Photonic Qubits

Photonic qubits encode a two-dimensional system in selected electromagnetic modes. Interferometers rotate a single photon’s mode amplitudes, optical links transport them, and detectors convert them into outcomes. Because passive optics supplies no deterministic interaction between separate dual-rail qubits, processors also use ancillary photons, conditional measurements, entangled resources, or matter-mediated interactions. Loss and the probability of delivering a usable resource are as important as its conditional fidelity.

This page develops discrete-variable photonic encodings and their hardware. Continuous-Variable Platforms and Bosonic Qubits discuss quadrature processing and oscillator encodings.

Required background. Hardware Overview distinguishes component and logical performance; Bits, Qubits, Qudits, and Modes defines the code subspace; Quantized Electromagnetic Modes supplies the field operators; Beam Splitters derives one- and two-photon interference.

Modes Come Before Qubits

A photon occupies a mode

A normalized wave-packet creation operator can be written schematically as

$$\begin{aligned} a_f^\dagger &= \int d\omega\, f(\omega)a^\dagger(\omega),\\ \int d\omega\,|f(\omega)|^2&=1. \end{aligned}$$

The label $f$ can include spatial profile, propagation direction, polarization, and other degrees of freedom in addition to frequency. The one-photon state is

$$\lvert1_f\rangle = a_f^\dagger\lvert\mathrm{vac}\rangle.$$

Two nominal photons interfere as required only to the extent that all unresolved degrees of freedom match. Arrival time, spectrum, polarization, spatial profile, chirp, and source history can carry which-source information. A logical rail label does not erase hidden distinguishability.

For two pure normalized modes $f$ and $g$, define

$$\mathcal M = |\langle f\vert g\rangle|^2, \qquad 0\le\mathcal M\le1.$$

For a balanced beam splitter with one ideal photon in each input and otherwise matched detection, the coincidence probability is

$$P_{\mathrm{coin}} = \frac{1-\mathcal M}{2}.$$

For independent mixed one-photon states, replace $\mathcal M$ by $\operatorname{Tr}(\rho_f\rho_g)$. Identical independent copies then measure modal purity, not necessarily unit overlap. Correlated inputs require their joint internal-mode state. Real Hong–Ou–Mandel visibility can also be affected by multiphoton emission, background, detector timing, imbalance, and normalization conventions; it is not itself a two-qubit gate fidelity.

The physical Hilbert space is larger than a qubit

Two bosonic modes $a$ and $b$ have Fock basis $\lvert n_a,n_b\rangle$ with unbounded occupations. A dual-rail qubit uses only the one-photon sector:

$$\begin{aligned} \lvert0_L\rangle &=\lvert1,0\rangle_{ab}, \\ \lvert1_L\rangle &=\lvert0,1\rangle_{ab}. \end{aligned}$$

The vacuum $\lvert0,0\rangle$ is a loss state. States such as $\lvert2,0\rangle$, $\lvert1,1\rangle$, and $\lvert0,2\rangle$ are multiphoton sectors, not valid states of this one-photon qubit. A physical description must retain them whenever source or detector imperfections can populate or confuse them.

The code projector is

$$P_{\mathrm{rail}} = \lvert1,0\rangle\langle1,0\rvert + \lvert0,1\rangle\langle0,1\rvert.$$

Conditional tomography after projecting with $P_{\mathrm{rail}}$ characterizes the surviving qubit. Unconditional performance must also report $\operatorname{Tr}(P_{\mathrm{rail}}\rho)$.

Encoding Families

Polarization

Choose two orthogonal polarizations of one spatial and temporal mode:

$$\begin{aligned} \lvert0_L\rangle&=\lvert H\rangle, \\ \lvert1_L\rangle&=\lvert V\rangle. \end{aligned}$$

Wave plates, polarization rotators, and polarization beam splitters implement preparation and analysis. Polarization is compact in free space and natural for many emitters and entangled-pair sources. In fibre or integrated circuits, birefringence, polarization-mode dispersion, stress, and coupling can rotate or dephase the basis. Active polarization tracking or conversion to another encoding may be required.

Polarization is still a dual-mode encoding: the rails are two orthogonal polarization modes rather than two paths. Photon loss takes both logical basis states to the same vacuum sector.

Path or spatial dual rail

Choose two waveguides or interferometer arms:

$$\begin{aligned} \lvert0_L\rangle&=\lvert1\rangle_a\lvert0\rangle_b, \\ \lvert1_L\rangle&=\lvert0\rangle_a\lvert1\rangle_b. \end{aligned}$$

A tunable Mach–Zehnder mixer with sufficient independent input/output phase controls implements an arbitrary one-qubit unitary, up to global phase. Path encoding maps naturally onto integrated photonics and local interferometers, but doubles the number of guided modes per qubit and makes path-dependent loss and phase drift logical errors.

