Compressed quantum metrology for the Ising Hamiltonian

We show how quantum metrology protocols that seek to estimate the parameters of a Hamiltonian that exhibits a quantum phase transition can be efficiently simulated on an exponentially smaller quantum computer. Specifically, by exploiting the fact that the ground state of such a Hamiltonian changes drastically around its phase-transition point, we construct a suitable observable from which one can estimate the relevant parameters of the Hamiltonian with Heisenberg scaling precision. We then show how, for the one-dimensional Ising Hamiltonian with transverse magnetic field acting on $N$ spins, such a metrology protocol can be efficiently simulated on an exponentially smaller quantum computer while maintaining the same Heisenberg scaling for the squared error, i.e., $O\left(N^{-2}\right)$ precision, and derive the explicit circuit that accomplishes the simulation.

Figure 1

Schematic representation of the compressed quantum circuit which can be used to measure $\langle \mathcal{B}\rangle (B,J)$. The initial state is the $(m+1)$-qubit state $\left|\mathrm{\Phi }\right.〉$, which is a product state of $m$ single-qubit states $\left|{\phi }_l\right.〉=\frac{1}{\sqrt{2}}\left(\left|0\right.〉+e^{i2\pi 2^{l-m}}\left|1\right.〉\right)$ for $0\le l\le m-1$ and one single-qubit state $\left|{+}_{\mathrm{y}}\right.〉$. The initial state is transformed by an orthogonal transformation $R^{\mathsf{T}}$, which is decomposed into more elementary operations in Fig. 2. The circuit ends with a measurement of the last qubit on the $Y$ basis.

Figure 2

Depiction of the decomposition of one Trotter step of the matrix $R^{\mathsf{T}}$ into more elementary operations. The gate $R^{\mathsf{T}}$, whose transpose is given in Eq. (37), consists of a product of $L+1$ similar steps. The unitary operation $A$, given in Eq. (38), can be implemented as a product of $m$ controlling operations, as shown in Fig. 3. Furthermore, the single-qubit gate $S_1$ depends on the unknown parameter $J$, which we wish to estimate. In Fig. 4 we show how this single-qubit gate can be implemented as a time evolution under the action of a two-qubit Ising Hamiltonian.

Figure 3

Decomposition of the unitary $A$ given in Eq. (40) as a product of $m$ controlled-$X$ gates and a single $X$ gate acting on the last qubit. This operation corresponds to the mapping of $\left|j\right.〉$ to $\left|j+1\mathrm{mod}\left(2N\right)\right.〉$ for any $j$. As $O\left(k\right)$ elementary gates can be used to implement a $k$-qubit controlled-$X$ operation [38], $A$ is implementable with $O\left(m^2\right)$ elementary gates.

Figure 4

The implementation of the single qubit $S_1(J,l)$ using an auxiliary qubit prepared in the state $\left|+\right.〉$ and the two-qubit gate $R_{\mathrm{XX}}(J,l)$, which can be implemented by letting two qubits evolve with interactions governed by a two-qubit Hamiltonian $H(B=0,J)$ for a time $\tau \left(l\right)=2l\mathrm{\Delta }/L$.