Photonic Qubits

A photonic qubit is a single particle of light carrying one bit of quantum information — not in how bright it is, but in which of two modes it occupies. Give a photon two orthogonal homes — horizontal versus vertical polarization, an upper versus a lower waveguide, an early versus a late time slot, two frequency bins — and a single excitation coherently shared between them is a genuine two-level system. The logical states $|0\rangle_L$ and $|1\rangle_L$ mean "the photon is here" versus "there," and the superposition $\alpha|0\rangle_L+\beta|1\rangle_L$ is one photon genuinely in both places at once, as single-photon interference confirms. Crucially the information lives in the one-photon sector: it is which mode, never how many photons, that encodes the bit. That choice of carrier has one overwhelming consequence. A photon has no charge and no rest mass, so it barely couples to anything — not to stray fields, not to the walls of a fibre, and, fatally for computing, not to other photons. Weak coupling makes photons the premier flying qubits: they travel at $c$ , hold their state across kilometres of fibre or the vacuum between satellites, and are the only sensible way to move entanglement between processors. But the same weak coupling means two photons pass through each other unaware, so passive linear optics can never make one photon's phase depend on another's presence — there is no native two-qubit gate. The reason photonic quantum computing exists at all is that a measurement can manufacture an effective nonlinearity. The 2001 KLM protocol showed that single-photon sources, beam-splitter networks, and photon-number-resolving detectors suffice for universal computation — probabilistically, with ancillas and feed-forward pushing the success probability toward one. And because the dominant error is photon loss, which ejects the carrier from the logical space rather than silently flipping it, photonic errors are heralded erasures with unusually forgiving thresholds. This article builds the picture from the field operators up.

Key takeaways

A photonic qubit encodes information in a degree of freedom of a single photon — polarization, spatial path, time-bin, or frequency — with the two logical states spanning a two-dimensional subspace of the one-photon Hilbert space.
Photons are near-decoherence-free flying qubits: they barely interact with the environment, travel at $c$ , and are the natural carriers for quantum communication, networking, and distributed entanglement.
Single-qubit gates are easy, deterministic, and ideally lossless — wave plates rotate polarization and beam splitters plus phase shifters realize any $SU(2)$ rotation on a path qubit.
The same weak interaction that gives long coherence forbids a deterministic two-qubit gate in passive linear optics; the KLM protocol supplies one probabilistically through measurement-induced nonlinearity.
Two-photon gates run on indistinguishability: the Hong–Ou–Mandel effect makes identical photons bunch, and its coincidence dip $P_c=\tfrac12(1-|\langle\psi_1|\psi_2\rangle|^2)$ is the resource meter.
The dominant error is photon loss, a heralded erasure that leaves the logical space entirely — not energy relaxation — which is why photonic architectures lean on erasure codes and quantum repeaters.
Key idea: A photonic qubit is a single photon coherently spread across two orthogonal field modes; single-qubit gates are passive linear optics, but entangling gates require either measurement-induced nonlinearity (KLM) or interaction with matter, making photons ideal flying qubits and the backbone of quantum networks.

Where this lives today

Photonic qubits are the only qubits that routinely leave the lab. They are the physical layer of quantum key distribution — polarization and time-bin encodings run BB84, decoy-state, and measurement-device-independent protocols over metropolitan fibre, and in 2017 the Micius satellite distributed entangled photon pairs between ground stations over a thousand kilometres apart, using exactly the loss and no-cloning physics of this article. In computing, the field has moved past gate-by-gate KLM to measurement-based architectures: PsiQuantum and Xanadu build fusion-based and cluster-state processors in which large entangled states are stitched together from small ones by Hong–Ou–Mandel-style fusion measurements, with loss handled as erasure inside the code. The 2020 Jiuzhang experiment sent dozens of indistinguishable photons through a 100-mode interferometer and sampled an output distribution governed by matrix permanents — classically intractable — to claim a photonic quantum advantage: the boson-sampling route that needs interference but not universality. And photons are the connective tissue of hybrid architectures: microwave-to-optical transduction lets a superconducting or trapped-ion processor emit a flying photonic qubit, so entanglement swapping at repeater nodes can knit matter qubits into a quantum internet. Everything here is load-bearing there — the indistinguishability that fusion gates consume, the erasure structure of loss the codes exploit, and the single-qubit ease that makes photonic control cheap.