The state

$$\alpha\lvert1,0\rangle +\beta\lvert0,1\rangle$$

depends on a relative optical phase. Stabilization and calibration must preserve that phase through every splitter, crossing, heater, switch, and coupling interface.

Time bin

Choose nonoverlapping early and late wave packets in one propagation channel:

$$\begin{aligned} \lvert0_L\rangle&=\lvert E\rangle, \\ \lvert1_L\rangle&=\lvert L\rangle. \end{aligned}$$

An unbalanced interferometer prepares and analyzes their coherent superposition. Time-bin encoding can be robust against slowly varying polarization transformations in fibre and supports temporal multiplexing. It requires detector timing sufficient to resolve bins, stable interferometric phase, matched dispersion, and active switching or delay when arbitrary single-qubit operations and feed-forward are needed.

Direct time-resolved detection requires spacing large enough to separate the wave packets and timing distributions at the desired error probability. Coherent processing also requires stable relative phase and suitable delay and switching windows. Closely spaced bins increase rate but tighten detector and modulator requirements; widely spaced bins increase delay and memory overhead.

Frequency bins and single rail

Frequency-bin qubits use two spectral modes and electro-optic or nonlinear frequency mixing. They can exploit dense wavelength multiplexing, but require phase-coherent frequency processors and careful treatment of dispersion and spectral leakage.

A single-rail qubit uses

$$\lvert0_L\rangle=\lvert0\rangle, \qquad \lvert1_L\rangle=\lvert1\rangle$$

of one mode. It uses fewer rails, but a coherent superposition of vacuum and one photon requires a phase reference, and photon loss maps $\lvert1\rangle$ into the logical $\lvert0\rangle$ rather than an orthogonal flag. Dual rail is often preferred when located vacuum is operationally useful.

Encoding conversion is an operation

A network may convert path to polarization, polarization to time bin, or an emitter wavelength to the telecom band. Such conversion is not merely a change of notation. It has efficiency, noise, bandwidth, phase, and mode-overlap specifications. The conversion must preserve arbitrary superpositions, not only transmit the two basis states separately.

Source Layer

What an ideal source promises

For a triggered discrete-variable architecture, the ideal instruction is:

On each accepted clock cycle, deliver one photon into a declared normalized mode at a declared reference plane, with no vacuum or multiphoton component.

No source is summarized by one number. A useful source record includes:

Quantity	Operational question
generation probability	how often is the desired excitation created per request?
extraction and coupling	how often does it reach the output reference plane?
number purity	how much vacuum and multiphoton contamination remains?
spectral purity	is the heralded photon in one mode or a mixture?
indistinguishability	do photons from relevant times and sources interfere?
timing	what are emission jitter, pulse duration, and repetition rate?
stability	do frequency, brightness, and mode remain calibrated over time?
scalability	can many sources be tuned, pumped, and packaged simultaneously?

A small $g^{(2)}(0)$ constrains multiphoton statistics under a measurement model. It does not determine extraction efficiency, mutual indistinguishability, spectral purity, or accepted-photon rate.

Deterministic emitters

Quantum dots, atoms, ions, molecules, color centers, and other localized emitters can approximate one photon per excitation cycle. An isolated two-level emitter excited once and allowed to decay without re-excitation emits at most one photon on that transition; finite driving pulses, cascades, or repumping can violate this ideal cycle model. A cavity or waveguide can enhance emission into a desired mode. Resonant excitation can improve coherence and reduce timing jitter.

The word deterministic should be attached to a reference plane. Exciting an emitter with near-unit probability is not the same as delivering a photon after branching, extraction, filtering, conversion, coupling, switching, and transport. Independent emitters must also be tuned into mutual spectral and temporal overlap without adding unacceptable noise.

Heralded pair sources

Spontaneous parametric down-conversion and spontaneous four-wave mixing produce correlated photon pairs. In a simple two-mode squeezed description,

$$\lvert\psi\rangle = \sqrt{1-\lambda^2} \sum_{n=0}^{\infty} \lambda^n\lvert n,n\rangle.$$

Here $0\le\lambda<1$; the pump phase has been absorbed into the mode definitions. Detecting one mode heralds the likely presence of its partner. At low gain, vacuum dominates; increasing gain raises the pair rate and the probability of unwanted multiple pairs. Filtering can improve spectral purity while lowering brightness and heralding efficiency. Number-resolving herald detectors can reject some multipair events but do not undo loss before the detector.

