Computational Complexity of Unitary and State Design Properties

We investigate unitary and state <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>t</a:mi></a:math>-designs from a computational complexity perspective. First, we address the problems of computing frame potentials that characterize (approximate) <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>t</c:mi></c:math>-designs. We present a quantum algorithm for computing frame potentials and establish the following: (1) exact computation can be achieved by a single query to a <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mi mathvariant="normal">#</e:mi><e:mrow><e:mrow><e:mrow><e:mi mathvariant="sans-serif">P</e:mi></e:mrow></e:mrow></e:mrow></e:mrow></e:math> oracle and is <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:mi mathvariant="normal">#</i:mi><i:mrow><i:mrow><i:mrow><i:mi mathvariant="sans-serif">P</i:mi></i:mrow></i:mrow></i:mrow></i:mrow></i:math>-hard; (2) for state vectors, deciding whether the frame potential is larger than or smaller than certain values is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:mrow><m:mrow><m:mi mathvariant="sans-serif">B</m:mi><m:mi mathvariant="sans-serif">Q</m:mi><m:mi mathvariant="sans-serif">P</m:mi></m:mrow></m:mrow></m:mrow></m:math>-complete, provided that the promise gap between the two values is inverse polynomial in the number of qubits; and (3) for both state vectors and unitaries, this promise problem is <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mrow><r:mrow><r:mrow><r:mi mathvariant="sans-serif">P</r:mi><r:mi mathvariant="sans-serif">P</r:mi></r:mrow></r:mrow></r:mrow></r:math>-complete if the promise gap is exponentially small. Second, we address the promise problem of deciding whether or not a given set is a good approximation to a design. Given a certain promise gap that could be constant, we show that this problem is <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mrow><v:mrow><v:mrow><v:mi mathvariant="sans-serif">P</v:mi><v:mi mathvariant="sans-serif">P</v:mi></v:mrow></v:mrow></v:mrow></v:math>-hard, highlighting the inherent computational difficulty of determining properties of unitary and state designs. We further identify implications of our results, including variational methods for constructing designs, diagnosing quantum chaos, and exploring emergent designs in Hamiltonian systems.