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arXiv:2310.20421 · IEEE Conference on Decision and Control (CDC)

Two-Stage Solution for Ancilla-Assisted Quantum Process Tomography: Error Analysis and Optimal Design

Shuixin Xiao, Yuanlong Wang, Daoyi Dong, Jun Zhang

Published Dec 13, 2023• 1 citation
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Abstract

Quantum process tomography (QPT) is a fundamental task to characterize the dynamics of quantum systems. In contrast to standard QPT, ancilla-assisted process tomography (AAPT) framework introduces an extra ancilla system such that a single input state is needed. In this paper, we extend the two-stage solution, a method originally designed for standard QPT, to perform AAPT. Our algorithm has$O(Md_{A}^{2}d_{B}^{2})$computational complexity where$M$is the type number of the measurement operators,$d_{A}$is the dimension of the quantum system of interest, and$d_{B}$is the dimension of the ancilla system. Then we establish an error upper bound and further discuss the optimal design on the input state in AAPT. A numerical example on a phase damping process demonstrates the effectiveness of the optimal design and illustrates the theoretical error analysis.

DOI
10.1109/cdc49753.2023.10383498

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Quantum process tomography (QPT) is a fundamental task to characterize the dynamics of quantum systems. In contrast to standard QPT, ancilla-assisted process tomography (AAPT) framework introduces an extra ancilla system such that a single input state is needed. In this paper, we extend the two-stage solution, a method originally designed for standard QPT, to perform AAPT. Our algorithm hasO(MdA2dB2)O(Md_{A}^{2}d_{B}^{2})O(MdA2​dB2​)computational complexity whereMMMis the type number of the measurement operators,dAd_{A}dA​is the dimension of the quantum system of interest, anddBd_{B}dB​is the dimension of the ancilla system. Then we establish an error upper bound and further discuss the optimal design on the input state in AAPT. A numerical example on a phase damping process demonstrates the effectiveness of the optimal design and illustrates the theoretical error analysis.
Quantum process tomography is a critical task for characterizing the dynamics of quantum systems and achieving precise quantum control. In this paper, we propose a two-stage solution for both trace-preserving and non-trace-preserving quantum process tomography. Utilizing a tensor structure, our algorithm exhibits a computational complexity ofO(MLd2)O(MLd^{2})O(MLd2)where d is the dimension of the quantum system andM,L (M≥d2,L≥d2)M, L~(M\geq d^{2}, L\geq d^{2})M,L (M≥d2,L≥d2)represent the numbers of different input states and measurement operators, respectively. We establish an analytical error upper bound and then design the optimal input states and the optimal measurement operators, which are both based on minimizing the error upper bound and maximizing the robustness characterized by the condition number. Numerical examples and testing on IBM quantum devices are presented to demonstrate the performance and efficiency of our algorithm.
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Quantum process tomography (QPT) is a critical task for characterizing the dynamics of quantum systems and achieving precise quantum control. In this paper, we propose an effective two-stage solution (TSS) for QPT in the natural basis. Our algorithm has O(d6) computational complexity where d is the dimension of the quantum system. We establish an error bound O(d5N)O\left( {\frac{{{d^5}}}{{\sqrt N }}} \right)O(N​d5​) where N is the copy number for each output state. The numerical examples compared with a convex optimization algorithm demonstrate the effectiveness of our algorithm.
Standard quantum process tomography on a ddd-dimensional input is performed by preparing several states of an input probe that then evolve under the action of the quantum channel corresponding to the progress. The final states of the probe are reconstructed by means of state tomography. An alternative is offered by ancilla-assisted process tomography: a single probe-ancilla state is used, and the correlations existing between probe and ancilla are exploited to fully reconstruct the information on the channel. In order for ancilla-assisted process tomography to be possible, the probe-ancilla input state does not need to be entangled, but still needs to have maximal operator Schmidt rank. Here we establish and analyze a framework for process tomography that interpolates between these two methods, aiming at exploiting any correlations that may exist between probe and ancilla to allow process tomography with as few input preparations as possible, when the probe-ancilla state may be operator-Schmidt-rank deficient. The main object of our investigation is the minimal number of initial local operations on the input probe for a given starting probe-ancilla state that are needed to allow process tomography. We prove that such a number scales inversely proportional to the operator Schmidt rank of the input probe-ancilla state for arbitrary local input processing. We also provide results for the case where this initial local processing is restricted to be a unitary rotation, in particular showing that the mentioned scaling is satisfied for pure input entangled states, and that in the case of mixed states there might be extreme cases where the fixed input probe-ancilla state provides information on a ≈d2/2\approx d^2/2≈d2/2-dimensional subspace of the operator space and a single additional input unitary rotation allows process tomography.
Quantum process tomography (QPT) methods aim at identifying a given quantum process. QPT is a major quantum information processing tool, since it especially allows one to characterize the actual behavior of quantum gates, which are the building blocks of quantum computers. The present paper focuses on the estimation of a unitary process. This class is of particular interest because quantum mechanics postulates that the evolution of any closed quantum system is described by a unitary transformation. Unitary processes have significantly fewer parameters than general quantum processes (22nqb vs. 24nqb−22nqb real independent parameters for nqb qubits). By assuming that the process is unitary we develop two methods that scale better with the size of the system. In the present paper, we stay as close as possible to the standard setup of QPT: the operator has to prepare copies of different input states. The properties those states have to satisfy in order for our method to achieve QPT are very mild. Therefore, we choose to operate with copies of 2nqb initially unknown pure input states. In order to perform QPT without knowing the input states, we perform measurements on half the copies of each state, and let the other half be transformed by the system before measuring them (each copy is only measured once). This setup has the advantage of removing the issue of systematic (i.e., same on all the copies of a state) errors entirely because it does not require the process input to take predefined values. We develop a straightforward analytical solution that first estimates the states from the averaged measurements and then finds the unitary matrix (representing the process) coherent with those estimates by using our analytical solution to an extended version of Wahba’s problem. This estimate may then be used as an initial point for a fine tuning algorithm that maximizes the likelihood of the measurements. Simulation results show the effectiveness of the proposed methods.
In a large-scale quantum network, quantum process tomography is a powerful tool for quantifying and verifying the quality of quantum channels. Within this framework, ancilla-assisted process tomography (AAPT) provides a significant advantage for in situ network metrology, as the same states used for entanglement distribution can also be used for tomography. Here, we perform AAPT within a reconfigurable three-node topology on the Washington DC metropolitan quantum network (DC-QNet), consisting of deployed fiber links up to 11.5 km in length. Using a source of polarization-entangled photon pairs at one node, single photons are routed to either of the two receiver nodes via optical switch, and correlations to the local (ancilla) qubits are leveraged to characterize all relevant links in the network. Additionally, AAPT performance is characterized in the presence of noise from coexistent classical synchronization signals, and the advantages of dynamic routing are shown theoretically for larger network topologies.