For this single-pair-mode model, let $q=\lambda^2$ and let the herald detector have efficiency $\eta_h$, with no dark counts. The pair distribution is $P_n=(1-q)q^n$, and an $n$-pair event clicks with probability $1-(1-\eta_h)^n$. Summing the geometric series gives

$$P_{\mathrm{click}}=\frac{q\eta_h}{1-q(1-\eta_h)}.$$

The conditional single-pair probability is

$$\Pr(n=1\mid\mathrm{click}) =(1-q) \times[1-q(1-\eta_h)].$$

For $q=0.05$ and $\eta_h=0.60$, a herald occurs in about $3.06\%$ of attempts, but only $93.1\%$ of those heralds arise from a single pair, even before signal loss. A herald is information about the state, not a guarantee of one photon. Parametric Down-Conversion treats multimode spectra, spectral purity, and more detailed herald measurements.

Multiplexing

If $M$ independent source bins each provide a valid heralded photon with probability $p$, the probability of at least one success is

$$P_{\ge1} = 1-(1-p)^M.$$

Multiplexing can turn heralded uncertainty into a higher delivery probability, but it consumes source modes, herald detectors, routing switches, delay, and classical decisions. If every selectable route has the same transmissivity $\eta_{\mathrm{sw}}$, independent of the herald, the delivery probability is bounded by

$$P_{\mathrm{del}} \le \eta_{\mathrm{sw}} \left[1-(1-p)^M\right].$$

For unequal routes, average their transmissions over the actual selection rule; a single arbitrary route efficiency is not an upper bound. False heralds, correlated pumps, multipair noise, and switch contention change the model.

Even ideal selection can have an optimum source count. In a balanced binary switch tree with $M=2^d$ inputs, every selected photon traverses $d$ switches. If each switch transmits $\eta_s$,

$$P_{\mathrm{del}}(d)=\eta_s^d[1-(1-p)^{2^d}].$$

For $p=0.10$ and $\eta_s=0.90$, this gives approximately $0.415$, $0.570$, and $0.478$ at $M=8,32,128$. More sources raise the chance of a herald but also increase every selected path’s loss in this particular tree. Different routing designs, bypasses, or storage strategies change the tradeoff.

Passive Linear Optics

Mode transformations

For $m$ lossless optical modes, a passive interferometer acts by a unitary matrix $U$:

$$b_j = \sum_{k=1}^{m} U_{jk}a_k, \qquad U^\dagger U=I.$$

This is the Heisenberg input–output relation for annihilation operators. The corresponding Schrödinger one-photon amplitudes transform with $U$: an input excitation in mode $k$ maps to $\sum_j U_{jk}|1_j\rangle$ in the output-mode basis. The conjugations differ if one instead writes an operator basis change for creation operators. Beam splitters and phase shifters synthesize any finite-dimensional passive spatial-mode unitary at a common carrier frequency; distinct frequency-bin mixing needs a driven converter. The transformation conserves total photon number:

$$\sum_j b_j^\dagger b_j = \sum_k a_k^\dagger a_k.$$

In the one-photon sector of two rails, this gives arbitrary one-qubit rotations. Large interferometers can implement mode permutations, Fourier transforms, state analyzers, sampling networks, and the optical parts of entangling measurements.

Real devices require an enlarged scattering matrix that includes loss ports. The implemented transfer matrix can depend on wavelength, polarization, temperature, drive history, and fabrication variation. A target unitary fidelity alone does not report insertion loss.

Why two-qubit gates are difficult

Passive linear optics transforms field modes linearly. It does not provide a deterministic interaction energy conditioned on two separate photons being present. Two identical photons can interfere, but interference alone does not make every desired dual-rail two-qubit unitary deterministic.

There are three broad responses:

1. use a strong optical or matter-mediated nonlinearity;
2. use ancillary photons, photodetection, and feed-forward to induce a measurement-conditioned gate;
3. avoid an online deterministic gate by preparing entangled resources and consuming them through measurements or fusions.

The Knill–Laflamme–Milburn result established that ideal single photons, passive optics, photodetection, ancillas, and feed-forward are sufficient for scalable universal quantum computation in principle. The effective nonlinearity comes from measurement and conditioning. Sufficiency does not mean low overhead: source probabilities, detector efficiencies, failed measurements, ancilla preparation, switching, and loss determine the architecture.

Bell-state analysis

For four equiprobable dual-rail Bell states, a fixed passive interferometer, vacuum ancillary modes, and ideal photon-number-resolving detection can unambiguously identify the input with probability at most $1/2$. This is the Calsamiglia–Lütkenhaus bound; its stated class excludes adaptive conditional dynamics. Adding vacuum modes or number resolution alone does not evade it. The bound concerns unambiguous success, not minimum-error guessing or discrimination of a restricted input ensemble.

Nonvacuum ancillas change the problem. For example, Grice’s construction reaches ideal success $3/4$ with an entangled ancillary photon pair and suitable number-resolving detection. More elaborate ancillary states approach unity with growing resources. Nonlinear interactions or an independently prepared entangled degree of freedom can also change the available measurement. Their preparation, loss, and detection costs belong in the success probability.

