1. Introduction

The superpositions of quantum states is at the heart of the basic postulates and theorems of quantum mechanics. Quantum superposition studies the application of quantum theory or phenomena, that is different from the classical world, leads to many other intriguing quantum phenomena such as quantum entanglement [1] and quantum coherence [2, 3]. It is a vital physical resource and has many important applications in quantum information processing (QIP) and quantum computation such as quantum algorithms [4, 5], quantum metrology [6], and quantum cryptography [7].

A quantum adder is a quantum machine adding two arbitrary unknown quantum states of two different systems onto a single system [8, 9]. How to generate a superposition of two arbitrary states has recently aroused great interest in the field of quantum optics and quantum information. For example, Refs. [8, 9] have proved that it is impossible to generate a superposition of two unknown states, but Ref. [9] proposed a method to probabilistically creating the superposition of two known pure states with the fixed overlaps. Reference [10] has shown that superpositions of orthogonal qubit states can be produced with unit probability, and Ref. [11] has demonstrated that the state transfer can be protected via an approximate quantum adder. Recently, the probabilistic creation of superposition of two unknown quantum states has been demonstrated experimentally in linear optics [12] and nuclear magnetic resonance (NMR) [13].

Circuit quantum electrodynamics (QED) consisting of superconducting qubits and microwave cavities are now moving toward multiple superconducting qubits, multiple three-dimensional (3D) cavities with greatly enhanced coherence time, making them particularly appealing for large-scale quantum computing [14–16]. For example, a 3D microwave cavity with the photon lifetime up to 2 s [17] and a transmon with a coherence time $\backsim$ 0.1 ms [18] have been recently reported in 3D circuit QED. Hence, 3D cavities are good memory elements, which can have coherence time at least four orders of magnitude longer than the transmons. By encoding quantum information in microwave cavities, many schemes have been proposed for synthesizing Bell states [19], NOON states [20–26], and entangled coherent states [27–29] of multiple cavities, implementing multi-qubit gates in a mult-cavity system [30–32], and realizing cross-Kerr nonlinearity interaction between two cavities [33, 34].

Three-dimensional circuit QED has emerged as a well-established platform for QIP and quantum computation [35–40], including creation of a Schrödinger cat state of a microwave cavity [36], preparation and control of a five-level transmon qudit [37], demonstration of a quantum error correction [38], realization of a two-mode cat state of two microwave cavities [39], and implementation of a controlled-NOT gate between multiphoton qubits encoded in two cavities [40]. Considering these advancements in 3D circuit QED, it is quite meaningful and necessary to implement a quantum adder in such systems.

In this paper, we present a probabilistic scheme to realize a quantum adder that creates a superposition of two unknown states by using a superconducting transmon qubit dispersively coupled to two 3D cavities. This circuit architecture has been experimentally demonstrated recently in [39]. Our protocol has the following features and advantages: (i) The superposition of two states are encoded in two three-dimensional cavities which have long coherence time rather than encoded in qubits. (ii) The states of cavities can be in arbitrary states, e.g., discrete-variable states or continuous-variable states. (iii) Our proposal can also be applied to other physical systems such as quantum dot-cavity system [41], natural atom-cavity system [42], superconducting circuits with other types of superconducting qubits (e.g., phase qubit [43], Xmon qubit [44], flux qubit [45]), hybrid circuits [46] for two nitrogen-vacancy center ensembles coupled to a single flux qubit [47].

The remaining of this paper is organized as follows. In Sec. 2, we review some basic theory of quantum adder that creates the superposition of two arbitrary states. Our experimental system and Hamiltonian are introduced in Sec. 3. In Sec. 4, we show the method to implementing a quantum adder in 3D circuit QED system. In Sec. 5, we give a brief discussion on the experimental implementation of a quantum adder with state-of-the-art circuit QED technology. Finally, Sec. 6 gives a brief concluding summary.

