Parameter estimation with mixed-state quantum computation

We present a quantum algorithm to estimate parameters at the quantum metrology limit using deterministic quantum computation with one bit. When the interactions occurring in a quantum system are described by a Hamiltonian $H=\theta H_0$, we estimate $\theta$ by zooming in on previous estimations and by implementing an adaptive Bayesian procedure. The final result of the algorithm is an updated estimation of $\theta$ whose variance has been decreased in proportion to the time of evolution under $H$. For the problem of estimating several parameters, we implement dynamical-decoupling techniques and use the results of single parameter estimation. The cases of discrete-time evolution and reference frame alignment are also studied within the adaptive approach.

Figure 1

DQC1 circuit for estimating the trace of a unitary $U$. The filled circle denotes that $U$ acts on the probe when the state of the ancilla (control qubit) is $\mid 1{⟩}_{\mathsf{a}}$, and $H$ is the Hadamard gate. $⟨{\sigma }_x^{\mathsf{a}}⟩$ and $⟨{\sigma }_y^{\mathsf{a}}⟩$ are the expectation values of the corresponding Pauli operators on the ancilla $\mathsf{a}$.

Figure 2

DQC1 circuit for the estimation of $\mathrm{tr}\left[W^{†}\right(T\left){\sigma }_{\mu ,j}W\right(T\left){\sigma }_{{\mu }^{\prime },j^{\prime }}\right]∕2^n\equiv ⟨{\sigma }_x^{\mathsf{a}}⟩$. Note that $⟨{\sigma }_y^{\mathsf{a}}⟩=0$ in this case. Since every unitary is controlled by $\mathsf{a}$ (filled circles), including the evolution operator $W\left(T\right)$, the execution of this algorithm may be unfeasible due to the nature of the problem.

Figure 3

Simplified version of the DQC1 circuit of Fig. 2. The operators ${}^{c}W\left(T\right)$ and $W^{†}\left(T\right)$ are avoided. The estimation of $\cos \left(2\theta T\right)$ or $\sin \left(2\theta T\right)$ [Eqs. (9, 10)] is performed using available resources only.

Figure 4

Elementary step of the frame alignment protocol. The $n$-qubit probe, whose state remains completely mixed, is exchanged between Alice and Bob. The Pauli products ${\sigma }_{\mu ,j}$ and ${\sigma }_{{\mu }^{\prime },j^{\prime }}$ appear in the decomposition of $H_1$ [see Eqs (9, 10)].