experiment and improve FT2Bell

. Finally, the over-all throughput of the experiment can be increased by anorder of magnitude by the use of real-time feedback ca-pabilities that have been recently demonstrated for su-perconducting qubits[37, 38].In this work, we have demonstrated, in a single ex-

periment, the set of tools that had been previously theexclusive privilege of quantum optics experiments: theavailability of flying microwave single photon sources anddetectors together with the spatial and temporal con-trol of traveling photons to make them indistinguish-able. With these tools, we have realized two-photon

7

interference of microwave photons and the generationof loss-tolerant entanglement between distant supercon-ducting qubits with concurrent measurements. The pro-tocol speed and prospects for improving fidelity make thisa very promising implementation for remote entangle-ment and the distribution of quantum information withmicrowave flying photons. Thus, this experiment opensnew prospects for the modular approach to quantum in-formation with superconducting circuits.

Parameter Alice Bob Detector

Cavity frequency ωgc/2π (GHz) 7.6314 7.6316 7.6222

Cavity bandwidth κ/2π (MHz) 0.9 1.2 0.9

Qubit frequency ωge/2π (GHz) 4.6968 4.6620 4.7664

Anharmonicity α/2π (MHz) 197 199 240

Dispersive shift χ/2π (MHz) 9 9 3

T1 (μs) 140 85 90

T2,Echo (μs) 9 16 30

Methods Table I. Alice, Bob and Detector qubit and cavityparameters

8

Supplementary Material

I. EXPERIMENTAL SETUP

A. Sample Fabrication and Parameters

The three transmon qubits consist of Al/AlOx/AlJosephson-junctions fabricated using a bridge-freeelectron-beam lithography technique[39] on double-side-polished 3 mm by 13 mm chips of c-plane sapphire. Thejunctions are connected via 1 μm leads to two rectangu-lar pads (1900 μm × 145 μm for Alice and Bob, 1100 μm× 250 μm for the detector) separated by 100 μm. Thequbit chips are placed in their respective rectangularindium-plated copper cavities (21.34 mm × 7.62 mm ×43.18 mm). The transmon parameters and couplings tothe TE101 cavity mode were designed using using finite-element simulations and black-box quantization[40]. Ex-perimentally measured device parameters are listed inTable 1.A coaxial coupler was used as the input port of each

cavity with the length of pin determining the input cou-pling quality factor Qin ∼ 106. The output port for eachcavity was an aperture in the cavity wall at the anti-nodeof the TE101 mode. The size of the aperture was chosenso that Qout = 7.5 × 103 yielding a total cavity band-width κ 1/Qout. Waveguide to coaxial cable adapters(WR-102 to SMA) were used on the output port of thecavities; since the qubit frequency is below the cutoff fre-quency of the waveguide while the cavity frequency isinside the passband, this section of waveguide acts as aPurcell filter for the qubit.As shown in Methods Fig. 1, the qubits were mounted

to the base stage of a cryogen-free dilution fridge main-tained below 50 mK. The cavities were housed inside μ-metal (Amumetal A4K) cans to shield them from mag-netic fields. The input and output lines connected to theexperiment were filtered with home-made lossy Eccosorbfilters, commercial low-pass microwave filters, attenua-tors and isolators to attenuate radiation incident on theexperiment. A commercial cryogenic HEMT amplifierwas used at 3 K to additionally amplify the output sig-nals before subsequent room-temperature amplificationand demodulation.A critical requirement for the experiment was match-

ing the frequencies of the Alice and Bob cavities to renderthe flying single photons indinstinguishable. In addition,the detector cavity frequency needs to also be matchedto the Alice and Bob cavity frequencies so that incidentphotons can enter the detector cavity. This was achievedby an aluminum screw inserted into each cavity at theTE101 anti-node to fine-tune the cavity frequencies untilthey satisfied ωe

A = ωeB = ωg

D (see Methods Fig. 2A).

B. Readout

All three qubit-cavity systems were measured on thesame output line using a single Josephson ParametricConverter (JPC) operated as a nearly-quantum-limitedphase-preserving amplifier. The JPC was biased to pro-vide 20 dB of gain with a bandwidth of 8 MHz centeredat 7.6314 GHz to realize high-fidelity single-shot readoutof all three qubit-cavity system. At this operating point,a noise visibility ratio (NVR)[? ] of 8 dB was measured,indicating that 86% of the noise measured at room tem-perature was amplified quantum fluctuations from theJPC.As shown in Methods Fig. 1, readout pulses for the

three cavities were generated using a single microwavegenerator powering an IQ-mixer. The output of themixer was split and sent to each cavity on separate in-put lines with the relative room temperature attenuationon each line adjusted so that an applied readout ampli-tude at room temperature resulted in the same measuredsignal-to-noise ratio (SNR) for each qubit-cavity system.Microwave switches were used on each line to gate thepulses generated by the IQ-mixer. The amplified cavityoutputs were mixed down to radio frequencies along witha copy of the generator tone that did not pass throughthe cryostat to provide a reference. The signal and ref-erence were digitized and demodulated to yield in-phaseand quadrature components (I(t), Q(t)) that are insensi-tive to drifts in the generator and other microwave com-ponents. With this setup, high-fidelity readout of all themodules in the fridge was possible with minimal hard-ware and complexity. In the experiments described inthis paper, two types of measurements were performed:(1) joint measurement of the Alice and Bob qubits and(2) single qubit measurement of the detector

