K-sparse Pure State Tomography with Phase Estimation

Abstract

Quantum state tomography (QST) for reconstructing pure states requires exponentially increasing resources and measurements with the number of qubits by using state-of-the-art quantum compressive sensing (CS) methods. In this article, QST reconstruction for any pure state composed of the superposition of $K$ different computational basis states of $n$ qubits in a specific measurement set-up, i.e., denoted as $K$-sparse, is achieved without any initial knowledge and with quantum polynomial-time complexity of resources based on the assumption of the existence of polynomial size quantum circuits for implementing exponentially large powers of a specially designed unitary operator. The algorithm includes $\mathcal{O}(2/|c_k{|}^2)$ repetitions of conventional phase estimation algorithm depending on the probability $|c_k{|}^2$ of the least possible basis state in the superposition and $\mathcal{O}(dK(logK{)}^c)$ measurement settings with conventional quantum CS algorithms independent from the number of qubits while dependent on $K$ for constant $c$ and $d$. Quantum phase estimation algorithm is exploited based on the favorable eigenstructure of the designed operator to represent any pure state as a superposition of eigenvectors. Linear optical set-up is presented for realizing the special unitary operator which includes beam splitters and phase shifters where propagation paths of single photon are tracked with which-path-detectors. Quantum circuit implementation is provided by using only CNOT, phase shifter and $-\pi /2$ rotation gates around X-axis in Bloch sphere, i.e., $R_X(-\pi /2)$, allowing to be realized in NISQ devices. Open problems are discussed regarding the existence of the unitary operator and its practical circuit implementation.

arXiv comment19 pages, 5 figures, edited v2

References in corpus (16)

• Supervised learning with quantum enhanced feature spaces
• Quantum machine learning in feature Hilbert spaces
• Quantum state tomography via compressed sensing
• Many-body quantum state tomography with neural networks
• Efficient quantum state tomography
• Reliable Quantum State Tomography
• Experimental quantum compressed sensing for a seven-qubit system
• Quantum Overlapping Tomography
• Quantum embeddings for machine learning
• Quantum state tomography via reduced density matrices
• Experimentally exploring compressed sensing quantum tomography
• Embedding Quantum Simulators for Quantum Computation of Entanglement
• Multiphoton Tomography with Linear Optics and Photon Counting
• Scalable estimation of pure multi-qubit states
• Simulation of wave-particle duality in multi-path interferometers on a quantum computer
• Semi-device-dependent blind quantum tomography