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Bad Qubits
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To reach accuracy
ϵ
\epsilon
ϵ
over time
t
t
t
, first-order Trotter needs
r
=
O
(
t
2
/
ϵ
)
r = O(t^2/\epsilon)
r
=
O
(
t
2
/
ϵ
)
steps. How does the accuracy (
ϵ
\epsilon
ϵ
) dependence of LCU, qubitization, and QSP compare?
They scale quadratically in
1
/
ϵ
1/\epsilon
1/
ϵ
— worse than Trotter
They scale only logarithmically in
1
/
ϵ
1/\epsilon
1/
ϵ
— exponentially better in accuracy
They have no dependence on
ϵ
\epsilon
ϵ
at all
They have exactly the same
1
/
ϵ
1/\epsilon
1/
ϵ
dependence as first-order Trotter
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