Heat Bath Algorithmic Cooled Quantum Otto Engines
Emre Köse, Selçuk Çakmak, Azmi Gençten, Iannis K. Kominis, Özgür E. Müstecaplıoğlu
HHeat Bath Algorithmic Cooled Quantum Otto Engines
Emre Köse, Selçuk Çakmak, Azmi Gençten, Iannis K. Kominis, and Özgür E. Müstecaplıo˘glu ∗ Department of Physics, Koç University, 34450 Sariyer, ˙Istanbul, Turkey Department of Physics, Ondokuz Mayıs University, 55139 Samsun, Turkey Department of Physics, University of Crete, 70013 Heraklion, Greece (Dated: January 7, 2019)We suggest alternative quantum Otto engines, using heat bath algorithmic cooling with partner pairingalgorithm instead of isochoric cooling. Liquid state nuclear magnetic resonance systems in one entropy sinkare considered as working fluids. Then, the extractable work and thermal efficiency are analyzed in detail forfour-stroke and two-stroke type of quantum Otto engines. The role of heat bath algorithmic cooling in thesecycles is to use a single entropy sink instead of two. Also, this cooling algorithm increases the power of enginesreducing the time required for one cycle.
I. INTRODUCTION
With the advances of miniaturized information and energydevices, the question of whether using a quantum system toharvest a classical resource can have an advantage over a classi-cal harvester has gained much attention in recent years [1–24].The argument is more or less settled in the case of quantuminformation devices, and the main challenge remained is theirefficient implementation. In typical quantum information de-vices, both the inputs and the algorithmic steps of operationare of completely quantum nature. In contrast, quantum en-ergy devices process incoherent inputs and they operate withthermodynamical processes being quantum analogs of theirclassical counterparts. The quantum superiority in such a ther-mal device reveals itself when the resource has some quantumcharacter, for example squeezing [25, 26], or when the har-vester has profound quantum nature, for example, quantumcorrelations [27, 28]. Studies of both cases are limited to ma-chine processes analogs of classical thermodynamical ones.Here we ask how can we use genuine quantum steps in themachine operation and if we can do so what are the quantumadvantages we can get.As a specific system to explore completely quantum stepsin thermal quantum device operation we consider an NMRsystem. Very recently NMR quantum heat engines becomeexperimentally available [29, 30]. Power outputs of these ma-chines are not optimized. One can use non-classical resources,though such a resource would not be natural and require somegeneration cost reducing overall efficiency. Alternatively onecan use dynamical shortcuts to speeding up adiabatic transfor-mations [31] but this would increase experimental complexity,and moreover in NMR thermalization is more seriously slowstep reducing the power output of the NMR machines.Here we, propose to replace adiabatic steps by SWAP opera-tions while the cooling step by an algorithmic cooling [32–45].By this way, NMR thermal device operation would closely re-semble an NMR quantum computer [46–49] albeit processinga completely noisy input. We remark that there will be only theclassical energy source as the input to the machine, while the ∗ [email protected] second heat bath required by the second law of thermodynam-ics for work production would be an effective one, engineeredby a spin ensemble, typical in the algorithmic cooling scheme.In addition to engine cycles, NMR systems were also proposedfor studies of single-shot thermodynamics [50].Our scheme allows us to provide at least one answer, in thecontext of NMR heat engines, to the question of how to imple-ment genuine quantum steps in quantum machine operationto harvest a classical energy source. In addition, our calcula-tions suggest that compared to the NMR engine with standardthermodynamical steps quantum algorithmic NMR engine pro-duce more power. The advantage comes from the replacingthe long-time isochoric cooling process with the more efficientand fast algorithmic cooling stage. We provide systematicinvestigation by first examining the case of standard Otto cycleas a benchmark then introduce the algorithmic cooling stage in-stead of isochoric cooling one, and finally introduce the SWAPoperation stages instead of the adiabatic transformations.The organization of the paper is as follows. We review thetheory and heat bath algorithmic cooling (HBAC) considering3-qubit NMR sample in Sec. II. The results and discussionsare given in Sec. III. In Sec. III A, efficiency, work, and poweroutput of four-stroke quantum Otto cycles cooled by HBACand isochoric stage are discussed by considering the sameparameters to extract work. Two-stroke type engine results arediscussed in Sec. III B. We conclude in Sec. IV. The details ofthe HBAC using partner pairing algorithm (PPA) are given inAppendix A. II. THE WORKING FLUID
Quantum Otto engines (QOEs) consist of quantum adiabaticand isochoric processes [31]. Three types of quantum Ottocycle is given in the literature as four-stroke, two-stroke, andcontinuous [1, 14]. In this article, four-stroke, and two-stroketype of QOEs are examined. The basic model of HBAC withPPA is considered instead of isochoric cooling. Implemen-tation of HBAC requires two sets of qubits; reset qubits andcomputational qubits. One of the computational qubits oper-ated as target qubit which is going to be cooled by applyingPPA while other computational qubits play a role in entropycompression [33]. This cooling process can be implemented a r X i v : . [ qu a n t - ph ] J a n with a minimal system composed of just 3-qubit [32–35]. Forthe four-stroke engine, we consider the target qubit as the work-ing fluid (see Fig. 1). As the working substance for two-stroke Figure 1. (Color online) Four-stroke quantum heat engine operatingin a single heat bath at temperature T . It has one isochoric heatingprocess, two adiabatic processes and one algorithmic cooling process. QOEs, we again chose the target qubit of the 3-qubit system,but with an extra qubit coupled to the target qubit (see Fig. 2).
