25. A transmon-based quantum half-adder scheme

Prog. Theor. Exp. Phys. 2015, 093A02 (16 pages)DOI: 10.1093/ptep/ptv122

A transmon-based quantum half-adder scheme

Dibyendu Chatterjee and Arijit Roy∗

Department of Electronics, West Bengal State University, Barasat, Kolkata, India 700 126∗E-mail: [email protected]

Received April 2, 2015; Revised June 16, 2015; Accepted July 26, 2015; Published September 18, 2015

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A four-level qubit system is applied to realize quantum half-adder operation. The half-adder cir-cuit is obtained in terms of a quantum CPHASE gate realized by the qubits comprised of fourenergy levels [C. P. Yang, Prog. Theor. Phys. 128, 587 (2012)], and such a CPHASE gate isdemonstrated using the transmon. Commonly, higher energy levels are very sensitive and areeasily perturbed by the noise sources. Compared to other qubit systems, the higher energy levelsof the transmon are less prone to noise such as charge noise, flux noise and other noises. Further,the order of the dephasing time of the higher energy levels (third and fourth energy levels) isnearly the same as that of the lower energy levels of the transmon when the ratio between theJosephson energy and the charging energy �1. A system of three transmons coupled to a sin-gle high quality-factor superconducting coplanar resonator is demonstrated to obtain two- andthree-qubit CPHASE gates which are in turn used to obtain the quantum half-adder operation.The main advantage of this quantum half-adder scheme is the reduction in the number of requiredelementary gates, leading to a significant increase in operational speed and robustness comparedto the other existing half-adder schemes. The operational time of a complete half-adder operationis ∼37 ns. The methods presented in this article can also be implemented for more complicatedquantum circuits.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index A61

1. Introduction

Superconducting qubits based on Josephson junctions are currently the most experimentallyadvanced solid-state qubits [1–11]. Superconducting qubits come in a variety of types, often clas-sified as charge [7,9–11], flux [3,4], or phase qubits [6]. The transmission-line shunted plasmaoscillation qubit, in short termed as transmon [7–9,12,13], is based on the Cooper pair box (CPB)which is the prototype of a qubit based on superconducting electronic circuits [11,14–16], and awealth of knowledge has been acquired on transmon-based quantum computing. The major advan-tages of the transmon are its scalability, charge insensitivity, increase of qubit–photon couplingstrength and longer dephasing times (T2) [7–9]. The quantum half-adder is one of the importantarithmetic operations in quantum computing. In fact, in the past decade, few schemes for realizinga quantum half-adder have been proposed [17–22]. These architectures involve implementation ofquantum gates using linear optics [17,18], molecular quantum trajectory [20], and nuclear magneticresonance [21]. On the other hand, single-qubit, two-qubit, and three-qubit operations [23–28], aswell as entanglement [26] and teleportation experiments [25], have been successfully demonstratedusing superconducting qubits.

Basic elementary operation actually uses two energy levels of a qubit; however, using more thantwo energy levels of a qubit system enhances the operational effectiveness and reduces the number of

© The Author(s) 2015. Published by Oxford University Press on behalf of the Physical Society of Japan.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/),which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

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elementary gates for a complicated operation, and hence reduces the operational time [29]. Further,Fedorov et al. implemented a transmon-based TOFFOLI gate using a three-level qubit system andclearly showed a significant reduction in the number of operations [24]. However, half-adder opera-tion using four energy levels of a qubit system has not been explored and its possible realization bya suitable solid state device, e.g. the transmon, is the objective of this study.

A quantum half-adder circuit consists of a TOFFOLI gate followed by a CNOT gate. In therealization of the half-adder circuit, the minimal requirement for qubits is three, where the twocontrol qubits of the TOFFOLI gate are used as the two inputs and after the operation the secondand third qubit measurements give the sum and carry output respectively. Using the conventionalgate-decomposing protocol, achieving half-adder operation is complicated because it will take30 steps, leading to speed compromise. The TOFFOLI operation in our scheme is decomposed intoa single-qubit Hadamard (H-gate) and three-qubit CPHASE gates. Similarly, the CNOT operationis decomposed into a single-qubit H-gate and two-qubit CPHASE gates [30]. These decompositionsare shown in Fig. 1.

Recently, it was theoretically shown that the CPHASE gate is achievable in a superconducting qubitsystem coupled to a cavity with a qubit comprised of four energy levels [31]. This proposal, alongwith the experimental design and realization of quantum operation by transmon, is the motivation todecompose the half-adder circuit as per our scheme. Here, it is to be noted that though the transmonuses four energy levels, we still specify the bits as qubit instead of qutrit or higher order term. Thisis because the first two energy levels are considered as the computational qubit states, |0〉 and |1〉and the second excited and third excited states (|2〉 and |3〉) are considered to perform the internaloperations, as well as for input and output in the form of |0〉 and |1〉 only.

