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On the moments of random quantum circuits and robust quantum complexity
Authors
Jonas Haferkamp
Publication date
1 June 2023
Publisher
View
on
arXiv
Abstract
We prove new lower bounds on the growth of robust quantum circuit complexity -- the minimal number of gates
C
δ
(
U
)
C_{\delta}(U)
C
δ
​
(
U
)
to approximate a unitary
U
U
U
up to an error of
δ
\delta
δ
in operator norm distance. More precisely we show two bounds for random quantum circuits with local gates drawn from a subgroup of
S
U
(
4
)
SU(4)
S
U
(
4
)
. First, for
δ
=
Θ
(
2
−
n
)
\delta=\Theta(2^{-n})
δ
=
Θ
(
2
−
n
)
, we prove a linear growth rate:
C
δ
≥
d
/
p
o
l
y
(
n
)
C_{\delta}\geq d/\mathrm{poly}(n)
C
δ
​
≥
d
/
poly
(
n
)
for random quantum circuits on
n
n
n
qubits with
d
≤
2
n
/
2
d\leq 2^{n/2}
d
≤
2
n
/2
gates. Second, for
δ
=
Ω
(
1
)
\delta=\Omega(1)
δ
=
Ω
(
1
)
, we prove a square-root growth of complexity:
C
δ
≥
d
/
p
o
l
y
(
n
)
C_{\delta}\geq \sqrt{d}/\mathrm{poly}(n)
C
δ
​
≥
d
​
/
poly
(
n
)
for all
d
≤
2
n
/
2
d\leq 2^{n/2}
d
≤
2
n
/2
. Finally, we provide a simple conjecture regarding the Fourier support of randomly drawn Boolean functions that would imply linear growth for constant
δ
\delta
δ
. While these results follow from bounds on the moments of random quantum circuits, we do not make use of existing results on the generation of unitary
t
t
t
-designs. Instead, we bound the moments of an auxiliary random walk on the diagonal unitaries acting on phase states. In particular, our proof is comparably short and self-contained.Comment: 13 pages, 1 figure, v2: modified main theorem due to a gap in v
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oai:arXiv.org:2303.16944
Last time updated on 02/04/2023