The Accuracy vs. Sampling Overhead Trade-off in Quantum Error Mitigation Using Monte Carlo-Based Channel Inversion

Abstract

Quantum error mitigation (QEM) is a class of promising techniques for reducing the computational error of variational quantum algorithms. In general, the computational error reduction comes at the cost of a sampling overhead due to the variance-boosting effect caused by the channel inversion operation, which ultimately limits the applicability of QEM. Existing sampling overhead analysis of QEM typically assumes exact channel inversion, which is unrealistic in practical scenarios. In this treatise, we consider a practical channel inversion strategy based on Monte Carlo sampling, which introduces additional computational error that in turn may be eliminated at the cost of an extra sampling overhead. In particular, we show that when the computational error is small compared to the dynamic range of the error-free results, it scales with the square root of the number of gates. By contrast, the error exhibits a linear scaling with the number of gates in the absence of QEM under the same assumptions. Hence, the error scaling of QEM remains to be preferable even without the extra sampling overhead. Our analytical results are accompanied by numerical examples. NOTATIONS• Deterministic scalars, vectors and matrices are represented by x, x, and X, respectively, whereas their random counterparts are denoted as x, x, and X, respectively. Deterministic sets, random sets, and operators are denoted as X , X, and X , respectively. • The notations 1 n , 0 n , 0 m×n , and I k , represent the ndimensional all-one vector, the n-dimensional all-zero vector, the m × n dimensional all-zero matrix, and the k × k identity matrix, respectively. The subscripts may be omitted if they are clear from the context. • The notation ∥x∥ p represents the ℓ p -norm of vector x, and the subscript may be omitted when p = 2. For matrices, ∥A∥ p denotes the matrix norm induced by the corresponding ℓ p vector norm. The notation 1/x represents the element-wise reciprocal of vector x. • The notation [A] i,j denotes the (i, j)-th entry of matrix A. For a vector x, [x] i denotes its i-th element. The submatrix obtained by extracting the i 1 -th to i 2 -th rows and the j 1 -th to j 2 -th columns from A is denoted as [A] i1:i2,j1:j2 . The notation [A] :,i represents the i-th column of A, and [A] i,: denotes the i-th row, respectively. • The notation A ⊗ B represents the Kronecker product between matrices A and B. The notation A ⊙ B denotes the Hadamard product between matrices A and B.