Linear optical quantum computing with photonic qubits

Linear optics with photon counting is a prominent candidate for practical quantum computing. The protocol by Knill, Laflamme, and Milburn [2001, Nature (London) 409, 46] explicitly demonstrates that efficient scalable quantum computing with single photons, linear optical elements, and projective measurements is possible. Subsequently, several improvements on this protocol have started to bridge the gap between theoretical scalability and practical implementation. The original theory and its improvements are reviewed, and a few examples of experimental two-qubit gates are given. The use of realistic components, the errors they induce in the computation, and how these errors can be corrected is discussed.

Figure 1

(Color online) The beam splitter with transmission amplitude $\cos \theta$.

Figure 2

(Color online) The polarizing beam splitter in different polarization bases. (a) The horizontal-vertical basis. (b) The diagonal basis.

Figure 3

(Color online) Decomposing an $N$-port unitary $\mathrm{U}\left(N\right)$ into SU(2) group elements, i.e., beam splitters and phase shifters. Moreover, this is an efficient process: the maximum number of beam splitters needed is $N(N-1)∕2$.

Figure 4

(Color online) Linear optical quantum computing simulation according to Cerf, Adami, and Kwiat. (a) Hadamard gate. (b) CNOT gate. The four two-qubit degrees of freedom are carried by which-path and polarization information. The dashed line indicates that there is no interaction between the crossing modes.

Figure 5

(Color online) Using cross-Kerr nonlinearities $\left(\tau \right)$ in optical information processing. (a) Single-photon quantum nondemolition measurement. The Mach-Zehnder interferometer is balanced such that the presence of a photon in the signal mode directs the probe field to the dark output port. (b) Single-photon CZ gate. When both photons in modes $a$ and $b$ are vertically polarized, the two-photon state acquires a relative phase. This results in an entangling gate that, together with single-photon rotations, is sufficient for universal quantum computing.

Figure 6

(Color online) The conditional phase gate (CZ). This gate uses two NS gates to change the relative phase of the two qubits: when both qubits are in the ∣1⟩ state, the two photons interfere on the 50:50 beam splitter $[{\cos }^2(\pi ∕4)=1∕2]$. The Hong-Ou-Mandel effect then ensures that both photons exit the same output mode and the NS gates induce a relative phase $\pi$. Upon recombination on the second beam splitter, this phase shows up only in the states where both qubits were in the ∣1⟩ state.

Figure 7

(Color online) The nonlinear sign (NS) gate according to Knill, Laflamme, and Milburn. The beam-splitter transmission amplitudes are ${\eta }_1={\eta }_3=1∕(4-2\sqrt{2})$ and ${\eta }_2=3-2\sqrt{2}$.

Figure 8

(Color online) The two equivalent versions of the NS gate by 174. Only two beam splitters are used, while the other resources are identical to the NS gate by Knill, Laflamme, and Milburn. The success probability of this gate is $(3-\sqrt{2})∕7$.

Figure 9

(Color online) The NS gate by Rudolph and Pan. Based on a vacuum detection of the first output port and a single vertically polarized photon on the second output port, the interferometer applies a NS gate to the input state. The success probability is also $(3-\sqrt{2})∕7$, which is close to the optimal value of $1∕4$.

Figure 10

(Color online) The Knill CZ gate based on two ancilla photons and two detected photons. The beam-splitter angles are $\theta =54.74°$ and $\varphi =17.63°$, such that the transmission amplitudes are given by $\cos \theta$ and $\cos \varphi$, respectively.

Figure 11

(Color online) The CNOT gate by 160. The two boxes I and II are parity gates in two complementary bases, where the detector measures in a complementary basis with respect to the polarizing beam splitter. The gate makes use of a maximally entangled ancillary state $\mid {\Phi }^{+}⟩$, which boosts the success probability up to one-quarter. The target ${\mid {\psi }_1⟩}_t$ and control ${\mid {\psi }_2⟩}_c$ input states will evolve into an entangled output state conditioned on the required detector signature.