The physical picture

Send a single photon into one port of a 50:50 beam splitter. Quantum mechanically it does not choose a path; it leaves in a coherent superposition of both output arms, and that superposition is a path qubit. The reflected arm carries a relative phase $i$ demanded by unitarity, so the output is $(|1,0\rangle+i|0,1\rangle)/\sqrt2$ — equal weight $1/\sqrt2\approx0.707$ in each rail. Recombine the arms on a second beam splitter and they interfere, with a visibility that measures how coherent the qubit stayed. Polarization tells the same story with the two-level structure built in: horizontal $|H\rangle$ and vertical $|V\rangle$ are orthogonal poles of a Bloch sphere, with diagonal, antidiagonal, and circular polarizations filling the equator, and a wave plate is nothing but a rotation of that sphere. The first figure holds both pictures — path and polarization — side by side, because they are the same abstract qubit wearing two costumes.

Figure

A single photon entering one port of a 50:50 beam splitter emerges as a path qubit — an equal superposition of the two output rails with amplitudes 1/√2 ≈ 0.707 and a reflection phase i — while the same two-state space is the Bloch sphere of a polarization qubit with horizontal and vertical poles; one photon, total number n̂ = 1, coherently shared between two orthogonal modes

The deep reason photons are superb messengers but poor computers is one and the same: they do not talk to anything. Weak coupling to the environment means almost no decoherence in transit, but weak coupling to each other means no native two-photon gate — you cannot make one photon's phase respond to another's presence without a material nonlinearity or a measurement. The workaround that linear optics offers is interference: when two identical photons meet at a beam splitter they refuse to leave by separate ports, an effect (Hong–Ou–Mandel) that is entirely a consequence of Bose statistics and perfect mode overlap. That interference is the raw material every probabilistic entangling gate is built from, and its quality is measured by the coincidence dip. The second figure is that measurement — the quantity an experimentalist actually watches to certify that their photons are indistinguishable enough to compute with.

Figure

The Hong–Ou–Mandel dip: the two-photon coincidence rate at the beam-splitter outputs, normalized to its distinguishable-photon value, falls to zero at zero relative delay τ = 0 when the photons are perfectly indistinguishable; the dip depth equals the visibility V = 1 for identical photons and V = 0 for fully distinguishable ones, and a vanishing coincidence rate is the certificate of the indistinguishability that every linear-optical gate consumes

The core equations

A photonic qubit is, first of all, a normalized vector in a two-dimensional slice of the one-photon Hilbert space. The physical content is entirely in what plays the role of the basis — polarization, path, time, or frequency — but the algebra is the universal qubit algebra.

|\psi\rangle=\alpha\,|0\rangle_L+\beta\,|1\rangle_L,\qquad |\alpha|^2+|\beta|^2=1 \qquad (1)

where: $|0\rangle_L,|1\rangle_L$ — the logical basis, the photon in mode 0 versus mode 1 (e.g. $|H\rangle,|V\rangle$ ); $\alpha,\beta$ — complex amplitudes. Reading it: this is an ordinary qubit, but its Bloch sphere is a physical space of optical modes, so single-qubit gates will turn out to be optical elements that rotate that space. The most useful realization for computing is dual-rail: two spatial modes, exactly one photon. Writing the states with creation operators makes the defining property visible — every logical state has total photon number one.

|0\rangle_L=\hat a_0^\dagger|\mathrm{vac}\rangle=|1,0\rangle,\quad |1\rangle_L=\hat a_1^\dagger|\mathrm{vac}\rangle=|0,1\rangle,\qquad \hat n=\hat a_0^\dagger\hat a_0+\hat a_1^\dagger\hat a_1=1 \qquad (2)

where: $\hat a_j^\dagger$ — the creation operator for optical mode $j$ ; $|\mathrm{vac}\rangle$ — the two-mode vacuum; $\hat n$ — total photon number. Reading it: because $\hat n=1$ on the whole logical space, anything that changes photon number — above all, loss — pushes the state out of the code space, where it can be detected as an erasure rather than mistaken for a bit flip. Single-qubit gates on a path qubit are built from two passive elements. A beam splitter mixes two modes; a phase shifter delays one. The beam splitter's mode transformation is the workhorse.

\begin{pmatrix}\hat b_0\\ \hat b_1\end{pmatrix}=\begin{pmatrix}\cos\phi & i\sin\phi\\ i\sin\phi & \cos\phi\end{pmatrix}\begin{pmatrix}\hat a_0\\ \hat a_1\end{pmatrix} \qquad (3)

where: $\hat a_j,\hat b_j$ — input and output mode operators; $\phi$ — the mixing angle, with $\phi=\pi/4$ giving a 50:50 splitter. Reading it: the matrix is unitary, so the transformation conserves photons and is lossless in principle; combined with a phase shifter $P(\varphi):\hat a_1\to e^{i\varphi}\hat a_1$ it generates every element of $SU(2)$ by the Euler decomposition $U=P(\alpha)\,BS(\beta)\,P(\gamma)$ . For polarization qubits the single-qubit gates are wave plates — birefringent crystals whose fast and slow axes retard one polarization relative to the other. A half-wave plate is a reflection of the Bloch sphere; a quarter-wave plate turns linear into circular polarization.