This limitation matters because Bell measurements drive teleportation, entanglement swapping, the untrusted relay in measurement-device-independent QKD, and many fusion gates. A network claim should report both conditional Bell-state fidelity and unconditional successful-analysis probability.

Computational Architectures

Circuit-model linear-optical computation

Circuit-model schemes compile logical gates into:

• deterministic one-qubit interferometers;
• probabilistic entangling gadgets;
• offline ancillary states;
• photon counting;
• active feed-forward and routing.

If an entangling gadget succeeds with probability $p_g$, a circuit containing $G$ such gadgets would have naive direct success $p_g^G$. Scalable schemes do not simply postselect the whole circuit. They use teleportation, offline resource preparation, multiplexing, encoding, and error correction so that failed attempts become located events that can be retried or decoded.

The price shifts into resource-state factories and optical depth. A gate list without ancilla photons, source attempts, detectors, switches, delays, and accepted-event rate is not a complete circuit resource estimate.

Cluster-state and one-way computation

A graph state can be written

$$\lvert G\rangle = \prod_{\{i,j\}\in E} CZ_{ij} \lvert+\rangle^{\otimes |V|}.$$

Measurement-Based Quantum Computation uses an entangled resource and a pattern of local measurements. Single-qubit measurements then consume vertices and propagate the logical state through the graph. Measurement bases can depend on earlier outcomes, and random Pauli byproducts are corrected physically or tracked in a Pauli frame.

This separates entangling-resource production from its consumption and aligns with destructive photodetection. A resource may be generated progressively; the full cluster need not coexist at one instant. It does not make resource generation free. A large useful cluster must be built from sources and probabilistic optical operations with enough redundancy to survive failed fusions and photon loss. Graph States develops the state and stabilizer structure.

Fusion-based computation

Fusion-based quantum computation treats small entangled resource states and projective entangling measurements as the primitive operations. A fusion consumes photons from resource states and attempts to connect their encoded graphs. Success, known failure, loss, and detector faults are passed directly into a fault-tolerant decoding problem.

In fusion-based quantum computation, the fusion outcomes provide both computational and error-correction information. The complete outcome map matters: a known fusion failure, missing photon, false detector event, and successful entangling measurement have different effects on the remaining qubits. Thresholds depend on the resource state, network, detector, noise, and decoder models. A theoretical loss threshold is not a measured loss allowance for every photonic device.

Detection and Feed-Forward

Detection is usually destructive

Optical photon counting typically absorbs the photon. A terminal basis measurement is natural; a nondestructive mid-circuit measurement generally requires a matter interaction, teleportation, or a larger optical construction. Measurement-based and fusion architectures embrace this destruction by making measured photons disposable resource qubits.

Two detector classes recur:

• a threshold detector distinguishes no click from one-or-more clicks;
• a photon-number-resolving detector distinguishes multiple registered photon counts over a declared mode and time gate; inefficiency still makes the incident photon number uncertain.

Number resolution can distinguish valid one-photon outcomes from collisions or some multiphoton faults. It does not restore photons lost upstream.

For ideal number state $\lvert n\rangle$, detector efficiency $\eta_d$, and no dark count, the no-click probability of a threshold detector is

$$P(0\mid n) = (1-\eta_d)^n.$$

With independent dark-click probability $d$ per gate,

$$P(0\mid n) = (1-d)(1-\eta_d)^n.$$

The full response also includes background, recovery, jitter, saturation, and mode selectivity. The Photon Counting page derives these detector models.

Feed-forward has a physical path length

Suppose the detection and the controlled photon start from synchronized reference events, with no existing path lead. The controlled photon must remain available while detection, signal propagation, electronics, and switching finish:

$$\tau_{\mathrm{delay}} \ge \tau_{\mathrm{det}} +\tau_{\mathrm{elec}} +\tau_{\mathrm{switch}} +\tau_{\mathrm{margin}}.$$

Here $\tau_{\mathrm{elec}}$ includes classical signal propagation. Any existing optical path lead reduces the additional delay needed; overlapping stages require their causal completion time rather than a double-counted sum. For group velocity $v_g$, the added propagation length is

$$L_{\mathrm{delay}} = v_g\tau_{\mathrm{delay}}.$$

Nanosecond decisions can require metres of waveguide or fibre. Longer delay adds propagation loss, dispersion, footprint, phase drift, and scheduling constraints. Fast feed-forward is therefore an optical and electronic co-design problem.

Pauli frames reduce physical switching

When a measurement induces a known Pauli byproduct, software can often update the interpretation of later operations rather than applying an immediate physical correction. This Pauli-frame tracking reduces switch depth and loss. It does not remove adaptive changes of non-Clifford measurement bases or all routing decisions.