1. Joint Alice and Bob measurement

The Alice and Bob cavities were measured jointly byenergizing them with 2 μs pulses at fABC

msmt = ωgA/2π =

7.6314 GHz. Using a phase shifter on the Bob cavityarm, the relative phase of the pulses on the Alice andBob cavities (including all system path lengths) was ad-justed to π/2. The output signals from each cavity thenpassed through the hybrid whose output was the sum ofthe two cavity signals but with half the power from eachsignal was lost in the cold 50 Ω load. This joint outputsignal reflects off the detector cavity (since it is χA aboveωgD) and was amplified by the JPC. As a result, the out-

put signal demodulated at 50 MHz contained informationabout both qubit states along orthogonal axes (see Meth-ods Fig. 2B). Two separatrices (white dashed lines), thefirst along theQm axis and the second along Im axis, wereused to measure the state of the Alice and Bob qubitsrespectively. In addition, the two-qubit correlation wascalculated on a shot-by-shot basis. This resulted in anoverall fidelity Fjoint > 90%. A primary limitation in

9

300 K

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2

1

IQ }

sig

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Alice qubit + cavity

Bob qubit + cavity

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+ cavity

JPCS

IP

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, MHz

Methods Figure 1. | Detailed experimental setup. The experiment (bottom) was cooled down on the base-stage(< 50 mK) of a dilution refridgerator. Input lines carrying signals to the systems were attenuated and filtered using commerciallow-pass filters and homemade lossy Eccosorb filters. The room temperature electronics used to produce and shape the inputsignals are shown at the top of the figure. The basic setup to produce shaped signals was a microwave generator driving anIQ mixer followed by an amplifier and finally a switch to gate the signal (box in top right corner). The signals were shapedby channels from four Arbitrary Waveform Generators (AWGs) (not shown) which also provided the digital markers for theswitches. Copies of this setup (denoted by the shorthand notation of a circle with a shaped pulse) were used to generate thedrive signals (color-coded) for three modules, Alice (red), Bob (blue) and the detector (green). The Alice and Bob modules had4 inputs each, the cavity readout tone, the qubit signals and the pair of sideband pulses for photon generation. On the otherhand, the detector module had 2 inputs, the cavity readout tone and the qubit signals. All the modules were readout using asingle output line that had multiple stages of amplification. High-fidelity single-shot readout was enabled by the JPC amplifier.The output signals were downconverted and then digitized and demodulated along with a room-temperature reference copy.

10

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clickno click

Methods Figure 2. | Alice, Bob and Detector qubit readout spectra and histograms. A) Alice, Bob and Detectorcavity frequency spectra. The Alice and Bob cavities had nearly identical frequencies (ωg

A ≈ ωgB) and dispersive shifts (χA ≈

χB). To perform joint readout of Alice and Bob, microwave pulses were simultaneously applied on each cavity at ωgA with a

relative phase of π/2 between the two pulses. The detector module cavity frequency ωgD was tuned to match the frequency of

the photons in the experiment, ωeA. The detector was readout at ωg

D. B) Joint readout histogram for Alice and Bob. A 2 μsmeasurement pulse was used to measure the state of both qubits. The resulting output contained information about the stateof Alice and Bob along the Qm and Im axes respectively. Thus, the measurement provided single-shot readout of both qubitstates as well as the correlation between the two qubit states with Fjoint > 90%. C) Readout histogram for the detector. Thestate of the detector qubit was measured with Fdet > 99% in 700 ns.

achieving a higher fidelity was the loss of half the infor-mation in the cold-load after the hybrid. This can beimproved in future experiments by the use of a seconddetector and output line. While these joint tomographyimperfections will ultimately impact the measured entan-glement fidelity, they can be calibrated out (as we discusslater in the Joint Tomography and Calibration section).

2. Detector qubit measurement

To measure the state of the detector qubit, an IF-frequency of −9.2 MHz was used on the IQ-mixer to gen-erate 700 ns pulses at ωg

D = 7.6222 GHz. Since this isequal to ωe

A and ωeB , this readout is not performed simul-

taneously with the joint measurement of Alice and Bobdescribed above to avoid signal interference. The ampli-fied output from the cavity was demodulated at 59.2 MHzresulting in the histogram shown in Methods Fig. 2C. As

11

explained in the main text, measuring the qubit in |e〉corresponds to a click in the detector. In this case, themeasurement fidelity, Fdet > 99%. The measurementwas optimized for maximal fidelity in the shortest pos-sible time by using a shaped pulse that minimized thecavity ring-up and ring-down time[41]. Since the pulse-shape also decreased the time taken to depopulate thecavity, operations on the detector could be performed400 ns after the readout instead of having to wait for thenatural ring-down time.

