McCutcheon, W., McMillan, A., Rarity, J. G., & Tame, M. S. (2018).Experimental demonstration of a measurement-based realisation of aquantum channel. New Journal of Physics, 20(3), [033019].https://doi.org/10.1088/1367-2630/aa9b5c

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New J. Phys. 20 (2018) 033019 https://doi.org/10.1088/1367-2630/aa9b5c

PAPER

Experimental demonstration of ameasurement-based realisation ofa quantum channel

WMcCutcheon1, AMcMillan1 , J GRarity1 andMSTame2

1 QuantumEngineering Technology Laboratory, Department of Electrical and Electronic Engineering, University of Bristol,WoodlandRoad, Bristol, BS8 1UB,United Kingdom

2 School of Chemistry and Physics, University of KwaZulu-Natal, Durban 4001, SouthAfrica

E-mail:markstame@gmail.com

Keywords: quantum simulation, photonics, measurement-based quantum computation, decoherence, open quantum system, cluster state,experiment

AbstractWe introduce and experimentally demonstrate amethod for realising a quantum channel using themeasurement-basedmodel. Using a photonic setup andmodifying the basis of single-qubitmeasurements on a four-qubit entangled cluster state, representative channels are realised for the caseof a single qubit in the formof amplitude and phase damping channels. The experimental resultsmatch the theoreticalmodel well, demonstrating the successful performance of the channels.We alsoshowhowother types of quantum channels can be realised using our approach. This work highlightsthe potential of themeasurement-basedmodel for realising quantumchannels whichmay serve asbuilding blocks for simulations of realistic open quantum systems.

Introduction

Themodelling and simulation of quantum systems is an important topic at present as it promises to open upinvestigations intomany new areas of science [1–3]. This includes exploring exotic states ofmatter [4],thermalisation and equilibration processes [5, 6], chemical reaction dynamics [7] and probing quantum effectsin biological systems [8, 9]. A number of approaches are currently being studied, using both classical andquantummethods.While classicalmethods are limited to specific conditions for efficient simulation ofquantum systems [10, 11], quantummethods have amuch larger scope, and a range of techniques have beendeveloped so far, such as analogue [1, 2], digital [12, 13], digital–analogue [14, 15], algorithmic [16–18] andembedded [19, 20], eachwith its own advantages and disadvantages.Mostmethods consider ideal quantumsystems, where the constituent elements are isolated from the outsideworld.However, realistic quantumsystems invariably interact with some environment [21].Work onmodelling and simulating such quantumsystems has seenmuch progress recently [22–24], andmay shed light on fundamental physical phenomena,including phase transitions in dissipative systems [25–27], thermalisation [28, 29] and using dissipation as aresource [30, 31]. In this context, the development of techniques to realise quantum channels [32, 33]representing the dynamics of realistic quantum systems has seen rapid growth—most notably for single qubits[34–41] and qudits [42–44]. So far, however, studies have been limited to the standard quantum circuitmodel [45].

A naturalmodel for simulating quantum systems is themeasurement-basedmodel [46–48], which has beenused to demonstrate the simulation of quantum computing on entangled resource states using only single-qubitmeasurements [49–55]. Themeasurement-basedmodel is an interestingmethod for simulating quantumsystems, as it can do this simply by carrying out quantum computing [56]. However, theremay also be thepossibility of going further by exploiting the structure of the entangled resource being used to reduce the overallcomplexity and put a given simulationwithin reach of current technology. Recently, thefirst steps in thisdirection have been taken theoretically [57]. Despite this potential, the realisation and simulation of realisticquantum systems using themeasurement-basedmodel has not yet been explored.

