Scalable projected entangled-pair state representation of random quantum circuit states

Classical simulation of a programmable quantum processor is crucial in identifying the threshold of a quantum advantage. We demonstrate the simple update of projected entangled-pair states (PEPSs) in the Vidal gauge that represent random quantum circuit states, which center around recent quantum advantage claims. Applied to square lattices of qubits akin to state-of-the-art superconducting processors, the PEPS representation is exact for circuit depths less than <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi mathvariant="script">D</a:mi><a:mi>tr</a:mi></a:msub><a:mo>=</a:mo><a:mrow><a:mi>β</a:mi><a:msub><a:mo form="prefix">log</a:mo><a:mn>2</a:mn></a:msub><a:mi>χ</a:mi></a:mrow></a:math>, where <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>χ</d:mi></d:math> is the maximum bond dimension and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mn>2</e:mn><e:mo>≲</e:mo><e:mi>β</e:mi><e:mo>≲</e:mo><e:mn>4</e:mn></e:mrow></e:math> depends on the choice of two-qubit gates, independent of the qubit number <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>n</f:mi></f:math>. We find the universal scaling behaviors of the state fidelity by treating large-scale circuits of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>n</g:mi><g:mo>≤</g:mo><g:msup><g:mn>10</g:mn><g:mn>4</g:mn></g:msup></g:mrow></g:math>, using <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>χ</h:mi><h:mo>≤</h:mo><h:mn>128</h:mn></h:mrow></h:math> on a conventional CPU. Our method has a polynomial scaling of computational costs with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>n</i:mi></i:math> for circuit depth <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi mathvariant="script">D</j:mi><j:mo>=</j:mo><j:mi>O</j:mi><j:mo>(</j:mo><j:mo form="prefix">log</j:mo><j:mi>n</j:mi><j:mo>)</j:mo></j:mrow></j:math> and is more advantageous than matrix product state approaches if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>n</m:mi></m:math> is large. This work underscores PEPSs as a scalable tool for benchmarking quantum algorithms with future potential for sampling applications using advanced contraction techniques.