Projective measurement of energy on an ensemble of qubits with unknown frequencies

In projective measurements of energy, a target system is projected to an eigenstate of the system Hamiltonian and the measurement outcomes provide the information of corresponding eigenenergies. Recently, it has been shown that such a measurement can be, in principle, realized without detailed knowledge of the Hamiltonian by using probe qubits. However, in this approach, the size of the dimension for the probe increases as we increase the dimension of the target system; also, individual addressability of every qubit is required, which may not be possible for many experimental settings involving large systems. Here, we show that a single probe qubit is sufficient to perform such a projective measurement of energy if the target system is composed of noninteracting qubits whose resonant frequencies are unknown. Moreover, our scheme requires only global manipulations where every qubit is subjected to the same control fields. These results indicate the feasibility of our energy-projection protocols.

Figure 1

A quantum circuit representing our protocol for projective measurement of energy. A combination of the cnot gates and free time evolution for a time $t$ provides the probe qubit with a phase information of the target systems. To avoid decoherence on the probe qubit, we perform a swap gate between the probe qubit and memory qubit (with a long coherence time) before and after the free evolution of the target system. To perform a Fourier basis measurement, we reset, rotate, and measure the probe qubit $L$ times, where $R_j$ denotes the unitary rotation whose angle is defined by the previous measurement outcomes: $R_j^{\prime }=|1\rangle \langle 1|-{\varphi }_j^{\prime }|0\rangle \langle 0|$ with ${\varphi }_j^{\prime }=\mathrm{Exp}[-2\pi i{\sum }_{k=2}^j{m}_{j-k}/2^k]$. The measurement outcomes reveal the energy eigenvalues of the target qubits and induce the energy projection.

Figure 2

The standard deviations ($\sigma$) of estimation against the number of measurements ($L$). The parameters are $g/2\pi =100$ kHz, $t\sqrt{N}=0.16$ ms, and ${\sigma }_{\mathrm{G}}/2\pi =1\mathrm{kHz}$, where $N$ denotes the number of the target qubits. In our scheme, $\sigma$ decreases exponentially with $L$.

Figure 3

The projection errors $(1-F)$ against the number of measurements $\left(L\right)$, where $F$ denotes a fidelity of the postmeasurement states. The projection error increases linearly with the number of the target qubits $N$. We use the same parameters as Fig. 2.

Figure 4

The purity of the state against the number of measurements. We use the same parameters as in Fig. 2. The purity approaches to the unity as we increase the number of the measurements. Each continuous line is a guide for the eyes.