A Framework For Estimating Amplitudes of Quantum State With Single-Qubit Measurement

TL;DR

This paper introduces a framework for estimating the amplitudes of an n-qubit quantum state using only single-qubit measurements, by solving nonlinear equations derived from measurement outcomes, which is more measurement-efficient than traditional methods.

Contribution

The authors propose a novel method that uses single-qubit measurements and nonlinear equation solving to estimate quantum state amplitudes, reducing measurement complexity.

Findings

Estimates all amplitude norms with  (4^n/elta^4) measurements.

Achieves total variation error with  (6^n/elta^4) measurements.

Obtains average L1 norm error with  (2^n/elta^4) measurements.

Abstract

We propose and analyze a simple framework for estimating the amplitudes of a given $n$-qubit quantum state $\ket{\psi} = \sum_{i=0}^{2^n-1} a_i \ket{i}$ in computational basis, utilizing a single-qubit measurement only. Previously, it was a common procedure that one could measure all qubits in order to collect measurement outcomes, from which one can estimate amplitudes of given quantum state. Here, we show that if restricting to single-qubit measurement, and one can perform measurement on arbitrary basis, then the measurement outcomes can be used to assist the finding of amplitudes in the usual computational, or Z basis. More concretely, such outcomes are capable of constructing a system of nonlinear algebraic equations, and by classically solving them, we obtain $\Tilde{a}_i$, which is the approximation to the corresponding amplitudes $a_i$, including both real and imaginary component. We then discuss our framework from a broader perspective. First, we show that estimating all (norms of) amplitudes to additive accuracy $\delta$, i.e., $| |\Tilde{a}_i - |a_i| | \leq \delta$ for all $i$, $\mathcal{O}(4^n/\delta^4)$ single-qubit measurements is sufficient. Second, we show that to achieve total variation $\sum_{i=0}^{2^n-1} | |\Tilde{a}_i|^2 - |a_i|^2| \leq \delta $, $\mathcal{O}(6^n/\delta^4)$ a single bit measurement is required. Finally, in order to achieve an average $L_1$ norm error $ \sum_{i=0}^{2^n-1} | |\Tilde{a}_i| - |a_i| |/2^n \leq \delta$, a single bit measurement $\mathcal{O}(2^n/ \delta^4)$ is needed.