Figure 2. (Color online) Two-stroke quantum heat engine operating ina single heat bath at temperature T . It has isochoric heating processand algorithmic cooling process happen at the same time. Then, oneadiabatic process using SWAP operation. In NMR systems, various 3-qubit models have been used forquantum information processing [51–54]. To make an applica-ble and realistic model for QOEs using these systems, a suitablesample and a well-designed procedure must be considered. Weused parameters of C -trichloroethylene (TCE) (see Table I)with paramagnetic reagent Cr(acac) in cloroform-d solution( CDCl ) in our numerical calculations, which are also exper-imentally used for HBAC in Refs. [34, 35]. Carbon-1 andCarbon-2 qubits are classified as the target and the compres-sion qubits. And, Hydrogen qubit is selected as the reset qubitbecause of the relaxation time, which is small compared to theother two qubits. To implement HBAC, the Hamiltonian of3-qubit in the lab frame can be written as [46] H (0) = − (cid:126) X i ω i I iz + (cid:126) X i = j J ij I iz I jz , i, j = {T , C , R} , (1)where, ω i = γ i B z is the characteristic frequency of the i th qubit, γ i is the nuclear gyromagnetic ratio and B z is the mag-netic field. T , C and R stand for target, compression and reset qubits, respectively. I z is the component of the spin angu-lar momentum of the i th qubit and J ij is the scalar isotropiccoupling strength between i th and j th qubits. Target- C1 Compression- C2 Reset- H γ/ π ω/ π τ
43 [s] 20 [s] 3.5 [s]
C1-C2 C1-H C2-H J/ π
103 [Hz] 9 [Hz] 200.8 [Hz]Table I. The first row in the table shows gyromagnetic ratio values ofCarbon1, Carbon2 and Hydrogen qubits. Considering the 500MhzNMR device, the corresponding characteristic frequencies are givenin the second row. Also, their experimental τ relaxiation times aregiven in the third row. Last row shows J-coupling strenght betweenthese qubits [34, 35]. III. RESULTS AND DISCUSSIONA. Four-stroke Heat Bath Algoritmic Cooled Quantum OttoEngine
Normally, four-stroke QOEs consist of two isochoric andtwo adiabatic stages. However, we consider one isochoricstage, one algorithmic cooling stage, and two adiabatic stages(see Fig. 1). The details of the four stroke cycle is described asfollows.