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In ourworkwe address this issue by introducing and experimentally demonstrating amethod for therealisation of a quantum channel that can be used to represent the dynamics of a realistic quantum systemusingthemeasurement-basedmodel.We demonstrate the simple case of a single qubit. To do this, wefind an efficientmapping from the circuitmodel to themeasurement-basedmodel for the simulation, which allows us toconsider the use of an entangled linear cluster state of only four qubitsmade from three photons—using thepolarisation degree of freedomof each photon as a qubit and the path degree of freedomof one of the photons asan additional qubit.Many previous photonic experiments using cluster states have employed only thepolarisation degree of freedom to carry out quantumprotocols [49–53, 55], however the use of other degrees offreedom to represent qubits in ‘hybrid’ cluster states has been considered in order to improve the state qualityand protocol results [58–62]. In recent work, a quantum error-correction code [63], a secret-sharing protocol[64] and a quantumalgorithm [54] have all been realised using four-photon cluster states consisting of bothpolarisation and path qubits. The setupwe use is similar to these experiments, however the overall goal isdifferent and the use of only three photons compared to four ensures we can achieve a high quality performancefor ourmeasurement-based realisation of a quantum channel.

Bymeasuring the qubits of our hybrid cluster state in a particular waywe are able to realise arbitrarydamping channels on a logical qubit residingwithin the cluster state. Themain advantage of thismeasurement-based approach over the standard circuitmodel [36–41] is that only the pattern ofmeasurements needs to bemodified in order to implement different systemdynamics. This is particularly useful in a photonic setting,where a reconfiguring of the basic optical elements is not required, both in bulk [49–55] and on-chip setups[66–68]. The experimental results obtainedmatch the theoretical expectations well and highlight the potentialuse of themeasurement-basedmodel as an alternative approach to realising quantum channels.

Experimental setup

The experimental setup is shown infigure 1(a). It generates a four-qubit linear cluster statemade of threephotons—three qubits are encoded in the polarisation degree of freedomof three photons using the basis

H V,ñ ñ{∣ ∣ }, and the fourth qubit is encoded in the path degree of freedomof one of the photons using the basisp p,1 2ñ ñ{∣ ∣ }. The photons are generated by spontaneous four-wavemixing in photonic crystalfibers (PCFs)

tailored to generate a spectrally separable naturally narrowband bi-photon state cross-polarised to the pump[69, 70]. The signal wavelength is 625 nmsl » and the idler wavelength is 876 nmil » when the PCF ispumped at 726 nmpl = . For the pump laser a 80MHz repetition rate femto-secondTi-Sapphire laser isfiltered through a 4F arrangement, with a spectralmask on the Fourier plane achieving the desired spectrumwith bandwidth 1.7 nmplD = tominimise parasitic non-linear effects which reduce photon purities [65], andsent to two PCF sources. One of the PCF sources (PCF 1) is arranged in a twisted Sagnac-loop configuration togenerate the polarisation entangled Bell pair HH VV s i

1

2 1 1ñ + ñ(∣ ∣ ) on the signal and idler photons s1 and i1,

respectively, for whichwe achieve Bell statefidelities of 0.89, limited by the spectral separability of the generatedphoton pairs [69]. The second source (PCF 2) is pumped in one direction onlywith the generated state

Figure 1.Experimental scheme for realising a quantum channel for a single qubit using themeasurement-basedmodel. (a)Experimental photonic setup, with photonic crystalfibers (PCFs), half-wave plates (HWPs), quarter-wave plates (QWPs), Soleil–Babinet (SB), polarizing beamsplitter (PBS), beamsplitter (BS), glass plate (GP) and dichroicmirror (DM). The setup generates a four-qubit cluster state between photons s1, s2 and i1, with the polarisation and path degree of freedomof photon s1 used to represent twoqubits. (b)Expectation values used for calculating the quality of the generated cluster state in terms of thefidelity. (c)Circuitmodel forsimulating an arbitrary single-qubit channel. (d)Measurement-based protocol for implementing the simulation of the channel and itsequivalent representation. (e) Scheme for generalizing the approach to a full open quantum system simulation for a single qubit,where rotations and/or interactions with other qubits (dotted lines) can be carried out stroboscopically.