 

Fig. 1 A diagram of the quantum adder. Here, ${|\psi 〉}_A$ and ${|\phi 〉}_B$ are the two arbitrary input pure states, while ${|\psi 〉}_A+{|\phi 〉}_B$ is the out superposition state with the referential state $|\chi 〉$.

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2. Basic theory of quantum adder

We now review some basic theory of quantum adder (Fig. 1) which can generate the superposition of two pure states. Assume that particles A and B are respectively initially in arbitrary pure states ${|\psi 〉}_A$ and ${|\phi 〉}_B$, and an ancilla particle T is prepared in an arbitrary superposition state $\left|\varphi {〉}_T={\alpha }_T\right|0{〉}_T+{\beta }_T|1{〉}_T$ with normalized coefficients α_T and β_T. We first implement a three-qubit controlled-SWAP gate such that the initial state $\left|\psi {〉}_A\right|\phi {〉}_B|\varphi {〉}_T$ of three qubits evolves into

Then we make projective measurements on the states $|\pm {〉}_T$ and $|\chi {〉}_B$ of qubits T and A, respectively. Here, $|\pm {〉}_T=\left(\left|0{〉}_T\pm \right|1{〉}_T\right)/\sqrt{2}$ and $|\chi {〉}_B$ is the referential state which satisfies $〈\chi |\psi {〉}_B\ne 0$ and $〈\chi |\phi {〉}_B\ne 0$. Accordingly, one obtains the following superposition state of qubit A

 

Fig. 2 (a) Schematic of a single transmon qubit dispersively coupled to two three-dimensional microwave cavities A and B. (b) Schematic diagram of transmon-cavity interaction. Cavity j is far-off resonant with the $|g**\leftrightarrow |e**$ ($|e**\leftrightarrow |f**$) transition of transmon qubit with coupling strength g_j ($\sqrt{2}{g}_j$) and detuning Δ_j (δ_j). Here, ${\mathrm{\Delta }}_j={\omega }_{eg}-{\omega }_j$ and ${\delta }_j={\omega }_{fe}-{\omega }_j$ ($j=A,B$).

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3. System and Hamiltonian

Superconducting qubits based on Josephson junctions are mesoscopic element circuits like artificial atoms, with multiple discrete energy levels whose spacings can be rapidly adjusted by varying external control parameters [44, 48–50]. Typically, a transmon [51] has weakly anharmonic multilevel structure and the transition between non-adjacent levels is forbidden or very weak.

Motivated by the experimental advances in 3D circuit QED, we here consider a circuit system consisting of a transmon qutrit (with three states $|g〉$, $|e〉$ and $|f〉$) capacitively coupled to two separate 3D superconducting cavities as shown in Fig. 2. The Hamiltonian to describe the microwave cavities coupled to the transmon reads $H=H_0+H_I$. The free Hamiltonian H₀ is given by

The coupling Hamiltonian H_I of the whole system is given by

To proceed, let’s turn to the the interaction picture with respect to $H_0={\omega }_{eg}\left|e〉〈e\right|+\left({\omega }_{eg}+{\omega }_{fe}\right)\left|f〉〈f\right|+{\omega }_A{a}^{†}a+{\omega }_B{b}^{†}b$. Then the Hamiltonian can be given by

Applying the large-detuning conditions $\left|{\mathrm{\Delta }}_A\right|\gg g_A$ and $\left|{\mathrm{\Delta }}_B\right|\gg g_B$ (i.e., $\left|{\delta }_A\right|\gg \sqrt{2}{g}_A$ and $\left|{\delta }_B\right|\gg \sqrt{2}{g}_B$), the Hamiltonian (5) changes to [52]