Isochoric Heating: ρ (0)th = e − βH (0) Z . (2)Here Z = Tr h e − βH (0) i is the partition function and β =1 /k B T . The initial density matrix of our working fluid can beexpressed by taking a partial trace of 3-qubit system ρ (0) T = Tr C , R h ρ (0)th i . (3) Adiabatic Compression:
Qubits are isolated from the heatbath and undergo finite-time adiabatic expansion. The adi-abatic processes of the cycle are assumed to be generatedby a time-dependent magnetic field [55, 56]. Here, H (0) at t = 0 is changed to H (1) at t = τ / by driving the initialmagnetic field as B z → B z / . The time evolution of thedensity matrix is governed by the Liouville-von Neumannequation ˙ ρ ( t ) = − [ H ( t ) , ρ ( t )] , and H ( t ) can be expressed as H ( t ) = H (0) + H drive , where H drive is given by H drive = (cid:126) X i ( ω i − ω i ) I iz sin (cid:18) πtτ (cid:19) . (4)Here, ω i = γ i B z / is the characteristic frequency at the endof the adiabatic stage. Up to this point, we used Hamiltonianin Eq (1) which is written for 3-qubit. However, to find thework done in this stage, we need to consider only the targetqubit. The local Hamiltonian for target qubit before adiabaticcompression can be written as H (0) T = − (cid:126) ω T I z . After theadiabatic compression, it will be H (1) T = − (cid:126) ω I z . The initialdensity matrix of the target qubit ( t = 0) is given in Eq (3). Thefinal density matrix of 3-qubit system at the end of adiabaticcompression is ρ (1) = ρ ( τ / . The density matrix of the targetqubit at the end of this process is ρ (1) T = Tr C , R (cid:2) ρ (1) (cid:3) . Then,the work performed by the working fluid is W = Tr h H (0) T ρ (0) T i − Tr h H (1) T ρ (1) T i . (5) Heat Bath Algorithmic Cooling:
In this part of the engine,we normally need a cold heat bath to cool down our target qubit.Instead of using a cold heat bath, the working fluid treated inthe same heat bath as isochoric heating. Thus, to cool down thetarget qubit HBAC is used. Details of the cooling mechanismgiven in the Appendix A. For each qubit, polarization is definedas (cid:15) i = P ↑ i − P ↓ i = tanh (cid:18) (cid:126) γ i B z k B T (cid:19) , (6)where, P ↑ i and P ↓ i denote the probability of up and downstates. In a closed quantum system, Shannon’s bound limitsthe polarization of single spin in a collection of equilibriumspin system. Using HBAC take advantage of the heat bathto cool the target qubit beyond Shannon’s bound [34]. As aresult, the polarization of the target qubit using HBAC becomeshigher than the polarization of the heat bath. After the first Shannon's Bound
Figure 3. (Color online) The polarization (cid:15) (dimensionless) calculatedby using the Eq. (7) for each rounds of algorithmic cooling. The targetand reset qubits polarizations calculated by taking into account theperfectly applied quantum logic gates in terms of several rounds ofthe PPA. The black line shows Shannon’s limit of polarization. Thepolarization of the target qubit has exceeded this limit after the firstiteration and the reset qubit stays under this limit.
SWAP operation before PPA, polarizations of the qubits are equal to each other. The value of their polarization is givenat zeroth iteration in Fig. 3 as ∼ . × − . Then, severalrounds of PPA are applied. The effective temperature of thetarget qubit is determined by Eq. (7) and plotted in Fig. 4. Thetarget qubit reached a polarization above the Shannon limit asa result of one round of PPA and target qubit cooled down to ∼ . After seven rounds of PPA, it almost reached to itsmaximum value as ∼ . × − and cool down to ∼ temperature. At the end of PPA, density matrix of target qubitis given by Eq. (A8) as ρ (2) T = Tr C , R h ρ (1 , AC i . (7) Shannon's Bound
Figure 4. (Color online) The effective temperature of target qubitcalculated by using the relation between temperature and polarizationgiven in Eq. 7. The black line shows the corresponding Shannon’slimit of temperature. The effective temperature of the target qubit hasexceeded this limit after the first iteration.