R(\theta)=\begin{pmatrix}\cos\theta & \sin\theta\\ \sin\theta & -\cos\theta\end{pmatrix},\qquad Q=\begin{pmatrix}1 & 0\\ 0 & i\end{pmatrix} \qquad (4)

where: $R(\theta)$ — a half-wave plate with fast axis at angle $\theta$ , reflecting the polarization about that axis; $Q$ — a quarter-wave plate ( $\pi/2$ retardance). Reading it: a half-wave plate followed by a quarter-wave plate realizes any $SU(2)$ rotation, so polarization single-qubit control is as complete and lossless as path control, with no active elements at all. Scaling up, an $m$ -mode interferometer of beam splitters and phase shifters realizes an arbitrary unitary on the mode operators — the Reck and Clements decompositions guarantee it, mesh by mesh.

\hat a_j\to\sum_{k=1}^{m}U_{jk}\,\hat a_k,\qquad U^\dagger U=\mathbb{1} \qquad (5)

where: $U$ — any $m\times m$ unitary; $m$ — the number of optical modes. Reading it: this is the entire computational power of passive linear optics — a single-particle unitary on the modes — and Derivation 3 shows why, on its own, it can never entangle independent photons. Two-photon interference is the one place linear optics reaches beyond single-particle physics, through Bose statistics. When two indistinguishable photons meet at a 50:50 splitter, the amplitudes for them to leave by separate ports cancel exactly.

|1,1\rangle_{ab}\xrightarrow{\ 50:50\ }\tfrac{1}{\sqrt2}\big(|2,0\rangle-|0,2\rangle\big),\qquad P_c=\tfrac12\big(1-|\langle\psi_1|\psi_2\rangle|^2\big) \qquad (6)

where: $|1,1\rangle_{ab}$ — one photon in each input port; $|2,0\rangle,|0,2\rangle$ — both photons bunched into one output; $P_c$ — the coincidence (one-each-port) probability; $\langle\psi_1|\psi_2\rangle$ — the overlap of the photons' internal modes. Reading it: perfectly indistinguishable photons ( $|\langle\psi_1|\psi_2\rangle|=1$ ) give $P_c=0$ — never a coincidence — while fully distinguishable ones give the classical $P_c=\tfrac12$ ; the depth of the dip is the resource that measurement-based gates spend. The chief enemy is loss. A lossy channel is modelled as a beam splitter that leaks the photon into an unmonitored mode, mapping the qubit onto the vacuum with probability $1-\eta$ .

\rho\to\eta\,\rho+(1-\eta)\,|\mathrm{vac}\rangle\langle\mathrm{vac}|,\qquad \eta=e^{-L/L_{\mathrm{att}}} \qquad (7)

where: $\eta$ — the survival (transmission) probability; $|\mathrm{vac}\rangle$ — the leakage state with the photon gone; $L$ — propagation distance; $L_{\mathrm{att}}\approx21.7\,\mathrm{km}$ — the $1/e$ attenuation length of standard fibre at $1550\,\mathrm{nm}$ (the $0.2\ \mathrm{dB/km}$ minimum). Reading it: transmission falls off exponentially with distance, and because the lost photon changes photon number it is a heralded erasure — the reason repeaters and erasure codes, not $T_1$ correction, define the photonic error-correction playbook. Finally, since passive optics cannot entangle photons, a deterministic two-qubit gate is impossible; the KLM protocol buys one probabilistically with ancilla photons and number-resolving detection.

P_{\mathrm{NS}}=\tfrac14,\qquad P_{\mathrm{CZ}}=\tfrac{1}{16}\ \text{(non-teleported)},\qquad P_{\mathrm{tel}}=\frac{n}{n+1} \qquad (8)

where: $P_{\mathrm{NS}}$ — success of the nonlinear-sign (NS) gate using two ancilla modes; $P_{\mathrm{CZ}}$ — controlled-sign built from two NS gates before teleportation; $P_{\mathrm{tel}}$ — success of a gate-teleported version using an $n$ -photon ancilla. Reading it: the bare gates fire only sometimes, but gate teleportation and feed-forward drive the success $n/(n+1)\to1$ as the ancilla grows — universality restored at the price of overhead (Worked example 2).