4. Experimental implementation of a quantum adder in 3D circuit QED

In this section we will show how to prepare a superposition state for arbitrary input states in 3D circuit QED with the above Hamiltonian with the weak anharmonicity. Let $\left|\psi {〉}_A={\displaystyle \sum }_{n=0}^{\infty }{c}_n\right|n{〉}_A$ and $\left|\phi {〉}_B={\displaystyle \sum }_{m=0}^{\infty }{d}_m\right|m{〉}_B$ be two arbitrary input pure states of microwave cavities A and B, respectively. Here, the Fock states of the cavities are expressed as $\left|n{〉}_A=\frac{{(a^{†})}^n}{\sqrt{n!}}\right|0{〉}_A$ and $\left|m{〉}_B=\frac{{(b^{†})}^m}{\sqrt{m!}}\right|0{〉}_B$. We assume that the level $|f〉$ of transmon is not occupied, thus the initial state of the transmon is

Accordingly, the effective Hamiltonian Eq. (7) reduces to

According to the Hamiltonian (9), the initial state $\left|\varphi {〉}_T\right|\psi {〉}_A|\phi {〉}_B$ of the total system at time t becomes

By solving the Heisenberg equations for $H_I^g$ and $H_I^e$, the dynamics of the operators $a^{†}$ and $b^{†}$ can be derived as

4.1. $|\mathrm{g}〉$ as the control state

By waiting for the evolution time $t=\pm \left(\frac{\pi }{2}+2k_1\pi \right)/\chi$, one has $a^{†}\left(t\right)=i{b}^{†}$ and $b^{†}\left(t\right)=i{a}^{†}$. That corresponds to for the level $|g〉$, an exchange of quantum states between two microwave cavities except for a phase shift $\pi /2$ conditioned on the photon number in cavities. When the condition $\mathrm{\Lambda }t=\mp \left(2\pi +2k_2\pi \right)$ is satisfied, we also have $a^{†}\left(t\right)=a^{†}$ and $b^{†}\left(t\right)=b^{†}$ for the level $|e〉$. Here, the sign $``+"$ or $``-"$ of the expression t depends on the detuning $\mathrm{\Delta }>0$ or $\mathrm{\Delta }<0$, and k₁ and k₂ are non-negative integers. Accordingly, we can obtain the following relationship between the Δ and anharmonicity α

When the conditions $\left|\chi \right|t=\left(\frac{\pi }{2}+2k_1\pi \right)$ and $\left|\mathrm{\Lambda }\right|t=2\pi +2k_2\pi$ are satisfied, the Eq. (10) changes to (see Eq. (26) of Appendix A)

We perform a $R_{\pi /2}^{ge}$ rotation on the transmon qubit that realizes the conversions $\left|g〉\to \right|+〉$ and $\left|e〉\to \right|-〉$ with $|+〉=\left(\left|e〉+\right|g〉\right)/\sqrt{2}$ and $|-〉=\left(\left|e〉-\right|g〉\right)/\sqrt{2}.$ Now we perform a projective measurement onto the state $|e〉$ or $|g〉$ of transmon qubit, the state (14) becomes

Then we perform another measurement on the cavity B in the referential state $|\chi {〉}_B$ that satisfies $〈\chi |\psi {〉}_B\ne 0$ and $〈\chi |\phi {〉}_B\ne 0$. Thus, one can obtain the following superposition state of cavity A

4.2. $|\mathrm{e}〉$ as the control state

By waiting for the evolution time $t=\mp \left(\frac{\pi }{2}+2k_1\pi \right)/\mathrm{\Lambda }$, one has $a^{†}\left(t\right)=-i{b}^{†}$ and $b^{†}\left(t\right)=-i{a}^{†}$. That corresponds to for the level $|e〉$, an exchange of quantum states between two microwave cavities except for a phase shift $3\pi /2$ conditioned on the photon number in cavities. Here the sign $``-"$ of the expression t corresponds to $\mathrm{\Delta }>0$ while the sign $``+"$ corresponds to $\mathrm{\Delta }<0$. When the condition $\chi t=\pm \left(2\pi +2k_2\pi \right)$ is satisfied, we also have $a^{†}\left(t\right)=a^{†}$ and $b^{†}\left(t\right)=b^{†}$ for the level $|g〉$. Accordingly, one has the following relationship between the detuning Δ

and anharmonicity α

When the conditions $\left|\mathrm{\Lambda }\right|t=\left(\frac{\pi }{2}+2k_1\pi \right)$ and $\left|\chi \right|t=2\pi +2k_2\pi$ are satisfied, the Eq. (10) changes to (see Eq. (28) of Appendix A)