Adiabatic Expansion:
In this process H (1) at t = 0 ischanged to H (0) at t = τ / by driving back the magneticfield as B z / → B z . The work performed by the target qubitin this process can be written as W = Tr h H (1) T ρ (2) T i − Tr h H (0) T ρ (3) T i . (8) Work and Power Output of Four-Stroke QOE:
The totalwork done by the working fluid at the end of adiabatic stagescan be found as W = W + W . Alternatively, total workcan also be calculated from the isochoric stage and algorith-mic cooling stage using W = Q in − Q out where, Q in =Tr h H (0) T ( ρ (3) T − ρ (0) T ) i and Q out = Tr h H (1) T ( ρ (2) T − ρ (1) T ) i are the heat released and absorbed in these stages, respectively.The efficiency of the cycle is determined by η = 1 − ω /ω T and for this engine η = 0 . . In Fig. 5 , we plot the heat re-leased and absorbed by the target qubit as per rounds of PPA.As the number of iterations increases, it can be observed thatthe absorbed heat increases more than the heat released. Afterthe first iteration, the target qubit absorbed ∼ × − J / mol of heat and released ∼ . × − J / mol of heat. As a result ∼ . × − J / mol work was performed. In order to see Rounds of Algorithmic Cooling . . . . Q [ J / m o l ] × − Q in Q out Figure 5. (Color online) Heat absorbed and released in a four-strokecycle by the target qubit in the isochoric process ( Q in ) and algorithmiccooling process ( Q out ) for per rounds of PPA. the difference in power caused by the algorithmic cooling andisochoric processes, we consider cold baths, corresponding tothe temperatures in Fig. 4, to simulate a quantum Otto cyclecooled by isochoric stage with same parameters. By this way,the work output of the quantum Otto cycle cooled by isochoricstage will be the same as the cycle cooled by HBAC (see Fig 6).As we can see from Fig. 6; while the work produced by thetarget qubit rapidly increases with the number of iterations atthe beginning, it remains constant after a certain iteration ofthe PPA. The reason for this behavior, the HBAC is able to coolthe target qubit up to a certain limit. In Fig. 5 , after the fourthiteration, the target qubit almost reaches the maximum valueit can absorb and release heat. Absorbed and released heatfrom qubit in the cycle at this iteration is ∼ . × − J / mol and ∼ . × − J / mol . Then, maximum work output ofthe cycle is ∼ . × − J / mol (see Fig. 6). The number Figure 6. (Color online) Work obtained, in a four-stroke cycle pernumber of iteration of PPA, from target qubits of one mol of TCEcooled by HBAC (green line). And work obtained from a mol ofqubits, which are cooled by isochoric stage to temperatures corre-sponding in Fig 4 (black line). of iteration of PPA is important for quantum heat engines.Because more iteration means that more relaxation of resetqubit and it increases the time required to complete enginecycle. Even if we increase the work output iterating more PPA,we may lose power output. For a quantum Otto cycle usingNMR system as working fluid, the adiabatic stages of the cycleare considered as fast compared to the isochoric stages [31].We estimate the power output of cycles considering isochoricstages and HBAC. A single number of iteration of PPA requirestwo reset process. Taking the number of iteration ’n’, we canwrite power output for quantum Otto cycle using HBAC as P = W/ ( τ T + τ R (2 n + 1)) , where, τ T and τ R are relaxationtimes of the Carbon1 and the Hydrogen qubits respectivelyfrom the Table I. For the cycle using isochoric process in thecooling stage, power output is P = W/ τ T . In Fig. 7, weplot power output for these engines. We see that the seconditeration of PPA gives maximum power output as ∼ . × − Watt/mol. For more number of iteration, this power output isgetting decrease. After the fifth iteration of PPA, the cycle
Figure 7. (Color online) Power output for a four-stroke cycle pernumber of iteration of PPA, from target qubit of one mol TCE, cooledby HBAC (red) and from a mol of qubits cooled by isochoric stage totemperatures corresponding in Fig 4 (black). using the isochoric cooling stage can dominate the engine us-ing the HBAC stage. Accordingly, the optimum choice of thenumber of rounds in HBAC is two for our four-stroke modelsystem. Such a choice optimizes the power output of the cycleyielding high power performance.
