Perfect Quantum State Transfer in a Superconducting Qubit Chain with Parametrically Tunable Couplings

Faithfully transferring the quantum state is essential for quantum information processing. Here we demonstrate a fast (in 84 ns) and high-fidelity (99.2%) transfer of arbitrary quantum states in a chain of four superconducting qubits with nearest-neighbor coupling. This transfer relies on full control of the effective couplings between neighboring qubits, which is realized only by our parametrically modulating the qubits without increasing circuit complexity. Once the couplings between qubits fulfill a specific ratio, perfect quantum state transfer can be achieved in a single step, and is therefore robust to noise and accumulation of experimental errors. This quantum state transfer can be extended to a larger qubit chain and thus adds a desirable tool for future quantum information processing. The demonstrated flexibility of the coupling tunability is suitable for quantum simulation of many-body physics, which requires different configurations of qubit couplings.

Figure 1

(a) Perfect QST. The quantum state is transferred from the first qubit ${\mathrm{Q}}_0$ to the last qubit ${\mathrm{Q}}_{N-1}$ in a chain when the couplings between neighboring qubits satisfy a specific ratio. (b) The five-qubit chain sample. Five cross-shaped transmon qubits (Xmons, ${\mathrm{Q}}_0$–${\mathrm{Q}}_4$) arranged in a linear array. Each qubit has independent $XY$ and $Z$ control (labeled as “$x$” and “$f$” respectively), and is coupled to a separate $\lambda /4$ resonator for simultaneous and individual read-out. (c) The operating regime for QST from ${\mathrm{Q}}_0$ to ${\mathrm{Q}}_3$, with ${\mathrm{Q}}_4$ being decoupled. Both state preparation and measurement of each qubit are performed at the red points (the idle points), which are the maximum-frequency spots (sweet spots) with the best coherence times for ${\mathrm{Q}}_0$, ${\mathrm{Q}}_2$, and ${\mathrm{Q}}_3$. ${\mathrm{Q}}_1$ is biased about 60 MHz below its sweet spot to avoid the unwanted cross talk between ${\mathrm{Q}}_1$ and ${\mathrm{Q}}_3$. The dashed line (the “operating” point) represents the mean operating frequency of each qubit during the QST experiment: for ${\mathrm{Q}}_0$, it is fixed at the dashed line; for ${\mathrm{Q}}_1$–${\mathrm{Q}}_3$, their frequencies are parametrically modulated around the dashed lines to achieve the required coupling $g_j^{\prime }$ between neighboring qubits.

Figure 2

(a) Tunable coupling $g_1^{\prime }$ between ${\mathrm{Q}}_0$ and ${\mathrm{Q}}_1$ as a function of ${\epsilon }_1$ with other qubits being decoupled. $g_1^{\prime }$ is extracted from the so-called chevron pattern. ${\mathrm{Q}}_0$ is prepared in $|e⟩$ and biased at a fixed point ${\omega }_{o0}$, while ${\mathrm{Q}}_1$, initially in $|g⟩$, is flux biased to oscillate sinusoidally as ${\omega }_1={\omega }_{o1}+{\epsilon }_1\sin ({\nu }_{1}t+{\phi }_1)$. Coherent excitation oscillation between ${\mathrm{Q}}_0$ and ${\mathrm{Q}}_1$ as a function of ${\nu }_1$ and time $t$ at fixed ${\epsilon }_1$ produces a chevron pattern. The full-scale oscillation is achieved when ${\nu }_1={\mathrm{\Delta }}_1={\omega }_{o1}-{\omega }_{o0}$, and its frequency gives the effective coupling $g_1^{\prime }$ (dots). Two typical chevron patterns are shown in the inset, with blue and red corresponding to $|g⟩$ and $|e⟩$, respectively, of ${\mathrm{Q}}_1$: the top-left pattern and the bottom-right pattern correspond to ${\epsilon }_1/2\pi =30$ MHz and ${\epsilon }_1/2\pi =80$ MHz, respectively. (b) Tunable coupling $g_2^{\prime }$ between ${\mathrm{Q}}_1$ and ${\mathrm{Q}}_2$ as a function of ${\epsilon }_2$, with other qubits being decoupled. Both ${\mathrm{Q}}_1$ (initially in $|e⟩$) and ${\mathrm{Q}}_2$ (initially in $|g⟩$) are parametrically driven as ${\omega }_1={\omega }_{o1}+{\epsilon }_1\sin ({\nu }_{1}t+{\phi }_1)$ and ${\omega }_2={\omega }_{o2}+{\epsilon }_2\sin ({\nu }_{2}t+{\phi }_2)$ simultaneously. Similarly, at ${\nu }_2={\mathrm{\Delta }}_2={\omega }_{o2}-{\omega }_{o1}$, the oscillation frequency gives the effective coupling $g_2^{\prime }$ (dots). The top-left and bottom-right insets show the chevron pattern with ${\epsilon }_2/2\pi =50$ MHz and ${\epsilon }_2/2\pi =140$ MHz, respectively. The red lines in (a),(b) are fitted with Eq. (4). The resulting $g_1/2\pi \approx g_2/2\pi \approx 19$ MHz is slightly larger than the value measured from a static case in which the chevron pattern is measured while the frequency of one qubit is tuned across the other qubit with no parametric modulation. The deviation is presumably due to imperfect deconvolution of the flux pulse (see Appendix pp4), so the effective ${\epsilon }_1$ and ${\epsilon }_2$ are slightly larger than applied.

