Papers › Computationally Efficient Quantum Expectation with Extended Bell Measurements
Computationally Efficient Quantum Expectation with Extended Bell Measurements
Ruho Kondo, Yuki Sato, Satoshi Koide, Seiji Kajita, Hideki Takamatsu
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Evaluating an expectation value of an arbitrary observable A∈ℂ^(2ⁿ×2ⁿ) through na\"ive 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 4ⁿ matrix elements into at most 2ⁿ⁺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(nᶜ), the number of groups for simultaneous measurement reduces to O(nᶜ⁺¹). In addition, when non-zero elements densely fill the band, the variance is O((nᶜ⁺¹/2ⁿ) tr(A²)), 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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