In this work, we focus on the realization of a three-qubit half-adder by superconducting qubitsystems based on the cavity quantum electrodynamics (QED) technique. The basic element ofthis superconducting qubit system is the transmon, which is a charge-based qubit. Instead ofother superconducting qubit elements such as SQUID, we have considered the transmon as thequbit element in our scheme because of its characteristics and plausible leading role in quan-tum information processing (QIP). In our scheme, the cavity acts as a quantum data bus whichcan provide a sequence of controlled operation by the qubits coupled to the cavity bus. Threeindependent transmons are coupled to the cavity. The qubits associated with each transmon arecomprised of four energy levels [7,27]. It is to be noted here that the half-adder operation canalso be achieved by using other superconducting qubits that have four energy levels by which theCPHASE gate can be constructed as proposed by Yang [31], e.g. SQUID, Flux, Phase qubit, etc.However, the transmon is chosen for our scheme because it can have a long dephasing time forhigher energy levels (third, fourth and above) also. On the other hand, the CPHASE gate usingfour energy levels has been demonstrated [31], and construction of a half-adder circuit using sucha CPHASE gate significantly reduces the number of operations. The concept to realize half-adderoperation utilizing a four-level qubit system is challenging, and the qubit operation using thetransmon along with the timing diagram is needed to understand the entire process of half-adderoperation.

The outline of this paper is as follows. The second section deals with the advantages of our scheme,and useful characteristics of the transmon for realization as a four-level qubit system [7]. The inter-action of the transmon with the cavity and the interaction with the external pulse are also described inthis section. In the third section, we demonstrate the operations of the CPHASE gate, CNOT gate, andH-gate. Section 4 contains a description of the half-adder operation, and in the section 5 we present

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(a)

(c)

(d)

(b)

Fig. 1. (a) Schematic circuit of a half-adder, where a and b represent inputs and sum and carry are the results[17]. (b) Equivalent circuit operations for the TOFFOLI gate. (c) The decomposition of the three-qubit con-trolled phase gate. (d) The CNOT gate is achieved by two-qubit CPHASE gate and Hadamard operations. Thecombination of circuits in (a), (b), (c) and (d) indicates that constructing a half-adder circuit requires seventwo-qubit CPHASE gates, 16 single-qubit H-gates and seven single-qubit phase shift gates. Namely, a total of30 basic gates are needed to build up a three-qubit half-adder circuit using the conventional gate-decomposingprotocol (for details, see Ref. [30]). Therefore, at least 30 operational steps will be required [31].

discussion of a possible experimental realization of our scheme. At the end, in the sixth section, wedraw conclusions from this work.

2. Transmon for our scheme

2.1. Important advantages of our scheme

The reduction in the number of elementary gates and the operational time are the main advantagesof our scheme. Apart from this, our scheme is also able to gain the important advantages of super-conducting qubit systems based on QED techniques [31], for example: (a) adjustment of qubit levelspacing is not required during the gate operations (thus decoherence, which is produced due to tun-ing the qubit level spacing, is avoided); (b) during the entire operation no photon detection is desired(thus the result of the photon-detection imperfection on the gate performance is avoided); (c) thescheme is tolerant of the unavoidable non-uniformity in device parameters as identical qubit–cavity

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coupling constants do not take any useful significance; (d) only resonant interaction is consideredand hence the gate operations can be performed at a faster speed.

2.2. Transmon characteristics

In cavity QED [32–37], atoms are coupled to photons inside a cavity with highly reflective walls.The cavity acts like a high finesse resonator which supports only a few narrow line-width standingwave modes. In circuit QED [28,38,39], the atom is replaced by superconducting qubits with anatom-like energy spectrum and the cavity by an electrical microwave resonator. The transmon is acomparatively new type of superconducting qubit which is able to fix the main weakness of the CPBby featuring an exponential gain in insensitivity to charge noise [7–9]. The design of the transmon isclosely related to the design of the CPB qubit. The transmon is actually a split Josephson junction witha large shunt capacitance. The split Josephson junction forms a dc-SQUID which allows for tuningof the Josephson energy EJ = EJ max| cos(πΦ/Φ0)| using the external magnetic flux Φ. The vitaldifference between CPB and the transmon is the large shunt capacitance which is responsible for thereduction in charge dispersion. It is clearly described below. The transmon is operated at significantlydifferent ratios of Josephson energy (EJ) to charging energy (EC). The effective Hamiltonian for acharge-based transmon qubit system can be reduced to a form identical to that of the CPB system[11,15,40,41],

H = 4EC(n − nG

)2 − EJ cos ϕ; (1)

where the first term indicates the energy associated with excess charges on the island and the secondterm is the energy associated with the Josephson coupling between the two islands. The symbolsn = −q/(2e) and ϕ = φ2e/� denote the number of Cooper pairs transferred between the islandsand the gauge-invariant phase difference between the superconducting electrodes respectively, andnG represents the offset charge which is tuned by the external gate voltage. The symbol ϕ represents acompact variable that satisfies ψ(ϕ + 2π) = ψ(ϕ), and its commutation relation with the conjugatevariables follows [ϕ, n] = −i .

