Transferring arbitrary d-dimensional quantum states of a superconducting transmon qudit in circuit QED

A qudit (d-level quantum system) has a large Hilbert space and thus can be used to achieve many quantum information and communication tasks. Here, we propose a method to transfer arbitrary d-dimensional quantum states (known or unknown) between two superconducting transmon qudits coupled to a single cavity. The state transfer can be performed by employing resonant interactions only. In addition, quantum states can be deterministically transferred without measurement. Numerical simulations show that high-fidelity transfer of quantum states between two superconducting transmon qudits (d ≤ 5) is feasible with current circuit QED technology. This proposal is quite general and can be applied to accomplish the same task with natural or artificial atoms of a ladder-type level structure coupled to a cavity or resonator.


Results
Quantum state transfer between two superconducting transmon qudits. Our system, shown in Fig. 1, consists of two superconducting transmon qudits 1 and 2 embedded in a 3D microwave cavity or coupled to a 1D resonator. In reality, the d involved in QIP may not be a large number. Thus, as an example, we will explicitly show how to transfer quantum states between two transmon qudits for d ≤ 5. We then give a brief discussion on how to extend the method to transfer arbitrary d-dimensional quantum states between two d-level transmon qudits for any positive integer d.
A transmon qudit has a ladder-type level structure 63 . We here label the d levels as ... − d 0 , 1 , 2 , and 1 ( Fig. 2 for d = 5). For a ladder-type level structure, the transition between adjacent levels is allowed but the transition between non-adjacent levels is forbidden or very weak. In the following, the transition frequency between two adjacent levels − l l 1 and of each qudit is labeled as ω (l − 1)l (l = 1, 2, …, d − 1). The initial phase, duration, and frequency of the pulses are denoted as {φ, t, ω}. For simplicity, we set the same Rabi frequency Ω for each pulse, which can be readily achieved by adjusting the pulse intensity. Here and below, qudit (qudits) refers to transmon qudit (qudits).
Case for d = 5. The five levels of qudits are labeled as 0 , 1 , 2 , 3 , and 4 (Fig. 2). Assume that qudit 1 is initially in an arbitrary quantum state ∑ = c l l l 0 4 1 (known or unknown) with level populations illustrated in Fig. 2(a), qudit 2 is initially in the ground state |0〉 2 , and the cavity is initially in the vacuum state |0〉 c . Here and below, c l is a normalized coefficient.
To begin with, the level spacings of the qudits need to be adjusted to have the cavity resonant with the |0〉 ↔ |1〉 transition of each qudit. The procedure for implementing the QST from qudit 1 to qudit 2 is described as follows: Step I. Let the cavity resonant with the |0〉 ↔ |1〉 transition of each qudit described by Hamiltonian (14) (see Section Methods below). According to Eq. (15), after an interaction time t 1 = π/ g ( 2 ), one has the state transformation Equation (2) shows that the population of the level |1〉 of qudit 1 is transferred onto the level |1〉 of qudit 2 [ Fig. 2(b)]. Step II. Apply a pulse of {π/2,π/2Ω,ω 12 } to qudit 1 while a pulse of {−π/2, π/2Ω, ω 12 } to qudit 2 [ Fig. 2  The color circles indicate the occupied energy levels. Each green arrow represents a classical pulse, which is resonant with the transition between the two neighbor levels close to each green arrow. In (e) and (g), the sequence for applying the pulses is from top to bottom, and the lower pulses are turned on after the upper pulses are switched off. In (i), the sequence for applying the pulses is from bottom to top, and the upper pulses are turned on after the lower pulses are switched off. For the details on the applied pulses, see the descriptions given in the text. Note that in (a-j), the left levels are for qudit 1 while the right levels are for qudit 2. For simplicity, we here consider the case that the spacings between adjacent levels become narrow as the levels move up, which is actually unnecessary.
Possible experimental implementation. For an experimental implementation, let us now consider a setup of two superconducting transmon qudits embedded in a 3D cavity. This architecture is feasible in the state-of-the-art superconducting setup as demonstrated recently in ref. 11. For simplicity, we consider QST between the two transmon qudits 1 and 2 for d ≤ 5. As an example, suppose that the state of qudit 1 to be transferred is: , 1 1 where ε 1 describes the unwanted off-resonant coupling between the cavity and the ↔ 1 2 transition of each qudit, which is given by ε σ = ∑ + . . where  g j is the coupling constant between the cavity and the |1〉 ↔ |2〉 transition of qudit j (j = 1, 2), ω c is the frequency of the cavity and σ =| 〉 〈 |.
where ε l describes the unwanted off-resonant couplings of the pulse with the − ↔ − l l 2 1 and ↔ + l l 1 transitions of each qudit, during the pulse resonant with the − ↔ l l 1 transition of each qudit [i.e., the pulse frequency is equal to ω (l − 1)l ]. Here, ε l is given by 63 . Note that the effect of the cavity-qudit interaction during the pulse application is also considered here, which is described by the ′ .
H I ,1 For a transmon qudit, the transition between non-adjacent levels is forbidden or very weak 63 . Thus, the couplings of the cavity/pulses with the transitions between non-adjacent levels can be neglected. In addition, the spacings between adjacent levels for a transmon qudit become narrow as the levels move up (Fig. 2). Therefore, the detunings between the cavity frequency and the transition frequencies for adjacent levels (e.g., levels |1〉 and |2〉, levels |2〉 and |3〉, levels |3〉 and |4〉, etc.) increase when the levels go up. As a result, when compared with the coupling effect of the cavity with the ↔ 1 2 transition, the coupling effect of the cavity with the transitions for other adjacent levels is negligibly small, which is thus not considered in the numerical simulation for simplicity. For similar reasons, when the pulse is resonant with the − ↔ l l 1 transition of each qudit, the coupling effect of the pulses with the transitions between other adjacent levels is weak and thus we only consider the effect of the coupling of the pulse with the two adjacent − ↔ − l l 2 1 and ↔ + l l 1 transitions. When the dissipation and dephasing are included, the dynamics of the lossy system is determined by the following master equation [ ] Here, κ is the photon decay rate of the cavity. In addition, γ − l l ( 1) , j is the energy relaxation rate of the level |l〉 for the decay path → − l l 1 and γ ϕl , j is the dephasing rate of the level |l〉 of qudit j (j = 1, 2).
The fidelity of the operation is given by  ψ ρ ψ = id id , where ψ id is the output state of an ideal system (i.e., without dissipation and dephasing considered), which is given by: The decoherence times of transmon qudits considered here are realistic because they are from the recent experimental report in ref. 11. In a realistic situation, it may be a challenge to obtain exact identical qudit-resonator couplings. Therefore, we consider inhomogeneous qudit-resonator couplings, e.g., g 1 = g and g 2 = 0.95 g.
We numerically calculate the fidelity of the entire operation based on the master equation. Figure 3(a,b,c) shows the fidelity versus g/2π and Ω/2π for QST between two qudits for d = 3, d = 4, and d = 5, respectively. From Fig. 3(a), one can see that for g/2π ∈ [2,8] MHz and Ω/2π ∈ [12,14] MHz, the fidelity can be greater than 98.8% for d = 3. When g/2π = 5.4 MHz and Ω/2π = 12.8 MHz, the fidelity value is the optimum with a value of ~99.6% for d = 3. As shown in Fig. 3(b), the value of the fidelity has a slow decline for d = 4. In Fig. 3(b) the optimal value for ~96.96% is obtained for g/2π = = 1.35 MHz and for Ω/2π = 17.00 MHz. While  drastically decreases for d = 5, a high fidelity ~90.32% is attainable with g/2π = 1.45 MHz and Ω/2π = 16.00 MHz [see Fig. 3(c)]. Note that the above values of the g and Ω are readily available in experiments [64][65][66][67] .
To investigate the effect of the pulse errors on the fidelity of the QST, we consider a small frequency error Aω, a small phase error Bφ, and a small duration error Ct of each pulse. The frequency, initial phase, and duration {ω, φ, t} of the pulses are thus modified as {ω + Aω, φ + Bφ, t + Ct}, where the ω, φ, and t can be found for each of the pulses applied during the QST, as described in Section Results. With this modification, we numerically calculate the fidelity and plot Fig. 4, which shows how the fidelity of the QST varies with parameters A, B, and C. The values of g and Ω used in Fig. 4 are the ones just mentioned above, corresponding to the optimum fidelities in Fig. 3 for d = 3, d = 4, and d = 5, respectively. Other parameters used in the numerical simulation for Fig. 4 are the same as those used in Fig. 3. Figure 4(a) shows that the effect of the pulse frequency error on the fidelity is negligibly small for A ∈ [−10 −4 , 10 −4 ], which corresponds to the pulse frequency error Aω ∈ [−10 −4 ω, 10 −4 ω]. Figure 4(b) shows that for d = 3 and d = 4, the fidelity is almost unaffected by the pulse phase error for B ∈ [−2 × 10 −2 , 2 × 10 −2 ]; and for d = 5 the fidelity has a small decrease for B ∈ [−5 × 10 −2 , 2 × 10 −2 ]. Figure 4(c) shows that the effect of the pulse duration error on the fidelity is negligible for C ∈ [−2 × 10 −2 , 2 × 10 −2 ] for d = 3, d = 4 and C ∈ [−5 × 10 −2 , 2 × 10 −2 ] for d = 5. These results indicate that compared to the phase error and the duration error, the fidelity is more sensitive to the pulse frequency error. From Fig. 4, one can see that the QST with high fidelity can be achieved for small errors in pulse frequency, phase, and duration. For a cavity with frequency ω c /2π = 4.97 GHz and dissipation time κ −1 used in the numerical simulation, the quality factor of the cavity is ∼ . × Q 4 7 10 5 . Note that three-dimensional cavities with a loaded quality factor Q > 10 6 have been implemented in experiments 64, 68 .

