Algorithm for the solution of the Dirac equation on digital quantum computers

A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator-splitting decomposition technique that allows for a mapping of the Dirac operator to a quantum walk supplemented by unitary rotation steps in spinor space. Every step of the splitting method is decomposed into sets of quantum gates. It is demonstrated that the algorithm has an exponential speed-up over the implementation of the same numerical scheme on a classical computer, as long as certain conditions are satisfied. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out to determine the resource requirements. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.

Figure 1

Circuit diagram for the rotation operator in spinor space. The other qubits of the register ($|i\rangle ,|j\rangle ,|k\rangle$) are not modified by this transformation.

Figure 2

Circuit diagram for the increment and decrement operators acting on qubits storing the data for a given dimension (the set of $n_a$ qubits). This implementation is borrowed from [60].

Figure 3

Circuit diagram that implements the local operator $Q_{\mathbf{A}}$. The bottom picture displays explicitly the decomposition of the matrices ${\mathcal{Q}}_{\mathbf{A}}$ and ${\mathcal{Q}}_{\mathbf{A}}^{†}$, appearing in the top picture.

Figure 4

Circuit diagram that evolves the wave function by one time step. The details of the spin rotation gates $H_a{|}_{a=x,y,z}$ are given in Fig. 1 and Eq. (46). The increment and decrement operators $I$ and $D$, respectively, are detailed in Fig. 2 and Eqs. (47, 48, 49, 50, 51, 52). The mass operator $Q_m$ is given in Eq. (54). Finally, the operator $Q_{\mathbf{A}}$ is displayed in Fig. 3 and is explicitly written in Eqs. (58, 59, 60, 61, 62, 63).

Figure 5

Circuit depth (number of gates) required to evolve the wave function by one time step as a function of the number of qubits. The number of qubits corresponds to $n_x$. It is assumed that $n_x=n_y=n_z$. The circuit width required is then given by $n_{\mathrm{total}}=4n_x+2$.

Figure 6

Quantum circuit for the inclusion of a magnetic field.

Figure 7

Definition and circuit diagram for the grouped uniformly controlled gate for $k+1$ qubits [68]. The half-black–half-white controls on the left-hand side imply a sum over all black and white controls with a different operator $U_{i,j,k}$.

Figure 8

Circuit diagram for the initialization of the wave function.

Figure 9

Circuit diagram for the computation of the autocorrelation function. The quantity $\langle ({\sigma }_x+i{\sigma }_y)\otimes \mathbb{I}\rangle$ is measured on the ancilla qubit.

Figure 10

Circuit diagram for the quantum implementation of the Feit-Fleck method. The gate $U_i:=U[(i-1)\mathrm{\Delta }t,i\mathrm{\Delta }t]$ advances the solution by $\mathrm{\Delta }t$ and its decomposition into quantum gates is given in Fig. 4. The gate $B_i$ is defined in Eq. (D10).

Figure 11

Circuit diagram for the implementation of the nonunitary operation. The upper qubit is an ancilla qubit prepared in the state $|0\rangle$. The measurement operator implements the projective measurement $|0\rangle \langle 0|$. If the measurement yields the state $|1\rangle$, the calculation has to be redone from the beginning.