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New J. Phys. 20 (2018) 033019 WMcCutcheon et al

H Hs i2 2ñ ñ∣ ∣ [70]. The idler photon, i2, serves as a heralding photon for the successful generation of the signalphoton, s2. After each PCF, the signal and idler photons are separated by dichroicmirrors (DM) and bandpassfilteredwithwidths 40nmand 10nm, respectively, to removeRaman noise. The signal photon s2 is rotated by ahalf-wave plate (HWP) into the state H V 21

2+ñ = ñ + ñ∣ (∣ ∣ )/ and overlappedwith the signal photon s1 at a

polarising beamsplitter (PBS), with the relative arrival time set by the pumpdelay so that 0tD .When onesignal photon exits each port of the PBS the heralded state is the three-qubit GHZ state in the polarisation basesof the photons s1, s2, and i1: HHH VVV s s i

1

2 1 2 1ñ + ñ(∣ ∣ ) . The quality of this ‘fusion’ operation is however,

limited by the spectral-temporal indistinguishability of the signal photons generated in each source, which canbemitigated to some extent by temperature tuning one of the sources, but limits thefidelity of the three-qubitGHZ state to 0.80±0.01 [71].

The three photons are collected into single-mode fibers, fromwhich s2 and i1 are sent straight totomography stages consisting of automatedQWPs andHWPs, followed by PBSs and pairs of single-photonavalanche photodiode detectors (APDs) capable of performing projectivemeasurements onto arbitrarypolarisation bases [73]. The signal photon s1 is path expanded to encode the fourth qubit. This entails a foldedMach–Zehnder interferometer (FMZI)with the anticlockwise and clockwise paths corresponding to theeigenstates of the path qubit, p1ñ∣ and p2ñ∣ [54].When the incoming photonmeets the PBS on entering the FMZI,it performs a controlled-not operation between the polarisation qubit and the path qubit of photon s1.With theaddition of aHWPbefore and after the PBS to performHadamard operations on the polarisation, the stategenerated is equivalent to a four-qubit linear cluster state

1

200 01 10 11 , 11 2 3 4yñ = + +ñ + + -ñ + - +ñ - - -ñ∣ (∣ ∣ ∣ ∣ ) ( )

wherewe havewritten all qubits in the computational basis and aHadamard operation has also been applied tothe polarisation of photon i1, performed at the tomography stage.Here, qubit 1 is represented by the polarisationof photon s1, qubit 2 by the path of photon s1, qubit 3 by the polarisation of photon s2 and qubit 4 by thepolarisation of photon i1. To achieve arbitrary projectivemeasurements for the polarisation qubit of photon s1,we use a tomography stage as described for photons i1 and s2. For the path qubit of photon s1, to achievecomputational basismeasurements we alternate blocking of the paths in the FMZI so that the population ofphotons in paths p1ñ∣ or p2ñ∣ can bemeasured after the paths aremerged on a 50:50 BS. Basismeasurements onthe equatorial plane of the Bloch sphere are achieved by imparting a relative phase between the paths in theFMZI using aGPmounted on an automated rotation stage, followed by theHadamard operation achieved bythe paths combining on the BS [54]. By using a dual PBS–BS cube for the FMZI the relative path length andtherefore phase between the paths can bemade relatively stable, leading to an interference visibility of 0.93withheralded single photons.

The cluster state yñ∣ is the state generated in our setup in the ideal case. However, due to the variousdominant sources of error discussed above, including spectral and spatial imperfections introduced during thefour-wavemixing process at the PCFs [65, 69–71], the fusion PBS between the signal photons [72], the pathexpansion [54] and to a lesser extent higher-order photon emissions and fibre inhomogeneity [71, 72], the actualstate generated is amixed state.We therefore first characterise the quality of the cluster state generated in oursetup. Thefidelity F Tr expr y y= ñá( ∣ ∣) quantifying the overlap between the experimental state expr and the ideal

