Quantum algorithm for obtaining the eigenstates of a physical system

We propose a quantum algorithm for solving the following problem: given the Hamiltonian of a physical system and one of its eigenvalues, how do we obtain the corresponding eigenstate? The algorithm is based on the resonance phenomenon. For a probe qubit coupled to a quantum system, the system exhibits resonance dynamics when the frequency of the probe qubit matches a transition frequency in the system. Therefore the system can be guided to evolve to the eigenstate with a known eigenvalue by inducing the resonance between the probe qubit and a designed transition in the system. This algorithm can also be used to obtain the energy spectrum of a physical system and can achieve even quadratic speedup over the phase estimation algorithm.

Figure 1

Quantum circuit for obtaining the eigenstates of a physical system. $U\left(\tau \right)$ is a time evolution operator driven by a Hamiltonian given in Eq. (1). The first line represents a probe qubit, the second line is an ancilla qubit, and the last $n$ qubits represent the quantum system.

Figure 2

The probability of the entire $(n+2)$-qubit system being evolved from the state $|{\mathrm{\Psi }}_0\rangle$ to the state $|{\mathrm{\Psi }}_1\rangle$ vs the evolution time $t$ and $E^{\prime }$. The parameter $\alpha =1$. Atomic units are used.

Figure 3

The same as in Fig. 2, except $\alpha =0$. Atomic units are used.

Figure 4

The variation of $P$ vs $E^{\prime }$ at $t=\frac{\pi }{2}\frac{1}{d^{1+\alpha }}$ for $\alpha =0,0.5,1$ while setting $d=0.01$. The black solid line shows the results for $\alpha =0$; the red dashed line shows the results for $\alpha =0.5$, and the blue dotted line shows the results for $\alpha =1$. Atomic units are used.

Figure 5

The variation of $P$ vs the evolution time $t$ for $\alpha =0,0.5,1$ while setting $d=0.1$ and $E^{\prime }=5$. The black solid line shows the results for $\alpha =0$; the red dashed line shows the results for $\alpha =0.5$, and the blue dotted line shows the results for $\alpha =1$. Atomic units are used.

Figure 6

The variation of the exponent $\alpha$ vs $d$ for $E^{\prime }=2,5,10,20$ (atomic units are used) while keeping $P=0.9$. The black squares show the results for $E^{\prime }=2$, the red circles show the results for $E^{\prime }=5$, the blue triangles show the results for $E^{\prime }=10$, and the cyan stars show the results for $E^{\prime }=20$.