Since charge noise is beautifully controlled by the transmon and reduces the charge dispersion,it provides stable higher energy levels for operation. In fact, this stability or less anharmonicity ofthe energy level is described as “sweet spot.” The “sweet spot” [16] is a special operating point of acharge qubit system where the sensitivity of the qubit to charge noise is reduced by biasing the systemto the charge degeneracy point at which nG is equal to an integer multiple of 0.5. This is shown inFig. 2, where we plot the eigenenergies Ek of the CPB Hamiltonian [Eq. (1)] as a function of theeffective offset charge nG. If the qubit is operated away from the charge degeneracy point the qubittransition frequency varies rapidly with gate charge (offset charge) nG, resulting in fast dephasingdue to random fluctuation in the local electrostatic potential. Since at “sweet spots” ∂Ek

∂nG= 0, the

qubit transition frequency remains unaffected by the contribution of linear noise. When the systemis biased at such a point, dephasing decreases and hence T2 increases significantly (from nanose-cond to microsecond range) [7]. Commonly, the charge qubit is operated at EJ/EC ∼ 1. However, ifwe increase the EJ/EC ratio, the charge dispersion decreases exponentially and the qubit transitionfrequency becomes stable with respect to the charge noise. This is achieved by the introduction of atransmon with EJ/EC � 1 for which the energy levels become increasingly flat, or in other words theenergy levels become independent of gate charge (see Fig. 2). Thus, as a result of higher EJ/EC(�1),effectively generates “sweet spot” everywhere, which in turn suppresses the sensitivity to chargenoise to a high degree. Under such a condition, the strong dependency of a particular bias point on

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EJ/EC = 1

Ek/

EC

nG nG nG

EJ/EC = 20 EJ/EC = 50

Fig. 2. Eigenenergies Ek (first four levels, k = 0, 1, 2, 3) of the CPB Hamiltonian [Eq. (1)] as a function ofthe effective offset charge nG for different ratios EJ/EC, in units of the charging energy EC. Similar energylevels can also be found in Ref. [27].

the device performance does not arise anymore [42]. To take the advantage of the “sweet spot,” theparameters EJ and EC should be estimated and designed appropriately. The probability of chargedispersion increases with the increase in energy levels (shown in Fig. 2a), but it also diminishesas EJ/EC increases for the upper energy levels (shown in Fig. 2b,c). So the dephasing time also

increases for the upper energy levels. As the dephasing time

(T2 ∼ �

A

∣∣∣ ∂Ei j∂ng

∣∣∣−1)

depends on∣∣∣ ∂Ei j∂ng

∣∣∣(where Ei j = E j − Ei ), it will increase as charge dispersion decreases. Therefore, for the third and

fourth energy level the charge dispersion does not vary noticeably(

for EJEG

≥ 50)

and hence thedephasing time remains of the same order as the first two energy levels. The parameter EC is atransmon design parameter and its appropriate value should be chosen in the transmon fabrication.Moreover, the ratio EJ/EC can be set to a desired value by tuning EJ by applying external flux.Thus the difference between the energy levels can be tuned by appropriate design as well as applyingexternal flux, and hence the resonant interaction with the cavity resonator becomes more versatile.Therefore the advantages such as high T2, low charge dispersion, versatility of interaction with thecavity, availability of less anharmonic higher energy levels etc. are achievable for our transmon-basedscheme over other similar qubit systems.

In short, in the realization of transmon-based quantum computing, the following importantcharacteristics are useful [7]:

1) An increase of the ratio of EJ/EC leads to an exponential decrease of the charge disper-sion, leading to higher stability of the transition frequency of a qubit with respect to chargenoise.

2) In comparison to CPB, the transmon can be operated for better charge-noise insensitivity. Also,for an EJ/EC range in which the charge-noise insensitivity is significantly improved, a trans-mon can have sufficiently large anharmonicity. This range provides the useful anharmonicenergy levels for quantum operation (Fig. 2).

3) For asymmetric Josephson junctions in the transmon, the flux dependence of ϕ0 (defined inRef. [7]) may allow for additional qubit control without involving the resonator by applyingac magnetic fields.

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(a) (b) (c)

Fig. 3. Diagram of four-level qubit systems [27,31]. The energy eigenvalues for the four levels |0〉, |1〉, |2〉,and |3〉 are denoted by E0, E1, E2, and E3, respectively. In (a), the level spacings satisfy E2 − E1 > E1 − E0,E3 − E2, and E3 − E2 < E1 − E0. In (b) the level spacings satisfy E1 − E0 > E2 − E1 > E3 − E2.In (c) the energy level spacings of the transmon satisfy the level spacing of the phase-qubit sys-tem (though it does not follow the requirement of a charge-based qubit system). However, the trans-mon is a charge-based qubit system and such a charge-based qubit system is suited to our proposal.In a transmon-based charge qubit system, when designed properly, the parameter EJ/EC provides similarenergy levels. Such an energy-level diagram is shown here [27,42]. Here, |g〉, |e〉, |f〉, and |h〉 are four lev-els having energy eigenvalues Eg, Ee, Ef, and Eh, respectively, where Ee − Eg > Ef − Ee > Eg − Ef [21].Here, in our description we replace |g〉, |e〉, |f〉, and |h〉 by |0〉, |1〉, |2〉, and |3〉, respectively.

