1Toyota Central R&D Labs., Inc., 41-1, Yokomichi, Nagakute, Aichi 480-1192, Japan
2Toyota Motor Corporation, 1 Toyota-Cho, Toyota, Aichi 471-8571, Japan
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Abstract
Evaluating an expectation value of an arbitrary observable $Ain{mathbb C}^{2^ntimes 2^n}$ through naïve Pauli measurements requires a large number of terms to be evaluated. We approach this issue using a method based on Bell measurement, which we refer to as the extended Bell measurement method. This analytical method quickly assembles the $4^n$ matrix elements into at most $2^{n+1}$ groups for simultaneous measurements in $O(nd)$ time, where $d$ is the number of non-zero elements of $A$. The number of groups is particularly small when $A$ is a band matrix. When the bandwidth of $A$ is $k=O(n^c)$, the number of groups for simultaneous measurement reduces to $O(n^{c+1})$. In addition, when non-zero elements densely fill the band, the variance is $O((n^{c+1}/2^n),{rm tr}(A^2))$, which is small compared with the variances of existing methods. The proposed method requires a few additional gates for each measurement, namely one Hadamard gate, one phase gate and at most $n-1$ CNOT gates. Experimental results on an IBM-Q system show the computational efficiency and scalability of the proposed scheme, compared with existing state-of-the-art approaches. Code is available at https://github.com/ToyotaCRDL/extended-bell-measurements.
Featured image: Measurement operators for each component of the matrix using the extended Bell measurements.
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Cited by
[1] Francisco Escudero, David Fernández-Fernández, Gabriel Jaumà, Guillermo F. Peñas, and Luciano Pereira, “Hardware-efficient entangled measurements for variational quantum algorithms”, arXiv:2202.06979.
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