Figure 12

(Color online) Schematic diagram of the experimental setup of the three-photon CNOT gate of 159. The gate starts by preparing the qubits with polarization rotations ${\lambda }_i$, followed by mixing the ancilla and control qubits on a polarizing beam splitter. The ancilla qubit is then mixing with the target qubit on the second polarizing beam splitter. The gate is implemented upon a threefold detector coincidence in the control, target, and ancilla modes. The polarization rotations ${\theta }_i$ are used to select different polarization bases.

Figure 13

(Color online) Experimental demonstration of the CNOT gate by 159. The probability of threefold coincidences as a function of the output qubit analyzers for all four computational basis states $HH$, $HV$, $VH$, and $VV$ in the input registers is shown. The experimental error in the gate is approximately 21%.

Figure 14

(Color online) Schematic diagram of the CNOT gate demonstrated by 153. (a) Concept of the two-qubit gate: The beam-splitter coefficients are ${\theta }_1=\pi ∕4$ and ${\theta }_2=\arccos (1∕\sqrt{3})$. (b) Translation circuit for converting polarization and dual-rail qubits. (c) Schematic of the experimental setup. Simultaneous detection of a single photon at each detector heralds the successful operation of the gate.

Figure 15

(Color online) Experimental demonstration of the CNOT gate by 153 in the logical qubit basis. The data are obtained by full state tomography of the output states.

Figure 16

(Color online) Schematic diagram of the four-photon CNOT gate by 49. A parametric down-conversion source is used to create the control and target input qubits in the spatial modes $a_1$ and $a_2$, as well as a maximally entangled ancilla pair in the spatial modes $a_3$ and $a_4$. Polarizing filters (Pol) can be used to destroy the initial entanglement in $a_1$ and $a_2$ if necessary.

Figure 17

(Color online) Experimental demonstration of the CNOT gate by 49. Fourfold coincidences for all combinations of inputs and outputs are shown.

Figure 18

Plot of the real part of the density matrix reconstructed from quantum process tomography for the input state $({\mid H⟩}_c-{\mid V⟩}_c){\mid V⟩}_t$. This shows the highly entangled singlet state of the form ${\mid H⟩}_c{\mid V⟩}_t-{\mid V⟩}_c{\mid H⟩}_t$

Figure 19

Plot of the (a) real and (b) imaginary parts of the reconstructed process matrix of the CNOT gate by 153. The ideal CNOT can be written as a coherent sum ${\widehat{U}}_{\mathrm{CNOT}}=\frac{1}{2}(\mathbb{1}\otimes \mathbb{1}+\mathbb{1}\otimes X+Z\otimes \mathbb{1}-Z\otimes X)$ of the tensor products of Pauli operators $\{\mathbb{1},X,Y,Z\}$ acting on control and target qubits, respectively. The input abscissae are (from left to right) $II$, $IX$, $IY$, $IZ$, $XI$, $XX$, $XY$, $XZ$, $YI$, $YX$, $YY$, $YZ$, $ZI$, $ZX$, $ZY$, and $ZZ$ while the output abscissae are (from left to right) $ZZ$, $ZY$, $ZX$, $ZI$, $YZ$, $YY$, $YX$, $YI$, $XZ$, $XY$, $XX$, $XI$, $IZ$, $IY$, $IX$, and $II$.

Figure 20

The CZ applied to two qubits inside a quantum circuit. If it fails, then the two-qubit states are lost.

Figure 21

The teleportation circuit. The state $\mid {\varphi }_j⟩$ is teleported via a Bell state $\mid {\Phi }^{+}⟩$ and a Bell measurement $B$. The binary variables $x$ and $z$ parametrize the outcome of the Bell measurement and determine which Pauli operator is applied to the output mode.

Figure 22

(Color online) The CZ gate using teleportation: here $\mid \psi ⟩=U_{\mathrm{CZ}}\mid {\varphi }_1{\varphi }_2⟩$. By commuting the CZ gate through the Pauli gates from the computational circuit to the teleportation resources, we have taken the probabilistic part off-line. We can prepare the teleportation channel (the shaded area, including the CZ) in many trials, without disrupting the quantum computation.