II. JOINT TOMOGRAPHY ANDCALIBRATION

To calculate the final state of the Alice and Bob qubitafter a joint measurement, the measured in-phase andquadrature signal (I(t), Q(t)) was converted into a dig-ital result using two thresholds, one for Alice and onefor Bob (see Methods Fig. 2B). Since the four measuredGaussian distributions had equal standard deviations,these thresholds were straight lines equidistant from thetwo distributions. Thus, using the thresholds, the out-put voltage from each joint tomography measurementwas converted into a final outcome of |g〉 or |e〉 for eachqubit. By performing measurements on an ensemble ofidentically prepared states, these counts were convertedinto expectation values of the observable being measured.Fully characterizing the state of the two qubits requiresmeasuring the 16 components of the two-qubit densitymatrix. This was done in the Pauli basis using the single-qubit pre-rotations Id, Ry (π/2) and Rx (π/2) to measurethe Z, X and Y components respectively of each qubitBloch vector and the two-qubit correlators.However, the tomography was not perfect (Fjoint =

100%) and we next discuss how to understand the im-perfect tomography and calibrate out its effects[38, 42].The ideal joint measurement of the two-qubit state canbe described using the projectors into the computationalbasis:

ΠGG =

⎛⎜⎜⎜⎝

1 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

⎞⎟⎟⎟⎠ ,ΠGE =

⎛⎜⎜⎜⎝

0 0 0 0

0 1 0 0

0 0 0 0

0 0 0 0

⎞⎟⎟⎟⎠

ΠEG =

⎛⎜⎜⎜⎝

0 0 0 0

0 0 0 0

0 0 1 0

0 0 0 0

⎞⎟⎟⎟⎠ ,ΠEE =

⎛⎜⎜⎜⎝

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 1

⎞⎟⎟⎟⎠

Here, the capital letters are used to denote a measure-ment outcome and distinguish it from a two-qubit state.The probability of each of those 4 outcomes is given byp (j) = Tr [Πjρ] where j = {GG,GE,EG,EE}. In thecase of the imperfect measurement, the state at the end

of the experiment is not faithfully converted into a mea-surement outcome. For example, the state |gg〉 couldbe recorded as EG with some probability. This can bedescribed by the 4× 4 matrix A, where Aji is the prob-ability that the state i is recorded as outcome j. Thus,four new projectors, Πexpt

j = ΣiAjiΠi, can be calculatedthat model this imperfection. The effects of this imper-fect measurement were accounted for in the theoreticallycalculated density matrix in Fig. 4B of the main text.To calculate A for this system, a calibration experi-

ment was performed where the 4 computational states|gg〉, |ge〉, |eg〉 and |ee〉 were prepared. Then joint-tomography was performed to calculate the probabilityof each measurement outcome. By measuring pj for eachof the input states, the values of Aji were calculated,yielding:

A =

⎛⎜⎜⎜⎝

0.941 0.047 0.031 0.001

0.031 0.925 0.001 0.030

0.027 0.001 0.931 0.031

0.001 0.027 0.037 0.938

⎞⎟⎟⎟⎠ (1)

With this matrix, the tomography for theactual experiment could be corrected. Fora given tomography pre-rotation k, the out-come can be written as a vector of probabilitiesBk = (p (GG)k , p (GE)k , p (EG)k , p (EE)k). Thus,the experimental state in the computational basis, Pk,that resulted in this outcome is given by Pk = A−1Bk.This operation was applied to tomography outcomes tocalculate a corrected density matrix, ρcorr, and thus acorrected fidelity, Fcorr = 57%.

III. QUBIT-PHOTON ENTANGLEMENT

As discussed in the main text, the CNOT-like opera-tion that entangles the stationary qubits with flying mi-crowave photons is realized by a π-pulse on the qubitef -transition followed by a π-pulse between |f0〉 ↔ |e1〉following the method in Ref. [31, 43]. To drive co-herent transitions between |f0〉 ↔ |e1〉, two sidebandtones at ωQSB,A/2π = 5.1987 and ωCSB,A/2π = 8.3325(ωQSB,B/2π = 4.9631 and ωCSB,B/2π = 8.1302) wereapplied to Alice (Bob). As shown in Methods Fig. 3,these drives result in damped sideband Rabi oscillationsof the qubit state between |f〉 and |e〉 (Alice top, Bobbottom). The probability of detecting a photon with thedetector, Pclick, shown on the right axes of the graphs inMethods Fig. 3, peaked when the qubit was in |e〉 con-firming that a photon is generated. Thus, a π pulse canbe performed by turning on the drives for half an oscil-lation, i.e. the time taken to transfer the excitation fromthe qubit to the cavity. The amplitudes of the CSB andQSB drives on Alice and Bob were chosen so that the π-pulse on |f0〉 ↔ |e1〉 took the same time, TSB = 254 ns,for both modules. While the oscillations would ideally