4) The coupling between cavity and transmon expressed by the coupling energies gi j , does notbecome small but in fact even increases compared to other superconducting qubits, and it iscrucial for utilizing the transmon system as an actual qubit.

5) Suitability of the relaxation time (T1) and the dephasing time (T2) for operation.

2.3. The transmon as a four-level qubit system

Unlike the natural atom, the transition between any two levels in a transmon is not constrained by thenatural selection rule and is different for superconducting artificial atoms [43,44]. For example, thetransition between two levels (e.g. the transition between the levels |0〉 and |2〉, |1〉 and |3〉, |0〉 and |3〉)is possible in transmon-based qubit system [31]. The eigenenergies of a transmon as a four-level qubitare shown in Fig. 2. In order to realize the half-adder operation, along with the above characteristics ofthe transmon, we follow the theoretical proposal by Yang [31] presented for superconductor qubits.The energy levels of such a four-level qubit are shown in Fig. 3 and are used in this scheme toobtain CPHASE, H-gate and CNOT operation [26]. We noticed that the transmon-based qubit canbe designed to obtain similar energy levels (four-level qubit) for quantum operation and thus we canobtain half-adder operation comprised of CPHASE, H-gate, and CNOT operation using the transmon.

The transition frequency between ground level |g〉 and the first excited state |e〉 of a transmon isgiven by

ωeg ≈√

8EC EJ(ϕ)/� − EC/�. (2a)

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The transition frequency between the first excited state |e〉 and the second excited state |f〉 of thetransmon is given by

ωfe ≈√

8EC EJ(ϕ)/� − 2EC/�. (2b)

Similarly, the transition frequency between the third level |f〉 and the fourth level |h〉 is given by

ωhf ≈√

8EC EJ(ϕ)/� − 3EC/�. (2c)

2.4. Transmon–cavity resonant interaction

The coupling of the superconducting qubit with the cavity fabricated on a microwave coplanarwaveguide follows the physics of cavity QED using natural atoms [34,45]. Among the four levels(see Fig. 3c) of a transmon qubit, the resonator is resonant with the transition between energy levels|f〉 and |h〉 of the transmon, whereas transition between any other two levels is highly detuned withthe cavity resonator. The interaction of transmon with cavity (i.e. the interaction picture) is describedby the interaction Hamiltonian, which is given by

H = �g(σfh + σhf

)(a + a†), (3)

where a† and a are the photon creation and annihilation operators of the cavity mode, g is the couplingconstant between the cavity mode and the |f〉 ↔ |h〉 transition of the system, and σfh = |f〉〈h|.

This transformation keeps the states |g〉|0〉c, |g〉|1〉c, |e〉|0〉c, |e〉|1〉c the same (the subscript c refersto cavity), since the Hamiltonian does not interact with the |g〉, |e〉 levels and |f〉|0〉c remains unaltered,whereas the states

|h〉|0〉c → −isin(gt)|f〉|1〉c + cos(gt)|h〉|0〉c

|f〉|1〉c → cos(gt)|f〉|1〉c − isin(gt)|h〉|0〉c.(4)

The coupling strength g may vary for different transmons used in the scheme and provide bet-ter flexibility and less complexity. The coupling between cavity and transmon is expressed by thecoupling energies �gi j .

Mathematically, the couplings gi j (coupling between |i〉 and | j〉 level) are determined by a prefactorcontaining the capacitance ratio β, the rms voltage of the local oscillator V 0

rms and by a matrix elementof the number operator for Cooper pairs (which depends on the energy ratio EJ/EC) and is expressedas [7]

gi j = 2βeV 0rms

�ωr. (5)

For simplicity, here we have considered that all the parameters and variables are identical for eachtransmon.

2.5. Transmon-pulse resonant interaction

The transition between the two levels shown in Fig. 2, |i〉 ↔ | j〉 of the system can be controlledby the corresponding resonant pulse ωi j , where the level |i〉 represents the lower energy level. Theinteracting Hamiltonian is given by

HI = �

(�i j e

iφ|i〉〈 j | + H.C.); (6)

where�i j and φ are the Rabi frequency and the initial phase of the pulse, respectively. Based on theHamiltonian expressed by Eq. (6), one can show that a pulse of duration t results in the following

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rotations:

|i〉 → cos�i j t |i〉 − ie−iφ sin�i j t | j〉| j〉 → cos�i j t | j〉 − ieiφ sin�i j t |i〉.

(7)

Note that the time required for state transformation given by Eq. (7) can be shortened by increasingthe Rabi frequency �i j . Three identical transmons A, B, C are considered in our scheme, and hence(�i j )A = (�i j )B = (�i j )C. The Rabi frequencies of the system are defined as [46]

�i j = mω2i jχi jε j f j (t)/�; (8)

where m = C ϕ2; ωi j = resonance frequency between two levels; χi j = 〈i |χ | j〉, being thenormalized matrix element of the |i〉 ↔ | j〉 transition; ε = applied field amplitude; and f j (t) is thedimensionless pulse envelope.