state yñ∣ can be obtained by decomposing the projector y yñá∣ ∣ into a summation of termsmade fromprojectorelements arising from the eigenvectors of tensor products of Pauli operators. Each term can then bemeasuredlocally, with the total expectation value of all the terms for expr giving thefidelity. There are a total of 15 terms[74], leading to afidelity of F 0.63 0.01= . The expectation values of the terms are shown infigure 1(b). Thepresence of genuinemultipartite entanglement, signifying that all qubits were involved in the generation of thestate, is confirmed as F 0.5> [74]. Improvements to the quality of our state could bemade by operating at areduced pumppower for the PCFs in order to suppress higher-order photon emissions from the four-wavemixing [71, 72]. However, this reduces the state generation rate and impacts on the data collection time. Bettermatching of the spectral profiles of the signal photons produced via four-wavemixing processes in separatePCFswould also improve the state quality as the PBS fusion operation relies on spectral indistinguishability ofthe photons [72].While the above factors would improve the state quality, the current fidelity value iscomparable to other photonic cluster state experiments and allows us to demonstrate a proof-of-principlerealisation of a quantum channel using themeasurement-basedmodel.

Results

We start our implementation by showing how the standard circuitmodel for realising a quantum channel ismapped to themeasurement-basedmodel. Infigure 1(c) the quantum circuit for carrying out an arbitrarycompletely positive trace-preserving (CPTP) channel for a single qubit Sr is depicted [34]. For simplicity, the

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New J. Phys. 20 (2018) 033019 WMcCutcheon et al

unitary operationsU d( ) andU j( ) at the start and end are not considered, as they are not needed for the specificexamples we demonstrate. They are local operations and if needed for a given channel they can be applied easily

in themeasurement-basedmodel [48]. In the circuit, the rotation Rcos 2 sin 2

sin 2 cos 2yq qq q

=-q

⎛⎝⎜

⎞⎠⎟,X is the Pauli xs

operator andM represents ameasurement in the computational basis. The circuit shown implements the

quantum channel K K K KS S S0 0 1 1 r r r +( ) † †, where theKraus operators are Kcos 0

0 cos0

ba

= ⎜ ⎟⎛⎝

⎞⎠ and

K0 sin

sin 01a

b= ⎜ ⎟⎛⎝

⎞⎠. The relations linking these operators to the rotations in the circuit are2 21g b a p= - +( ) and 2 22g b a p= + -( ) . If themeasurement outcomeM of the ancilla qubit is 0,

then the operatorK0 is applied and if it is 1, then anX operation is applied to the systemqubit in order for theoperatorK1 to be applied [34]. Taking into account that both outcomes can occur for the ancilla qubitmeasurement, the system is put into a summation of the two processes.We stress that this procedure is capableof simulating arbitrary single-qubit channels of which there exist a continuous family. In this workwewilldemonstrate 3 different channels: amplitude damping, phase damping and a channel we callβ damping, anexample extremal channel characterised by simultaneous amplitude damping and phase damping occurring inperpendicular bases. For the first two channels it is convenient to set the parameters in the circuit as

ecos t1 2a = h- -( ) and 0b = , where η is an effective damping rate and t is the simulation time desired.Amplitude damping is then implemented naturally by the circuit. On the other hand, phase damping does notrequire theX operation from the ancillameasurement outcome 1 to be applied. For the third channel wefixαand choose specific values ofβ.We nowmap the circuitmodel to themeasurement-basedmodel and show thata four-qubit entangled cluster state is all that is needed to carry out the simulation.While we do not claim thatourmapping is optimal, in that itmay be possible to do some elements of the simulation using only a three-qubitcluster state, the efficientmappingwe present puts the simulationwithin reach of our setup and allows us toexperimentally demonstrate the fundamental workings of ameasurement-based approach.