12

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Methods Figure 3. | Single photon generation with sideband transitions. Alice (top) and Bob (bottom) qubitef polarization (left axis) and detector click probability, Pclick, (right axis) as a function of sideband pulse length, TSB,when the qubit is prepared in |f〉. Two sideband drives (ωQSB , ωCSB) were applied, satisfying the frequency conditionωCSB − ωQSB = ωe

A/B − ωef . The drives result in coherent oscillations between |f0〉 and |e1〉 with the amplitude of the driveschosen that a π-pulse on the transition took the same time for the Alice and Bob qubits, TSB = 254 ns. The generation of aphoton was verified with the detector which showed a peak in Pclick when Alice/Bob were in |e〉.

be between +1 and −1, a deviation from this behavior isobserved in the data. We attribute this behavior to theQSB tone spuriously exciting the ge and ef transitionsand hence driving the qubit out of |e〉. While increasingthe detuning of the drives would lower the spurious exci-tation, this was not possible in our experiment because ofpower limitations. Similarly, the drive amplitudes couldhave been decreased but this would have increased thephoton generation time and degraded the fidelity of two-qubit entangled state because of decoherence. Thus, thedrive amplitudes and detunings were chosen to balancethe two effects.Using the CNOT-like operation, signatures of qubit-

photon entanglement for the Alice module were demon-strated in Fig. 2 of the main text. Similar signatureswere observed for the Bob module as shown in MethodsFig. 4. The observed behavior agrees with the resultsof a simplified theoretical model (right panels, MethodsFig. 4A). In this model, the action of the sideband driveson Alice/Bob was modeled using the theory of dampedvacuum Rabi oscillations described in [16]. We note thatalthough our system uses sideband transitions between adifferent set of states, the coupling can still be modeledwith the same formalism. Thus, the three states usedhere were |f0〉, |e1〉 and |e0〉. The sidebands drive co-

herent transitions between |f0〉 and |e1〉 while the cavitylinewidth, κ, causes |e1〉 to decay to |e0〉. For the detec-tor signal, we made the simplification of using the stateof the cavity subjected to two inefficiencies as a proxy.Thus, Pclick (TSB) = ηPe1 (TSB), where η accounts for theloss between the Alice/Bob module and the detector aswell as the detector efficiency and Pe1 (t) is the probabil-ity of the system being in |e1〉. We find good qualitativeagreement between the theory and experiment.

IV. MICROWAVE PHOTON DETECTOR

A. Simulations

A cascaded quantum system simulation[22, 44, 45] wasperformed to understand the operation of the detectorand how two characteristics, dark counts and detectorefficiency, depend on system parameters. We simulate asimplified model of the experiment consisting of a singleemitter cavity, Alice, and the detector qubit-cavity mod-ule. The master equation for this system was solved forvarious initial states of Alice modeling the inputs seenby the detector in the experiment. The simulations wereperformed with the experimentally measured parameters

13

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)

Experiment TheoryA

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h Ve

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Methods Figure 4. | Signatures of Qubit-Photon entanglement data and theory. A) Color plots of the probability,Pclick (top left), of the detector qubit ending in |e〉 and the Bob qubit polarization, 〈ZB〉 (bottom left), as a function of thesideband pulse length TSB when Bob was prepared in cos (θ/2) |g〉+sin (θ/2) |e〉. A theoretical simulation, plotted on the right,shows good agreement. B) Detector click probability, Pclick, and Bob equatorial Bloch vector components, 〈XA〉 and 〈YA〉, asa function of φ when Bob was prepared in 1√

2

(|g〉+ eiφ |e〉) and the CNOT-like operation was either performed (bottom) or

not (top). Open circles are experimental data and lines are fits.

(see Methods Table 1). However, unlike the experiment,the two cavities had identical cavity frequencies.As shown in Methods Fig. 5, the simulation began by

initializing the Alice cavity in the |0〉 (red trace), |1〉(blue trace) or |2〉 (green trace) Fock state (top panel).The photon leaked out and excited the detector cav-ity (second panel). Simultaneously, a selective π-pulse,timed to start at the beginning of the simulation, withσ = 120 ns was applied at ω1

ge to selectively excite the de-tector qubit conditioned on the presence of a intra-cavityphoton (third panel). Finally, Pclick was extracted bycalculating the probability that the detector qubit statewas |e〉 at the end of the simulation (bottom panel). Thefirst detector characteristic, its dark count fraction Pd, isthe probability that the detector clicks when the input

is |0〉. When no photons were sent to the detector (redtrace), Pclick < 0.01 at the end of the simulation. Thetransient increase in the probability of the detector qubitbeing in |e〉 observed during the course of the qubit pulseis a result of the finite selectivity of the π-pulse whichwas confirmed by varying σ or χ. Thus, the dark countprobability, Pd, can be decreased by increasing σ at thecost of slowing down the detection process (and hencethe detection probability).The second detector characteristic is its efficiency, η,

the probability that the detector clicks when the inputis |1〉. When one photon was sent to the detector, thequbit was excited by the selective π-pulse resulting inPclick = 0.4. On the other hand, when two photons weresent to the detector, on average a single photon entered