Derivation AThe single-photon path qubit from a beam splitter

The goal is to show explicitly that a single photon incident on one port of a 50:50 beam splitter emerges as an equal-weight superposition of the two output rails — the optical realization of a Hadamard on a path qubit — and that this superposition is a genuine one-particle state, not a statistical mixture. We start from a single excitation in one input mode, apply the beam-splitter transformation to its creation operator, and read the result as a logical operation. The lesson is that single-qubit gates on photonic qubits are exact, deterministic, and ideally lossless, because they are unitary rearrangements of the modes rather than interactions.

Step 1. Write the input as $|\psi_{\mathrm{in}}\rangle=\hat a^\dagger|\mathrm{vac}\rangle=|1,0\rangle_{ab}$ — one photon in port $a$ , vacuum in port $b$ , which is logical $|0\rangle_L$ in a path encoding. Why: expressing the state through the creation operator lets the beam splitter act by simple substitution, since a linear optical element is defined by how it transforms mode operators, not state vectors.

Step 2. Apply the 50:50 beam-splitter relation to the operator, $\hat a^\dagger\to\tfrac{1}{\sqrt2}(\hat c^\dagger+i\,\hat d^\dagger)$ , the $\phi=\pi/4$ case of equation (3) with a $\pi/2$ phase on the reflected arm. Why: unitarity forces the reflected amplitude to carry a relative factor $i$ ; without it the transformation would not preserve the commutation relations, and probability would not be conserved.

Step 3. Act on the vacuum: $|\psi_{\mathrm{out}}\rangle=\tfrac{1}{\sqrt2}(\hat c^\dagger+i\,\hat d^\dagger)|\mathrm{vac}\rangle=\tfrac{1}{\sqrt2}(|1,0\rangle_{cd}+i\,|0,1\rangle_{cd})$ . Why: linearity of the field operators carries the single excitation intact into a coherent sum over the two output rails — the photon is not split into halves, it is one photon whose amplitude now spans both arms.

Step 4. Read it as a gate: up to the removable phase $i$ (undone by a phase shifter on rail $d$ ), this is $\tfrac{1}{\sqrt2}(|0\rangle_L+|1\rangle_L)$ , the action of a Hadamard on $|0\rangle_L$ . Why: recombining the two rails on a second beam splitter reproduces single-photon interference — the empirical proof that the intermediate state was a true superposition and the qubit stayed coherent.

Go deeper · Derivation B · 1 minHong–Ou–Mandel interference and the meaning of indistinguishability

Derivation BHong–Ou–Mandel interference and the meaning of indistinguishability

The goal is to derive that two perfectly indistinguishable photons entering opposite ports of a 50:50 beam splitter never leave by separate ports — the coincidence amplitude cancels identically — and then to quantify what happens when the photons are only partly alike, producing the coincidence dip whose depth is the visibility. Physically this is Bose bunching: identical bosons prefer to share an output mode, absolutely so when the overlap is perfect. It matters because every measurement-based photonic gate consumes exactly this interference, so its visibility audits how well a gate can possibly work.

Step 1. Take one photon in each input, $|\psi\rangle=\hat a^\dagger\hat b^\dagger|\mathrm{vac}\rangle$ , with identical internal (spectral and temporal) modes. Why: the identical internal mode is the entire hypothesis — it is what lets us treat the two photons with the same pair of spatial operators and expose the interference, and its later relaxation is what quantifies distinguishability.

Step 2. Substitute the beam-splitter relations $\hat a^\dagger\to(\hat c^\dagger+\hat d^\dagger)/\sqrt2$ and $\hat b^\dagger\to(\hat c^\dagger-\hat d^\dagger)/\sqrt2$ to get $\tfrac12(\hat c^\dagger+\hat d^\dagger)(\hat c^\dagger-\hat d^\dagger)|\mathrm{vac}\rangle$ . Why: the coincidence outcome corresponds to the cross term $\hat c^\dagger\hat d^\dagger$ , so writing the product out lets us track precisely the amplitude for one photon in each output.

Step 3. Expand: the cross terms $\hat c^\dagger\hat d^\dagger-\hat d^\dagger\hat c^\dagger=0$ cancel because the operators commute, leaving $\tfrac{1}{\sqrt2}(|2,0\rangle-|0,2\rangle)$ . Why: the two paths to a coincidence — both transmitted, both reflected — carry opposite signs and equal magnitude, so they interfere destructively and the coincidence probability is exactly zero; only bunched outcomes survive.