B. Two-Stroke Heat-Bath Algorithmic Cooled Quantum OttoEngine
We also investigate two-stroke QOE that is proposed in Refs.[1, 14]. To construct it, we need to consider two qubits donatedas S and T as the working fluid. First, two qubits are isolatedfrom each other. One of the qubits contacts with a heat bathat temperature T until it reaches to equilibrium. Our purposeis to cool the other qubit within the same heat bath utilizingalgorithmic cooling. Hence we need another two qubits forthis process as compression qubit and reset qubit. This partis considered as two isochoric processes for the heat engine.Second, two qubits decouple from the heat bath. Then, theSWAP operation is performed between these two qubits (seeFig. 8). Local Hamiltonian of the qubit S and qubit T can be Figure 8. Quantum circuit demonstrating the two stroke heat bathalgorithmic cooled quantum Otto engine. |Si stands for qubit S and |T i for qubit T . Here, isochoric heating demonstrated as R operationand applied only one time on qubit S . For qubit T , PPA applied andit is given in Appendix A. After these two process finished SWAPoperation applied between two qubits and cycle is completed withadiabatic process. written as H S = − (cid:126) ω S I z and H T = − (cid:126) ω T I z , respectively.And density matrix of qubit S , at the end of the isochoric stageis given by ρ (0) S = e − βH S /Z . Using the PPA given in theAppendix A, we can write the density matrix of qubit T at theend of HBAC as ρ (0) T = Tr C , R h ρ (1 , AC i . It is assumed that thecoupling between S and T qubits are small compered to ω S and ω T . The density matrix of the total working fluid can beexpressed as ρ (0) S , T ≈ ρ (0) S ⊗ ρ (0) T . (9)In the adiabatic process, a SWAP gate is applied to exchangethe states of S and T qubits ρ (1) S , T = SWAP (cid:16) ρ (0) S , T (cid:17) SWAP † . (10)At the end of the cycle, density matrices of individual qubitsbecome ρ (1) S = Tr T h ρ (1) S , T i , ρ (1) T = Tr S h ρ (1) S , T i . (11)Heat absorbed by the qubit S calculated as follows Q in = Tr h H S ρ (0) S i − Tr h H S ρ (1) S i , (12)and the heat released by the qubit T is Q out = Tr h H T ρ (1) T i − Tr h H T ρ (0) T i . (13)The net work is evaluated by W = Q in − Q out , with efficiency η = 1 − ω T /ω S . To get a positive work, the frequency ofqubit S needs to be greater than the frequency of the qubit T ( ω S > ω T ). In addition, T > T T ( ω S /ω T ) needs to be satisfied. Here T T is the temperature of target qubit in HBAC.Positive work conditions then can be expressed as ω T < ω S < ω T TT T . (14)Fig. 9 shows the relation between the work output and ω S , fordifferent number of iterations. When the number of iterationof PPA is increased, the best work output is also increased.However, after some number of iteration, it remains almost Figure 9. (Color online) Positive work obtained in 2 strokes HBAC-QOE as a function of ω S . The legend of the plot shows the numberof rounds of PPA applied to qubit T and direction of the arrow showsan increase in work output from 1st iteration to 8th iteration. constant, because of the limitation of PPA to cool qubit T . Itis found that for 580 MHz, we almost get the best work outputat 5th and more iterations as ∼ . × − J/mol. But thisfrequency does not give us the best work output at 1st iteration,430 MHz gives. For 1st iteration with 430 MHz frequencywe get ∼ . × − J/mol work output. The adiabatic stageis evaluated by a SWAP operation, which is fast compared tothe HBAC. Relaxation time of the S qubit may be estimatedby considering spectral density functions J ( ω, t c ) , where t c is the correlation time [48, 57]. If we look for the optimumpower output, the ω S is close to the frequency of the R qubit.Considering environmental effects are the same, t c may beassumed to be close for the R and S qubit. As a result, wecan say that the S qubit is thermalized until the HBAC processis complete. In addition, using isochoric cooling instead ofHBAC to cool target qubit requires more time up to the 5thiteration of PPA, as we have shown in the Sec. III A. Thus,estimation of the power output depends on the relaxation timeof reset qubit and the number of rounds of PPA. Then, we canwrite power output as P = W/τ R (2 n + 1) . The relaxationtime of Hydrogen ( τ R ) is given in Table I. We plot the poweroutput of the cycle in Fig. 10. When the number of iterationsincreased, the power output is decreased despite the increase inthe work output. The optimum value given by 1st iteration with430 MHz frequency of S qubit as ∼ . × − Watt/mol. Ifwe compare these results to four-stroke QOEs both cooled byHBAC and isochoric stage, we see that two stroke cycle givesmore work and power output. The efficiency of the cycle as
Figure 10. (Color online) Power output obtained in a two-strokequantum Otto cycle for the different number of iterations of PPA as afunction of ω S . The legend of the plot shows the number of roundsof PPA applied to qubit T and the direction of the arrow shows anincrease in the power output from 8th iteration to 1st iteration. a function of ω S is given above. Taking the optimum valuefor the power output at 1st iteration and 430 MHz frequencyof S qubit, the efficiency of the cycle is 0.7, which also moreefficient than four-stroke QOEs. IV. CONCLUSIONS
We have investigated the possible quantum Otto engines con-sidering HBAC instead of isochoric cooling. In conventionalNMR setups do not let to change strength (huge) magneticfield along the z direction. In order to solve this restriction,NMR setup can be modified to change strong magnetic fieldvia gradient coils such as Magnetic Resonance Imaging (MRI)system. In addition, the sample always in a single entropy sinkin NMR systems. This is the main problem to design QOEs,which requires two heat bath to extract work from NMR qubits.Using HBAC allowed us to cool the working fluid in a singleheat bath. Here we specifically showed this cooling processcan be implemented to four-stroke and two-stroke QOEs. Theisochoric cooling process of the cycle takes too much timecompared to the HBAC. Comparing our results with a singlespin NMR heat engine, utilizing the HBAC to QOEs improvesthe power output of the cycles up to a certain iteration of PPA.
