Experimental simulation of shift operators in a quantum processor

The ability to implement quantum operations plays a fundamental role in manipulating quantum systems. Creation and annihilation operators which transform one quantum state into another by adding or subtracting a particle are crucial in constructing the quantum description of many-body quantum theory and quantum field theory. Here we present a quantum algorithm to perform the creation and annihilation operators by the linear combination of unitary operations associated with a two-qubit ancillary system. Our method can realize shift operators akin to creation and annihilation operators simultaneously in the subspace of the whole system. A prototypical experiment was performed with a four-qubit liquid-state nuclear magnetic resonance processor, demonstrating the algorithm via full-state tomography. With a postselected probability of about 50%, the shift operators are realized with a fidelity above 96%. Moreover, our method can be employed to quantum random walk in an arbitrary initial state. With the prosperous development of quantum computing, our work provides a quantum control technology to implement nonunitary evolution in a near-term quantum computer.

Figure 1

Quantum circuit of realizing addition and subtraction operators. $|\mathrm{\Psi }\rangle$ denotes the initial state of the work system, and $|00\rangle$ is the initial state of the auxiliary system. The squares represent unitary operations, and the circles represent the state of the controlling qubit. Unitary operations $U_0, U_1, U_2$, and $U_3$ are activated only when the auxiliary qubit is $|00\rangle , |01\rangle , |10\rangle$, and $|11\rangle$, respectively. We “read out” the output of the auxiliary system and construct the quantum density matrix of the work system. We can realize the addition operator for the auxiliary system in states $|00\rangle$ and $|01\rangle$ and subtraction operator for the auxiliary system in states $|10\rangle$ and $|11\rangle$.

Figure 2

Molecular structure and Hamiltonian parameters of ${}^{13}\mathrm{C}$-labeled trans-crotonic acid. ${\mathrm{C}}_1, {\mathrm{C}}_2, {\mathrm{C}}_3$, and ${\mathrm{C}}_4$ are used as four qubits. The chemical shifts and $J$ couplings (in Hz) are listed by the diagonal and off-diagonal elements, respectively. $T_2$ (in seconds) are also shown at the bottom.

Figure 3

NMR sequence to realize the addition and subtraction operators. In the first step, we use a spatial averaging method to prepare the pseudopure state. In this part, the purple rectangles stand for the unitary operations which rotate ${45}^{\circ }$ with a coordinate axis. Similarly, the green rectangles rotate ${90}^{\circ }$. The white rectangles stand for the unitary operations which rotate with the special angle with the $X$ axis. The blue rectangles stand for the unitary operations which rotate ${180}^{\circ }$ with the $Y$ axis. The double straight lines stand for the gradient field in the $z$ direction. In the second step, the radio-frequency pulses during this procedure are optimized using the GRAPE technology to realize the controlled operator. In the third step, we measure the first two qubits and use the quantum tomography technology to obtain the final density matrix. The brown squares are the auxiliary pulse which can be used for quantum tomography.

Figure 4

The spectrum of the thermal equilibrium state and PPS. (a) The spectrum of the thermal equilibrium state. The $x$ axis is the relative frequency, and the $y$ axis is the amplitude in arbitrary units. Since the qubits are all carbon atoms, all signals of these four qubits can be detected through the carbon channel in NMR experiments. (b) The spectrum of spin C1 in the thermal equilibrium state. This is an expanded view of spin C1 in (a). From plot, we can observe more information in detail. The quantum state of the remaining spins (C2, C3 and C4) is as indicated based on J12, J13, and J14 shown in Fig. 2. (c) The spectrum of the PPS. As we know, the ground state $|0000\rangle$ cannot be detected directly in NMR, and we can apply $R_y^i(\pi /2)$ to it to obtain a single peak of the $i\mathrm{th}$ spin. In our case, only the peak in the spectrum of C1 can be detected by applying the $R_y^1(\pi /2)$ pulse. (d) The spectrum of spin C1 in the PPS. This is also an expanded view of spin C1 in (c). It is clear that the peaks standing for the quantum state except $|0\rangle$ are close to zero.

Figure 5

The density matrix of the initial quantum state and final state (in the normalized units). (a) The initial quantum state $|01\rangle$. (b) The final state when the measurement of ancillary qubits is $|00\rangle$. (c) The final state when the measurement of ancillary qubits is $|10\rangle$. The left part is the real part of the density matrix, and the right part is the imaginary part in (b) and (c).

Figure 6

The density matrix of the initial quantum state and final state (in the normalized units). (a) The initial quantum state $\frac{|01\rangle +|10\rangle }{2}$. (b) The final state when the measurement of ancillary qubits is $|00\rangle$. (c) The final state when the measurement of ancillary qubits is $|10\rangle$. The left part is the real part of the density matrix, and the right part is the imaginary part in (b) and (c).

Figure 7

The real part of the density matrix of the initial quantum state and final state (in the normalized units). (a) The initial quantum state is $\left(|00\rangle +|01\rangle +|10\rangle +|11\rangle \right)/2$. We apply the $U_0$ operator on the initial state; the final state is shown in (b). The results of the work system are shown in (c) when the ancillary qubit subspace is $|00\rangle$. Following the same setups as in the statement above, we realize subtraction operator on the initial state, as shown in (e) when the ancillary qubit subspace is $|10\rangle$. Similarly, we can perform ${\widehat{J}}^{†}$ with the ancillary qubit $|01\rangle$, as shown in (d), and $\widehat{J}$ with the ancillary qubit $|11\rangle$, as shown in (f).

Figure 8

The simulation of a quantum random walk when the initial state is $|01000000\rangle$ (128). The $x$ axis represents the quantum state, from $|00000000\rangle$ (0) to $|11111111\rangle$ (255). The $y$ axis represents the measurement probabilities (in the normalized units). The solid circles are the probabilities of each state. The statistical distribution can be observed by the line linked by the nonzero circles.

Figure 9

The simulation of a quantum random walk when the initial state is $\frac{|01000000\rangle +|01000001\rangle }{2}$ (128, 129). The $x$ axis represents the quantum state, from $|00000000\rangle$ (0) to $|11111111\rangle$ (255). The $y$ axis represents the measurement probabilities (in the normalized units). The solid circles are the probabilities of each state. The statistical distribution can be observed by the line linked by all the circles.