Loss and Error Budgets

Optical loss composes multiplicatively

If a photon traverses a fixed route of scalar, mode-independent pure-loss components with transmissivities $\eta_1,\ldots,\eta_K$, then

$$\eta_{\mathrm{path}} = \prod_{k=1}^{K}\eta_k.$$

In decibels,

$$\mathcal L_{\mathrm{dB}} = -10\log_{10}\eta_{\mathrm{path}},$$

so component losses add. Coupling, propagation, bends, crossings, filters, switches, delays, conversion, and detector inefficiency must all be assigned to explicit reference planes.

If $N$ required photons independently survive with probability $\eta$, the all-photon probability is

$$P_{\mathrm{all}}=\eta^N.$$

This exponential is why a conditional fidelity after detecting all photons can coexist with a poor unconditional rate.

Unequal rail loss also filters the surviving state

On the dual-rail code space, phase-compensated attenuation has survival Kraus operator

$$K_s=\sqrt{\eta_a}|0_L\rangle\langle0_L| +\sqrt{\eta_b}|1_L\rangle\langle1_L|.$$

The other Kraus operators map each rail to vacuum with amplitudes $\sqrt{1-\eta_a}$ and $\sqrt{1-\eta_b}$. Conditional on survival,

$$\rho_s=\frac{K_s\rho K_s^\dagger} {\operatorname{Tr}(K_s\rho K_s^\dagger)}.$$

Only when $\eta_a=\eta_b$ does this equal the input state for every $\rho$. With unequal rail attenuation, observing survival supplies information about the logical basis and changes the amplitude balance. Thus an ideal symmetric erasure channel is more restrictive than loss that merely leaves an observable vacuum flag.

Loss can be an erasure, but not always

For a dual-rail qubit known to contain one photon, loss produces the orthogonal vacuum state $\lvert0,0\rangle$. A complete terminal measurement with well-characterized detectors can label no click as an erasure. Inside a network, however, the absence may be discovered only later, and inefficient detectors or dark counts can confuse loss with measurement outcomes.

The useful record includes:

• probability of loss;
• probability that loss is flagged;
• spatial and temporal resolution of the flag;
• false-erasure and missed-erasure rates;
• operations attempted after the unknown loss time;
• correlations from common coupling or switching failures.

Erasure and Loss Channels derives the channel-level distinction among flagged erasure, unflagged loss, postselected survival, and bosonic attenuation.

Mode mismatch exposes hidden information

Imperfect overlap lowers two-photon interference and can entangle logical rails with unobserved spectral or temporal modes. If those modes are traced out, the logical state dephases or mixes. The error may vary with source pair, time separation, temperature, and route. Characterizing one adjacent-pulse Hong–Ou–Mandel dip does not establish long-term mutual indistinguishability across a source array.

Multiphoton contamination is not simple loss

A pair source can produce two pairs, and an imperfect emitter can re-excite. Threshold detectors may accept a multiphoton event as an ordinary click. The result can create false heralds, incorrect fusion outcomes, or unmodeled correlations. Number-resolving detectors and source monitoring help only under their calibrated efficiency and response.

Phase, polarization, and switching errors

Path and time-bin qubits depend on relative phase. Polarization qubits depend on controlled birefringence. Active switches and modulators can add insertion loss, finite extinction, ringing, thermal crosstalk, and time-dependent phase. These are logical errors even when no photon is lost.

Correlated faults

Photonic errors are not automatically independent. One pump fluctuation can affect many pair sources; one thermal phase shifter can drift neighboring interferometers; one fibre connector can remove both rails; detector crosstalk can correlate outcomes; a clock error can shift an entire time-bin frame. Fault-tolerance analyses need measurements of these shared causes.

Network Compatibility

Flying qubits are a native strength

Photonic qubits can leave a processor without a separate quantum transducer when their wavelength and encoding already match the transmission channel. Telecom-band photons are compatible with low-loss standard fibre; visible or near-infrared emitters may require quantum frequency conversion. Interconnects and Transduction develops source-to-target channels and the converter metrics used when the carrier, frequency band, or encoding changes.

For fibre attenuation $\alpha$ in decibels per unit length and distance $L$,

$$\eta_{\mathrm{fib}}(L) = 10^{-\alpha L/10}.$$

Direct transmission therefore decreases exponentially with distance. A memory-based repeater uses heralded elementary links, memories, and entanglement swapping to change end-to-end rate scaling. It does not violate loss; it distributes attempts across segments and waits for successful links.

Encoding choice changes channel engineering

Encoding	Network advantage	Network burden
polarization	simple analyzers and free-space interfaces	fibre birefringence and polarization tracking
time bin	robust to slow polarization drift	unbalanced interferometers, timing and dispersion
frequency bin	wavelength multiplexing and routing	coherent frequency conversion and spectral phase
path	natural on chip	two physical channels and path-phase stability off chip

Conversion between a chip’s path encoding and a fibre’s polarization or time-bin encoding can be useful, but the converter’s efficiency and process fidelity must both be measured.