We perform a $R_{\pi /2}^{ge}$ rotation on the transmon qubit that realizes the conversions $\left|g〉\to \right|+〉$ and $\left|e〉\to \right|-〉$ with $|+〉=\left(\left|e〉+\right|g〉\right)/\sqrt{2}$ and $|-〉=\left(\left|e〉-\right|g〉\right)/\sqrt{2}.$ Now we perform a projective measurement onto the state $|e〉$ or $|g〉$ of transmon, the state (18) becomes

Then we perform another measurement on the cavity B in the referential state $|\chi {〉}_B$ that satisfies $〈\chi |\psi {〉}_B\ne 0$ and $〈\chi |\phi {〉}_B\ne 0$. Thus, one can obtain the following superposition state of cavity A

Equation (16) or (20) shows that a superposition of two states is prepared in the cavity-transmon system. In order to maintain the prepared state, the level spacings of the transmon need to be rapidly (within 1-3 ns [44, 48–50]) adjusted so that the transmon is decoupled from cavities A and B after the desired superposition state is generated. Alternatively, to have the cavities coupled or decoupled from the transmon, one also can tune the frequencies of cavities that because the rapid (with a few nanoseconds) tuning of cavity frequency has been reported in experiments [53–55].

It should be mentioned that when the level $|e〉$ of transmon is not occupied, i.e., $|\varphi {〉}_T=\sin \theta \left|g〉+\cos \theta \right|f〉$, the above superposition state (16) or (20) can also be prepared for $|g〉$ or $|f〉$ as the control state.

5. Possible experimental implementation

Recent experimental results for the 3D circuit system demonstrate the great promise of quantum computation and QIP. For an experimental implementation, our setup of two superconducting 3D cavities coupled to a transmon has been demonstrated recently by [39].

Taking into account the effect of transmon weak anharmonicity, the dissipation and the dephasing, the dynamics of the lossy system is governed by the Markovian master equation

The generation efficiency can be evaluated by fidelity $\mathcal{F}=\sqrt{@@{\psi }_{\text{id}}|\rho |{\psi }_{\text{id}}**}$, where $|{\psi }_{\text{id}}**$ is the ideal target state. It is obvious that how to realize the controlled-SWAP gate is the key of our proposal. So we mainly consider the impurity introduced in this process. In fact, the initial state preparation and measurement of transmon and cavities can be relatively accurately performed [39, 40, 56, 57], thus it is also reasonable for us not to consider the initial state preparation, the rotations and measurement impurites on the fidelity. Accordingly, the ideal target state is given by Eq. (14) or Eq. (18) for the case of (i) the state $|g〉$ acts as a control state or (ii) the state $|e〉$ acts as a control state. The corresponding parameters used are: (i) $k_1=1,k_2=0,$ i.e., $\mathrm{\Delta }=-9\alpha =-1.035$ GHz, and (ii) $k_1=1,k_2=0,$ i.e., $\mathrm{\Delta }=-9\alpha =-1.035$ GHz, respectively. In addition, the input state of the transmon-cavity system is $\left|\varphi {〉}_T\right|\psi {〉}_A|\phi {〉}_B$. The initial state of cavities $|\psi {〉}_A$ and $|\phi {〉}_B$ can be arbitrary states such as discrete-variable states or continuous-variable states. In the following, we choose the cavities are initially in the coherent states, i.e., $\left|\psi {〉}_A=\right|\alpha {〉}_A$ and $\left|\phi {〉}_B=\right|-\beta {〉}_B$ with $\alpha =\beta =0.1$. Thus, the input state of the transmon-cavity system is $\left(\sin \theta \left|g〉+\cos \theta \right|e〉\right)\left|\alpha {〉}_A\right|-\beta {〉}_B$.