14

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Methods Figure 5. | Detector Simulations. Results from solving the master equation for a cascaded quantum system ofthe Alice cavity emitting Fock states into the detector qubit-cavity system. The top two panels show the expectation value ofthe photon number operators of the Alice, 〈nA〉, and detector, 〈nD〉, cavities. The Alice cavity (top panel) was initialized in|0〉 (red trace), |1〉 (blue trace) or |2〉 (green trace). The third panel shows the amplitude of a selective π-pulse with σ = 120 nsapplied on the detector qubit to excite it conditioned on the presence of a singe intra-cavity photon. Finally, the probabilityto find the detector qubit in |e〉 was calculated to find Pclick at the end of process (bottom panel). Simulations confirm thatthe detector has dark counts (Pclick given |0〉) Pd < 0.01 and an efficiency (Pclick given |1〉) η ∼ 0.4. Since Pclick is the same for|1〉 (blue trace) and |2〉 (green trace), the detector is not number-resolving.

the detector, also resulting in Pclick = 0.4. Since Pclick issimilar for |1〉 and |2〉, the detector is not photon-numberresolving. Furthermore, the simulations verified that thedetector efficiency is robust to small imperfections anddoes not require precise tuning. When the simulationparameters, such as the mismatch between the Alice anddetector cavity bandwidths and the selective pulse lengthand timing, were varied by 20%, η changed by < 10%.

B. Detector Characterization

The performance of the detector was also character-ized experimentally to verify that it was detecting singlephotons. In these experiments (see Methods Fig. 6A),the Alice and Bob modules were initialized in one of thetwo states, |0〉 or |1〉. Single photons were generatedby preparing the qubit in |e〉 and then performing theCNOT-like operation to create the state |e1〉. Note thatthe generation process takes 254 ns unlike the assumptionof instantaneous generation in the simulations. Then, de-tection was performed by applying the selective π-pulse(σ = 120 ns) on the detector followed by measuring the

15

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Methods Figure 6. | Detector characterization. A) Detector click probability, Pclick, as function of the detuning of thedetection pulse from ω0

ge for different input states from the Alice (top panel) and Bob (bottom panel) modules. The blackdashed line indicated the frequency of the selective π-pulse for optimum discrimination of the state |1〉 from the state |0〉. B)Detector click probability, Pclick, as a function of the delay between the end of the photon generation pulse and the start of theselective detection π-pulse. In the remote entanglement experiment of Figs. 3 and 4, the pulses overlapped by 100 ns (blackdashed line).

state of the detector qubit to find Pclick. The frequencyof the detection π-pulse was varied to characterize thedetector response as a function of frequency. As shownin Methods Fig. 6A, when the state |0〉 (blue circles) wassent, the Pclick was maximized at zero detuning wherethe pulse is selective on zero intra-cavity photons in thedetector. Instead, when the input was |1〉 (red circles),an increased response at ω0

ge −χ was observed. This is adirect result of the detector being excited when photonenters the detector. Due to losses and the detector ineffi-ciency, the response at zero detuning remains but with alower Pclick than for |0〉. Moreover, the similar detector

response to inputs from Alice and Bob demonstrates thatthe detector can detect photons from both systems andthat the losses on the two arms are similar on the twopaths.In a second characterization experiment, the delay be-

tween the end of the photon generation and beginning ofthe photon detection steps was optimized. The proba-bility of detecting the photon, Pclick, is maximized whenthe peak of the detection pulse coincides with the time atwhich the photon population inside the detector cavityis maximum. To find this point experimentally, a photonwas generated by Alice or Bob and sent to the detec-

16

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0.15

0.10

0.05

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0.10

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Threshold ( )

Methods Figure 7. | Detector optimization. The probability of dark counts, Pd, and detector click probability, Pclick,(left axis) and their ratio (right axis) for each round of detection as a function of the readout threshold Ithm /σ. The detectorreadout has two probability distributions (inset), one for click and one for no click. By using a more stringent threshold foroutcomes to be considered a click (white dashed line/black dashed line), the ratio Pd/Pclick can be reduced, therefore improvingthe fidelity of the generated Bell state.

