Efficient mutual magic and magic capacity with matrix product states
Poetri Sonya Tarabunga, Tobias Haug
Abstract
Stabilizer Rényi entropies (SREs) probe the non-stabilizerness (or “magic”) of many-body systems and quantum computers. Here, we introduce the mutual von-Neumann SRE and magic capacity, which can be efficiently computed in time O(N\chi^3) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> for matrix product states (MPSs) of bond dimension \chi <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>χ</mml:mi> </mml:math> . We find that mutual SRE characterizes the critical point of ground states of the transverse-field Ising model, independently of the chosen local basis. Then, we relate the magic capacity to the anti-flatness of the Pauli spectrum, which quantifies the complexity of computing SREs. The magic capacity characterizes transitions in the ground state of the Heisenberg and Ising model, randomness of Clifford+T circuits, and distinguishes typical and atypical states. Finally, we make progress on numerical techniques: we design two improved Monte-Carlo algorithms to compute the mutual 2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>2</mml:mn> </mml:math> -SRE, overcoming limitations of previous approaches based on local update. We also give improved statevector simulation methods for Bell sampling and SREs with O(8^{N/2}) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msup> <mml:mn>8</mml:mn> <mml:mrow> <mml:mi>N</mml:mi> <mml:mi>/</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> time and O(2^N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> memory, which we demonstrate for 24 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>24</mml:mn> </mml:math> qubits. Our work uncovers improved approaches to study the complexity of quantum many-body systems.