Themeasurement-basedmodel involvesmaking single-qubitmeasurements on a cluster state in order tocarry out logic operations on quantum information encodedwithin. For cluster states two types ofmeasurements allow logic operations to be performed: (i)measuring a qubit j in the computational basis allowsit to be disentangled and removed from the cluster, leaving a smaller cluster of the remaining qubits, and (ii) inorder to perform logic gates, qubitsmust bemeasured in the equatorial basis B ,j j ja a a= ñ ñ+ -( ) {∣ ∣ }, where

e0 1 2ji

ja ñ = ñ ña

-∣ (∣ ∣ ) , for 0, 2a pÎ ( ]. Thismeasurement on qubit j, initially in the logical state fñ∣ ,results in propagation of the state to qubit j 1+ with the operations Rx

szHs a applied. Here, the rotation

R iexp 2z zas= -a ( ) has been applied alongwith aHadamard operation,H, and a PauliX operation dependenton the outcome s from themeasurement 3.

Using the cluster state generated in our experiment, the input states corresponding to the ancilla qubit +ñ∣and systemqubit Sr = +ñá+∣ ∣are naturally encoded on qubits 1 and 4, respectively, as shown infigure 1(d). Forthe ancilla qubit, note that in the circuitmodel shown infigure 1(c), thefirst gate, Ry

2 1g , is applied to an initial

state 0ñ∣ . This gate can be decomposed into a product of several gates: R R Rz z z2 2 21H Hp g p- . Taking the first two

operations of the gate, we have R 0z2H ñ = + ñp- ∣ ∣ . Therefore the remaining operations that need to be carried

out on the ancilla qubit using the cluster state are R Rz z2 2 1Hp g . By including the controlled-X (CX) gate between

the ancilla and system, and the subsequent gate Ry2 2g , the total operation for the remainder of the circuit for both

system and ancilla is given by R R Ry z z2 2 22 1H CZ H H Ä Äg p g( ) ( ), where the system is the first qubit and the

ancilla is the second.Here, we have decomposed the CX gate as H CZ H Ä Ä( ) ( ). Thefirst two operations,Rz

2 1H g , are implemented bymeasuring qubit 1 of the cluster state in the basis B 21 1g( ), which propagates thelogical ancilla to qubit 2 of the cluster. The next two operations, Rz

2H p , are implemented bymeasuring qubit 2of the cluster state in the basis B 22 p( ), which propagates the ancilla to qubit 3. TheCZ gate is then naturallyapplied as the logical qubit of the ancilla now resides on qubit 3 and the logical qubit of the system resides onqubit 4—the edge linking qubits 3 and 4 is aCZ gate. Thefinal two operations, Ry

2 2Hg , are incorporated into themeasurement basis of the ancilla qubit on qubit 3, which is normallymeasured in the computational basis. Thus,we have outcomes 0, 1{ }of the ancilla in the circuit corresponding to the outcomes of themeasurementbasis R R0 , 1y y

2 22 2H Hñ ñg g{ ( ) ∣ ( ) ∣ }† † .In themeasurement-basedmodel it is important to include the unwanted Pauli byproduct operators that act

on the logical qubits due to the randomnature of the outcomes ofmeasurements of qubits in the cluster.Including the byproductsmakes the logical operations fully deterministic [48]. The byproducts can bepropagated right to the end and incorporated into the finalmeasurements of the system and ancilla. For theancilla, the byproducts lead to themeasurement basis R R0 , 1z

sxs

y zs

xs

y2 21 2 2 1 2 2H Hs s s sñ ñg g{ ( ) ∣ ( ) ∣ }† † for qubit 3, and the

3For a detailed introduction to one-wayQC, see [48, 49].

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New J. Phys. 20 (2018) 033019 WMcCutcheon et al

basis of qubit 2must bemodified to B 1 2s2

1p-(( ) ). Here, si corresponds to themeasurement outcome for qubiti. For the system, the byproduct operation is z

sxs2 3s s , where the xs frommeasurementM in the circuit has been

included. All the operations from the circuitmodel have nowbeenmapped into themeasurement-basedmodeland it is clear that a four-qubit linear cluster state is sufficient for simulating the action of an arbitrary channel ona single qubit.