Step 4. Relax perfect overlap by giving the photons internal states $|\psi_1\rangle,|\psi_2\rangle$ ; the coincidence probability becomes $P_c=\tfrac12(1-|\langle\psi_1|\psi_2\rangle|^2)$ , so the dip depth is the visibility $V=|\langle\psi_1|\psi_2\rangle|^2$ . Why: only the indistinguishable part of the two-photon amplitude interferes, weighted by the mode overlap squared; distinguishable photons ( $\langle\psi_1|\psi_2\rangle=0$ ) recover the classical $P_c=\tfrac12$ , and the measured dip therefore reads off indistinguishability directly.

Go deeper · Derivation C · 1 minWhy passive linear optics cannot entangle separate photonic qubits

Derivation CWhy passive linear optics cannot entangle separate photonic qubits

The goal is to prove the structural no-go that motivates the entire KLM programme: any photon-number-conserving linear-optical network maps a product state of independent single photons to another product state, so no passive optics can produce a Bell state of two separate photonic qubits. The physics is that a passive network is a single-particle unitary — it rearranges each photon's amplitude among the modes but never couples the presence of one photon to the phase of another. We write each qubit as a superposition of creation operators, push them through the network, and see the product structure survive, leaving nonlinearity or measurement as the only escape.

Step 1. Write two independent photons as a product of creation-operator superpositions, $|\psi\rangle=\big(\sum_j c_j\hat a_j^\dagger\big)\big(\sum_k d_k\hat a_k^\dagger\big)|\mathrm{vac}\rangle$ . Why: each factor is one photon spread over modes; the joint state's separability is encoded precisely in the fact that it is a product of two such single-photon operators.

Step 2. Apply the network, which acts linearly on every mode operator, $\hat a_j^\dagger\to\sum_l U_{jl}\hat a_l^\dagger$ , so each factor becomes a new single-photon superposition $\sum_l c_l'\hat a_l^\dagger$ and likewise for the second. Why: linearity is the whole story — a passive element transforms operators one at a time and cannot generate a term proportional to $\hat a^\dagger\hat a^\dagger$ that would bind the two photons together.

Step 3. Conclude that the output is again a product $\big(\sum_l c_l'\hat a_l^\dagger\big)\big(\sum_m d_m'\hat a_m^\dagger\big)|\mathrm{vac}\rangle$ — two independent single-photon wavefunctions, never an entangled Bell state. Why: with the product form preserved, the reduced state of either qubit stays pure, so by definition there is no entanglement across the cut, no matter how large or clever the interferometer.

Step 4. Identify the only two exits: a genuine Kerr-type photon–photon nonlinearity (astronomically weak at the single-photon level), or measurement — post-selecting on ancilla-detector outcomes that project onto an entangled component, which succeeds only probabilistically. Why: KLM takes the second road, trading determinism for feasibility, and this trade is the defining feature of linear-optical quantum computing.

Limiting cases

Photonic qubits behave in sharply different ways depending on whether optics is lossless, whether measurement is invoked, and whether the photons are truly identical.

Case	Result	Meaning
Lossless passive optics	$\hat a_j\to\sum_k U_{jk}\hat a_k,\ U\in U(m)$	Any network on $m$ modes is a single-particle unitary; single-qubit gates are exact and deterministic, but no entangling gate between separate photonic qubits can arise (Derivation 3).
Measurement-induced nonlinearity	$P_{\mathrm{NS}}=\tfrac14,\ P_{\mathrm{tel}}=\tfrac{n}{n+1}$	Adding ancilla photons and number-resolving detection gives an effective controlled-sign gate, but only probabilistically; teleportation with an $n$ -photon ancilla pushes success toward one at growing cost.
Lossy channel (erasure regime)	$\eta=e^{-L/L_{\mathrm{att}}},\ L_{\mathrm{att}}\approx21.7\,\mathrm{km}$	Transmission decays exponentially; the lost photon is a heralded erasure with favourable thresholds, motivating repeaters rather than $T_1$ -style correction.
Distinguishable photons	$V=\|\langle\psi_1\|\psi_2\rangle\|^2<1$	When interfering photons differ in spectral or temporal mode the Hong–Ou–Mandel visibility drops below one, degrading every two-photon gate that relies on interference.

