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We study a sensor network of distributed Mach-Zehnder interferometers (MZIs)for the parallel (simultaneous) estimation of an arbitrary number $d \geq 1$ ofphase shifts. The scheme uses a squeezed-vacuum state that is split between $d$modes by a quantum circuit (QC). Each output mode of the QC is the input of oneof $d$ MZIs, the other input of each MZI being a coherent state. In particular,${\it i}$) we identify the optimal configuration of the sensor network thatallows the estimation of any linear combination of the $d$ phases with maximalsensitivity. The protocol overcomes the shot-noise limit and reaches Heisenbergscalings with respect to the total average number of particles in the overallprobe state, the multiphase estimation only requiring local photocounting.Furthermore, the parallel multiphase estimation overcomes optimal separablestrategies for the estimation of any linear combination of the phases: thesensitivity gain being a factor $d$, at most. Viceversa, ${\it ii}$) given aspecific QC, we identify the optimal linear combination of the phases thatmaximizes the sensitivity and show that results are robust against randomchoices of the QC. Our scheme paves the ways to a variety of applications indistributed quantum sensing.

Distributed Quantum Sensing with Squeezed-Vacuum Light in a Configurable Network of Mach-Zehnder Interferometers