3. Realization of H-gate, CNOT-gate, and TOFFOLI-gate by transmon

The entire half-adder operation is decomposed using H-gate, CNOT-gate and TOFFOLI-gate.The realizations of these gates by transmon are described below.

3.1. Realization of H-gate

The Hadamard transformation |0〉 → (|1〉 + |0〉)/√2 and |1〉 → (|1〉 − |0〉)/√2 can be achieved byapplying a pulse of frequency ω = ω01 or ω10, phase φ = −π/2 and duration tH = π/4�01 [47].

3.2. Realization of CNOT-gate

The CNOT operation is shown in Fig. 1d. This operation is achieved by a sequence of operations:H-gate operation on target bit → two-qubit CPHASE gate → H-gate operation on target bit. Amongthese transformations, the operation of the H-gate was just described. The operation of the two-qubitCPHASE gate is achieved by the following three steps [48].

Step 1: For transmon A

(a) Apply a pulse ω = ω31, phase φ = −π/2 and duration t1,a = π/2�13.(b) Wait for time t1,b = π/2gA (cavity mode resonantly interacting with the |2〉 ↔ |3〉 transition

of transmon A).(c) Apply a pulse ω = ω20, phase φ = −π/2 and duration t1,c = π/2�02.

Mathematically, the result of the above steps is expressed as follows:

|1〉A|0〉c

|0〉A|0〉c

(a)−→ |3〉A|0〉c

|0〉A|0〉c

(b)−→ i |2〉A|1〉c

|0〉A|0〉c

(c)−→ i |0〉A|1〉c

|2〉A|0〉c(9)

Step 2: For transmon B

(a) Apply a pulse ω = ω21, phase φ = −π/2 and duration t2,a = π/2�12.(b) Wait for time t2,b = π/gB (cavity mode resonantly interacting with the |2〉↔ |3〉 transition of

transmon B).(c) Apply a pulse ω = ω21, phase φ = π/2 and duration t2,c = π/2�12.

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The result of this step is expressed as follows:

|0〉B|0〉c

|1〉B|0〉c

|0〉B|1〉c

|1〉B|1〉c

(a)−→|0〉B|0〉c

|2〉B|0〉c

|0〉B|1〉c

|2〉B|1〉c

(b)−→|0〉B|0〉c

|2〉B|0〉c

|0〉B|1〉c

−|2〉B|1〉c

(c)−→|0〉B|0〉c

|1〉B|0〉c

|0〉B|1〉c

−|1〉B|1〉c

(10)

Step 3: For transmon A

(a) Apply a pulse ω = ω20, phase φ = π/2 and duration t3,a = π/2�02.(b) Wait for time t3,b = π/2gA (cavity mode resonantly interacting with the |2〉 ↔ |3〉 transition

of transmon C).(c) Apply a pulse ω = ω31, phase φ = −π/2 and duration t3,c = π/2�13.

Mathematically, the effect of this step is expressed as follows:

|0〉A|1〉c

|2〉A|0〉c

(a)−→ −|2〉A|1〉c

|0〉A|0〉c

(b)−→ i |3〉A|1〉c

|0〉A|0〉c

(c)−→ i |1〉A|1〉c

|0〉A|0〉c(11)

Based on the obtained results expressed by Eqs. (9)–(11), we can find the following operations onthe states of the two-qubit CPHASE gate:

|00〉|0〉c

|01〉|0〉c

|10〉|0〉c

|11〉|0〉c

Step 1−−−→|20〉|0〉c

|21〉|0〉c

i |00〉|1〉c

i |01〉|1〉c

Step 2−−−→|20〉|0〉c

|21〉|0〉c

i |00〉|1〉c

−i |01〉|1〉c

Step 3−−−→|00〉|0〉c

|01〉|0〉c

|10〉|0〉c

−|11〉|0〉c

(12)

3.3. Realization of TOFFOLI-gate

The TOFFOLI-gate operation is shown in Fig. 1b. This operation consists of the operations in thesequence: H transformation on target bit → three-qubit CPHASE gate → H transformation on targetbit. Out of these operations we will now describe the operation of the three-qubit CPHASE gate. Thethree-qubit CPHASE operation consists of five steps [31]. These steps are as follows.Step 1: For transmon A

(a) Apply a pulse ω = ω31, phase φ = −π/2 and duration t1,x = π/2�13.(b) Wait for time t1,y = π/2gA (cavity mode resonantly interacting with the |2〉 ↔ |3〉 transition

of transmon A).(c) Apply a pulse ω = ω20, phase φ = −π/2 and duration t1,z = π/2�02.

Step 2: For transmon B

(a) Apply a pulse ω = ω20, phase φ = −π/2 and duration t2,x = π/2�02.(b) Wait for time t2,y = π/2gB (cavity mode resonantly interacting with the |2〉↔ |3〉 transition

of transmon B).(c) Apply a pulse ω = ω30, phase φ = π and duration t2,z = π/2�03.

