Figure 1

Bell-state analyzer for noninteracting electrons, consisting of three $50/50$ beam splitters (dashed horizontal lines), four mirrors (solid horizonal lines), two local spin rotations (Pauli matrices ${\sigma }_x$ and ${\sigma }_z$), and three charge detectors (squares). The charge detectors may operate either as electrometers (counting the occupation $q_i=0,1,2$ in an arm) or as parity meters (measuring $p_i=q_i\mathrm{m}\mathrm{o}\mathrm{d}2$). The first charge detector can identify the spin singlet state $|{\Psi }_0\rangle$, which is the only one of the four Bell states (2, 3, 4) to show ($p_1=0$). Since $(\mathbb{1}\otimes {\sigma }_z)|{\Psi }_1\rangle =-|{\Psi }_0\rangle$, the second charge detector can identify $|{\Psi }_1\rangle$ when $p_2=0$. Finally, since $(\mathbb{1}\otimes {\sigma }_x{\sigma }_z)|{\Psi }_2\rangle =|{\Psi }_0\rangle$, the third charge detector can identify the two remaining states $|{\Psi }_2\rangle$ (when $p_3=0$) and $|{\Psi }_3\rangle$ (when $p_3=1$).Reuse & Permissions

Figure 2

Gate that converts a charge parity measurement to a spin parity measurement. The shaded box at the right represents the circuit shown at the left. A pair of electrons is incident in arms $a$ and $b$. A polarizing beam splitter (double dashed line) transmits spin up and reflects spin down. A charge detector records bunching ($p=0$) or antibunching ($p=1$) and passes the electrons on to a second polarizing beam splitter. If each electron at the input is in a spin eigenstate $|\uparrow \rangle$ or $|\downarrow \rangle$, then output equals input and $p$ measures the spin parity ($p=1$ if the two spins are aligned, and $p=0$ if they are opposite). The gate can be used to encode a qubit $|\uparrow \rangle$ as the two-particle state $|\uparrow \rangle |\uparrow \rangle$ and $|\downarrow \rangle$ as $|\downarrow \rangle |\downarrow \rangle$. For that purpose the input consists of the qubit to be encoded in arm $a$ plus an ancilla in arm $b$ in the state $\left(\right|\uparrow \rangle +|\downarrow \rangle )/\sqrt{2}$. The output is the required two-particle state in arms $c$ and $d$ for $p=1$. For $p=0$ it becomes the required state after a spin-flip (${\sigma }_x$) operation on the electron in arm $d$.Reuse & Permissions

Figure 3

Deterministic cnot gate for noninteracting electrons. Each shaded box contains a pair of polarizing beam splitters and a charge detector, as described in Fig. 2. The four Hadamard gates $H=({\sigma }_x+{\sigma }_z)/\sqrt{2}$ rotate the spins entering and leaving the second box. The input of the cnot gate consists of the control and target qubits plus an ancilla in the state $\left(\right|\uparrow \rangle +|\downarrow \rangle )/\sqrt{2}$. The spin of the ancilla is measured at the output. The outcome of that measurement together with the two parities $p_1,p_2$ measured by the charge detectors determine which operations ${\sigma }_c,{\sigma }_t$ one has to apply to control and target at the output in order to complete the cnot operation. For the control, ${\sigma }_c={\sigma }_z$ if $p_2=0$ while ${\sigma }_c=\mathbb{1}$ if $p_2=1$. For the target, ${\sigma }_t={\sigma }_x$ if the ancilla is down and $p_1=1$, or if the ancilla is up and $p_1=0$. Otherwise, ${\sigma }_t=\mathbb{1}$. The calculation is given in Ref. 30.Reuse & Permissions