Efficient quantum state tomography (1101.4366v1)

Abstract: Quantum state tomography, the ability to deduce the state of a quantum system from measured data, is the gold standard for verification and benchmarking of quantum devices. It has been realized in systems with few components, but for larger systems it becomes infeasible because the number of quantum measurements and the amount of computation required to process them grows exponentially in the system size. Here we show that we can do exponentially better than direct state tomography for a wide range of quantum states, in particular those that are well approximated by a matrix product state ansatz. We present two schemes for tomography in 1-D quantum systems and touch on generalizations. One scheme requires unitary operations on a constant number of subsystems, while the other requires only local measurements together with more elaborate post-processing. Both schemes rely only on a linear number of experimental operations and classical postprocessing that is polynomial in the system size. A further strength of the methods is that the accuracy of the reconstructed states can be rigorously certified without any a priori assumptions.

• The paper introduces scalable methods that leverage matrix product states to exponentially reduce complexity in traditional quantum state tomography.
• The paper details two schemes: one using unitary operations on local subsystems and another employing local measurements with polynomial post-processing.
• The paper ensures rigorous certification of reconstructed states, providing essential tools for advancing large-scale quantum computing experiments.

Efficient Quantum State Tomography: A Summary

This paper presents an exploration into quantum state tomography, addressing its inherent complexity when scaling to larger quantum systems. Due to the exponential growth of parameters necessary to describe quantum states, traditional quantum state tomography becomes computationally infeasible beyond a small number of components. The authors propose scalable methods for quantum state tomography that significantly reduce the complexity and resource requirements compared to traditional approaches.

Key Contributions

1. Reduction in Complexity: The authors propose methods that utilize matrix product states (MPS) as a more efficient representation for certain classes of quantum states commonly encountered in quantum information and many-body physics. These methods achieve exponential improvements in complexity over traditional tomography methods by leveraging MPS, which can express complex quantum states with only a polynomial number of parameters relative to system size.
2. Two Tomographic Schemes:
   • Unitary and Local Measurements: The first scheme involves applying unitary operations to a constant number of subsystems. This technique requires sophisticated unitary control over adjacent qudits to achieve efficient state reconstruction within desired accuracy.
   • Local Measurements with Polynomial Post-Processing: The second scheme only requires local measurements and employs post-processing techniques. It eliminates the need for unitary control, relying instead on computational intensity in post-processing to reconstruct the state.
3. Certified Accuracy: Both schemes offer mechanisms to rigorously certify the accuracy of reconstructed states, independent of prior assumptions on the state in the laboratory. The certification process includes quantifying errors at each step, ensuring the fidelity and the practical applicability of the reconstructed states.

Implications and Future Directions

The use of matrix product states linked to efficient quantum state tomography is significant not only for simplifying the verification and benchmarking of large-scale quantum devices but also for facilitating advancements in quantum computing and quantum information processing. These methods address previously existing barriers to scaling quantum technology, thus supporting experimental endeavors involving many qubits systems.

For future developments, these methodologies could extend to capture mixed quantum states, handling more complex physical systems such as higher-dimensional arrays and offering viable strategies for tomographic applications in prospective quantum computing environments. Also, exploring the applicability and extension of these techniques to other tensor network states, such as PEPS or MERA, could broaden their utility in quantum physics.

Conclusion

This paper represents a pivotal step towards making the tomography of large quantum systems feasible by reducing the reliance on classical computational resources through smart quantum-to-classical reduction strategies. By effectively harnessing MPS and linear number of operations, the authors provide an essential toolkit for the experimental realization of complex quantum systems, which is increasingly critical as the quantum industry edges towards practical and scalable applications.