Step 3: For transmon C

(a) Apply a pulse ω = ω21, phase φ = −π/2 and duration t3,x = π/2�12.(b) Wait for time t3,y = π/gC (cavity mode resonantly interacting with the |2〉↔ |3〉 transition of

transmon C).(c) Apply a pulse ω = ω21, phase φ = π/2 and duration t3,z = π/2�12.

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Step 4: For transmon B

(a) Apply a pulse ω = ω30, phase φ = π and duration t4,x = π/2�03.(b) Wait for time t4,y = π/2gB (cavity mode resonantly interacting with the |2〉↔ |3〉 transition

of transmon B).(c) Apply a pulse ω = ω20, phase φ = π/2 and duration t4,z = π/2�02.

Step 5: For transmon A

(a) Apply a pulse ω = ω20, phase φ = π/2 and duration t5,x = π/2�02.(b) Wait for time t5,y = π/2gA (cavity mode resonantly interacting with the |2〉 ↔ |3〉 transition

of transmon A).(c) Apply a pulse ω = ω31, phase φ = −π/2 and duration t5,z = π/2�13.

The effect of above five operational steps on the three transmons can be expressed as:

|000〉|0〉c

|001〉|0〉c

|010〉|0〉c

|011〉|0〉c

|100〉|0〉c

|101〉|0〉c

|110〉|0〉c

|111〉|0〉c

Step 1−−−→

|200〉|0〉c

|201〉|0〉c

|210〉|0〉c

|211〉|0〉c

i |000〉|1〉c

i |001〉|1〉c

i |010〉|1〉c

i |011〉|1〉c

Step 2−−−→

|220〉|0〉c

|221〉|0〉c

|210〉|0〉c

|211〉|0〉c

i |000〉|0〉c

i |001〉|0〉c

i |010〉|1〉c

i |011〉|1〉c

Step 3−−−→

|220〉|0〉c

|221〉|0〉c

|210〉|0〉c

|211〉|0〉c

i |000〉|0〉c

i |001〉|0〉c

i |010〉|1〉c

−i |011〉|1〉c

Step 4−−−→

|200〉|0〉c

|201〉|0〉c

|210〉|0〉c

|211〉|0〉c

i |000〉|1〉c

i |001〉|1〉c

i |010〉|1〉c

−i |011〉|1〉c

Step 5−−−→

|000〉|0〉c

|001〉|0〉c

|010〉|0〉c

|011〉|0〉c

|100〉|0〉c

|101〉|0〉c

|110〉|0〉c

|111〉|0〉c

(13)

Interaction between the energy levels when transmons interact with the cavity mode or with thepulses during TOFFOLI-gate and CNOT-gate operation is shown in Fig. 4 (the timing diagram ofthese operations is discussed later). Here, the first two states |0〉 and |1〉 of a transmon are not involvedin the operation and remain unaffected by the cavity mode, no matter whether or not a photon ispopulated in the cavity. In such a case, when the operation is performed on a given transmon, thedecoupling of the other two transmons from the cavity mode cannot be made. This fact in turn pro-vides huge advantages to the scheme, e.g. reduction in operational time, reduction in noise, as wellas simplicity in transmon design and fabrication.

4. Realizing the half-adder

In order to realize the half-adder operation, we would like to recall the sequential operation ofTOFFOLI- and CNOT-gates to add two single qubits (see Fig. 1a for the circuit diagram). Further,the entire half-adder operation is expressed in terms of the Bloch sphere representation and is shownin Fig. 5.

In contrast to all other half-adder schemes where two-level qubits are used, our scheme uses four-level transmon-based qubits. The sequential operations of transmons by applying different pulses ina time sequence realize different rotations and changes of state, and such changes of state generatevarious operations like Hadamard transform, controlled phase shift, etc. Using the operations in aparticular manner we have obtained the half-adder operation. The various sequential operations for

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Fig. 4. Illustration of transmons interacting with the cavity mode or with the pulses during the TOFFOLI-gate(CPHASE I) and CNOT-gate (CPHASE II) operation. For superconducting qubit systems, the level spacingbetween two certain levels (the levels |2〉 and |3〉 here) can be made to be the same for other qubits in thesystem (by appropriately designing the transmons), such that the transition between these two levels for eachof the other two qubits is resonant with the cavity mode. For any other two energy levels, the level-spacingdifferences for transmon A, B, C are maintained. Such level-spacing parameters arise due to the non-uniformityof the transmon parameters.

the half-adder on all possible input states are given below.

|000〉|0〉c

|001〉|0〉c

|010〉|0〉c

|011〉|0〉c

|100〉|0〉c

|101〉|0〉c

|110〉|0〉c

|111〉|0〉c

Hadamardoperation onthird qubit−−−−−−−→

(|000〉 + |001〉)√2

|0〉c

(|000〉 − |001〉)√2

|0〉c

(|010〉 + |011〉)√2

|0〉c

(|010〉 − |011〉)√2

|0〉c

(|100〉 + |101〉)√2

|0〉c

(|100〉 − |101〉)√2

|0〉c

(|110〉 + |111〉)√2

|0〉c

(|110〉 − |111〉)√2

|0〉c

Three-qubitCP operation−−−−−−−→

(|000〉 + |001〉)√2

|0〉c

(|000〉 − |001〉)√2

|0〉c

(|010〉 + |011〉)√2

|0〉c

(|010〉 − |011〉)√2

|0〉c

(|100〉 + |101〉)√2

|0〉c

(|100〉 − |101〉)√2

|0〉c

(|110〉 − |111〉)√2

|0〉c

(|110〉 + |111〉)√2

|0〉c

Hadamardoperation onthird qubit−−−−−−−→

|000〉|0〉c

|001〉|0〉c

|010〉|0〉c

|011〉|0〉c

|100〉|0〉c

|101〉|0〉c

|111〉|0〉c

|110〉|0〉c

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(a)

