Pauli error estimation via Population Recovery
Steven T. Flammia, Ryan O apos Donnell
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-qubit channel to precision <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03F5;</mml:mi></mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x2113;</mml:mi><mml:mi mathvariant="normal">&#x221E;</mml:mi></mml:msub></mml:math> using just <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>&#x03F5;</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mi>log</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&#x03F5;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&#x03F5;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x2264;</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>4</mml:mn></mml:math>.We then consider the case where the noise channel is close to the identity, meaning that the no-error outcome occurs with probability <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>&#x03B7;</mml:mi></mml:math>. In the regime of small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B7;</mml:mi></mml:math> we extend our algorithm to achieve multiplicative precision <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn><mml:mo>&#x00B1;</mml:mo><mml:mi>&#x03F5;</mml:mi></mml:math> (i.e., additive precision <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03F5;</mml:mi><mml:mi>&#x03B7;</mml:mi></mml:math>) using just <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mrow class="MJX-TeXAtom-OPEN"><mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msup><mml:mi>&#x03F5;</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>&#x03B7;</mml:mi></mml:mrow></mml:mfrac><mml:mrow class="MJX-TeXAtom-CLOSE"><mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&#x03F5;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> applications of the channel.