Memory must cover a waiting-time distribution

Matter qubits can store entanglement while photons attempt another link. For independent attempts of duration $\tau_a$ and success probability $p$, let $N$ be the number of attempts until success. Then

$$\mathbb E[N]=\frac1p,\qquad \Pr(N>n)=(1-p)^n.$$

The mean waiting time $\tau_a/p$ is not a deadline guarantee. For $p=0.01$, the mean is 100 attempts, but 299 attempts are needed for at least $95\%$ success. Memory quality must be averaged over the actual waiting distribution, including communication and the decay incurred by an earlier successful link. If $\tau_a$ already includes herald return time, do not add that time again. Multiple links, finite buffers, deadlines, and retries alter this elementary model; Quantum Memories treats the write–store–read channel.

Network-compatible does not mean network-complete

A high-fidelity chip-to-chip channel conditional on arrival establishes state preservation for surviving photons. A useful modular computer additionally needs low loss, synchronized sources, entangling protocols, memories or buffers, feed-forward, and a code that tolerates failed links. A QKD network distributes secret classical keys; it is not automatically a network of coherent quantum processors.

Infrastructure and Integration

Free-space, fibre, and integrated optics

Free-space optics offers flexible access and rapid prototyping. Fibre offers low-loss guided transmission and mature connectors. Integrated photonics offers dense, stable interferometers, repeatable lithography, and electrical control. Large systems often combine all three.

Integration changes the error budget rather than deleting it. Relevant constraints include:

• propagation loss and sidewall scattering;
• fibre-to-chip and chip-to-chip coupling;
• waveguide crossings, bends, and polarization handling;
• fabrication variation in splitters, resonators, and sources;
• thermal or electro-optic tuning power;
• optical pump rejection near single-photon detectors;
• superconducting detector temperature and cryogenic wiring;
• yield, screening, packaging, repair, and calibration at wafer scale.

Calling an optical qubit room-temperature hardware because it propagates at room temperature can be misleading. Sources, memories, frequency converters, and high-efficiency superconducting detectors may require cryogenic systems.

Calibration is a mode graph

An interferometer mesh has many coupled phases and losses. Calibrating each phase shifter separately may fail when thermal crosstalk or shared electrical drivers are active. Sources must be calibrated against filters and detector gates; detectors against arrival-time and wavelength distributions; switches against the routes they actually serve.

A predictive optical model combines complex transfer matrices, source mode states, losses, detector response, and their control dependence. Tests outside the fitted settings reveal whether it predicts operations that were not used for calibration.

Synchronization is a first-class resource

Independent photons must meet within their coherence time. Clock distribution, pump timing, emitter jitter, path delay, dispersion, and detector time tags all contribute. Temporal multiplexing increases mode count but also increases the scheduler’s responsibility to track which bins contain valid resources.

Throughput and Resource Preparation

Accepted-event rate

Let the clock rate be $f_{\mathrm{rep}}$ and the probability that a complete trial is accepted be $p_{\mathrm{acc}}$. Then

$$R_{\mathrm{acc}} = f_{\mathrm{rep}}p_{\mathrm{acc}}.$$

For a simple $N$-photon event,

$$p_{\mathrm{acc}} \sim \left( p_{\mathrm{src}}\eta_{\mathrm{path}}\eta_d \right)^N p_{\mathrm{logic}},$$

where $p_{\mathrm{logic}}$ contains the success probabilities of required fusions, heralds, and acceptance tests. This factorization is only a baseline: source correlations, multiplexing, detector saturation, and route contention can alter it.

Conditional fidelity $F_{\mathrm{cond}}$ and $R_{\mathrm{acc}}$ answer different questions. Multiplying them does not generally produce a universal utility metric. Report both, together with output distribution, uncertainty, and the task’s success criterion.

Resource factories

Cluster and fusion architectures continuously consume small entangled states. A useful factory metric is not only resource-state fidelity but the rate at which usable states reach the fusion network:

$$R_{\mathrm{res}} = f_{\mathrm{rep}} p_{\mathrm{gen}} p_{\mathrm{verify}} \eta_{\mathrm{deliver}}.$$

Here $p_{\mathrm{verify}}$ is acceptance conditional on generation and $\eta_{\mathrm{deliver}}$ is delivery conditional on that acceptance; losses already included in one factor must not be counted again. Destructive tomography characterizes an ensemble and consumes its samples. A per-instance herald must leave the resource photons available, for example by measuring ancillas rather than destroying the intended output.

The system must balance factory rate, buffer depth, fusion demand, detector recovery, and decoder latency. Overproduction wastes photons and detector bandwidth; underproduction stalls the logical clock.

