How many quantum queries are required to determine the coefficients of a degree-
d
polynomial in
n
variables? We present and analyse quantum algorithms for this multivariate polynomial interpolation problem over the fields
F
q
,
R
and
C
. We show that
k
C
and
2
k
C
queries suffice to achieve probability 1 for
C
and
R
, respectively, where
k
C
=
⌈
(
1
/
(
n
+
1
)
)
(
n
+
d
d
)
⌉
except for
d
=2 and four other special cases. For
F
q
, we show that ⌈(
d
/(
n
+
d
))(
n
+
d
d
) ⌉ queries suffice to achieve probability approaching 1 for large field order
q
. The classical query complexity of this problem is (
n
+
d
d
) , so our result provides a speed-up by a factor of
n
+1, (
n
+1)/2 and (
n
+
d
)/
d
for
C
,
R
and
F
q
, respectively. Thus, we find a much larger gap between classical and quantum algorithms than the univariate case, where the speedup is by a factor of 2. For the case of
F
q
, we conjecture that
2
k
C
queries also suffice to achieve probability approaching 1 for large field order
q
, although we leave this as an open problem.