Pauli error estimation via Population Recovery

Abstract

Motivated by estimation of quantum noise models, we study the problem of learning a Pauli channel, or more generally the Pauli error rates of an arbitrary channel. By employing a novel reduction to the "Population Recovery" problem, we give an extremely simple algorithm that learns the Pauli error rates of an n-qubit channel to precision ϵ in ℓ∞ using just O(1/ϵ2)log⁡(n/ϵ) applications of the channel. This is optimal up to the logarithmic factors. Our algorithm uses only unentangled state preparation and measurements, and the post-measurement classical runtime is just an O(1/ϵ) factor larger than the measurement data size. It is also impervious to a limited model of measurement noise where heralded measurement failures occur independently with probability ≤1/4.We then consider the case where the noise channel is close to the identity, meaning that the no-error outcome occurs with probability 1−η. In the regime of small η we extend our algorithm to achieve multiplicative precision 1±ϵ (i.e., additive precision ϵη) using just O(1ϵ2η)log⁡(n/ϵ) applications of the channel.

Quantum advantages for Pauli channel estimation

“…where λ := {λ b } b is called the Pauli eigenvalues [23,25]. These two sets of parameters, p and λ, are related by the Walsh-Hadamard transform…”

“…Both p and λ are physically interesting parameters: The Pauli error rates are directly related to the error thresholds in fault-tolerant quantum computation [29,30] and have been the quantities of interest for many quantum benchmarking protocols [16,17,[23][24][25]; The Pauli eigenvalues quantify how well a Pauli observable is preserved through the noise channel (hence also known as Pauli fidelities) and have applications in quantum error mitigation (see, e.g., Ref. [31]).…”

“…Because of the important role played by Pauli channels, it is of great interest to study the estimation of these objects in an efficient and practical way. Despite a long line of research [18][19][20][21][22][23][24][25][26], the ultimate sample complexity for Pauli channel estimation has not yet been fully characterized.…”

The learnability of Pauli noise

“…where λ := {λ b } b is called the Pauli fidelities or Pauli eigenvalues [10,13,35]. These two sets of parameters, p and λ, are related by the Walsh-Hadamard transform…”

“…Characterizing quantum noise is an essential step in the development of quantum hardware [1,2]. Remarkably, despite recent progress in both gate-level and scalable noise characterization methods [3][4][5][6][7][8][9][10][11][12][13][14][15][16], the full characterization of the noise channel of a single CNOT/CZ gate remains infeasible. This is unlikely to be caused by limitations of existing benchmarking algorithms.…”