Postselection and fault tolerance

Discarding every shot with a missing photon can demonstrate a conditional protocol, but its all-photon acceptance often decreases exponentially. A scalable proposal must instead account for missing resources through heralded retry, encoded redundancy, recovery, or decoding. Tree clusters, parity encodings, topological cluster states, and fusion networks address different versions of this problem.

The tolerable loss is not a platform constant. It depends on where losses occur, which fusion outcomes are known, source correlations, dark counts, mode mismatch, code geometry, and decoding. Logical error suppression should be compared at increasing code resources with the same physical noise model and with preparation rate and total elapsed time included.

For current component records and open scaling questions, see Quantum Optics Frontiers and Quantum Hardware Scaling.

Worked Six-Photon Acceptance Calculation

Consider a notional six-photon fusion event clocked at $f_{\mathrm{rep}}=1\,\mathrm{MHz}$. Each source bank delivers its required photon with probability $p_{\mathrm{src}}=0.80$. Each photon then traverses twelve components of transmissivity $\eta_c=0.995$ and is measured by a detector with efficiency $\eta_d=0.95$. The event requires two fusions, with joint ideal success probability $p_f^2=0.25$ conditional on all required photons and detector registrations. This is a stipulated independent example, not a universal factorization of a fusion network.

1. Propagation and detection

The single-photon path-and-detector efficiency is

$$\begin{aligned} \eta_1 &= \eta_c^{12}\eta_d \\ &= (0.995)^{12}(0.95) \\ &\approx0.895. \end{aligned}$$

All six delivered photons survive and register with probability

$$\eta_1^6\approx0.5124.$$

2. Source simultaneity

All six sources deliver in the same cycle with probability

$$p_{\mathrm{src}}^6 =(0.80)^6 \approx0.262.$$

This factor would be absent from a fidelity reported only after six photons were detected.

3. Fusion acceptance

The two ideal fusions both succeed with probability

$$p_f^2 =(0.50)^2 =0.25.$$

Under the independent baseline, the accepted-event probability is

$$\begin{aligned} p_{\mathrm{acc}} &= p_{\mathrm{src}}^6 \eta_1^6 p_f^2 \\ &\approx (0.262144)(0.512393)(0.25) \\ &\approx0.03358. \end{aligned}$$

The idealized accepted rate is therefore

$$R_{\mathrm{acc}} \approx 3.36\times10^4\,\mathrm{s}^{-1}.$$

Dead time, route contention, rejection, and decoder barriers can reduce the rate of valid events. False heralds can instead increase raw acceptance while lowering conditional quality; they must be separated from genuine success.

4. Feed-forward storage

Suppose detection, electronics, switching, and margin require $35\,\mathrm{ns}$, $20\,\mathrm{ns}$, $10\,\mathrm{ns}$, and $5\,\mathrm{ns}$ respectively. Then

$$\tau_{\mathrm{delay}} \ge 70\,\mathrm{ns}.$$

For $v_g=2.0\times10^8\,\mathrm{m\,s^{-1}}$,

$$L_{\mathrm{delay}} \ge 14\,\mathrm{m}.$$

This simple calculation explains why electronic latency becomes metres of fibre or waveguide and why feed-forward loss belongs in the architecture.

Common Pitfalls

Treating a weak coherent pulse as a single photon. Attenuation reduces mean occupation but retains vacuum and multiphoton components. Photon counting and mode-overlap measurements probe different source properties.

Reading conditional fidelity as delivery probability. A nearly perfect accepted state can occur rarely. State quality, event validity, and accepted events per second answer different questions.

Equating mode count with qubit count. Modes may be empty, continuously variable, ancillary, or part of a sampling problem. A logical qubit also requires a specified encoding and recovery procedure.

Assuming room-temperature propagation removes cryogenics. Waveguides may operate at room temperature while emitters, memories, or superconducting detectors require cooling.

Exercises

1. Check a one-photon mode transformation

A balanced beam splitter maps

$$a^\dagger \longmapsto \frac{c^\dagger+d^\dagger}{\sqrt2}.$$

Find the output of $\lvert1,0\rangle_{ab}$ and verify normalization.

Solution

Since

$$\lvert1,0\rangle_{ab} = a^\dagger\lvert\mathrm{vac}\rangle,$$

the output is

$$\frac{ \lvert1,0\rangle_{cd} +\lvert0,1\rangle_{cd} }{\sqrt2}.$$

The output modes are orthogonal, so the norm is

$$\frac12+\frac12=1.$$

This is a deterministic one-qubit rail rotation, not a deterministic two-qubit entangling gate.

2. Infer mode overlap from a coincidence dip

Under the ideal balanced-beam-splitter model, a two-photon experiment measures $P_{\mathrm{coin}}=0.035$ at zero delay. Estimate $\mathcal M$.

