Computationally Efficient Quantum Expectation with Extended Bell Measurements

Evaluating an expectation value of an arbitrary observable A ∈ C 2 n × 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 ( n d ) 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 ) t r ( 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.