Witnessing Quantum Resource Conversion within Deterministic Quantum Computation Using One Pure Superconducting Qubit

Deterministic quantum computation with one qubit (DQC1) is iconic in highlighting that exponential quantum speedup may be achieved with negligible entanglement. Its discovery catalyzed a heated study of general quantum resources, and various conjectures regarding their role in DQC1’s performance advantage. Coherence and discord are prominent candidates, respectively, characterizing nonclassicality within localized and correlated systems. Here we realize DQC1 within a superconducting system, engineered such that the dynamics of coherence and discord can be tracked throughout its execution. We experimentally confirm that DQC1 acts as a resource converter, consuming coherence to generate discord during its operation. Our results highlight superconducting circuits as a promising platform for both realizing DQC1 and related algorithms, and experimentally characterizing resource dynamics within quantum protocols.

Figure 1

DQC1 model. Initially the ancilla qubit is prepared in a pure ground state and the register qubits are prepared in a maximally mixed state. Coherence in the quantum system is then prepared in the ancilla qubit by a Hadamard gate and converted into discord by a controlled operation $U$. Measurements of $⟨{\sigma }_x⟩$ and $⟨{\sigma }_y⟩$ on the ancilla qubit give the real and imaginary parts of $\mathrm{Tr}(U)/2^n$, respectively.

Figure 2

Experimental quantum circuit to measure the conversion between coherence and quantum correlations with a DQC1 model. (a) The whole process can be divided into three parts: state preparation, DQC1 algorithm, and joint tomography measurement. The maximally mixed state (${\sum }_{k=0}^7|k⟩⟨k|$) of the initial registers is deterministically generated through a quantum channel construction based on QND measurements of the ancilla and adaptive control of the ancilla-register system. The $R_y^{\pi /2}$ operation to create input coherence in the DQC1 algorithm and the following $R_{x(y)}^{\pi /2}$ operations before $M_1$ in the joint tomography are all generated by GRAPE to compensate the extra phases corresponding to different register states due to the dispersive interaction during these gates, such that the gate operations are independent of the states of the register qubits. The controlled-$U$ gate is realized through an appropriate ancilla-register interation time and is used to convert coherence to discord. To characterize the ancilla-register system, joint tomography is performed by correlating the ancilla tomography and subsequent register Wigner tomography. In the joint tomography, the two $\pi /2$ rotations $R_{un,y}^{\pi /2}$ before and after the controlled $\pi$ phase gate $C_{\pi }$ are unconditional gates with a Gaussian envelope of $\sigma =5\mathrm{ns}$. (b) Real part of the reconstructed density matrix (truncated to maximum photon number state $N_{\max }=9$) of the initial register qubits with a fidelity of 0.977.

Figure 3

DQC1 algorithm output. The real and imaginary parts of the normalized trace $\mathrm{Tr}(U)/8$ are measured for different $\phi$. All data (points) are averaged with over $10^7$ measurements and error bars correspond to 1 standard deviation. The solid lines show theoretical expectations. Crosses present the simulated results including all decoherence channels. Experimental results agree well with both theory and simulation even with the presence of decoherence processes from both ancilla and register qubits.

Figure 4

The coherence consumption $\mathrm{\Delta }C({\rho }_A)$ and discord production $D({\tilde{\rho }}_{AR})$ as a function of phase $\phi$ in the DQC1 model. (a) Theoretical expectations are shown with solid lines. Dots give the experimental results. In the experiment, each point in the joint Wigner functions has been averaged over 3000 single-shot joint ancilla and register measurements, and the standard deviation in $\mathrm{\Delta }C({\rho }_A)$ and $D({\tilde{\rho }}_{AR})$ are estimated by bootstrapping on the measured joint Wigner functions [67]. Crosses present the simulated data including all decoherence channels with both the ancilla and register qubits. The measured $\mathrm{\Delta }C({\rho }_A)$ agrees fairly well with the theoretical one with a small gap at the middle plateau. This gap comes from the imperfect generation and measurement of the initial coherence $C({\rho }_A)=0.894$ while the ancilla state fidelity is 0.993 to the ideal state $(|g⟩+|e⟩)/\sqrt{2}$ after the $R_y^{\pi /2}$ operation. The measured $D({\tilde{\rho }}_{AR})$, in excellent agreement with the theoretical expectation, captures all the important features as in the theoretical curve and is unambiguously lower than $\mathrm{\Delta }C({\rho }_A)$ as expected. (b) The fidelity of the measured ${\tilde{\rho }}_{AR}$ compared to the ideal ones with an average of 0.96.