Guaranteed recovery of quantum processes from few measurements

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 ${\ell }_2$-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.

45+11 pages, several figures and plots. v2: Improved version, section on approximate designs added. v3: Small changes and incomplete funding information corrected. v4: Robustness property clarified; published version

References in corpus (28)

• The density-matrix renormalization group in the age of matrix product states
• Direct Fidelity Estimation from Few Pauli Measurements
• Quantum Process Tomography: Resource Analysis of Different Strategies
• Evenly distributed unitaries: on the structure of unitary designs
• Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators
• Realization of the quantum Toffoli gate with trapped ions
• Comparing Experiments to the Fault-Tolerance Threshold
• Approximate unitary $t$-designs by short random quantum circuits using nearest-neighbor and long-range gates
• Both Toffoli and Controlled-NOT need little help to do universal quantum computation
• Quantum process tomography of unitary and near-unitary maps
• Efficient unitary designs with nearly time-independent Hamiltonian dynamics
• Process POVM: A mathematical framework for the description of process tomography experiments
• Compressed sensing quantum process tomography for superconducting quantum gates
• Recovering quantum gates from few average gate fidelities
• The Clifford group fails gracefully to be a unitary 4-design
• Experimentally exploring compressed sensing quantum tomography
• Efficient Quantum Pseudorandomness
• Mixing properties of stochastic quantum Hamiltonians
• Representations of the multi-qubit Clifford group
• Simpler semidefinite programs for completely bounded norms
• Improving compressed sensing with the diamond norm
• Phase Retrieval Using Unitary 2-Designs
• Statistically efficient tomography of low rank states with incomplete measurements
• Provable quantum state tomography via non-convex methods
• Accurate and robust unitary transformation of a high-dimensional quantum system
• Statistical analysis of low rank tomography with compressive random measurements
• Note on the saturation of the norm inequalities between diamond and nuclear norm
• On the complexity of approximating the diamond norm

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