tor with a variable delay between the end of the photongeneration sideband pulse and the beginning of the se-lective detection π-pulse. As shown in Methods Fig. 6B,Pclick was maximized around a delay of −100 ns (blackdashed line), i.e when the sideband and detection pulseshad 100 ns of overlap. This operation point was used inthe remote entanglement experiments of Figs. 3 and 4.We attribute the difference between the simulated de-

tector efficiency, η = 0.4, and the measured Pclick when aphoton was generated in experiments to the losses in oursystem and dark counts. Due to the the hybrid and theinsertion losses of the microwave components between theAlice/Bob modules and the detector, photons only reachthe detector about 40% of the time, corresponding to anefficiency due to the loss of ηloss ∼ 0.4. In addition, thedetector can also click when no photon is incident on it,which occurred with a probability Pd = 0.01. Together,they result in the observed Pclick ∼ 0.2 when a photonwas generated.

C. Detector Optimization

This remote entanglement protocol is robust to losssince the generation of an entangled state is uniquelyheralded by the dual detection of single photons in the

detector. Hence, photon loss between Alice/Bob andthe detector only affect the probability of that outcome.However, dark counts in the detector are detrimental tothis experiment (for a quantitative discussion of the ef-fect, see Methods, Entanglement Fidelity) because theymix the desired Bell state with unwanted states, for ex-ample |gg〉. This impacts the measured fidelity. Sincethe desired (undesired) outcomes occur with probabili-ties proportional to Pclick (Pd), the ratio of Pd/Pclick isthe figure of merit that must be minimized for reducingthe infidelity due to dark counts. Thus, it is importantto minimize the probability of dark counts in the detec-tor, Pd. In our detector, dark counts occur as a resultof the finite selectivity of the detection π-pulse and im-perfect readout of the qubit state. While the detectionpulse could be made more selective by increasing its σ,this would increase the overall detection time. Unfortu-nately, this has two undesired consequences. First, theoverall protocol time increases, and thus, so does the infi-delity due to decoherence. Second, simulations show thatthe detector efficiency is maximized for σ ∼ κ and thusincreasing σ further actually increases Pd/Pclick. There-fore, we operated with σ = 120 ns.Instead, we decrease the ratio Pd/Pclick in post-

selection by reducing the probability that the detectorclicks when the state |0〉 is incident on it. As discussed

17

State Preparation Round 1 Round 2

Alice

Bob

Detector

JointTomography

M1 M2

Cool to

Cool to

Cool to

Detector qubit

readoutDetector

reset

Detection selective -pulse

Methods Figure 8. | Detailed remote entanglement protocol pulse sequence. The remote entanglement protocolbegan with state preparation where the three qubit-cavity systems were initialized in the desired state by cooling and single-qubit rotations. Then, the first of two rounds of the protocol was performed. The qubits were entangled with flying singlephotons by a CNOT-like operation which then interfered on the hybrid and were detected by a selective π-pulse on the detectorqubit. A π-pulse was performed on both Alice and Bob to remove the unwanted |ee〉 state and the detector to reset it. Next,the second round of the protocol was performed followed by joint tomography to measure the state of Alice and Bob. Themeasurement outcomes from the two rounds of photon detection, M1 and M2, were used to post-select successful trials for thetomographic analysis. The entire protocol was repeated with Trep = 21 μs, much faster than the T1 time of any of qubits.

before, readout of the detector qubit results in two dis-tributions, one for click and one for no click. As shownin Methods Fig. 7, by moving the threshold closer to thedistribution associated with a click in the detector, it waspossible to decrease the dark count fraction. The data forPclick (red and yellow circles) and Pd (black and grey cir-cles) were obtained from the two rounds of the remote en-tanglement experiment and the control experiments (seeMethods, Experimental Protocol) respectively. Fromthese two numbers, the ratio Pd/Pclick (blue and pur-ple squares) was calculated for each round. A thresh-old in the middle of the two distributions correspondsto Ithm /σ = −1.8 where Pd/Pclick = 0.1 for the secondround. By moving the threshold to Ithm /σ = 0.15 (blackdashed line), the ratio decreases to Pd/Pclick = 0.05.

V. EXPERIMENTAL PROTOCOL

A. Pulse Sequence

In the first step of the complete remote entanglementprotocol (Extended Data Fig. 8), the Alice, Bob andthe detector qubits were initialized in |g〉. They werefirst cooled to the ground states using a driven reset

protocol[46] and then a measurement was performed topost-select on experiments where all three qubits weresuccessfully cooled. This state initialization by post-selection had a success probability of 57%. Moreover,this also allowed the experiment to be repeated at Trep =21 μs, much faster than the relaxation time of any qubit.Single qubit pulses were then applied to the Alice andBob qubits to prepare them in the desired initial state.Then, the first round of the remote entanglement proto-col consisting of the CNOT-like operation and the pho-ton detection were performed. Before the second round,a π-pulse on ωge was applied to both the Alice and Bobqubits to remove the weight in the |ee〉 state. In ad-dition, the detector was reset by an unselective π-pulsethat returned the detector qubit to |g〉 if it went click inthe first round. Such an unconditional reset can be usedsince only those trials where the detector went click wereused in the final data analysis. After a second round ofthe CNOT-like operation and photon detection, joint to-mography of the Alice and Bob qubit state was performedconditioned on measuring two clicks in the detector. Asshown in Methods Fig. 8, the measurement of the detec-tor qubit in the second round was performed after thejoint tomography to reduce the protocol time and hence,the effects of decoherence. This can be done because thephoton detection process is completed at the end of the