Common trapthe photon and the mode are not the same qubit

It is tempting, and wrong, to conflate two encodings that look superficially similar. A dual-rail qubit is one photon shared between two modes — $|0\rangle_L=|1,0\rangle$ , $|1\rangle_L=|0,1\rangle$ — and it is a true qubit whose logical states share the same photon number, so loss ejects it cleanly to the detectable $|0,0\rangle$ outside the code space. A single-rail (Fock) qubit instead encodes the bit in the photon number of one mode, $|0\rangle_L=|0\rangle$ versus $|1\rangle_L=|1\rangle$ . That distinction is not cosmetic: in the single-rail case loss maps $|1\rangle_L\to|0\rangle_L$ inside the logical space, a silent, uncorrectable error rather than a heralded erasure, and a superposition $\alpha|0\rangle_L+\beta|1\rangle_L$ requires coherence between states of different photon number — fragile, and forbidden outright by photon-number superselection whenever a fixed phase reference is unavailable. Nearly every practical linear-optical architecture therefore uses dual-rail (or polarization, its close cousin) precisely so that the dominant error is an erasure. Do not assume a "photonic qubit" is robust to loss until you have checked which encoding is meant.

Worked examples

1 · Photon loss over optical fibre and why it heralds (fully worked)

A polarization qubit is sent through $L=100\ \mathrm{km}$ of standard single-mode fibre at the $1550\ \mathrm{nm}$ telecom minimum, where the attenuation is $0.2\ \mathrm{dB/km}$ . Find the attenuation length, the survival probability, the useful rate from a $10\ \mathrm{MHz}$ single-photon source, and explain why the loss is an erasure.

Step 1. Attenuation length: the $1/e$ length relates to the dB figure by $L_{\mathrm{att}}=10/(\alpha_{\mathrm{dB}}\ln 10)=10/(0.2\times2.3026)=\mathbf{21.7\ km}$ . This is the distance over which the survival probability falls by a factor $e$ .

Step 2. Survival probability over $100\ \mathrm{km}$ : $\eta=e^{-L/L_{\mathrm{att}}}=e^{-100/21.7}=e^{-4.61}=\mathbf{1.0\times10^{-2}}$ , i.e. $1\%$ . The dB bookkeeping agrees: $0.2\times100=20\ \mathrm{dB}$ of loss is $10^{-20/10}=10^{-2}$ . Only one photon in a hundred reaches the far end.

Step 3. Useful rate: a $10\ \mathrm{MHz}$ source delivers $10^7$ photons per second, so $\eta\times10^7=\mathbf{1.0\times10^{5}\ s^{-1}}$ survive. Double the distance to $200\ \mathrm{km}$ and $\eta=10^{-4}$ ; at $1000\ \mathrm{km}$ , $\eta=10^{-20}$ — direct transmission is hopeless, which is exactly why quantum repeaters exist. Loss, not decoherence, sets the reach of a photonic link.

Step 4. Why it heralds: in dual-rail or polarization encoding the surviving qubit always has $\hat n=1$ , whereas a lost photon gives $\hat n=0$ — the state $|0,0\rangle$ outside the logical space. A quantum non-demolition measurement of total photon number (or, in practice, a coincidence/heralding scheme) flags the loss without reading the logical bit. The error is therefore a known-location erasure, whose fault-tolerance threshold is far kinder than that of an undetected Pauli error. Result: $L_{\mathrm{att}}=21.7$ km, $\eta=1\%$ over 100 km, $10^5$ heralded photons/s, loss = erasure.

2 · The cost of a KLM entangling gate

Estimate the resources a KLM controlled-sign gate needs. Start from the nonlinear-sign gate, build the bare CZ, then use gate teleportation to reach a $99\%$ success probability, and comment on the overhead.

Step 1. The nonlinear-sign (NS) gate applies a $\pi$ phase to the two-photon component using two ancilla modes and post-selection; it succeeds with $P_{\mathrm{NS}}=\mathbf{1/4}$ . This single probabilistic non-linearity is the seed of every linear-optical gate.

Step 2. A controlled-sign gate in the original KLM construction uses two NS gates on the interfering rails, so the bare, non-teleported success is $P_{\mathrm{CZ}}=(1/4)^2=\mathbf{1/16}\approx6\%$ . Firing a gate once in sixteen tries is far too lossy to chain — a naive circuit of many gates would essentially never succeed.

