Probabilistic eigensolver with a trapped-ion quantum processor

Preparing the eigenstate, especially the ground state, of a complex Hamiltonian is of great importance in quantum simulations. Many proposals have been introduced and experimentally realized, among which are quantum variational eigensolver and heat-bath algorithmic cooling, with the former hindered by local minima and the latter lacking of complex system Hamiltonians. Here we introduce a dissipative quantum-classical hybrid scheme, the probabilistic eigensolver. The scheme repeatedly uses an ancilla qubit to acquire information on the system, based on which it postselectively lowers the average energy of the system. The optimal reduction is achieved through classical optimization with a single variational parameter. We describe the implementation of the probabilistic eigensolver with trapped-ion systems and demonstrate the performance by numerically simulating the ground-state preparation of several paradigmatic models, including the Rabi and the Hubbard models. We believe the scheme would enrich the functionalities of universal quantum simulators and be useful as a module for various quantum-computation tasks.

Figure 1

Scheme for preparing the first-excited state of an $N$-dimensional Hamiltonian. First, we apply ${\widehat{U}}_0$ and measure the ancilla qubit. Then we use postselection to remove the ground-state contribution from the initial state $\left|{\varphi }_0\right.〉$. After that, we apply the cooling unitary ${\widehat{W}}_{\gamma }\left({\tau }_k\right)$ and subsequent ancilla measurement $k$ times, until we reach convergence.

Figure 2

Scheme for obtaining the optimal value of $\tau$ for the $k\mathrm{th}$ stage in the variational approach to the probabilistic eigensolver. First, prepare the state at the $(k-1)\mathrm{st}$ stage by applying the effective cooling $k-1$ times with optimized parameters $\{{\tau }_1,{\tau }_2,\cdots ,{\tau }_{k-1}\}$. Then, search for the optimal value ${\tau }_k$ with the help of a classical optimization algorithm. Specifically, the quantum simulator estimates the average energy and feeds it into the classical algorithm, which determines whether the optimal value is reached or alternatively the next trial value for $\tau$.

Figure 3

Quantum circuits for the trapped-ion toolkit. (a) Scheme for implementing the unitary $exp(-\frac{i\varphi }{2}{\widehat{\sigma }}_{i_1}^{\mathrm{x}}{\widehat{\sigma }}_{i_2}^{\mathrm{x}}{\widehat{\sigma }}_{\mathrm{A}}^{\mathrm{x}})$ by using single-qubit rotations and global Mølmer-Sørensen gates $\mathrm{MS}=exp(-\frac{i\pi }{4}{\sum }_{i<j}{\widehat{\sigma }}_i^x{\widehat{\sigma }}_j^x)$, where the summation runs over all possible pairs of qubits. Here, the single-qubit rotations are defined as $R_{\alpha }\left(\varphi \right)=exp(-\frac{i\varphi }{2}{\widehat{\sigma }}^{\alpha })$ with $\alpha =x, y$, or $z$ and we use $R_{\alpha }\left(\frac{\pi }{2}\right)\equiv R_{\alpha }$ for simplicity. (b) Scheme for implementing the unitary operator $exp[-\frac{i\varphi }{2}(\widehat{a}+{\widehat{a}}^{†}){\widehat{\sigma }}_i^{\mathrm{x}}{\widehat{\sigma }}_{\mathrm{A}}^{\mathrm{y}}]$ using MS gates, single-qubit gates, and the analog block ${\widehat{U}}_{\mathrm{R}}\left(\varphi \right)$ involving the $m\mathrm{th}$ motional mode.

Figure 4

Probabilistic ground-state preparation of a single harmonic mode. (a) Average energy $\langle \widehat{H}\rangle$ (black dots) and the overall success probability $P_{\mathrm{suc}}\left(k\right)$ (blue dots) as functions of the number of iterations $k$ for the fixed-step protocol. The normalized step size $\tau$ is fixed to be 0.3. (b) Average energy $\langle \widehat{H}\rangle$ (red dots over the black curve) and the overall success probability $P_{\mathrm{suc}}\left(k\right)$ (red dots over the blue curve) as functions of the number of iterations $k$ for the optimized variational protocol. Black dots represent results for all the different trial steps that are needed until reaching convergence for the optimum ${\tau }_k$, which are $\{0.8487,0.5044,0.9919,0.9919,0.9910,0.7430,0.4194,0.9881\}$.

Figure 5

Probabilistic ground-state preparation of the quantum Rabi model with ${\omega }_0=1.2, \omega =0.8$, and $g=1.0$. The energy is in units of $g$ and the number of stages is denoted by $k$. The number of Trotter steps is chosen as $r=3$. (a) Fixed-step probabilistic state preparation. The step size is chosen as ${\tau }_{\mathrm{fix}}=0.3$. (b) Variational probabilistic state preparation. The total number of trial steps is denoted by $n$ (black dots). The trial steps for different stages are discriminated by different background colors. Red dots represent the energy and overall probability for different stages using the optimized $\tau$ values $\{0.4762,0.9839,0.9032,0.5575\}$.

Figure 6

Probabilistic ground-state preparation of (a) two-site and (b) three-site Hubbard models. The black dots indicate the average energy of the trial states after successfully projecting the ancilla qubit to the reference state ${\left|0\right.〉}_{\mathrm{A}}$, while the blue dots represent the overall success probability. Red dots represent the average energy and overall success probability after the $k\mathrm{th}$ stage, which is done by applying the ${\widehat{W}}_{\gamma }\left(\tau \right)$ unitary with the optimized ${\tau }_k$, using $r=3$ Trotter steps. The optimized taus are $\{0.3692,0.3959,0.3684,0.3959\}$ and $\{0.2694,0.3661,0.3247,0.3973,0.2726,0.3959\}$ for the two-site and three-site models, respectively.

Figure 7

Noisy analysis. (a) The average fidelity between the ideal and noisy processes for the quantum circuit in Fig. 3. The horizontal axis is the single-qubit dephasing rate per multiqubit gate. (b) The excited-state probability $P_{1,k}$ after $k$ successful steps of cooling, when the probabilistic eigensolver is subjected to a depolarizing environment characterized by the error rate $\delta =0.1$. The input state is chosen as the completely mixed state, and the cooling strength is characterized by ${\varphi }_1=\pi /8, \pi /4$, and $\pi /2$.

Figure 8

Cooling of a two-level system under dephasing. The system Hamiltonian is the one in Eq. (C5) with $b_x=b_z=1$. (a) Average energy after a one cooling step versus the effective evolution time $\tau$. The initial state is $\left|0_S{0}_A\right.〉$ and the average is taken over the postselected final state. The applied cooling unitary $W_{\gamma }\left(\tau \right)$ is depicted in the inset. Solid markers are results obtained in the ideal situation while hollow ones are obtained with dephasing noise. The error rate per MS gate is $p=0.01$. (b) Average energy (solid red) and ground-state fidelity (dashed blue) versus the dephasing strength $p$.