Quantum theory allows the traversing of multiple channels in a superposition of different orders. When the order in which the channels are traversed is controlled by an auxiliary quantum system, various unknown parameters of the channels can be estimated by measuring only the control system, even when the state of the probe alone would be insensitive. Moreover, increasing the dimension of the control system increases the number of simultaneously estimable parameters, which has important metrological ramifications. We demonstrate this capability for simultaneously estimating both unitary and noise parameters, including multiple parameters from the same unitary such as rotation angles and axes and from noise channels such as depolarization, dephasing, and amplitude damping in arbitrary dimensions. We identify regimes of unlimited advantages, taking the form of
p2 smaller variances in estimation when the noise probability is $1\ensuremath{-}p$, for both single and multiparameter estimations when using our schemes relative to any comparable scheme whose causal order is definite.
Indefinite causal order is known to offer enhancements in quantum metrology, notably including an unbounded advantage in the measurement of a geometric phase of a harmonic oscillator. This advantage, however, is specific to the infinite dimensional setting, and its finite dimensional analogue remains elusive, with recent findings suggesting that advantages in finite dimensions may be fundamentally limited to bounded constant factors. Here we show that, in fact, arbitrarily large advantages arise for finite dimensional systems in the finite sample regime. Specifically, we establish an unbounded separation between definite and indefinite causal order in the estimation of a geometric phase associated to two sets of
N displacements generated by discrete position and momentum operators on a
d-dimensional quantum system: for any given constant
R, there exist values of
N and
d=Ω(N2) such that a strategy with indefinite order uses an initial probe with
R times less energy than the probe required by every strategy with definite order achieving the same mean squared error, whenever the number of measurement shots
ν is bounded as
ν=O(exp(πd/16)/poly(d)). In other words, indefinite order offers an energy saving that grows arbitrarily large with the parameters of the problem. To prove this result, we establish an approximate Weyl relation for discrete Gaussian wavepackets, which is of independent technical interest.