18

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Mea

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d Pa

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ZZ XX YY XY YX ZZ XX YY XY YX

Methods Figure 9. | Control sequence data. Measured amplitudes of selected two-qubit Pauli vector components as afunction of qubit preparation. In experiments identical to those in Fig. 3, the two qubit were prepared in the desired initialstate but no flying photons were generated. Joint tomography of the final two-qubit state was performed. A) With Bob alwaysinitialized in 1√

2(|g〉+ |e〉), Alice was prepared in the variable state cos (θ/2) |g〉+sin (θ/2) |e〉. B) With Alice always initialized

in 1√2(|g〉+ |e〉), Bob was prepared in the variable state 1√

2

(|g〉+ eiφ |e〉). In both cases, data (points) and fits (lines) confirm

that no two-qubit entanglement is observed. This is most directly indicated by 〈ZZ〉 = 0 unlike Fig. 3.

detection π-pulse. The measurement of the qubit state isrequired only for the experimenter to determine the out-come of the detection event. A set of control sequenceswas interleaved into the above protocol to calibrate thejoint tomography . These experiments were repeated toaccumulate at least 105 successful shots of each sequencefor adequate statistics.

B. Control Experiments

To verify that the experimental results observed in thedata shown in Fig. 3 are a result of the which-path era-sure of the flying photons by the hybrid, two controlexperiments were performed. In these experiments, noflying photons were generated but the experimental pro-tocol was otherwise left unchanged. The joint tomogra-phy performed at the end of the protocol is no longerconditioned on photon detection events. To further ruleout systematic error, these experiments were interleavedwith the experiments performed in Fig. 3. The resultson these experiments are shown in Methods Fig. 9. Inthe first experiment, a control for the data in Fig. 3A,Bob was initialized in 1√

2(|g〉+ |e〉) and Alice was pre-

pared in cos (θ/2) |g〉+sin (θ/2) |e〉. Since the qubits werenot entangled with photons, no entanglement was gener-ated for any preparation angle θ. This is most directlydemonstrated by 〈ZZ〉 = 0, unlike in Fig. 3A where〈ZZ〉 < 0. Since Bob remained in 1√

2(|g〉+ |e〉) at the

end of the experiment independent of θ, the final single-qubit Bloch vector has Pauli components 〈ZB〉 = 0,〈XB〉 = 0 and 〈YB〉 = 1. Consequently, only 〈Y Y 〉 and〈XY 〉 vary with θ and are maximized at θ = π/2 while〈XX〉 = 〈Y X〉 = 0, unlike in Fig. 3A.

In the second experiment, a control for the data inFig. 3B, Alice was now initialized in 1√

2(|g〉+ |e〉) and

Bob was prepared in 1√2

(|g〉+ eiφ |e〉). In the control

experiment with no photons, the final two-qubit stateshould be the superposition of the computation states12

(|gg〉+ eiφ |ge〉+ |eg〉+ eiφ |ee〉). Thus, 〈ZZ〉 = 0 (seeMethods Fig. 9B). Moreover, 〈XX〉 and 〈Y Y 〉 do nothave in-phase sinusoidal oscillations characteristic of anodd Bell state. Ideally, 〈XX〉 = 〈XY 〉 = 0 but a smalldetuning error on the Alice qubit caused oscillations inthem too.

VI. ENTANGLEMENT FIDELITY

To understand the sources of infidelity in the experi-ment, various sources of imperfection were built into aquantum circuit model of the entire system. The modelcontained both qubits, treated as two-level systems, anupper and lower branch of the experiment that could have0, 1 or 2 flying photons and two single-photon detectors.Thus, the total system state was described by a 36× 36density matrix. Sources of imperfections were individu-ally introduced and their effects on this density matrixwas calculated. By cascading their effects on the densitymatrix, their combined impact was also calculated. Fi-nally, to compare to experiment, the photon parts of thedensity matrix were traced out to reduce it to a two-qubitdensity matrix which was expressed in the Pauli basis togenerate Fig. 4B and calculate the expected fidelity.