(b)

(c)

(d)

(e)

(f)

Fig. 5. The Bloch sphere representation of the half-adder operation. In this figure, (a) and (b), (c) and (d), and(e) and (f) represent the first, second, and third qubits, respectively, and their states during the operations. Theoperational sequences are labeled I to VII. Here, I represents the initial state (which could be either |0〉 or |1〉).IV is the state after TOFFOLI operation. One can see that when the first two qubits are in state |1〉 (representedby (b) and (d) in the figure), then the state of the third qubit (represented by (e) or (f) in the figure) flips, butotherwise remains unaffected. Again, the sequences V to VII represent the CNOT operation between the firsttwo qubits, where one can see that the state of the second qubit (represented by (c) or (d) in the figure) flipsonly when the first qubit state is described by (b).

Hadamardoperation onsecond qubit−−−−−−−→

(|000〉 + |010〉)√2

|0〉c

(|001〉 + |011〉)√2

|0〉c

(|000〉 − |011〉)√2

|0〉c

(|001〉 − |011〉)√2

|0〉c

(|100〉 + |110〉)√2

|0〉c

(|101〉 + |111〉)√2

|0〉c

(|101〉 − |111〉)√2

|0〉c

(|100〉 − |110〉)√2

|0〉c

Two-qubitCP operation

(first two qubits)−−−−−−−−−→

(|000〉 + |010〉)√2

|0〉c

(|001〉 + |011〉)√2

|0〉c

(|000〉 − |011〉)√2

|0〉c

(|001〉 − |011〉)√2

|0〉c

(|100〉 − |110〉)√2

|0〉c

(|101〉 − |111〉)√2

|0〉c

(|101〉 + |111〉)√2

|0〉c

(|100〉 + |110〉)√2

|0〉c

Hadamardoperation onsecond qubit−−−−−−−→

|000〉|0〉c

|001〉|0〉c

|010〉|0〉c

|011〉|0〉c

|110〉|0〉c

|111〉|0〉c

|101〉|0〉c

|100〉|0〉c

(14)

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(a)

(b)

Fig. 6. (a) Schematic diagram of possible experimental design of a cavity quantum half-adder by transmonwhere l1 is the length of the transmon, l2 is the spacing between two adjacent strip-lines, and the length of thecavity resonator is L (=wavelength of the applied field). (b) Transmon and CPB [49].

Four-level qubits speed up the operation, as the number of operational steps reduces in this half-adderoperation, and reduce the operational cost, as the effective number of elementary gates is reduced asshown by the operation above.

5. Scheme for possible experimental realization

In this section, we have briefly described the important aspects for experimental realization of ourscheme. A sketch of the half-adder circuit as a transmon–cavity system is shown in Fig. 6 for exper-imental realization. Details of the realization of transmon-based quantum computing can be foundin Refs. [7,49].

The total operational time required for our proposed half-adder scheme can be given by

τ = TOFFLI operation + CNOT operation

= 2tH +5∑

i=1

ti,x + ti,y + ti,z + 2tH +3∑

j=1

t j,a + t j,b + t j,c. (15)

The time τ should be much shorter than the relaxation time T1 and the dephasing time T2 of level |3〉(note that the levels |1〉 and |2〉 have a longer decoherence time than level |3〉), so that the decoherencedue to spontaneous decay and the dephasing process of the qubit systems is negligible compared tothe duration of operation. Also, the time τ needs to be much shorter than the lifetime of the cavityphoton (which is given by κ−1 = Q/2πνc, where Q is the quality factor of the cavity and νc is thecavity field frequency), so that the decay of the cavity photon can be neglected during the operation.These requirements can be met by designing the transmon system appropriately. For example, one candesign the transmons to have longer energy relaxation time and dephasing time, such that τ � T1, T2;and choose a high-Q cavity such that τ � κ−1. In actual experiment, there are many componentsand sources of relaxation time. Relaxation due to spontaneous emission, the Purcell effect, dielectriclosses, relaxation due to quasiparticle tunneling, relaxation due to flux coupling and coupling tospurious modes, balancing, and other noise sources are all responsible for the relaxation time T1.On the other hand, charge noise, flux noise, critical current noise, dephasing due to quasiparticletunneling and EC noise are all responsible for the dephasing of the transmon qubit. A theoreticalestimation indicates that the condition τ � T1, T2 for level |3〉 will be fully satisfied by the transmonwhen EC/EJ � 1.

