Variational Entanglement-Assisted Quantum Process Tomography with Arbitrary Ancillary Qubits

Quantum process tomography is a pivotal technique in fully characterizing quantum dynamics. However, exponential scaling of the Hilbert space with the increasing system size extremely restrains its experimental implementations. Here, we put forward a more efficient, flexible, and error-mitigated method: variational entanglement-assisted quantum process tomography with arbitrary ancillary qubits. Numerically, we simulate up to eight-qubit quantum processes and show that this tomography with $m$ ancillary qubits ($0\le m\le n$) alleviates the exponential costs on state preparation (from $4^n$ to $2^{n-m}$), measurement settings (at least a 1 order of magnitude reduction), and data postprocessing (efficient and robust parameter optimization). Experimentally, we first demonstrate our method on a silicon photonic chip by rebuilding randomly generated one-qubit and two-qubit unitary quantum processes. Further using the error mitigation method, two-qubit quantum processes can be rebuilt with average gate fidelity enhanced from 92.38% to 95.56%. Our Letter provides an efficient and practical approach to process tomography on the noisy quantum computing platforms.

Figure 1

(a) Structure of the VEAPT with arbitrary ancillary qubits. The scheme consists of the preparation of input entanglement states, the evolution of the unknown channel and the controllable process, and the classical processing unit. The classical part is composed of cost function evaluation, classical optimization, and parameter updating. (b) Structure of the VEAPT with full ancillaries. It is a specialized version of VEAPT in that the $V(\overrightarrow{\theta })$ process can be transposed to the ancillary space by quantum ricochet property when the ancillary qubits are full. (c) Scheme of the error mitigation method. The first stage is the aligning operation by directly setting the circuit as $I$. The second stage is the normal VEAPT optimizing with target $U$ and encoding $V(\overrightarrow{\theta })$. (d) Photonic chip structure and experimental realization. When the ancillary qubits are arbitrary, the laser is inputted at the first layer of the beam splitter network (denoted as ①). When the ancillaries are full, it pumps in the second layer (denoted as ②). In the encoding circuit part, we give an illustration of the VEAPT with full ancillaries [the dashed box denotes $U$ and the dotted box denotes $V^T(\overrightarrow{\theta })$]. The red colored phase shifters are for one-qubit process encoding, while the blue colored ones are for the two-qubit case. More details are provided in the Supplemental Material [32].

Figure 2

Numerical simulations on variational quantum circuit simulator for RQC with different circuit depth $d$ and Heisenberg $XXZ$ spin chain time evolution quantum processes up to eight-qubit cases. It is obvious that the VEAPT method significantly decreases the total number of measurements compared to standard QPT (blue starred line) when the qubit number scales. Furthermore, in the inset graph, we present the graph in the logarithmic axis and obtain a clear and enlarged measurement gap (at least a 1 order of magnitude reduction).

Figure 3

Experimental demonstration of the VEAPT method. (a) VEAPT with full ancillary qubits on one-qubit (triangle) and two-qubit (circle) cases. The cost function $f(\overrightarrow{\theta })$ (red dotted lines) and the average gate fidelity $F_{\mathrm{avg}}$ (blue solid lines) gradually converge. Minimum $f(\overrightarrow{\theta })$ and maximum $F_{\mathrm{avg}}$ are listed in the table. (b) Two-qubit VEAPT with one ancillary qubit and two input states. The red dotted line is the overall cost function, while the yellow and purple ones show the corresponding infidelities between the output state and the two entangled states, respectively. (c) Error mitigation results on two-qubit cases, where the blue circle and red triangle lines are different trials with and without error mitigation, respectively. The final maximum $F_{\mathrm{avg}}$ is further enhanced to 95.56% after the error mitigation operation. (d) Two-qubit VEAPT with one ancillary qubit under different levels of shot noise. The vertical axis denotes the average cost function values among the past 10 epochs. In the inset graph, it is clear that longer integration time $t$ with more photons achieves better performance. We used the BFGS algorithm in (a) and (c), while the SPSA algorithm is employed in (b) and (d).