Epsilon-Nets, Unitary Designs, and Random Quantum Circuits

• Published in: IEEE transactions on information theory Vol. 68; no. 2; pp. 989 - 1015
• Main Authors: Oszmaniec, Michal, Sawicki, Adam, Horodecki, Michal
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.02.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Circuit design; compilation of quantum gates; Diamond; Diamonds; epsilon nets; Expanders; Gates; Hilbert space; Layout; Logic gates; Mathematical analysis; Polynomials; Quantum channels; Quantum circuit; Quantum computing; Quantum phenomena; random quantum circuits; unitary channels; Unitary designs
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2021.3128110

Epsilon-nets and approximate unitary <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-designs are natural notions that capture properties of unitary operations relevant for numerous applications in quantum information and quantum computing. In this work we study quantitative connections between these two notions. Specifically, we prove that, for <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> dimensional Hilbert space, unitaries constituting <inline-formula> <tex-math notation="LaTeX">\delta </tex-math></inline-formula>-approximate <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-expanders form <inline-formula> <tex-math notation="LaTeX">\epsilon </tex-math></inline-formula>-nets for <inline-formula> <tex-math notation="LaTeX">t\simeq \frac {d^{5/2}}{ \epsilon } </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\delta \simeq \left ({\frac { \epsilon ^{3/2}}{d}}\right)^{d^{2}} </tex-math></inline-formula>. We also show that for arbitrary <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">\epsilon </tex-math></inline-formula>-nets can be used to construct <inline-formula> <tex-math notation="LaTeX">\delta </tex-math></inline-formula>-approximate unitary <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-designs for <inline-formula> <tex-math notation="LaTeX">\delta \simeq \epsilon t </tex-math></inline-formula>, where the notion of approximation is based on the diamond norm. Finally, we prove that the degree of an exact unitary <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula> design necessary to obtain an <inline-formula> <tex-math notation="LaTeX">\epsilon </tex-math></inline-formula>-net must grow at least as fast as <inline-formula> <tex-math notation="LaTeX">\frac {1}{ \epsilon } </tex-math></inline-formula> (for fixed dimension) and not slower than <inline-formula> <tex-math notation="LaTeX">d^{2} </tex-math></inline-formula> (for fixed <inline-formula> <tex-math notation="LaTeX">\epsilon </tex-math></inline-formula>). This shows near optimality of our result connecting <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-designs and <inline-formula> <tex-math notation="LaTeX">\epsilon </tex-math></inline-formula>-nets. We apply our findings in the context of quantum computing. First, we show that that approximate t-designs can be generated by shallow random circuits formed from a set of universal two-qudit gates in the parallel and sequential local architectures considered in (Brandão et al., 2016). Importantly, our gate sets need not to be symmetric (i.e., contains gates together with their inverses) or consist of gates with algebraic entries. Second, we consider compilation of quantum gates and prove a non-constructive Solovay-Kitaev theorem for general universal gate sets. Our main technical contribution is a new construction of efficient polynomial approximations to the Dirac delta in the space of quantum channels, which can be of independent interest.