ACKNOWLEDGMENTS
The authors thank to M. Paternostro for fruitful discussions.
Appendix A1. Heat Bath Algorithmic Cooling - Partner PairingAlgorithm(PPA)
We consider heat bath algorithmic cooling (HBAC) withpartner pairing algorithm (PPA) (see Fig. 11), which uses quan-tum information processing to increase the purification levelof qubits in NMR systems [32, 33]. Before starting PPA, the
Figure 11. Quantum circuit demonstating partner pairing algorithmfor 3-qubit system given in Ref. [33]. |T i , |Ci and |Ri stands fortarget, compression and reset qubits respectively. R process meansthe relaxation of reset qubit. In PPA, the first reset process appliedonly one time. Then, the iteration part consists of SWAP and 3-bitcompression operations applied n times. individual density matrices of the target, the compression andthe reset qubits are ρ (1) T = Tr C , R (cid:2) ρ (1) (cid:3) , ρ (1) C = Tr T , R (cid:2) ρ (1) (cid:3) , ρ (1) R = Tr T , C (cid:0) ρ (1) (cid:1) respectively and ρ (1) is the initial den-sity matrix of 3-qubit system. The reset has small relaxationtime compared to the target ( τ R (cid:28) τ T ) and the compression( τ R (cid:28) τ C ) qubit, where τ T , τ C and τ R are given in Table I re-spectively. In each step of HBAC, R operation is applied to thereset qubit to thermalize it with the heat bath ( ρ (1) R → ρ (1) R ,th ) at temperature T. Scalar couplings between qubits are smallcompared to ω i values. If we write the total density matrix ofthree qubits as a tensor product of the individual states, thefidelity of the density matrix in Eq. 2 and this product densitymatrix numerically is found to be close to 1. Thus, at first step,the density matrix of 3-qubit system can be written as a tensorproduct of these states ρ (1 , AC = ρ (1) T ⊗ ρ (1) C ⊗ ρ (1) R ,th , (A1)where, 1st index of ρ (1 , AC stands for stage of the cycle and2nd index from 0 to 5 indicates the state after each process ofHBAC. After the reset qubit regains its polarization, a unitarySWAP operator is used to exchange polarizations of the targetand the reset qubit. ρ (1 , AC = SWAP T , R (cid:16) ρ (1 , AC (cid:17) SWAP †T , R (A2)The states of the target and compression qubits become ρ (1 , T = Tr C , R h ρ (1 , AC i and ρ (1 , C = Tr T , R h ρ (1 , AC i respec-tively. After the unitary evolution, PPA can be applied n times,which is given as follows1. The reset qubit is thermalized with the heat bath. Thedensity matrix of the 3-qubit system becomes ρ (1 , AC = ρ (1 , T ⊗ ρ (1 , C ⊗ ρ (1) R ,th (A3)2. SWAP is applied to change polarization between thecompression and the reset qubit, such that ρ (1 , AC = SWAP C , R (cid:16) ρ (1 , AC (cid:17) SWAP †C , R (A4)Then, the states of the target and the compressionqubits becomes ρ (1 , T = Tr C , R h ρ (1 , AC i and ρ (1 , C =Tr T , R h ρ (1 , AC i .3. The reset qubit is regained its polarization. The state of 3-qubit system is expressed as ρ (1 , AC = ρ (1 , T ⊗ ρ (1 , C ⊗ ρ (1) R ,th . (A5)4. To lower the entropy of the target qubit and to increasethe entropy of the reset qubit 3-bit compression gateapplied to the density matrix. ρ (1 , AC = COMP (cid:16) ρ (1 , AC (cid:17) COMP † (A6)where COMP is the unitary operation of compressiongate composed of unitary operators as two control-not-not gates and a Toffoli gate, which can be expressedas COMP = [CNotNot][Toffoli][CNotNot] . 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