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(a)

(b)

(c)

Fig. 7. Timing diagram of the entire half-adder operation. A sequence of pulses of different widths fortransmons A, B, C are required for the proposed operation. CPHASE I: (a) t1,x = π/2�13, t1,y = π/2gA,t1,z = π/2�02, t5,x = π/2�02, t5,y = π/2gA, t5,z = π/2�13 (b) t2,x = π/2�02, t2,y = π/2gB, t2,z = π/2�03,t4,x = π/2�03, t4,y = π/2gB, t4,z = π/2�02 (c) t3,x = π/2�12, t3,y = π/gC, t3,z = π/2�12. CPHASEII: (a) t1,a = π/2�13, t1,b = π/2gA, t1,c = π/2�02, t3,a = π/2�02, t3,b = π/2gA, t3,c = π/2�13 (b)t2,a = π/2�12, t2,b = π/gB, t2,c = π/2�12. Before and after CPHASE operation, tH = π/4�01 pulse appliedon the target qubit for Hadamard operation. In this timing diagram, we have assumed (�13)transmon A =(�13)transmon B = (�13)transmon C, (�02)transmon A = (�02)transmon B = (�02)transmon C, (�03)transmon A =(�03)transmon B = (�03)transmon C, (�12)transmon A = (�12)transmon B = (�12)transmon C, and g1 = g2 = g3.

Let us consider the experimental possibility of realizing our scheme for the three-qubit half-adderusing three identical transmons coupled to a resonator (see Fig. 6a). As a rough estimation, assumeg1 ≈ g2 ≈ g3 = g and all the Rabi frequencies �i j are nearly the same (see Fig. 7). The qubit–resonator coupling strengths g/2π ∼ 80 MHz could be reached for a superconducting qubit system,coupled to a one-dimensional standing-wave coplanar waveguide transmission resonator [49]. Withthe choice of� ∼ 10 g, one obtains τ ∼ 37 ns, which is much shorter than min{T1, T2} [7], and thusthe requirements for T1 and T2 are met. In addition, consider a resonator with frequency νc ∼ 6.7 GHz(e.g. in Ref. [49]) and photon decay rate κ/2π = 6.8 MHz (as in Ref. [49]), which is much longerthan the operation time τ here. The operation can be performed faster, or in other words the time τcan be shorten, by increasing g/2π and �. Considering a worst case scenario (for noise level anddesign parameters), a rough estimation of the operational time τ is made as follows:

τ = 2tH +5∑

i=1

ti,x + ti,y + ti,z + 2tH +3∑

j=1

t j,a + t j,b + t j,c

= π

2�+ 3π

g+ 5π

�+ π

2�+ 2π

g+ 3π

�= 5π

g+ 9π

�= 5π

g+ 9π

10g≈ 3 × 2π

g

= 3 × 1

80 MHz≈ 37 ns.

The timing diagram of the entire half-adder operation along the various time components in the aboveexpression Eq. (15) is shown in Fig. 7.

Figure 7 shows the pulse sequence of the complete half-adder operation by considering the idealpulse sequences to represent the operational scheme schematically. In practice, this type of pulse

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creates noises in the third and fourth level operation of the transmon. For actual experimental imple-mentation of this half-adder, a sequence of microwave and flux pulses are applied to the qubit.Gaussian-shaped DRAG-pulses [50,51] should be used to prevent population of the third and fourthlevels and to reduce phase errors during the single-qubit operations.

It is to be noted here that the higher transmon levels may suffer from stronger dissipation, whichin turn may reduce the fidelity of this operation; however, application of Gaussian-shaped DRAG-pulses may reduce the error drastically. In an experiment, there are other factors, e.g. decoherence,pulse broadening, level shifting by application of microwave pulse for Rabi oscillation, etc., thatwill tend to reduce the overall fidelity of the scheme, and therefore these factors should be controlledwithin the experimental limits; a comparative study on fidelity is beyond the scope of our work. Sucha comparative study could be undertaken with a solid numerical simulation that takes into accountthe full generalized Jaynes–Cummings model.

6. Conclusion

A transmon-based solid state scheme is proposed for quantum half-adder operation and the entireoperation of the scheme is described step by step. Three independent transmons representing threequbits of the half-adder circuit with each qubit associated with four energy levels are consideredin the scheme. With the use of the four levels of the transmon, the states |0〉 and |1〉 of the qubitsystems not involved in the operation are not affected by the cavity mode, no matter whether or nota photon is populated in the cavity. For this reason, our scheme results in faster operation and thespeed of operation can be controlled by changing the transmon parameters. The important parametersfor experimental realization of the proposed scheme are discussed. The advantages of our scheme,in short, are: i) it is a solid state scheme, and is scalable and more compact than the other existinghalf-adder scheme; ii) it is less noisy than the other scheme: the transmon is a very low charge noiseand flux noise device compared to other qubit devices; iii) most importantly, it has a low operationaltime (∼37 ns) compared to the other system due to the four-level operation, which in turn reducesthe operational cost compared to the other systems. Our approach to implementing the half-adderoperation would be useful for more complicated quantum operations, and also shows the importanceof the transmon in quantum computation.

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