Figure 23

Near-deterministic teleportation according to Knill, Laflamme, and Milburn. The input state $\mid \varphi ⟩=\alpha \mid 0⟩+\beta \mid 1⟩$ is teleported to the $m\text{th}$ outgoing mode, where $m$ is the number of detected photons in the measurement of the $(n+1)$-point quantum Fourier transform. Note that $\mid \psi ⟩$ is a single-rail state; 0 and 1 denote photon numbers here.

Figure 24

The five-photon ancillary scheme for near-deterministic teleportation. The two rows correspond to two terms in the superposition state $\mid t_5⟩$ that can yield two detected photons (including the unknown input state $\alpha \mid 0⟩+\beta \mid 1⟩$). The first six columns are detected, while the last five columns correspond to the freely propagating modes.

Figure 25

A typical cluster state. Every circle represents a logical qubit, and the vertices represent CZ operations. A quantum computation proceeds by performing single-qubit measurements on the left column of qubits, thus removing them from the cluster and teleporting the quantum information through the cluster state. The vertical links induce two-qubit operations.

Figure 26

Different cluster and graph states. (a) A linear cluster of five qubits. (b) A cluster representing the Bell states. (c) A four-qubit GHZ state. This state can be obtained by an $X$ measurement of the central qubit in (a). (d) A general GHZ state.

Figure 27

(Color online) Using the “hyperentanglement” of the polarization and which-path observables, a single photon spans a four-dimensional Hilbert space $\{\mid H,1⟩,\mid H,2⟩,\mid V,1⟩,\mid V,2⟩\}$. A simple 50:50 beam-splitter and polarization rotation then furnishes a deterministic transformation from the computational basis to the Bell basis.

Figure 28

A typical three-qubit quantum computational circuit.

Figure 29

The computational circuit of Fig. 28 in terms of the physical implementation by 236. This is reminiscent of the cluster-state model of quantum computing. The open and solid dots represent the polarization and which-path degrees of freedom, respectively.

Figure 30

(Color online) How to add a link to the Yoran-Reznik chain. This will create the state given in Eq. (47) with $n+1=5$. The four vertically connected gray circles represent the probabilistic CZ gate. Note that we need two of them.

Figure 31

(Color online) Two types of fusion operator: (a) The type-I fusion operator employs a polarization beam splitter (PBS1) followed by the detection $D$ of a single output mode in the 45° rotated polarization basis. This operation determines the parity of the input mode with probability $1∕2$. (b) The type-II fusion operator uses a diagonal polarization beam splitter (PBS2), detects both output modes, and projects the input state onto a maximally entangled Bell state with probability $1∕2$.

Figure 32

The two probabilistic gates that complete a universal set. (a) The $Z_{\pi ∕2}$ gate uses a deterministic single-photon rotation and a single type-II fusion gate. (b) The CNOT gate uses one type-I and one type-II fusion gate. Both gates also need a parity measurement, which is implemented by ${\sigma }_z$ measurements on the individual physical qubits.

Figure 33

The function $n\left(t\right)$ in arbitrary units in the domain $-\pi ⩽t⩽\pi$ for different values of $\mu$ and $N=100$: (a) $\mu =0.001$, (b) $\mu =0.01$, (c) $\mu =0.05$, and (d) $\mu =0.49$.

Figure 34

The $G^{\left(2\right)}\left(\tau \right)$ for the InAs quantum dot single-photon source. Note that the variable $T$ in the text is here replaced with $\tau$. From 188.

Figure 35

The Hong-Ou-Mandel effect for the InAs quantum dot single-photon source. From 188.

Figure 36

The Raman process in a three-level atom. A classical pump field drives the transition $4{}^{2}S_{1∕2}\to 4{}^{2}P_{1∕2}$ off-resonantly, thus generating a photon in the cavity mode. The level $4{}^{2}P_{1∕2}$ is adiabatically eliminated and hence never populated.

Figure 37

Photon-loss-tolerant cluster states. (a) We can measure the Pauli operator $X$ on the shaded (lost) qubit by measuring all the adjacent qubits in the $Z$ basis. (b) Planting a cluster tree using two adjacent $X$ measurements in order to do a single-qubit measurement in the basis $A$.