Learning t-doped stabilizer states
Lorenzo Leone, Salvatore F. E. Oliviero, Alioscia Hamma
Abstract
In this paper, we present a learning algorithm aimed at learning states obtained from computational basis states by Clifford circuits doped with a finite number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi></mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>-gates. The algorithm learns an exact tomographic description of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi></mml:math>-doped stabilizer states in terms of Pauli observables. This is possible because such states are countable and form a discrete set. To tackle the problem, we introduce a novel algebraic framework for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi></mml:math>-doped stabilizer states, which extends beyond <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>-gates and includes doping with any kind of local non-Clifford gate. The algorithm requires resources of complexity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>poly</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>t</mml:mi></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> and exhibits an exponentially small probability of failure.