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

We propose and analyze a simple framework for estimating the amplitudes of a given n-qubit quantum state ketψ = sum_i=0^2n-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 δ, i.e., | |Tilde{a}_i - |a_i| | leq δ for all i, mathcal{O}\(4ⁿ/δ⁴\) single-qubit measurements is sufficient. Second, we show that to achieve total variation sum_i=0^2n-1 | |Tilde{a}_i|² - |a_i|²| leq δ, mathcal{O}\(6ⁿ/δ⁴\) a single bit measurement is required. Finally, in order to achieve an average L₁ norm error sum_i=0^2n-1 | |Tilde{a}_i| - |a_i| |/2ⁿ leq δ, a single bit measurement mathcal{O}\(2ⁿ/ δ⁴\) is needed.