Abstract
Quantum process tomography is the task of reconstructing unknown quantum channels from measured data. In this work, we introduce compressed sensing-based methods that facilitate the reconstruction of quantum channels of low Kraus rank. Our main contribution is the analysis of a natural measurement model for this task: We assume that data is obtained by sending pure states into the channel and measuring expectation values on the output. Neither ancillary systems nor coherent operations across multiple channel uses are required. Most previous results on compressed process reconstruction reduce the problem to quantum state tomography on the channel's Choi matrix. While this ansatz yields recovery guarantees from an essentially minimal number of measurements, physical implementations of such schemes would typically involve ancillary systems. A priori, it is unclear whether a measurement model tailored directly to quantum process tomography might require more measurements. We establish that this is not the case.Technically, we prove recovery guarantees for three different reconstruction algorithms. The reconstructions are based on a trace, diamond, and ℓ2-norm minimization, respectively. Our recovery guarantees are uniform in the sense that with one random choice of measurement settings all quantum channels can be recovered equally well. Moreover, stability against arbitrary measurement noise and robustness against violations of the low-rank assumption is guaranteed. Numerical studies demonstrate the feasibility of the approach.
Highlights
Recent years have seen significant advances in the precise control of quantum systems
We consider the task of reconstructing a quantum channel T ∈ completely positive and trace preserving (CPT)(Cn) from measurement data of the form (1): The unknown channel receives pure states |ψi ψi | as input and subsequently expectation values of observables Ai are measured for the output state; see Figure 2 for a pictorial description
We have proven that quantum processes can be reconstructed from an essentially optimal number of expectation values without the requirement of ancillary quantum systems
Summary
Recent years have seen significant advances in the precise control of quantum systems. Complex quantum states of systems with an increasing number of degrees of freedom can be prepared and manipulated with high accuracy In this development, it is important to have tools at hand that allow for a complete characterization of state or process that are being realized in a given experimental setup. We note that full quantum process tomography is distinct from certification protocols or coarser characterization schemes like randomized benchmarking (which reports only a single number: a certain error rate). While this makes the latter type of protocols much cheaper to implement, only process tomography allows one to understand in precisely which way a quantum gate deviates from its specification.
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