Solution

Using

$$P_{\mathrm{coin}} = \frac{1-\mathcal M}{2},$$

gives

$$\mathcal M = 1-2(0.035) = 0.93.$$

The inferred overlap is $93\%$ under the stated ideal model. Beam-splitter imbalance, multiphoton events, background, detector timing, and correction conventions must be included before identifying this number with intrinsic source indistinguishability.

3. Calculate multiplexed delivery

Eight independent source bins each succeed with probability $p=0.10$. Find the probability that at least one succeeds. If the selected switch route has transmissivity $0.85$, find the upper-bound delivery probability.

Solution

The at-least-one probability is

$$\begin{aligned} P_{\ge1} &= 1-(1-0.10)^8 \\ &= 1-0.9^8 \approx0.570. \end{aligned}$$

Including only the stated route loss gives

$$P_{\mathrm{del}} \le (0.85)(0.570) \approx0.485.$$

The bound omits herald efficiency, multipair rejection, switch contention, timing, and output coupling.

4. Convert an optical loss budget

A path contains $1.2\,\mathrm{dB}$ fibre coupling loss, $0.8\,\mathrm{dB}$ filtering loss, and $0.5\,\mathrm{dB}$ switching loss. Find total transmission.

Solution

Losses add in decibels:

$$\mathcal L_{\mathrm{tot}} = 1.2+0.8+0.5 = 2.5\,\mathrm{dB}.$$

Therefore

$$\eta_{\mathrm{path}} = 10^{-2.5/10} \approx0.562.$$

About $56.2\%$ of incident photons survive this path under the scalar-loss model. Polarization, spectrum, and temporal mode quality require separate checks.

5. Interpret a no-click event

A threshold detector has efficiency $\eta_d=0.92$ and dark-click probability $d=10^{-5}$ per gate. Find $P(0\mid1)$ and $P(0\mid2)$.

Solution

The model gives

$$\begin{aligned} P(0\mid1) &= (1-d)(1-\eta_d) \\ &\approx (0.99999)(0.08) \approx0.079999, \end{aligned}$$

and

$$\begin{aligned} P(0\mid2) &= (1-d)(1-\eta_d)^2 \\ &\approx (0.99999)(0.08)^2 \approx0.00640. \end{aligned}$$

A no-click record does not prove vacuum. Detector inefficiency leaves a substantial missed-one-photon probability.

6. Size a feed-forward delay

Detection, electronics, switching, and margin require $12\,\mathrm{ns}$, $18\,\mathrm{ns}$, $8\,\mathrm{ns}$, and $7\,\mathrm{ns}$. Find the minimum delay and the corresponding fibre length for $v_g=2.0\times10^8\,\mathrm{m\,s^{-1}}$.

Solution

The delay is

$$\tau_{\mathrm{delay}} \ge 12+18+8+7 = 45\,\mathrm{ns}.$$

The propagation length is

$$\begin{aligned} L_{\mathrm{delay}} &= (2.0\times10^8) (45\times10^{-9})\,\mathrm{m} \\ &= 9.0\,\mathrm{m}. \end{aligned}$$

This is a latency lower bound. Connectors, routing, dispersion compensation, and extra scheduling slack can increase physical length and loss.

7. Count naive fusion success

A small graph resource requires four independent fusions, each succeeding with probability $1/2$. Find the direct success probability. Explain why a scalable architecture should not build one full resource by blind postselection.

Solution

The direct probability is

$$\left(\frac12\right)^4 = \frac{1}{16} = 0.0625.$$

Blindly restarting the entire construction wastes every successful partial fusion. Scalable proposals use small verified resources, heralded outcomes, recycling where possible, multiplexing, redundant graphs, and decoders that interpret known fusion failures and loss.

8. Follow unequal rail attenuation

Send $|+_L\rangle=(|0_L\rangle+|1_L\rangle)/\sqrt2$ through the phase-compensated pure-loss channel with $\eta_a=0.90$ and $\eta_b=0.10$. Find the survival probability and the squared fidelity of the surviving state with $|+_L\rangle$. Is this an undistorted erasure channel?

Solution

The survival probability is $(\eta_a+\eta_b)/2=0.50$. The normalized survivor is

$$|\psi_s\rangle= \frac{\sqrt{\eta_a}|0_L\rangle+\sqrt{\eta_b}|1_L\rangle} {\sqrt{\eta_a+\eta_b}},$$

so

$$\begin{aligned} F&=|\langle+_L|\psi_s\rangle|^2\\ &=\frac{(\sqrt{\eta_a}+\sqrt{\eta_b})^2}{2(\eta_a+\eta_b)}=0.80. \end{aligned}$$

Survival alone has not preserved the logical state: the two amplitudes were filtered differently. For equal rail transmissions the normalized survivor would be unchanged. A compensating attenuator could equalize them, but would further lower the success probability.

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