UBC Theoretical Computer Science Seminar: Xingyu Zhou

We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most $t$ samples.  For sufficiently small $\eps$, estimating an unknown state on $C^d$ of rank at most $r$ to trace norm error $\eps$ with constant success probability requires, and is achievable with,

$$\Theta\left(\frac{dr}{\eps^2}\max\left\{1,\frac r{\sqrt t}\right\}\right)$$

samples.  The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is
nonadaptive.  Thus joint measurements on at most $t$ samples improve the complexity of algorithms making single-sample measurements by at most a factor $\sqrt t$. Further, measuring order $r^2$ samples jointly is necessary and sufficient to attain the unrestricted collective rate.

For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on $t$ samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound.  For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement.  An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.

The talk will not assume significant background in quantum computing.