Discussion
We have presented a method to deterministically transfer arbitrary d-dimensional quantum states (known or unknown) between two superconducting transmon qudits, which are coupled to a single cavity or resonator. As shown above, only a single cavity or resonator is needed, thus the experimental setup is very simple and the inter-cavity crosstalk is avoided. The state transfer can be performed by employing resonant interactions only. In addition, no measurement is required. Numerical simulation shows that high-fidelity transfer of quantum states between two transmon qudits for (d ≤ 5) is feasible with current circuit-QED technology. This proposal can be extended to transfer an arbitrary d-dimension quantum state between "ladder-type level structure" natural atoms (e.g., Rydberg atoms) or other artificial atoms (e.g., superconducting Xmon qudits, phase qudits, quantum dots), by employing a single cavity only.
The number of pulses can be reduced at a cost of using more than one cavity coupled to the qudits. However, the QST experimental setup will become complicated and the inter-cavity crosstalk is an issue, if two or more cavities are employed instead of a single cavity. Realistic QIP may not involve a large d. To the best of our knowledge, none of experimental works on QIP with qudits of d > 3 has been reported. In this sense, we think that this work is of interest. We hope this work will stimulate experimental activities in the near future.

Methods
Hamiltonian and time evolution. Consider two qudits 1 and 2 coupled by a cavity. The cavity is resonant with the transition between the two levels |0〉 and |1〉 of each qudit. In the interaction picture, the Hamiltonian is given by (in units of ħ = 1)