Shor’s quantum algorithm using electrons in semiconductor nanostructures

Shor’s factoring algorithm illustrates the potential power of quantum computation. Here, we present and numerically investigate a proposal for a compiled version of such an algorithm based on a quantum-wire network by exploiting the potential of fully coherent electron transport assisted by the surface acoustic waves. Specifically, a nonstandard approach is used to implement, in a simple form, the quantum circuits of the modular exponentiation execution for the simplest instance of Shor’s algorithm, that is, the factorization of $N=15$. The numerical procedure is based on a time-dependent solution of the multiparticle Schrödinger equation. The near-ideal algorithm performance and the large estimated fidelity indicate the efficiency of the protocol implemented, which also is almost insensitive to small destabilizing effects during quantum computation.

Figure 1

Left: Outline of the quantum circuit for quantum factorization of 15 for $C=11$, using the inverse QFT. [8] Right: Outline of the quantum circuit for quantum factorization of 15 for $C=11$, not using the inverse QFT.

Figure 2

Left: Outline of the quantum circuit for quantum factorization of 15 for $C=2$, using the inverse QFT and evaluating $2^x\hspace{0.5em}\mathrm{mod}\hspace{0.5em}15$ in the function register. Right: Outline of the quantum circuit for quantum factorization of 15 for $C=2$, not using the inverse QFT and evaluating $\log {}_2[2^x\hspace{0.5em}\mathrm{mod}\hspace{0.5em}15]$ in the function register. The two circuits correspond to the ones examined in Ref. [7].

Figure 3

Left: Sketch of the physical system used to factorize $N=15$ with $C=11$, corresponding to the quantum circuit displayed in the right panel of Fig. 1. The beam splitter $R_x$ of the qubit $x_0$ mimics the initialization procedure, that is, the equal splitting of electron wave function of the qubit $x_0$ in the two wires. The next two sets of gates, first $R_x{\mathit{TR}}_x$ acting on $\{x_0,y_1\}$ and then $R_x{R}_0{\mathit{TR}}_1{R}_x$ on $\{x_0,y_3\}$, play the role of two cnot gates creating the maximum entanglement of the qubits $\{x_0,y_1,y_3\}$ in the modular exponentiation step. For sake of clarity the qubits $y_0$ and $y_2$ evolving trivially during the computation have not been reported. Right: Sketch of the physical system used to factorize $N=15$ with $C=2$, corresponding to the quantum circuit displayed in the right panel of Fig. 2. In the initialization procedure two gates $R_x$ act on the couple of the argument register qubits $\{x_0,x_1\}$ and split each of them in an equal superposition of the $|0\rangle$ and $|1\rangle$ states. The modular exponentiation consists of two networks $R_x{\mathit{TR}}_x$, each operating onto $\{x_0,y_0\}$ and $\{x_1,y_1\}$ mimicking the action of two cnot gates. Note that here, for brevity, the phases of the quantum gates involved in the networks and explicitly indicated in Eqs. (16) and (18), are omitted.

Figure 4

The density matrix of the qubits $\{x_0,y_1,y_3\}$ evaluated at three different time steps: (a) input, (b) initialization procedure, and (c) after the modular exponentiation. Note that these matrices have been obtained from the full density matrices of the electrons of the qubits $\{x_0,y_1,y_3\}$ by integrating over the variables defining the position of three carriers along the channels. Here the moduli of the density matrix elements are plotted.

Figure 5

Density matrix ${\rho }_{\{x_0,x_0^{\prime }\}}(\mathbf{y},\mathbf{y})$ of the electron of qubit $x_0$ after the modular exponentiation. The diagonal elements describe the density probability of finding the electron at the point $\mathbf{y}$ along wire 0 or 1. Here the moduli of ${\rho }_{\{x_0,x_0^{\prime }\}}(\mathbf{y},\mathbf{y})$ are plotted. Note that the curves reported in the left panels refer to the left ordinate axes, while the ones reported in the right panels refer to the right ordinate axes.

Figure 6

The density matrix of the qubits $\{x_0,x_1,y_0,y_1\}$ evaluated at three different time steps: (a) input, (b) initialization procedure, and (c) after the modular exponentiation. Note that these matrices have been obtained from the full density matrices of the electrons of the qubits $\{x_0,x_1,y_0,y_1\}$ by integrating over the variables defining the position of three carriers along the channels. Here the moduli of the density matrix elements are plotted.

Figure 7

Density matrix ${\rho }_{\{x_0,x_0^{\prime },x_1,x_1^{\prime }\}}(\mathbf{y},\mathbf{y},{\mathbf{y}}^{\prime },{\mathbf{y}}^{\prime })$ of the couple of electrons of qubit $\{x_0,x_1\}$ after the modular exponentiation. The diagonal elements describe the density probability of finding the two electrons at the points $\mathbf{y}$ and ${\mathbf{y}}^{\prime }$ along wire 0 or 1. Here the moduli of ${\rho }_{\{x_0,x_0^{\prime },x_1,x_1^{\prime }\}}(\mathbf{y},\mathbf{y},{\mathbf{y}}^{\prime },{\mathbf{y}}^{\prime })$ are plotted.