We numerically simulate the fidelity of the operation by solving the master Eq. (21). Since the transmon qubit relaxation time $T_1=75\mu s$ and the transmon qubit dephasing time $T_2=45\mu s$ have been achieved in similar 3D circuit system [39], we set ${\gamma }_{\phi e}^{-1}=15\mu$s, ${\gamma }_{\phi f}^{-1}=10\mu$s, ${\gamma }_{eg}^{-1}=50\mu$s, ${\gamma }_{fe}^{-1}=25\mu$s, ${\gamma }_{fg}^{-1}=100\mu$s, In addition, we set ${\kappa }_A^{-1}=15\mu$s, ${\kappa }_B^{-1}=10\mu$s [39]. For cavities with frequency ${\omega }_c=5.4$ GHz [39] and dissipation times ${\kappa }_A^{-1}$ and ${\kappa }_B^{-1}$ used in the numerical simulation, the quality factors of the cavity A and B are $Q_A\sim 5.4\times {10}^5$ and $Q_B\sim 8.1\times {10}^5$. Note that three-dimensional cavities with a loaded quality factor $Q>{10}^6$ have been reported in experiments [18, 35]. According to Ref. [39], we choose the transmon anharmonicity $\alpha ={\omega }_{eg}-{\omega }_{fe}=115$ MHz in our numerical simulation. In the following, we choose the dispersive shift χ = 1 MHz that because this value of χ is readily available in experiments [39]. Accordingly, one can obtain the coupling strength $g\backsim$32 MHz (easy available in experiments [18, 35]).

 

Fig. 3 Fidelity $\mathcal{F}$ versus θ and by taking the unwanted inter-cavity crosstalk into account for g_AB =0, 0.01g, 0.1g. The parameters used in the numerical simulation are referred in the text.

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Figures 3(a) and 3(b) display the effect of the unwanted inter-cavity crosstalk on the fidelity, respectively. The effect of the inter-cavity crosstalk can be taken into account by adding a Hamiltonian of the form $H_{AB}=g_{AB}\left(a^{†}b+a{b}^{†}\right)$ in H_I, where g_AB is the inter-cavity crosstalk coupling strength. Figures 3(a) and 3(b) show fidelity $\mathcal{F}$ versus θ with $\theta \in \left[0,2\pi \right]$ for $g_{AB}=0,0.01g,0.1g$, which consider the case of (i) the state $|g〉$ acts as a control state and (ii) the state $|e〉$ acts as a control state, respectively. From the Figs. 3(a) and 3(b) we can see that the effect of the crosstalk on the fidelity is negligibly small for $g_{AB}=0.01g,0.1g$. Moreover, we calculates the average fidelities are approximately (i) $98.13\%$, $97.82\%$, and $98.03\%$, (ii) $97.88\%$, $97.79\%$, and $97.77\%$ for $g_{AB}=0,0.01g,0.1g$, respectively. In the following, we choose the inter-cavity crosstalk coupling strength $g_{AB}=0.1g$. This coupling strength condition is easily satisfied by the present circuit QED technology [32, 58].

 

Fig. 4 Fidelity $\mathcal{F}$ versus c and θ by taking into account the inhomogeneity in transmon-cavity interaction. Here g_A = g and g_B = cg with c ∈ [0.95, 1.05].