Figure 3

(a) Experimental sequence for QST. Initially, all four qubits are at their idle points, with only ${\mathrm{Q}}_0$ prepared in an arbitrary state by $R_{\hat{n}}^{\theta }$. Then step pulses are used to change the qubits from their idle spots to the operating points. ${\mathrm{Q}}_1$–${\mathrm{Q}}_3$ are modulated to achieve the required coupling ratio for perfect QST. The operation is for various times $t$, followed by step pulses to return all qubits to their idle points for the final-qubit-state read-out. (b) Time evolution of the qubit excited-state populations. The populations are measured simultaneously at different $t$ for the case with ${\mathrm{Q}}_0$ initially in $|e⟩$ as an example to calibrate the QST time. The dashed red lines are numerically simulated results with the measured parameters based on the Hamiltonian in Eq. (3). The population evolutions of ${\mathrm{Q}}_1$ and ${\mathrm{Q}}_2$ are noisier because in our implementation ${\mathrm{Q}}_1$ and ${\mathrm{Q}}_2$ are affected by multiple modulations, which lead to more high-order oscillations and cross talk. The slight deviation between simulation and experimental results mainly comes from the high-order terms and imperfect read-out calibration matrix due to the unwanted cross talk between qubits. At $t=84$ ns, QST from ${\mathrm{Q}}_0$ to ${\mathrm{Q}}_3$ is achieved.

Figure 4

(a)–(c) Dependence of phase ${\varphi }_s$ of the transferred state on ${\phi }_1$, ${\phi }_2$, and ${\phi }_3$ of the modulation pulses. The dependences of ${\varphi }_s$ on ${\phi }_1$ and ${\phi }_2$ are linear but with opposite slopes as expected. The dependence of ${\varphi }_s$ on ${\phi }_3$ deviates from a linear curve due to an extra phase accumulation. (d) QST process fidelity. Process tomography is used to benchmark the QST performance and is plotted as a function of the number of transfers, which is controlled by our setting the operating times $t=(4n+1)\times 84$ ns, equivalent to performing $1,5,9,\dots$ transfer processes. The phase of the transferred state on ${\mathrm{Q}}_3$ is deterministic and can be well controlled by adjustment of the phases in the sinusoidal modulation pulses as shown in (a)–(c). The process fidelities presented are all based on the transferred states after proper phase adjustments. The parameters ${\epsilon }_i$ and ${\nu }_i(i=1,2,3)$ are critical for high-fidelity QST and thus are further optimized by the function fminsearch in matlab. Each point is averaged $10000$ times and the error bars corresponding to one standard deviation are obtained from ten repeated experiments. The red curve is a fit based on $F=A{P}^m+0.25$ with $A=0.737$ and $P=0.992$, demonstrating nearly perfect QST. The bottom-left inset shows the $\chi$ matrix after one QST and the top-right inset shows the $\chi$ matrix after $105$ QSTs. The bar height and color correspond to the amplitude and phase of the $\chi$-matrix element, respectively.

