K-sparse Pure State Tomography with Phase Estimation
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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 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 𝒪(d K (log K)^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 - π / 2 rotation gates around X-axis in Bloch sphere, i.e., R_X(- π / 2), allowing to be realized in NISQ devices. Open problems are discussed regarding the existence of the unitary operator and its practical circuit implementation.
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