محاسبات کوانتومی - Quantum-Computing
عنوان انگلیسی مقاله:
Epsilon-Nets, Unitary Designs, and Random Quantum Circuits
ترجمه فارسی عنوان مقاله:
شبکه های اپسیلون، طرح های واحد و مدارهای کوانتومی تصادفی
ieee - ieee Transactions on Information Theory;2022;68;2;10:1109/TIT:2021:3128110
Michal Oszmaniec; Adam Sawicki; Michal Horodecki
Epsilon-nets and approximate unitary t-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 d dimensional Hilbert space, unitaries constituting
δ-approximate t-expanders form -nets for t d5/2 and δ
3d/2 d2. We also show that for arbitrary t, -nets can be used
to construct δ-approximate unitary t-designs for δ t, where
the notion of approximation is based on the diamond norm.
Finally, we prove that the degree of an exact unitary t design
necessary to obtain an -net must grow at least as fast as 1 (for
fixed dimension) and not slower than d2 (for fixed ). This shows
near optimality of our result connecting t-designs and 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 twoqudit 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 nonconstructive 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.]
Index Terms: Unitary designs, epsilon nets | random quantum circuits | compilation of quantum gates | unitary channels.
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