K-sparse Pure State Tomography with Phase Estimation
- URL: http://arxiv.org/abs/2111.04359v2
- Date: Mon, 15 Nov 2021 09:13:10 GMT
- Title: K-sparse Pure State Tomography with Phase Estimation
- Authors: Burhan Gulbahar
- Abstract summary: Quantum state tomography (QST) for reconstructing pure states requires exponentially increasing resources and measurements with the number of qubits.
QST reconstruction for any pure state composed of the superposition of $K$ different computational basis states of $n$bits in a specific measurement set-up is presented.
- Score: 1.2183405753834557
- License: http://creativecommons.org/licenses/by/4.0/
- 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.
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