Step 3. Gate teleportation fixes this: prepare the gate offline on an $n$ -photon entangled ancilla and teleport it onto the data, succeeding with $P_{\mathrm{tel}}=n/(n+1)$ . Solving $n/(n+1)=0.99$ gives $n=99$ , so $\mathbf{n\approx99}$ ancilla photons per gate reach $99\%$ . Because failures are heralded (a detector pattern announces them), a failed teleportation only demands a retry, not a lost computation.

Step 4. Overhead: each teleportation needs a $2n$ -photon entangled resource state and a $2n$ -mode interferometer, and preparing that ancilla is itself probabilistic — so the photon budget scales roughly as $n^2$ , thousands of photons per near-deterministic gate. This staggering cost is why modern linear-optical computing abandoned gate-by-gate teleportation for measurement-based cluster-state and fusion architectures that amortize the randomness across the whole computation. Result: NS = 1/4, bare CZ = 1/16, and $n\approx99$ ancilla photons for 99% via teleportation.

Physics of the portrait

The photonic qubit is where the quantum theory of light meets quantum information: it took the discovery that two photons could interfere with each other — not just with themselves — to reveal indistinguishability as a computational resource.

Origins · 1 minLeonard Mandel and how this was discovered

Leonard Mandel

1927–2001 · quantum optics and the interference of single photons

Leonard Mandel was, with Roy Glauber and Emil Wolf, one of the architects of modern quantum optics — the physicist who turned "what is a photon, operationally?" into decisive table-top experiments. Born in Berlin in 1927 and educated in London, he spent most of his career at the University of Rochester, where his group built the experimental foundations of nonclassical light: photon antibunching, sub-Poissonian statistics, and the correlations that separate a stream of single photons from a dim classical beam. His most celebrated result came in 1987, when he and his students Chung Ki Hong and Zheyu Ou fired two photons at opposite faces of a beam splitter and found that, when the photons were made truly identical, the coincidence rate fell to zero — the two particles always left together. This Hong–Ou–Mandel effect is pure quantum interference of two particles, a manifestation of Bose statistics with no classical analogue, and its coincidence dip became the standard yardstick of photon indistinguishability. That yardstick is no curiosity: every linear-optical entangling gate, from KLM to today's fusion-based photonic processors, runs on exactly the interference Mandel first isolated, and the depth of his dip sets the ceiling on how well such a gate can work. His textbook with Wolf, Optical Coherence and Quantum Optics (1995), remains a standard reference; he died in 2001, having seen his effect become the beating heart of a new way to compute.

History

1905

— Einstein proposes the light quantum to explain the photoelectric effect

1927

— Dirac quantizes the electromagnetic field, giving light a mode structure

1963

— Glauber builds the quantum theory of optical coherence

1987

— Hong, Ou and Mandel observe two-photon interference in the coincidence dip

2001

— Knill, Laflamme and Milburn prove linear optics suffices for universal quantum computing

Going further

Texts. Nielsen & Chuang, Quantum Computation and Quantum Information, §1.4, §7.4 (physical implementations and optics) · Preskill, Lecture Notes for Physics 219, Ch. 9 · Kok et al., "Linear optical quantum computing with photonic qubits", Rev. Mod. Phys. 79, 135 (2007) · Knill, Laflamme & Milburn, "A scheme for efficient quantum computation with linear optics", Nature 409, 46 (2001) · Hong, Ou & Mandel, "Measurement of subpicosecond time intervals between two photons by interference", Phys. Rev. Lett. 59, 2044 (1987) · Mandel & Wolf, Optical Coherence and Quantum Optics (1995). The key numerical checks in this article are: the optical Hadamard follows directly from the 50:50 beam-splitter relation; the HOM coincidence cancellation yields $P_c=\tfrac12(1-|\langle\psi_1|\psi_2\rangle|^2)$ ; and the fibre figures $L_{\mathrm{att}}=10/(0.2\ln10)=21.7\ \mathrm{km}$ and $\eta=e^{-100/21.7}=1\%$ follow from the standard $0.2\ \mathrm{dB/km}$ attenuation at $1550\ \mathrm{nm}$ . The KLM success probabilities $P_{\mathrm{NS}}=1/4$ , $P_{\mathrm{CZ}}=1/16$ , and teleportation scaling $n/(n+1)$ are standard results. All device figures are standard tabulated values.

Prerequisites. The DiVincenzo Criteria · Linear-Optical Quantum Computing. Leads to. The KLM Protocol · Trapped-Ion Qubits · Superconducting Qubits.

Ask the tutor: Want to trace how a heralded Bell-state measurement turns two non-interacting photons into an entangled pair, and why that same trick powers KLM gates, quantum repeaters, and fusion-based photonic computing?

