Computationally Efficient Quantum Expectation with Extended Bell Measurements

Ruho Kondo1, Yuki Sato1, Satoshi Koide1, Seiji Kajita1, and Hideki Takamatsu2

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 $A\in{\mathbb C}^{2^n\times 2^n}$ 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.

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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", Physical Review Applied 20 3, 034044 (2023).

[2] Matteo Ippoliti, "Classical shadows based on locally-entangled measurements", Quantum 8, 1293 (2024).

[3] Bojia Duan and Chang-Yu Hsieh, "Hamiltonian-based data loading with shallow quantum circuits", Physical Review A 106 5, 052422 (2022).

[4] Yuki Sato, Ruho Kondo, Satoshi Koide, and Seiji Kajita, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 168 (2023) ISBN:979-8-3503-4323-6.

[5] Yuki Sato, Hiroshi C. Watanabe, Rudy Raymond, Ruho Kondo, Kaito Wada, Katsuhiro Endo, Michihiko Sugawara, and Naoki Yamamoto, "Variational quantum algorithm for generalized eigenvalue problems and its application to the finite-element method", Physical Review A 108 2, 022429 (2023).

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