K-sparse Pure State Tomography with Phase Estimation

TL;DR

This paper introduces a polynomial-time quantum state tomography method for pure states with K-sparsity, utilizing phase estimation and specific measurement setups, reducing resource requirements compared to traditional exponential methods.

Contribution

It presents a novel quantum tomography algorithm for K-sparse pure states that operates efficiently without prior knowledge, leveraging phase estimation and specialized unitaries.

Findings

Achieves polynomial-time reconstruction of K-sparse pure states.

Requires fewer measurement settings independent of the number of qubits.

Provides practical optical and circuit implementations for the proposed method.

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 $\mathcal{O}(2 \, / \, \vert c_{k}\vert^2)$ repetitions of conventional phase estimation algorithm depending on the probability $\vert c_{k}\vert^2$ of the least possible basis state in the superposition and $\mathcal{O}(d \, K \,(log K)^c)$ 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 $- \pi \, / \, 2$ 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.