Is a photonic qubit just a very dim laser pulse?

No, and the difference is the whole subject. A dim laser pulse is a coherent state — a superposition of different photon numbers, with classical phase and Poissonian statistics. A photonic qubit is a genuine single-photon state in the one-photon sector, and its superposition is over modes (which polarization, which path), not over photon numbers. You can tell them apart: a true single photon shows antibunching (it never triggers two detectors at once at a beam splitter), while a coherent pulse does not. This matters for security and gates — attenuating a laser to "less than one photon on average" leaves a residual multi-photon component that opens the photon-number-splitting attack in QKD and spoils interference-based gates, which is why heralded and on-demand single-photon sources are worth the trouble.

If single-qubit gates are lossless and photons barely decohere, why is photonic computing hard?

Because the easy part and the hard part share a root cause. Photons are wonderful to move and rotate precisely because they interact with almost nothing — but that same non-interaction means two photons cannot condition each other, so there is no deterministic two-qubit gate. Everything hard flows from patching that hole: you manufacture an effective interaction by measurement (KLM), which is probabilistic and demands ancillas, feed-forward, and heralding; you must make photons indistinguishable to sub-percent tolerances so the interference works; and you must fight photon loss, the dominant error, which grows exponentially with path length. Photonic quantum computing is not limited by decoherence in the usual sense — it is limited by the difficulty of building deterministic entanglement out of intrinsically non-interacting particles.

What is the difference between polarization, path, time-bin, and frequency encoding?

They are four physical realizations of the same abstract qubit, chosen to suit the channel. Polarization ( $|H\rangle,|V\rangle$ ) is simplest to manipulate — wave plates give any single-qubit gate — and is ideal for free-space and satellite links, but drifts in ordinary fibre. Path (dual-rail) uses two waveguides and is the native language of integrated photonic chips, where beam splitters and phase shifters are lithographic. Time-bin encodes the bit in an early-versus-late arrival window, is robust to fibre transmission, and is the workhorse of long-distance QKD. Frequency-bin uses two colours to pack many qubits into one spatial mode, attractive for multiplexing. Because they are the same qubit, the right interferometer converts between them — which is how a chip's path qubit becomes a time-bin qubit for the fibre and back.

Why must interfering photons be indistinguishable, and how indistinguishable is enough?

Two-photon gates work by interference, and interference only occurs between amplitudes that are otherwise identical. If the two photons differ in any degree of freedom you are not measuring — arrival time, spectrum, spatial mode, residual polarization — those differences label which photon went where, the amplitudes stop overlapping, and the Hong–Ou–Mandel cancellation weakens in proportion to the mode overlap squared, $V=|\langle\psi_1|\psi_2\rangle|^2$ . Since a gate's entangling action is exactly the interfering part, a visibility of $V$ caps the achievable gate fidelity. "Enough" is set by the error-correction threshold: fault-tolerant linear optics wants indistinguishability well above $99\%$ , which is why single-photon-source development obsesses over spectral purity and why a deep, clean HOM dip is the first thing a photonics group measures.

How do photons get entangled at all if they cannot interact?

By measurement, not interaction. You interfere two photons on a beam splitter and then detect in a way that erases which-path information — a partial Bell-state measurement. Conditioned on a particular detector pattern, the surviving photons are projected into an entangled state, even though no photon ever felt a force from another. This is the engine of KLM gates, of entanglement swapping in repeaters, and of the fusion operations that build photonic cluster states. The price is that it is probabilistic: the heralding pattern only appears some of the time, and you either post-select on success or use feed-forward to keep going. Alternatively, entanglement can be created at the source — spontaneous parametric down-conversion emits polarization-entangled photon pairs directly — but even there the "interaction" happened inside a nonlinear crystal, not between the two flying photons afterward.

Where do photonic qubits fit among trapped ions and superconducting circuits?

They play complementary roles. Trapped ions and superconducting circuits are excellent stationary qubits — they hold state and support deterministic gates, but do not travel. Photons are the opposite: hopeless at sitting still and interacting, but unmatched at carrying quantum information across a room, a city, or between satellites. So the mainstream vision is hybrid: matter qubits compute and store, while photonic qubits are the flying interconnect that links processors and distributes entanglement via microwave-to-optical transduction and photon–matter entanglement at repeater nodes. The exception is all-photonic computing (PsiQuantum, Xanadu), betting that measurement-based cluster-state architectures and erasure-tolerant codes can turn photons' probabilistic gates and loss into a scalable, room-temperature machine on their own.