Unitary <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>-Designs from Random Number-Conserving Quantum Circuits

Local random circuits scramble efficiently and, accordingly, have a range of applications in quantum information and quantum dynamics. With a global U(1) charge, however, the scrambling ability is reduced; for example, such random circuits do not generate the entire group of number-conserving unitaries. We establish two results using the statistical mechanics of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>k</a:mi></a:math>-fold replicated circuits. First, we show that finite moments cannot distinguish the ensemble that local random circuits generate from the Haar ensemble on the entire group of number-conserving unitaries. Specifically, the circuits form a <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>k</c:mi><c:mi>c</c:mi></c:msub></c:math>-design with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>k</e:mi><e:mi>c</e:mi></e:msub><e:mo>=</e:mo><e:mi>O</e:mi><e:mo stretchy="false">(</e:mo><e:msup><e:mi>L</e:mi><e:mi>d</e:mi></e:msup><e:mo stretchy="false">)</e:mo></e:math> for a system in <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>d</i:mi></i:math> spatial dimensions with linear dimension <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>L</k:mi></k:math>. Second, for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>k</m:mi><m:mo>&lt;</m:mo><m:msub><m:mi>k</m:mi><m:mi>c</m:mi></m:msub></m:math>, we derive bounds on the depth <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>τ</o:mi></o:math> required for the circuit to converge to an approximate <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>k</q:mi></q:math>-design. The depth is lower bounded by diffusion <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>k</s:mi><s:msup><s:mi>L</s:mi><s:mn>2</s:mn></s:msup><s:mi>ln</s:mi><s:mo stretchy="false">(</s:mo><s:mi>L</s:mi><s:mo stretchy="false">)</s:mo><s:mo>≲</s:mo><s:mi>τ</s:mi></s:math>. In contrast, without number conservation <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>τ</w:mi><w:mo>∼</w:mo><w:mrow><w:mi>poly</w:mi></w:mrow><w:mo stretchy="false">(</w:mo><w:mi>k</w:mi><w:mo stretchy="false">)</w:mo><w:mi>L</w:mi></w:math>. The convergence of the circuit ensemble is controlled by the low-energy properties of a frustration-free quantum statistical model which spontaneously breaks <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>k</ab:mi></ab:math> U(1) symmetries. We conjecture that the associated Goldstone modes set the spectral gap for arbitrary spatial and qudit dimensions, leading to an upper bound <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>τ</cb:mi><cb:mo>≲</cb:mo><cb:mi>k</cb:mi><cb:msup><cb:mi>L</cb:mi><cb:mrow><cb:mi>d</cb:mi><cb:mo>+</cb:mo><cb:mn>2</cb:mn></cb:mrow></cb:msup></cb:math>.