To generalise thismethod to a full simulation of a single-qubit quantum system, onemight also like toinclude interactionwith additional systems or a rotationwhile it is being subject to damping. In this case, theinteraction/rotation and damping could be split up into smaller time steps and carried out stroboscopically asthe logical qubit propagates along a larger cluster state (taking into consideration the passage of byproductsthrough the corresponding circuit), as highlighted infigure 1(e). Furthermore, the simulation of channels withmemory effects could be included by conditioning future time steps on the outcome of the ancillameasurement,s3, or initially entangling the ancilla qubits in order to introduce correlations in the environmental degrees offreedom [32].

We now characterise the performance of themeasurement-based approach for phase damping using thecluster state generated in our setup.We choose the basis states of our qubit to simulate that of a two-level system:

g e,ñ ñ{∣ ∣ }. For this, we use the convention H e0ñ = ñ « ñ∣ ∣ ∣ and V g1ñ = ñ « ñ∣ ∣ ∣ , and combine the damping

rate and time into a single quantity,Γ, with the correspondence e1 t 2- G = h .We then choose five differentdamping values: 0, 0.25, 0.5, 0.75, 1G = { }. These values determine the parameters cos 11a = - G- ( ) and

0b = , which are inserted into the formulas for 1g and 2g to obtain the angles for themeasurements of qubits inthe cluster. For each value ofΓ, we carry out quantumprocess tomography [75] by encoding the probe states gñ∣ ,

eñ∣ , +ñ∣ and g i ey1

2+ ñ = ñ + ñ∣ (∣ ∣ ), and performquantum state tomography on the output of the channel for

each probe state [73]. From this informationwe reconstruct the processmatrixχ for the channel, defined by therelation E ES i j ij i S j, r c r= å( ) †, with the operators Ei forming a complete basis for theHilbert space,

E X Y Z, , ,i = { } [45]. The probe state +ñ∣ is naturally encoded into the cluster state, whereas the probe state

y+ ñ∣ is encoded using aQWPonphoton i1, and the probe states gñ∣ and eñ∣ are encoded using a polariser. Infigures 2(a)–(c)we show theχmatrices for the simulation of phase damping for 0G = , 0.5 and 1, respectively,for the case ofmeasurement outcomes s 01 = and s 02 = . The left column in each corresponds to theexperiment,χ, and the right column the theoretically expected ideal case, idc . One can see that the process

matricesmatchwell, with processfidelities defined as F Tr Tr Trp id id2cc c c c= ( ) ( ) ( ) [76] equal to

0.71±0.03, 0.89±0.03 and 0.93±0.03, respectively. Infigure 3(a)we show Fp for all values ofΓ simulated.The left hand side (blue columns) shows the case of s 01 = and s 02 = , whichwe call ‘no feed forward’ (no-FF),while the right hand side (red columns) shows the case of s 01 = and s 12 = feed forward (FF), chosen as anexample of when byproducts are produced and the necessary rotations are applied to Sr , which are incorporatedinto themeasurements during the state tomography.

It can be seen infigure 3 that there is little difference in the processfidelities of the no-FF and FF cases, whichindicates that there is notmuch bias in the implementation of the channel due to themeasurement outcomes ofqubits in the cluster state. As FF operations are needed tomake the channels fully deterministic in themeasurement-basedmodel, the results show that the channels can be carried out deterministically andwithconsistent performance.While themain quantifier of howwell the channels perform can be taken to be theprocessfidelities shown infigure 3, theχmatrices shown in figure 2 help visualise what the channels are doing inthe Pauli operator basis. As an additional complementary plot, infigure 2(d)we show the effect of the channel onthe Bloch sphere for 1G = . The Bloch sphere is squashed into a cigar shape along the z-axis as expected [45].

Figure 2.Quantumprocessmatrices for the realisation of a phase damping channel. (a) 0G = . (b) 0.5G = . (c) 1G = . (d)Blochsphere representation showing the effect of the channel at 1G = . In panels (a)–(c) the left column is the experimental result and theright column is the ideal case, with the top row corresponding to the real part and the bottom row to the imaginary part of the elementsof thematrix.