Figure 5

Details of wiring and circuit components. The experimental device consists of five cross-shaped transmon qubits (Xmons, ${\mathrm{Q}}_0$–${\mathrm{Q}}_4$) [24, 46, 47] arranged in a linear array with nearly identical nearest-neighbor coupling strengths. Each qubit has independent $XY$ and $Z$ controls, which are properly attenuated and low-pass filtered. A common transmission line is coupled to separate $\lambda /4$ resonators for individual read-outs of the qubits. Two four-channel AWGs are used to fully manipulate the four qubits (${\mathrm{Q}}_0$–${\mathrm{Q}}_3$) to realize the QST. The fifth qubit, ${\mathrm{Q}}_4$, is biased with a dc source at a low frequency (less than 4 GHz) and is nearly completely decoupled from the first four qubits. The master AWG provides two pairs of sideband modulations at different frequencies, in combination with a qubit generator and a read-out generator as local oscillators (LOs), allowing $XY$ control and read-out of the qubits, respectively. The $XY$ control signal is divided by a four-way power divider, and the outputs are connected to the respective qubit $XY$ control lines through separate rf switches. These switches provide selective control of individual qubits. The slave AWG, triggered by the master AWG, is use to achieve $Z$ control of the qubits through individual flux-bias lines. A JPA at 10 mK with a gain of more than 20 dB and a bandwidth of about 260 MHz is used as the first stage of amplification, allowing high-fidelity single-shot measurements of the qubits. A high-electron-mobility-transistor (HEMT) amplifier at 4 K and an amplifier at room temperature are also used before the down-conversion of the read-out signal to the applied sideband frequencies with the same read-out generator as the LO. Part of the read-out signal does not go through the dilution refrigerator and is used as a reference to lock the phase of the returning read-out signal from the device for greater measurement stability.

Figure 6

Gain profile of the JPA. The JPA is singly pumped at a frequency of 6.951 GHz. The dashed lines represent the read-out frequencies for ${\mathrm{Q}}_0$–${\mathrm{Q}}_3$, respectively. The maximum gain is greater than 20 dB and the bandwidth is about 260 MHz.

Figure 7

Read-out histograms. (a)–(d) Histograms for ${\mathrm{Q}}_0$–${\mathrm{Q}}_3$, respectively. Each histogram is measured separately with a total count of $90000$, while the other qubits are in their thermal steady states. Blue curves are for an initial thermal steady state and red curves are for an initial thermal steady state followed by the corresponding $\pi$ rotation.

Figure 8

(a),(b) The longitudinal and transverse relaxation of the transferred state with an initial state of $|e⟩$ and $(|g⟩+|e⟩)/\sqrt{2}(|+⟩\mathrm{state})$, respectively. These data are part of the data presented in Fig. 4. Dots are experimental data. Lines are exponential fits, giving the decay times of the transferred state $T_1=11.8\mu \mathrm{s}$ and $T_2^{\ast }=11.5\mu \mathrm{s}$, respectively.

Figure 9

Simulated process fidelity $F$ with and without qubit thermal populations (0.02 on average) as a function of the transfer number $m$. Dots are simulated data. Similarly to Fig. 4, fits with $F=A{P}^m+0.25$ give a fidelity difference of only 0.03% (0.9920 vs 0.9923). In the simulation, we choose $T_2^{\ast }=11.5\mu \mathrm{s}$ for all qubits and the measured $T_1$ for each qubit at the operating point as in Table 1.

Figure 10

Qubit response to fast flux bias. The red curve corresponds to the uncorrected response. The blue curve corresponds to the corrected response after deconvolution. The inset shows the pulse sequence of the experiment.