Quantum State Designs with Clifford-Enhanced Matrix Product States

Nonstabilizerness, or “magic,” is a critical quantum resource that, together with entanglement, characterizes the nonclassical complexity of quantum states. Here, we address the problem of quantifying the average nonstabilizerness of random matrix product states (RMPSs). RMPSs represent a generalization of random product states featuring bounded entanglement that scales logarithmically with the bond dimension <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mi>χ</a:mi></a:math>. We demonstrate that the stabilizer Rényi entropies converge to that of Haar-random states as <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><d:mi>N</d:mi><d:mo>/</d:mo><d:msup><d:mi>χ</d:mi><d:mi>α</d:mi></d:msup></d:math>, where <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><g:mi>N</g:mi></g:math> is the system size and the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><j:mi>α</j:mi></j:math> are integer exponents. This indicates that MPSs with a modest bond dimension are as magical as generic states. Subsequently, we introduce the ensemble of Clifford-enhanced matrix product states (<m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><m:mrow><m:mi mathvariant="script">C</m:mi></m:mrow><m:mrow><m:mi>MP</m:mi><m:mi>Ss</m:mi></m:mrow></m:math>), built by the action of Clifford unitaries on RMPSs. Leveraging our previous result, we show that <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><q:mrow><q:mi mathvariant="script">C</q:mi></q:mrow><q:mrow><q:mi>MP</q:mi><q:mi>Ss</q:mi></q:mrow></q:math> can approximate quantum state 4-designs with arbitrary accuracy. Specifically, for a constant <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><u:mi>N</u:mi></u:math>, <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><x:mrow><x:mi mathvariant="script">C</x:mi></x:mrow><x:mrow><x:mi>MP</x:mi><x:mi>Ss</x:mi></x:mrow></x:math> become close to 4-designs, with a scaling as <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><bb:msup><bb:mi>χ</bb:mi><bb:mrow><bb:mo>−</bb:mo><bb:mn>2</bb:mn></bb:mrow></bb:msup></bb:math>. Our findings indicate that combining Clifford unitaries with polynomially complex tensor-network states can generate highly nontrivial quantum states.