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New J. Phys. 20 (2018) 033019 WMcCutcheon et al

Infigures 4(a) and (b)we show theχmatrices for the simulation of amplitude damping for 0.5G = and1G = . Theχmatrix for 0G = is the same as the phase damping channel. The processfidelities for these

channels are 0.76±0.03 and 0.66±0.02. The full range of processfidelities is given infigure 3(b) for the no-FFand FF cases. Infigure 4(d)we show the effect of the channel on the Bloch sphere for 1G = . The Bloch sphere issquashed into a cigar shape, similar to the phase damping case, but at the same time it is gradually pushed towardthe basis state gñ∣ , as expected [45]. Infigure 4(c)we show an exampleχmatrix for the casewhen 0b ¹ , whichwe call the ‘β channel’. Here, we have set 0.3a = and 1.2b = . The corresponding processfidelity is0.70±0.03. Infigure 3(c)we show the process fidelities for other non-zeroβ values, both in the no-FF (lefthand side) and FF cases (right hand side). Theβ channel results show that themeasurement-basedmethod canbe used for simulating non-standard quantum channels representing realistic quantum systemdynamics.

Figure 3.Process fidelities for the realisation of phase, amplitude andβ damping channels. (a)Phase damping. (b)Amplitudedamping. (c)β damping. In all, thefirst column corresponds to the case s 01 = and s 02 = for the outcomes of themeasurements ofthe qubits in the cluster, no feed forward (no-FF), while the second column corresponds to the case s 01 = and s 12 = , feed forward(FF), with appropriate byproduct operator applied to the output.

Figure 4.Quantumprocessmatrices for the realisation of an amplitude damping andβ channel. (a) 0.5G = for amplitude damping.(b) 1G = for amplitude damping. (c) 0.3a = and 1.2b = for rotated phase damping. (d)Bloch sphere representation showing theeffect of the amplitude damping channel at 1G = . In panels (a)–(c) the left column is the experimental result and the right column isthe ideal case, with the top row corresponding to the real part and the bottom row to the imaginary part of the elements of thematrix.

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New J. Phys. 20 (2018) 033019 WMcCutcheon et al

Discussion

In this workwe experimentally demonstrated amethod for the realisation of quantum channels using themeasurement-basedmodel for the simple case of a single qubit.Wemapped the circuitmodel to themeasurement-basedmodel and showed that an entangled linear cluster state of only four qubitsmade fromthree photons is sufficient. Bymeasuring the qubits of the cluster state wewere able to simulate differentquantum channels, including amplitude and phase damping, on a logical qubit residingwithin the cluster state.The experimental resultsmatch the theoretical expectations well.We also briefly discussed how to extend themethod to implement a full simulation of a single-qubit quantum system that would include rotationswhile thedecoherence takes place. Our results highlight the potential use of themeasurement-basedmodel as analternative approach to simulating realistic quantum systems. Futurework could look intowhether a smallercluster state of only two or three qubits can also be used for demonstrating specific quantum channels. Inaddition, it would be interesting to see how extra qubits provide extended functionality and flexibility.Furthermore, one could extend themodel to qudits, collectivemultiqubit channels and evenmemory effects.

Acknowledgements

Thisworkwas supported by theUK’s Engineering and Physical Sciences ResearchCouncil (EP/L024020/1), EUFP7 grant 600838QWAD, the SouthAfricanNational Research Foundation and the SouthAfricanNationalInstitute for Theoretical Physics.

Data availability

All relevant data are available from theUniversity of Bristol database, under theDOI: http://doi.org/10.5523/bris.3u8qqpz713uo02cc2aji8g1qr7

ORCID iDs

AMcMillan https://orcid.org/0000-0002-0031-2640MSTame https://orcid.org/0000-0002-5058-0248

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