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Figures 4(a) and 4(b) take into account the inhomogeneity in transmon-cavity interaction for the case of (i) the state $|g〉$ acts as a control state and (ii) the state $|e〉$ acts as a control state, respectively. Figure 4 displays the fidelity versus c and θ, where we set $g_A=g$ and $g_B=cg$ with $c\in \left[0.95,1.05\right]$. Figure 4(a) shows that for $0.98\le c<1$ and $1<c\le 1.02$, the effect of the inhomogeneity on the fidelity is very small. Moreover, one can see that for $0.97\le c\le 1.03$, the fidelity can be greater than 95.61%. Figure 4(b) displays that the fidelity is almost unaffected by the inhomogeneity for $0.95\le c\le 1.05$.

 

Fig. 5 Fidelity $\mathcal{F}$ versus d for θ = π/4 by considering the unequal cavity-transmon frequency detuning.

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In a realistic situation, it may be a challenge to obtain identical cavity-transmon frequency detuning. Thus, we consider the the effect of the unequal detuning on the fidelity in Figs. 5(a) and 5(b) for the case of (i) the state $|g〉$ acts as a control state and (ii) the state $|e〉$ acts as a control state, respectively. We set ${\mathrm{\Delta }}_A=\left(1+d\right){\mathrm{\Delta }}_B$ with $d\in \left[-0.05,0.05\right]$. Here, we choose $\theta =\pi /4$. One can find that the fidelities in Fig. 5(a) and Fig. 5(b) are, respectively greater than 94.18% and 95.83%, and the optimal fidelities can reach 97.98% and 97.68%.

 

Fig. 6 Fidelity $\mathcal{F}$ versus ϵ for θ = π/4 by considering the effect of the operation time errors.

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The operational time is estimated as$\backsim 1.25\mu$s or $1\mu$s for the case of (i) or (ii). To investigate the effect of the operation time errors on the fidelity, we consider a small operation time error $ϵt$ for the case of (i) and (ii) in Fig. 6(a) and 6(b). Here, t is the optimal operation time and $\left(ϵ+1\right)t$ ($ϵ\in \left[-0.05,0.05\right]$) is the actual operation time. Figure 6(a) or 6(b) shows that the effect of the operation time error on the fidelity is small for $-0.015\le ϵ\le 0.02$ and $-0.015\le ϵ\le 0.025$.

 

Fig. 7 Fidelity $\mathcal{F}$ versus f for θ = π/4 by considering the effect of the anharmonicity errors.

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We also consider the effect of the anharmonicity α or the detuning Δ errors on the fidelity. We consider a small deviation $f\alpha$ for the case of (i) and (ii). Accordingly, $\mathrm{\Delta }=-9\left(1+f\right)\alpha$ with $f\in \left[-0.05,0.05\right]$. Figures 7(a) and 7(b) indicate that the fidelity is negligibly affected by the anharmonicity or the detuning errors for $f\in \left[-0.05,0.05\right]$.

The simulations above show that the quantum adder that generates a superposition of two unknown states with high fidelity can be achieved for small errors in the undesired the weak anharmonicity, inter-cavity crosstalk, the inhomogeneity and detuning, etc.

6. Conclusion

We have devised a quantum adder based on 3D circuit QED that is a good candidate for quantum information processing, computation and simulation. The 3D microwave cavities dispersively interacting with the transmon to effectively create a superposition state of two arbitrary states encoded in two cavities. The initial state of each cavity is arbitrarily that selected as the discrete-variable state or the continuous-variable state. Our proposal also can be applied to other types of qubits such as natural atoms [42] and artificial atoms (other superconducting qubits (e.g., phase qubits [43], Xmon qubits [44], flux qubit [45]), NV centers [47], and quantum dots [41]). The Numerical simulations imply that the high-fidelity generation of a superposition state of two cavities is feasible with the current circuit QED technology. Finally, our proposal provides a way for realizing a quantum adder in a 3D circuit QED system, and such quantum machine may find other applications in quantum information processing and computation. We hope this work would stimulate experimental activities in the near future.