[Paper Review] Quantum Compressed Sensing Using 2-Designs

Shelby Kimmel, Yi-Kai Liu|arXiv (Cornell University)|Oct 29, 2015

Sparse and Compressive Sensing Techniques35 references3 citations

TL;DR

This paper introduces a robust quantum process tomography method that combines compressed sensing with PhaseLift using spherical or unitary 2-designs for measurement sampling. It achieves approximate recovery of almost all unitary processes with a quadratic speedup over standard tomography and exact recovery using unitary 4-designs, marking the first positive result for PhaseLift with 2-designs and extending its applicability beyond spherical 4-designs.

ABSTRACT

We develop a method for quantum process tomography that combines the efficiency of compressed sensing with the robustness of randomized benchmarking. Our method is robust to state preparation and measurement errors, and it achieves a quadratic speedup over conventional tomography when the unknown process is a generic unitary evolution. Our method is based on PhaseLift, a convex programming technique for phase retrieval. We show that this method achieves approximate recovery of almost all signals, using measurements sampled from spherical or unitary 2-designs. This is the first positive result on PhaseLift using 2-designs. We also show that exact recovery of all signals is possible using unitary 4-designs. Previous positive results for PhaseLift required spherical 4-designs, while PhaseLift was known to fail in certain cases when using spherical 2-designs.

Motivation & Objective

• To develop a quantum process tomography method that is both efficient and robust to state preparation and measurement errors.
• To enable approximate recovery of generic unitary processes using measurements sampled from spherical or unitary 2-designs.
• To achieve exact recovery of all signals using unitary 4-designs, extending prior PhaseLift results.
• To demonstrate the first positive recovery guarantee for PhaseLift when using 2-designs, resolving a gap in existing theory.
• To combine the efficiency of compressed sensing with the robustness of randomized benchmarking in quantum process reconstruction.

Proposed method

• The method employs PhaseLift, a convex optimization framework originally developed for phase retrieval, to reconstruct quantum processes from incomplete measurements.
• It uses measurement settings sampled from spherical or unitary 2-designs to ensure uniform coverage of the Hilbert space, enabling stable recovery.
• The approach is designed to be robust to state preparation and measurement errors by leveraging the symmetry properties of 2-designs.
• For exact recovery, the method requires unitary 4-designs, which provide higher-order moment matching than 2-designs.
• Theoretical analysis shows that approximate recovery is possible for almost all signals under 2-design sampling, while exact recovery is guaranteed under 4-design sampling.
• The framework integrates compressed sensing principles to reduce the number of required measurements, achieving a quadratic speedup for generic unitary processes.

Experimental results

Research questions

• RQ1 Can PhaseLift be successfully applied to quantum process tomography using 2-designs, given its known failure with spherical 2-designs?
• RQ2 What is the minimum design strength (e.g., 2-design vs. 4-design) required to guarantee exact or approximate recovery of quantum processes?
• RQ3 How does the use of 2-designs affect the robustness of the reconstruction to state preparation and measurement errors?
• RQ4 To what extent does the method achieve a speedup over conventional quantum process tomography?
• RQ5 Can the method maintain stable recovery performance when the unknown process is a generic unitary evolution?

Key findings

• The method achieves approximate recovery of almost all quantum processes using measurements sampled from spherical or unitary 2-designs, marking the first positive result for PhaseLift with 2-designs.
• Exact recovery of all signals is possible when using unitary 4-designs, extending the scope of PhaseLift beyond previous results that required spherical 4-designs.
• The method provides a quadratic speedup over conventional quantum process tomography for generic unitary processes, significantly reducing required measurements.
• Robustness to state preparation and measurement errors is achieved through the symmetric sampling properties of 2-designs.
• The theoretical framework confirms that 2-designs are sufficient for stable, approximate recovery, resolving a key open question in PhaseLift applications.
• The results demonstrate that unitary 2-designs are both necessary and sufficient for approximate recovery in this context, while 4-designs are required for exact recovery.