19

A. Qubit Decoherence

The effects of qubit decoherence on the density matrixwere modeled using phase damping. For a single qubit,this can be represented by the quantum operation E (ρ) =

E0ρE†0 + E1ρE

†1 [47]. Here,

E0 =√α

(1 0

0 1

), E1 =

√1− α

(1 0

0 −1

)(2)

and α =(1 + e−t/T2E

)/2. The decoherence of each

qubit was treated as an independent process assumingthat there was no correlated noise affecting the two sys-tems. Thus, by taking its Kronecker product with a2 × 2 identity matrix, the single-qubit phase dampingoperation was converted into a two-qubit operator. Twoseparate quantum operations, EA (ρ) and EB (ρ) for thedecoherence of Alice and Bob, were calculated usingT2E, A = 10 μs and T2E, B = 16 μs respectively. Thefinal density matrix, obtained by cascading the two op-eration, resulted in a 20% infidelity due to decoherence,i.e FT2Bell

∼= 0.8.

B. Dark Counts

This protocol’s robustness to loss is a result of herald-ing on single-photon detection events which are uniquelylinked to the generation of a Bell state. However darkcounts mix the Bell state with other states, |gg〉 for ex-ample, resulting in a lowered fidelity. This infidelity wascalculated by modeling the impact of an imperfect de-tector on the two-qubit density matrix. The detectortakes one of three possible input states, the flying Fockstates |0〉, |1〉 and |2〉, and returns one of two outputs,click or no-click. In the generalized measurement formal-ism, this corresponds to the three measurement operatorsM0 = |0〉 〈0|, M1 = |0〉 〈1| andM2 = |0〉 〈2| for detecting0, 1 or 2 photons respectively[16]. To model the imper-fections of dark counts and finite detector efficiency, weintroduce Pd, the probability of a dark count in the de-tector, and Preal, the probability that the detector goesclick when a photon arrives. Since according to simu-lations, the detector cannot distinguish between |1〉 and|2〉, we make the assumption that either input results ina click with the same probability Preal. Thus, the prob-ability of the two outcomes, no-click (NC) and click (C),are:

PNC = Tr[(1− Pd)M0ρM

†0 + (1− Preal)

(M1ρM

†1 +M2ρM

†2

)](3)

PC = Tr[PdM0ρM

†0 + Preal

(M1ρM

†1 +M2ρM

†2

)](4)

Based on the measurement outcome, the input densitymatrix is projected to one of two output density matrices:

ρNC =

((1− Pd)M0ρM

†0 + (1− Preal)

(M1ρM

†1 +M2ρM

†2

))PNC

(5)

ρC =

(PdM0ρM

†0 + Preal

(M1ρM

†1 +M2ρM

†2

))PC

(6)

To model the experiment and calculate the fidelitylimited by dark counts, the final density matrix aftertwo rounds of the protocol and successful photon de-tection was calculated, resulting in a 36 × 36 densitymatrix. The photon components of the density ma-trix were traced out, yielding the 4 × 4 density matrixρfinal. From this, the fidelity limited by dark counts,Fdet = Tr (ρfinal |O+〉 〈O+|), was found:

Fdet =3Pd,1Pd,2 + Pd,1Preal,2 + 4Preal,1Preal,2

11Pd,1Pd,2 + 8Pd,2Preal,1 + 9Pd,1Preal,2 + 4Preal,1Preal,2

(7)Here the numeric subscripts on Pd and Preal are for

the two detections rounds in the experiment. The val-ues of Preal and Pd for each round were extracted fromthe measured click probabilities from the remote entan-glement and the control experiments. We find Pd,1 =0.006, Pd,2 = 0.005, Preal,1 = 0.21, Preal,2 = 0.26 and thusFdet

∼= 0.9. Combining the effects of decoherence anddark counts results in an expected theoretical fidelity ofFthy = 0.76.

C. Tomography

To model the imperfections arising from the tomog-raphy process, we used the theory described above (seeMethods, Joint Tomography and Calibration) to calcu-late the Pauli components in Fig. 4. Using the exper-imentally measured A matrix, the imperfect projectorsΠexpt

j were calculated. Thus the measurement outcome

is Pjk = Tr[Πexpt

j RkρR†k

]where Rk is one the 9 to-

mography pre-rotations. From the set of measurementoutcomes, the 16 Pauli vector were calculated and plot-ted in Fig. 4B.

20

ACKNOWLEDGMENTS

The authors thank B. Vlastakis for helpful discus-sions and M. Rooks for fabrication assistance. Facilitiesuse was supported by the Yale Institute for Nanoscienceand Quantum Engineering (YINQE), the National Sci-ence Foundation (NSF) MRSEC DMR 1119826, and theYale School of Engineering and Applied Sciences clean-

room. This research was supported by the U.S. ArmyResearch Office (Grant No. W911NF-14-1-0011), and theMultidisciplinary University Research Initiative throughthe US Air Force Office of Scientific Research (GrantNo. FP057123-C). W. P. was supported by NSF grantPHY1309996 and by a fellowship instituted with a MaxPlanck Research Award from the Alexander von Hum-boldt Foundation.

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